denis serre systems of conservation laws 1_ hyperbolicity, entropies, shock waves 1999.pdf

Systems ofConservation Laws 1:

Hyperbolicity, Entropies, Shock Waves

CAMBRIDGE UNIVERSITY PRESS

DENIS SERRETranslated by

I . N. SNEDDON

Systems of Conservation Laws 1

Systems of conservation laws arise naturally in several areas of physics and chemistry. Tounderstand them and their consequences (shock waves, finite velocity wave propagation)properly in mathematical terms requires, however, knowledge of a broad range of topics.This book sets up the foundations of the modern theory of conservation laws describingthe physical models and mathematical methods, leading to the Glimm scheme. Buildingon this the author then takes the reader to the current state of knowledge in the subject. Inparticular, he studies in detail viscous approximations, paying special attention to viscousprofiles of shock waves. The maximum principle is considered from the viewpoint ofnumerical schemes and also in terms of viscous approximation, whose convergence isstudied using the technique of compensated compactness. Small waves are studied usinggeometrical optics methods. Finally, the initial–boundary problem is considered in depth.Throughout, the presentation is reasonably self-contained, with large numbers of exercisesand full discussion of all the ideas. This will make it ideal as a text for graduate courses inthe area of partial differential equations.

Denis Serre is Professor of Mathematics at the Ecole Normale Superieure de Lyon and was

a Member of the Institut Universitaire de France (1992–7).

This page intentionally left blank

Systems ofConservation Laws 1

Hyperbolicity, Entropies, Shock Waves

DENIS SERRE

Translated by

I. N. SNEDDON

PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

Originally published in French by Diderot as Systèmes de lois de conservation I: hyperbolicité, entropies, ondes de choc and © 1996 Diderot

First published in English by Cambridge University Press 1999 as Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves

English translation © Cambridge University Press 1999 This edition © Cambridge University Press (Virtual Publishing) 2003

First published in printed format 1999

A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 58233 4 hardback

ISBN 0 511 00900 3 virtual (netLibrary Edition)

To Paul and Fanny

Contents

Acknowledgments page xiIntroduction xiii

1 Some models 11.1 Gas dynamics in eulerian variables 11.2 Gas dynamics in lagrangian variables 81.3 The equation of road traffic 101.4 Electromagnetism 111.5 Magneto-hydrodynamics 141.6 Hyperelastic materials 171.7 Singular limits of dispersive equations

2 Scalar equations in dimension d = 1 2

191.8 Electrophoresis 22

52.1 Classical solutions of the Cauchy problem 252.2 Weak solutions, non-uniqueness 272.3 Entropy solutions, the Kruzkov existence theorem 322.4 The Riemann problem 432.5 The case of f convex. The Lax formula 452.6 Proof of Theorem 2.3.5: existence 472.7 Proof of Theorem 2.3.5: uniqueness 512.8 Comments 572.9 Exercises 60

3 Linear and quasi-linear systems 683.1 Linear hyperbolic systems 693.2 Quasi-linear hyperbolic systems 79

vii

viii Contents

3.3 Conservative systems 803.4 Entropies, convexity and hyperbolicity 823.5 Weak solutions and entropy solutions 863.6 Local existence of smooth solutions 913.7 The wave equation 101

4 Dimension d = 1, the Riemann problem 1064.1 Generalities on the Riemann problem 1064.2 The Hugoniot locus 1074.3 Shock waves 1114.4 Contact discontinuities 1164.5 Rarefaction waves. Wave curves 1194.6 Lax’s theorem 1224.7 The solution of the Riemann problem for the p-system 1274.8 The solution of the Riemann problem for gas dynamics 1324.9 Exercises 143

5 The Glimm scheme 1465.1 Functions of bounded variation 1465.2 Description of the scheme 1495.3 Consistency 1535.4 Convergence 1565.5 Stability 1615.6 The example of Nishida 1675.7 2× 2 Systems with diminishing total variation 1745.8 Technical lemmas 1775.9 Supplementary remarks 180

5.10 Exercises 182

6 Second order perturbations 1866.1 Dissipation by viscosity 1876.2 Global existence in the strictly dissipative case 1936.3 Smooth convergence as ε→ 0+ 2036.4 Scalar case. Accuracy of approximation 2106.5 Exercises 216

7 Viscosity profiles for shock waves 2207.1 Typical example of a limit of viscosity solutions 2207.2 Existence of the viscosity profile for a weak shock 2257.3 Profiles for gas dynamics 229

Contents ix

7.4 Asymptotic stability 2307.5 Stability of the profile for a Lax shock 2357.6 Influence of the diffusion tensor 2427.7 Case of over-compressive shocks 2457.8 Exercises 250

Bibliography 255Index 261

Acknowledgments

This book would not have seen the light of day without a great deal of help. Firstof all that of the Institut Universitaire de France, by whom I was engaged, whoassisted me by giving me the time and the freedom necessary to bring the first draftto a conclusion. Later my colleagues at the Ecole Normale Superieure de Lyon gavesimilar support by accepting my release from normal duties for a considerable timeso that I should be able to concentrate on this book. Finally and above all to mystudents, former students and friends, who have believed in using this work, whohave supported me by discussing it often and have read it in detail. Their interest hasbeen the most powerful of stimulants. I owe a considerable debt to Sylvie Benzoni,who has read the greater part of this book and whose severe criticism has constantlyled me to improve the text.

I give heartfelt thanks also to Pascale Bergeret, Marguerite Gisclon, FlorenceHubert, Christophe Cheverry, Herve Gilquin, Arnaud Heibig, Peng Yue Jun, JulienMichel and Bruno Sevennec for their collaboration. Finally certain persons havetaught me about topics which I did not properly know: Jean-Yves Chemin,Constantin Dafermos, Heinrich Freistuhler, David Hoff, Sergiu Klainerman,Ling Hsiao, Tai-Ping Liu, Guy Metivier and Roberto Natalini.

xi

Introduction

The conservation laws that are the subject of this work are those of physics ormechanics, when the state of the system considered is a field, that is a vector-valued function (x, t) �→ u(x, t) of space variables x = (x1, . . . , xd ) and of thetime t . The domain � covered by x is an open set of R

d , with in general 1 ≤ d ≤ 3.The scalar components u1, . . . , un of u are variables dependent on x and t : if � isbounded and in the absence of any exchange with the exterior,1 the mean state ofthe system

u := 1

|�|∫

�

u(x, t) dx

is independent of the time and the system tends to a homogeneous equilibriumu ≡ u as the time increases. The fact that we speak of the mean indicates that theset U of admissible values of the field u is a convex set of R

n .A conservation law is a partial differential equation

∂ui

∂t+ divx �qi = gi ,

where gi (x, t) represents the density (per unit volume) of the interaction with ex-ternal fields. Among these fields, we can even find some which depend on u; forexample the conservation of momentum of an electrically neutral continuum canbe written

∂ρvi

∂t+ divx (ρvi �v − �T i ) = ρGi ,

where ρ is the mass density (ρ is one of the components of u), T = ( �T 1, . . . , �T d ) isthe strain tensor, �v is the velocity and �G the gravity field. Hence, in general we shall

1 The boundary ∂� is thus impermeable and insulated, for example electrically, in short, there is no interactionwith a field other than u.

xiii

xiv Introduction

have gi (x, t) = hi (x, t, u(x, t)) where hi is a known function: here gi = u1Gi−1

since ρ is u1. We call �h the sources of the system.An equivalent formulation of a conservation law is given by an integral condition,

which expresses the physical balance for the quantity represented by ui in anarbitrary part ω of �:

d

dt

∫ω

ui (x, t) dx +∫

∂ω

�qi (x, t) · ν(x) dx =∫

ω

gi (x, t) dx

where ν(x) denotes the outward unit normal at a point x on the boundary of ω. Thevector field �qi is thus the flux of the variable ui :

flux of mass if ui is the mass density,flux of energy if ui is the energy density per unit volume,electric current if ui is the electric charge density, . . . .

The third formulation of a conservation law is also the most practical for finding thenew equation when we have to effect a change of variables. We define a differentialform αi of degree d in �× (0, T ) by

αi := uidx1 ∧ dx2 ∧ · · · ∧ dxd − qi1dt ∧ dx2 ∧ · · · ∧ dxd

+ · · · (−)dqid dt ∧ dx1 ∧ · · · ∧ dxd−1.

The conservation law is then written

dαi = gi dt ∧ dx1 ∧ · · · ∧ dxd .

This way of looking at the problem suggests that other conservation laws have anatural form dα = β whereα is a differential form of degree p, not necessarily equalto d. This is the case of Maxwell’s electromagnetic equations or the Yang–Millsequations for which p = 2 and d = 3.

In this form, the conservation laws are intangible, in so far as the scales of time,length, velocity . . . are compatible with a representation of the system by fields.2

However, the description of the evolution of the state of the physical system ispossible only if the system of equations

∂t ui + divx �qi = gi , 1 ≤ i ≤ n,

is closed under the state laws:

qi := Qi [u; ε].

These laws, in which ε denotes one or several dimensionless parameters,3 describe

2 Quantum effects are therefore excluded, but relativistic effects can, in general, be taken into account.3 Such as the inverse of a Reynolds number, a mean free path, a relaxation time.

Introduction xv

in an empirical manner the behaviour of a continuum put into a given homogeneousstate u ∈ U. For example, a fixed mass of gas, in a prescribed volume and at animposed temperature, exerts on the boundary a force whose density per unit surfacearea (the pressure) is constant and depends only on the thermodynamic parametersand on the nature of the gas; however, the complete ranges of time and of the spacevariables are not allowable and in certain cases, recourse must be had to a statisticaldescription or to molecular dynamics. Care must always be taken, specially whenan asymptotic analysis is being made, to ensure the validity of the model beingused.

The description of the road traffic on a highway shows that the state law dependsas much on human sciences as on physics: the average speed of vehicles is a functionof the traffic density and reflects the average behaviour of human beings (the drivers)and depends on circumstances; again, there is nothing absolute about it as it variesaccording to material conditions (the reliability and security of the vehicles, thequality of circulation lanes), and the regulations in force and even the culture of acountry.

The point common to all the models studied here is the fact that fi :u �→ Qi [u, 0]is an ordinary local function: its value at (x, t) depends only on that of u at this point.By an abuse of notation, we may therefore write fi [u](x, t) = fi (x, t, u(x, t)) and,on this occasion, fi denotes a function defined on U and with values in R, which,in general, will be regular. In first approximation, the evolution of the system canbe deduced from the knowledge of its initial state u0(·) = (u01, . . . , u0n) given on�, in the solution of the system of non-linear partial differential equations

∂t ui + divx fi (u) = gi (x, t, u), 1 ≤ i ≤ n, x ∈ �, t > 0, (0.1)

augmented by the appropriate boundary conditions. This is the mixed problemwhich when � = R

d we rather call the Cauchy problem.The apparent simplicity of these equations contrasts with the difficulty of the

problems encountered when solving the Cauchy problem or the mixed problem, asmuch from the theoretical point of view as from that of numerical analysis. Thesetwo ways of considering the Cauchy problem are equally interesting and difficult.However, the present work is devoted only to the theoretical aspects, in particularbecause the numerical part is covered by a number of very good books. Let us citeat least those of Leveque [62], Godlewski and Raviart [34], Richtmyer and Morton[86], Sod [98], and Vichnevetsky and Bowles [109].

To illustrate the mathematical difficulties, let us say that there is not a satisfactoryresult concerning the existence of a solution of the Cauchy problem. For givenregular initial data,4 there exists a regular solution, but only during a finite time

4 Let us say of class Hs , with s > 1+ d/2 with the result that u0 is of class C 1.

xvi Introduction

(Theorem 3.6.1) inversely proportional to ∇xu0. Since, beyond a certain time,discontinuities in u must develop, this theorem is not satisfactory for applications.The results which concern weak solutions (those which have a chance of beingdefined for all time) are limited to the scalar case (n = 1) or to the one-dimensionalcase (d = 1)! Again in this latter case the restrictions themselves are severe: theglobal existence in time (Theorem 5.2.1) is known if the total variation of u0 issufficiently small (if n = 2, only if the product TV(u0)‖u0‖∞ is supposed small).There is there a threshold effect, for a local result does not exist where the timeof existence would depend on the scale of the data. This question is discussed byTemple and Young [105], who have obtained recently a result of this type for thesystem of gas dynamics.5 For bounded initial data, but of arbitrary size, the situationis worse; only 2× 2 systems (i.e. with n = 2 and d = 1) and related systems (seeChapter 12) have been tackled by the method of compensated compactness, underrestrictive hypotheses and for results of relatively poor quality. Among these, theTemple systems gain from a suitable theory (see Chapter 13) in large part becausethey are a faithful generalisation of the scalar laws of conservation.

The appearance of discontinuities in finite time has led specialists in functionspaces to pay particular attention to spaces such as L∞ or BV (functions of boundedvariation). It is in one or the other space that existence theorems have been obtainedin one-dimensional space. The reason for their success is that these are algebras,which permits the treatment of the rather strong non-linearity of the equations.However, the work of Brenner [4], which is concerned with linear systems, showsthat these spaces cannot be adapted to the multi-dimensional case. To the contrary,the spaces would have to be of Hilbert type, at least to be constructed on L2. We arethus in the presence of a paradox which has up to the present not been resolved: tofind a function space which is an algebra, probably constructed on L2 and whichcontains enough discontinuous functions.

The study of discontinuous solutions, called weak solutions, makes use of theintegral form of the conservation laws, in the equivalent form below:6∫∫

�×R+

(ui

∂ϕ

∂t+ fi (u) · ∇xϕ

)dx dt +

∫�

ui0ϕ(·, u) dx =∫∫

�×R+giϕ dx dt

for every test function ϕ, of class C∞ and with compact support in �×R

+. We showeasily the equivalence with the partial differential equation ∂i ut + divx fi (u) = gi

everywhere u is of class C1. On the other hand, when u is of class C

1 on both sidesof a hypersurface ∈ R

d+1, with the boundary values u+(x, t) on one side andu−(x, t) on the other, the integral formulation expresses a transmission condition,

5 Their work is based on the particular structure of the system of gas dynamics and cannot be extended to systemsof general conservation laws, by reason of an estimation due to Joly, Metivier and Rauch [47].

6 The eventual boundary conditions have not been taken into account here, so as not to overburden the formulae.

Introduction xvii

called the Rankine–Hugoniot condition:

ν0[ui ]+d∑

α=1

να[ fiα(u)] = 0

where (x, t) �→ (ν0, . . . , νd ) is a normal vector field to . This formula suggeststhat the role of the sources gi in the propagation of discontinuities is negligible.This is the reason why these terms are omitted very frequently in this work.

A quick look at the systems of the form (0.1) suggests that they govern re-versible phenomena, at least when g≡ 0: if u is a solution so also is the functionu(x, t) := u(−x,−t). This is obvious if u is of class C

1, it is also true for a weaksolution. Nevertheless it is known that thermodynamics modelled by the equationsof Euler which are the archetype of systems of conservation laws is the centre of ir-reversible processes. This paradox is bound to the lack of uniqueness of the solutionof the Cauchy problem in the framework of weak solutions. The regular solutionsare effectively reversible, but the discontinuous solutions are not. We attack therea central question of the theory: how to separate the wheat from the chaff, the so-lutions observed in nature (called ‘physically admissible’) from those that are onlymathematical artefacts? There are two major types of reply to this question.

The first is descriptive and concerns only piecewise continuous solutions whosediscontinuities occur along regular hypersurfaces of �×R

+. These discontinuitiesare physically admissible if they obey a causality principle: the state of the systemcannot contain more information than it has at the initial instant.7 Mathematically,we consider the coupled system formed, on the one hand, partly of the conservationlaws and partly of the hypersurface and, on the other hand, of the Rankine–Hugoniotcondition, seen as an evolution equation for the location of the discontinuity. It isthus a free boundary problem, which can be transformed to a mixed problem ina fixed domain. We demand that this mixed problem be well-posed for increasingtime. In dimension d = 1, an equivalent condition, at least if the amplitude of thejump in u is moderate, is the shock criterion of Lax, formed of four inequalities,described in §4.3. In a higher dimension, the characterisation of the admissiblediscontinuities, much more complex, is explained in Chapter 14. There are prin-cipally two kinds of ‘good’ discontinuities, according as the Lax inequalities arestrict or two among them are equalities. Only the first type, called shock waves,are irreversible. The second type, reversible, bear the name contact discontinuities.Concerning thermodynamics, A. Majda [74, 73] has shown (see Chapter 14) thatthe shock waves are, in general, stable (that is, that the mixed problem introducedabove is locally well-posed), while the contact discontinuities (the vortex sheets)

7 or that the boundary conditions provide.

xviii Introduction

are strongly unstable.8 This instability a la Hadamard is a stone in the garden ofthe mechanics of fluids; it renders Euler’s equations unsuited to the prediction offlows and casts doubt on this model for thermodynamics.9

The second reply is of more general significance but manifests itself less prac-tically in applications. To begin with is the criticism of the approximation madeabove. To simplify the matter, let us suppose ε to be a scalar. It is reasonable toreplace Qi [u; ε] by fi (u) where the solution varies moderately, but it is debatablewhere it varies greatly. Now most often, the solution of the real problem (denoted byuε) is regular, of class C

1, but varies rapidly in the neighbourhood of a discontinuityof u. Typically this neighbourhood has a width of the order of ε and the gradientof uε must be of the order of 1/ε. In this narrow zone, what has been neglected isof the same order of magnitude as fi (u). We have in fact

Qi [v; ε]− fi (v) ∼ εBi (v)∇xv,

where B(v) is a tensor with four indices. A description of the evolution more faithfulthan (0.1) is therefore

∂t uεi + divx fi (u

ε) = ε divx (Bi (uε)∇xuε), 1 ≤ i ≤ n. (0.2)

The tensor B is such that the Cauchy problem for (0.2) is well-posed for ε > 0 andincreasing time.10 It represents, according to the case, the effect of a viscosity, thatof thermal conduction, the Joule effect, . . . . In the model of road traffic, where thescalar u is the density of the vehicles, it represents the faculty of anticipation of thedrivers as a function of the flow of traffic in the vicinity of their vehicle; it is thisanticipation that causes irreversibility.

The system (0.2) is irreversible. This is the essential difference from (0.1), whichis expressed quantitatively as follows. The undisturbed system is in general com-patible, for the regular solutions, with a supplementary conservation law11

∂t E(u)+ divx F(u) = dE(u) · g(x, t, u) (0.3)

where E : U → R is strictly convex. This can always be reduced to the case inwhich E has positive values. The equation (0.3) then yields an a priori estimate of

8 save in dimension d = 1 where the contact discontinuities are stable.9 It is not difficult to see that the vortex sheets necessarily appear, if d > 2, as a by-product of the interaction of

multi-dimensional shocks. The case d = 1 is less clear.10 It would be wrong nevertheless to believe that the system (0.2) is parabolic, that is, that the operator v �→

divx B(v)∇xv is elliptic. Its symbol is generally positive but not positive definite.11 It is principally in this setting that this work is placed.

Introduction xix

u in a Sobolev–Orlicz space via the differential equation12

d

dt

∫�

E(u(x, t)) dx =∫

�

dE(u) · g(x, t, u) dx .

This allows us to control the value of the positive expression

E (t) :=∫

�

E(u(x, t)) dx .

For example, in the absence of sources, E (t) remains equal to E (0), which dependsonly on the initial condition. Of course, this calculation, in which we differentiatecomposite functions (for example E ◦ u), does not have a rigorous basis for weaksolutions. On the other hand, the solution of the perturbed problem is in generalregular and satisfies the equation

∂t E(uε)+ divx �F(uε) = dE(uε) · (ε divx (B(uε)∇xuε)+ g(x, t, uε)).

If g ≡ 0 and if the boundary values are favourable, we obtain at the best

E′(t) = −ε

∫�

(D2E(uε)∇xuε · B(uε)∇xuε) dx

which is negative for all the realistic examples. But, above all, the right-hand sidedoes not tend to zero with ε, because we integrate an expression of the order ofε−1 (because of ε(∇xuε)2) over a zone whose measure is of the order of ε. Thedecay of E , certainly preserved by passage to the limit when ε → 0+, thus will bestrict in the presence of discontinuities. A criterion of the admissibility of solutionsis thus ∫∫

�×R+(E(u)∂tϕ + F(u) · ∇xϕ) dx dt +

∫R

E(u0)ϕ(·, 0) dx ≥ 0 (0.4)

for every positive test function ϕ ∈ D (�×R), with equality for a classical solutionof (0.1). On the level of the discontinuities, (0.4) is translated as the jump condition13

ν0[E(u)]+d∑

α=1

να[Fα(u)] ≤ 0. (0.5)

The equality in (0.5) is in general incompatible with the Rankine–Hugoniot condi-tion except where this concerns the contact discontinuities.

12 To simplify the exposition, no account has been taken of the boundary conditions. For example, the readercould assume that �F · ν is null on the boundary.

13 We remark that this condition is independent of the orientation of .

xx Introduction

A function E like that introduced above carries the name, in so far as it is amathematical object, of entropy. By extension, we call a function E , not necessarilyconvex, in a conservation law compatible with (0.1), also an entropy. The vectorfield �F is called the entropy flux associated with E . Again, this terminology isdue to thermodynamics, as the form of the equations of motion in lagrangian14

variables is a system of conservation laws compatible with a supplementary lawin which E is equal to −S, the opposite of physical entropy of the fluid. Thischange of sign, which renders convex that which is concave and conversely, isa cultural difference between mathematicians and physicists. For the physicist,the entropy has a tendency to increase, while for the mathematician the oppositeholds. In the eulerian representation, in which � is a domain in physical space, thedifference is still more marked, as E corresponds to−ρS. Despite this, the historicallink between the physical theory and mathematics has led to the inequality (0.4)being called the entropy condition, when a system of conservation laws can bemodelled on something other than the flow of a fluid. The weak solutions whichsatisfy (0.4) are called entropy solutions. By extension and in an improper manner,we again speak of entropy conditions with regard to the Lax shock condition,principally because in thermodynamics, the Lax inequalities express the fact that theentropy S, constant when it follows a particle, in fact grows when it crosses a shockwave.

The mechanism of dissipation, which makes E decrease and renders the evolutionirreversible, is so central that it would not make sense to study theoretically (0.1),in isolation. This necessarily leads to the algebraic notion of a hyperbolic systemin the linear case, but the understanding of the non-linear case calls for as muchattention to be paid to the (partially) parabolic (0.2). This is why this book isnot entitled Hyperbolic systems of conservation laws. Chapters 6, 7 and 15 areprincipally devoted to parabolic systems and these are involved in a significant wayin Chapters 8 and 9.

Some references This volume owes a great deal to those which preceded it, inparticular that of Majda [75]; this, at the same time short and profound, remainsan essential reference and the energy which animates it gives birth to a sense ofvocation. It is the only one to deal with nearly all the topics which deal withmulti-dimensional or asymptotic problems. It is with this that we have tried to dealhere, with more detail but less animation. Although dealing with many subjects, thiswork does not go on as long on classical problems as more specialised works. Thus,the reader who wishes to deepen his knowledge of the Riemann problem should read

14 In these variables, a particle is represented by a fixed value of the variable x . This is therefore not a ‘spacevariable’ as strictly defined.

Introduction xxi

the text of Ling Hsiao and Tong Zhang [46]. The global methods, based on the Con-ley index, for studying the viscosity profiles, are found in Smoller [97]. A systematicstudy of the propagation and the interaction of non-linear waves is greatly devel-oped by Whitham [112]; see also the monograph of Boillat [2]. For questions con-cerning the mechanics of fluids, with the description of multi-dimensional shocks,Courant and Friedrichs [11] should be consulted. Various types of singular pertur-bations (models of combustion, the incompressible limit) and the stability of multi-dimensional shocks are presented in Majda [75]. For mixed problems in a (partly)parabolic context, a good reference is Kreiss and Lorenz [55]. In the lecture notesby Hormander [45] is found a simple presentation of the blow-up mechanisms fora general system (not necessarily rich) as well as the global (or nearly global) exis-tence for a perturbation of the wave equation in dimension d ≥ 2. The notes of Evans[20] give a view of the methods utilising weak convergence, which goes beyondmere compensated compactness. The decay of entropic solutions to N-waves is thesubject of the memoir of Glimm and Lax [33]. Concerning the Cauchy problem andmixed problems for linear equations there are many references; let us, at least, citeIvrii [48], Sakamoto [87] and again Kreiss and Lorenz [55]. The quasi-linear mixedproblem in dimension d = 1, which includes the free boundary problems, is sys-tematically studied in Li Ta-Tsien and Yu Wen-ci [65]. The geometrical aspects ofthe conservation laws, especially affine and convex, are the subject of the memoir ofSevennec [93].

The way the chapters of this book are ordered is merely an indication to thereader, since the chapters depend little on one another. The core of the theory isconstituted by Chapters 2, 3, 4, 6 and 14. For a postgraduate course in which theaim is the solution of the Riemann problem for gas dynamics, Chapters 2 and 4are indispensable but are not enough to give an advanced student a representativepicture of the subject.

In spite of its length, this work does not pretend to be exhaustive. It leads to ablind alley on several questions, of which some are important. Uniqueness is themost important of these; the reason is that it is a matter of a subject which is muchless advanced than that of existence (with, however, recent progress by A. Bressan),and on which we could not give a synthetic view. Likewise, this book does not tacklequestions which touch on ‘pathology’: systems not strictly hyperbolic have otherconditions for the admission of shock waves. At times, it has been mathematicalrigour that has been neglected (with the hope that it is not too frequent for thetaste of the reader): above all an attempt has been made to be the most descriptivepossible, giving perhaps too many criteria and formulae, asymptotic analysis, andnot enough proofs. Some new results will be found (few enough and none major inall cases) and lists of exercises which should satisfy those who believe in acquiringinsight by the solution of examples. In spite of this range of descriptive material,

xxii Introduction

there is not a word on the phenomenon of relaxation, nor on kinetic formulations,and not more on the description by N-waves of behaviour in large times, threeimportant subjects of the theory. Perhaps, if the occasion arises, a future edition . . .

Lyon, November 1995

. . . cela peut durer pendant tres longtemps,si l’on ne fait pas d’omelette avant!

(Robert Desnos, Chantefables)

1

Some models

1.1 Gas dynamics in eulerian variables

Let us consider a homogeneous gas (all the molecules are identical with mass m)in a region �, whose coordinates x = (x1, . . . , xd ) are our ‘independent’ variables.From a macroscopic point of view, it is described by its mass density ρ, its mo-mentum per unit volume �q and its total energy per unit volume E . In a sub-domainω containing at an instant N molecules1 of velocities �v1, . . . , �vN respectively, wehave ∫

ω

ρ dx = Nm,

∫ω

�q dx = mN∑

j=1

�v j

from which it follows that �q = ρ�v, �v being the mean velocity of the molecules.2

Likewise, the total energy is the sum of the kinetic energy and of the rotational andvibrational energies of the molecules:

∫ω

E dx = 1

2m

N∑j=1

‖�v2‖ +N∑

j=1

(e j

v + e jR

)

where e jv and e j

R are positive. For a monatomic gas, such as He, the energy of rotationis null. The energy of vibration is a quantum phenomenon, of sufficiently weakintensity to be negligible at first glance. Applying the Cauchy–Schwarz inequality,we find that

1

2m

N∑j=1

‖�v j‖2 ≥ m

2N

∥∥∥∥N∑

j=1

�v j

∥∥∥∥2

1 N is a very large number, for example of the order of 1023 if the volume of ω is of the order of a unit, but theproduct mN is of the order of this volume.

2 This can be suitably modified if there are several kinds of molecules of different masses.

1

2 Some models

which gives

∫ω

E dx ≥ 1

2

(∫ω

ρ dx

)−1∥∥∥∥∫

ω

�q dx

∥∥∥∥2

+N∑

j=1

(e j

v + e jR

)

≥ 1

2

(∫ω

ρ dx

)−1∥∥∥∥∫

ω

ρ�v dx

∥∥∥∥2

.

This being true for every sub-domain, we can deduce that the quantity E/ρ− 12‖�v‖2

is positive. It is called the specific internal energy (that is per unit mass) and wedenote it by e; we thus have

E = 1

2ρ‖�v‖2 + ρe,

where the first term is (quite improperly) called the kinetic energy of the fluid. Forthe sequel it should be remembered that the internal energy can be decomposedinto two terms ek+ ef where ek is kinetic in origin and ef is due to other degrees offreedom of the molecules.

The law of a perfect gas

A perfect gas obeys three hypotheses:

the vibration energy is null,the velocities at a point (x, t) satisfy a gaussian distribution law

a exp(−b‖ · −�v‖2)

where a, b and �v are functions of (x, t) (of course, �v is the mean velocityintroduced above),

the specific internal energy is made up among its different components pro ratawith the degrees of freedom.

Comments (1) The gaussian distribution comes from the theorem of Laplace thatconsiders the molecular velocities as identically distributed random variables whenN tends to infinity. It is also the equilibrium distribution (when it is called ‘maxwell-ian’) in the Boltzmann equation, when it takes into account the perfectly elasticbinary collisions.

(2) Several reasons characterise the gaussian as being the appropriate law. On theone hand, its set is stable by composition with a similitude O of R

d (χ �→ χ ◦ O)and by multiplication by a scalar (χ �→ λχ ). On the other, the components of thevelocity are independent identically distributed random variables.

1.1 Gas dynamics in eulerian variables 3

(3) The hypothesis of the equi-partition of energy is pretty well verified whenthere are a few degrees of freedom, for example for monatomic molecules (He),diatomic molecules (H2, O2, N2) or rigid molecules (H2O, CO2, C2H2, C2H4). Themore complex molecules are less rigid; they thus have more degrees of freedom,which are not equivalent from the energetic point of view.

(4) The equi-partition takes place also among the translational degrees of free-dom. If the choice is made of an orthonormal frame of reference, each componentv

jα − vα of the relative velocity is responsible for the same fraction ekα = ek/d in

the energy of kinetic origin.

Let β be the number of non-translational degrees of freedom. The hypothesis ofequi-partition gives the following formula for each type of internal energy:

ek1 = · · · = ekd = 1

dek, eR = β

dek

and thus e = (d + β)ek1.

The pressure p is the force exerted per unit area on a surface, by the gas situatedon one side of it.3 Take as surface the hyperplane x1 = 0, the fluid being at rest(�v ≡ 0, a and b constants). Let A be a domain of unit area of this hyperplane. Theforce exerted on A by the gas situated to the left is proportional to the number M ofparticles hitting A per unit time, multiplied by the first component I1 of the meanimpulse of these.4 On the one hand, M is proportional to the number N of particlesmultiplied by the mean absolute speed (the mean of |vα

1 |) in the direction x1. Onthe other hand, NI1 is proportional to ρw2

1, that is to ρe1k. Nothing in this argumentinvolves explicitly the dimension d and we therefore have p = kρe1k, where k isan absolute constant. A direct calculation in the one-dimensional case yields theresult k = 2. Introducing the adiabatic exponent

γ = d + b + 2

d + b

there results the law of perfect gases

p = (γ − 1)ρe.

The most current adiabatic exponents are 5/3 and 7/5 if d = 3, 2 and 5/3 ifd = 2 and 3 if d = 1. In applications air is considered to be a perfect gas for whichγ = 7/5.

3 In this argument, the surface in question is not a boundary, since it would introduce a reflexion and wouldeventually distort the gaussian distribution.

4 This mean is not null as it is calculated solely from the set of molecules for which vj1 > 0.

4 Some models

The Euler equations

The conservation laws of mass, of momentum and of energy can be written

∂tρ + divx (ρ�v) = 0,

∂t (ρvi )+ divx (ρvi �v)+ ∂i p =d∑

j=1

∂ j Ti j , 1 ≤ i ≤ d,

∂t E + divx ((E + p)�v) =d∑

j=1

∂ j (vi Ti j )− divx �q

where T − pId is the stress tensor and �q the heat flux. In the last equation, twoterms represent the power of the forces of stress. The conservation of the kineticmoment ρ�v∧ x implies that T is symmetric. We have seen that T is null for a fluidat rest and also when it is in uniform motion of translation. The simplest case isthat in which T is a linear expression of the first derivatives ∇x �v, the coefficientsbeing possibly functions of (ρ, e). The principle of frame indifference implies theexistence of two functions α and β such that

Ti j = α(ρ, e)

(∂vi

∂x j+ ∂v j

∂xi

)+ β(ρ, e)(divx �v)δ j

i (1.1)

which clearly introduces second derivatives into the above equations. The tensor Trepresents the effects of viscosity and the linear correspondence is Newton’s law. Ifα and β are null the conservation laws are called Euler’s equations. In the contrarycase they are called the Navier–Stokes equations.

Likewise, the heat flux is null if the temperature θ (defined later as a thermo-dynamic potential) is constant. The simplest law is that of Fourier, which can bewritten

�q = −k(ρ, e)∇xθ,

with k ≥ 0.For a regular flow, a linear combination of the equations yields the reduced system

∂tρ + div(ρ�v) = 0,

∂tvi + �v · ∇xvi + ρ−1∂i p = ρ−1 div(Ti .),

∂t e + �v · ∇xe + ρ−1 p div �v = ρ−1

(∑i, j

Ti j∂ jvi − div �q)

.

Let us linearise this system in a constant solution, in a reference frame in which thevelocity is null:

∂t R + ρ div �V = 0,

∂t Vi + ρ−1(pρ∂i R + pe∂iχ ) = ρ−1(α�Vi + (α + β)∂i div �V ),

∂tχ + ρ−1 p div �V = ρ−1k�(θρ R + θeχ ).


The last equation can be transformed to

∂t (θρ R + θeχ )+ λ div �V = kθe

ρ�(θρ R + θeχ ).

A necessary condition for the Cauchy problem for this linear system to be well-posed is the (weak) ellipticity of the operator

(R, �V , ξ ) �→ (0, α� �V + (α + β)∇ div �V , kθe�ξ )

which results in the inequalities

kθe ≥ 0, α ≥ 0, 2α + β ≥ 0. (1.2)

The entropy

In the absence of second order terms, the flow satisfies

p(∂tρ + �v · ∇ρ) = ρ2(∂t e + �v · ∇e)

which suggests the introduction of a function S(ρ, e), without critical point, suchthat

p∂S

∂e+ ρ2 ∂S

∂ρ= 0.

Such a function is defined up to composition on the left by a numerical function:if h: R → R and if S works, then h ◦ S does too, provided that h′ does not vanish.Such a function satisfies the equation

(∂t + �v · ∇)S = 0,

as long as the flow is regular, this signifies that S is constant along the trajectories5

of the particles. On taking account of the viscosity and of the thermal conductivity,it becomes

ρ(∂t + �v · ∇)S = Se

∑i, j

(Ti j∂ jvi )+ div(k∇θ ),

that is to say

∂t (ρS)+ div(ρS�v) = Se

(1

2α∑i, j

(∂iv j + ∂ jvi )2 + β(div �v)2

)+ Se div(k∇θ).

Free to change S to −S, we can suppose that Se is strictly positive. The namespecific entropy is given to S. The effect of the viscosity is to increase the integral

5 We refer to the mean trajectory.

6 Some models

of ρS. The second law of thermodynamics states that the thermal diffusion behavesin the same sense, that is that ∫

ω

Se div(k∇θ) dx ≥ 0

if there is no exchange of heat across ∂ω (Neumann condition ∂θ/∂ν = 0).Otherwise, this integral is compensated by these exchanges. In other terms, afterintegration by parts, we must have∫

ω

k∇θ · ∇Se dx ≤ 0,

without restriction on ω. Thus∇θ ·∇Se must be negative at every point and naturallyfor every configuration. It is then deducted that θ is a decreasing function of Se. Freeto compose θ on the left with an increasing function,6 there is no loss of generalityif we assume that θ = 1/Se, which gives the thermodynamic relation

θ dS = de + pd

(1

ρ

), θ ≥ 0,

in which 1/θ appears as an integrating factor of the differential form de+ pd(1/ρ).For a perfect gas are chosen as usual θ = e and S = log e − (γ − 1)log ρ.

Barotropic models

A model is barotropic if the pressure is, because of an approximation, a functionof the density only. There are three possible reasons: the flow is isentropic or it isisothermal, or again it is the shallow water approximation.

For a regular flow without either viscosity or conduction of heat (that makes upmany of the less realistic hypotheses), we have (∂t + �v · ∇)S = 0: S is constantalong the trajectories. If, in addition, it is constant at the initial instant, we haveS= const. As Se > 0, we can invert the function S(· , ρ): we have e = E (S, ρ),with the result that also p is a function of (S, ρ). In the present context, p must bea function of ρ alone and similarly this is true of all the coefficients of the system,for example α and β. The conservation of mass and that of momentum thus forma closed system of partial differential equations (here again we have taken accountof the newtonian viscosity7):

∂tρ + div(ρ�v) = 0,

∂t (ρvi )+ div(ρvi �v)+ ∂i p(ρ) = div(α(∇vi + ∂i �v))+ ∂i (β div �v)·6 This does not affect Fourier’s law, as k is changed with the result that the product k∇θ is not.7 One more odd choice!


The equation of the conservation of energy becomes a redundant equation.8 Weshall use it as the ‘entropy’ conservation law of the inviscid model. We call this theisentropic model:

∂tρ + div(ρ�v) = 0,

∂t (ρvi )+ div(ρvi �v)+ ∂i p(ρ) = 0.

Its mathematical entropy is the mechanical energy 12ρ(‖�v‖2 + e(ρ)), associated

with the ‘entropy flux’ ρ( 12‖�v‖2+e(ρ))�v+ p(ρ)�v. For a perfect gas, the hypothesis

S = const., states that eγ−1 = cρ and furnishes the state law p = κργ . This, then,is called a polytropic gas.

The isothermal model is reasonable when the coefficient of thermal diffusion islarge relative to the scales of the time and space variables. For favourable boundaryconditions, the entropic balance gives

d

dt

∫�

ρS dx ≥ −∫

�

k∇θ · ∇Se dx =∫

�

k‖∇θ‖2

θ2dx .

According to the conservation laws, we can add to ρS an affine function of thevariables (ρ, ρ�v, E) in the preceding inequality. Meanwhile, experience shows thatthe mapping (ρ, ρ�v, E) �→ ρS is concave.9 We can thus choose an affine functionη0 with the result that η := ρS+η0 is negative. If the domain � is the whole spaceR

d , the fluid being at rest at infinity, we can also take η to be null at infinity. Finally∫�

k‖∇θ‖2

θ2dx ≤ −

∫�

ηt=0 dx ·

The right-hand side is a datum of the problem, supposed finite. If k is large, we seethat it is all right to approach θ by a constant; that it is a constant and not a functionof time is not clear but is currently assumed. Again, the pressure and the viscositybecome functions of ρ only, and the conservation of mass and that of momentumform a closed system: the mechanical energy is taken as the mathematical entropyof the system. For a perfect gas, e = θ is constant, with the result that the state lawis linear: p = κρ.

The isothermal approximation is reasonable enough in certain regimes, because,for a gas, for instance, the thermal effects are always more significant than theviscous effects. A general criterion regarding these approximations is however thatthe shocks of the barotropic models are not the same as those of the Euler equations:the Rankine–Hugoniot condition is different.

8 Or rather incompatible, if we have included the newtonian viscosity.9 In fact, this concavity is the condition for the Cauchy problem of the linearised Euler equations to be well-posed.

It no longer holds if we model a fluid with several phases.

8 Some models

The third barotropic model describes the flow in a shallow basin, that is, in onewhose horizontal dimensions are great with respect to its depth. The domain � isthe horizontal projection of the basin: we thus have d = 1 or d = 2. The fluidis incompressible with density ρ0. We do not take the vertical displacements intoaccount. The variables treated are the horizontal velocity (averaged over the height)�v(x, t) and the height of the fluid h(x, t). The pressure is considered to be the integralof the hydrostatic pressureρ0gz where z is the vertical coordinate. We therefore havep = ρogh2/2. The conservation of mass and that of momentum give the system

∂t (ρ0h)+ div(ρ0h�v) = 0,

∂t (ρ0hvi )+ div(ρ0vi �v)+ 1

2g∂i (ρ0h2) = 0, 1 ≤ i ≤ d.

Comments Dividing by ρ0, we recover the isentropic model of a perfect gas forwhich γ = 2.

We have not taken into account the effects of viscosity and this is an error: they areresponsible for a boundary layer on the base of the basin which implies a resistanceto the motion. That resistance makes itself manifest in the model by a source termin the second equation of the form − f (h, |�v|)vi , with f > 0.

One way of obtaining these equations from the Euler equations is to integrate thelatter with respect to z (but not x). We then make the hypothesis that certain meansof products are the products of means, that is that the vertical variations in ρ and �vare weak.

The relativistic models of a gas, though much more complicated than those whichhave preceded, are also those of systems of conservation laws. We shall not give adetailed presentation here. By way of an example, we shall consider the simplestamong those systems: a barotropic fluid, isentropic, one-dimensional and in specialrelativity; the conversation of mass and that of momentum give

∂t

(p + ρc2

c2

v2

c2 − v2+ ρ

)+ ∂x

((p + ρc2)

v

c2 − v2

)= 0,

∂t

((p + ρc2)

v

c2 − v2

)+ ∂x

((p + ρc2)

v2

c2 − v2+ p

)= 0.

For more general models the reader should consult Taub [102].

1.2 Gas dynamics in lagrangian variables

Writing the equations of gas dynamics in lagrangian coordinates is very complicatedif d ≥ 2; in addition it furnishes a system which does not come into the spirit ofthis book. This is why we limit ourselves to the one-dimensional case (d = 1). We

1.2 Gas dynamics in lagrangian variables 9

shall make a change of variables (x, t) �→ (y, t) which depends on the solution.The conservation law of mass

ρt + (ρv)x = 0

is the only one which makes no appeal to any approximation. It expresses thatthe differential form α := ρ dx − ρv dt is closed and therefore exact.10 We thusintroduce a function (x, t) �→ y, defined to within a constant by α = dy. We havedx = v dt + τ dy, where we have denoted by τ = ρ−1 the specific volume (whichis rather a specific length here).

Being given another conservation law ∂t ui + ∂xqi = 0, which can be writtend (qi dt − ui dx) = 0, we have that

d((qi − uiv) dt − uiτ dy) = 0,

that is

∂t (uiτ )+ ∂y(qi − uiv) = 0.

The system, written in the variables (y, t), is thus formed of conservation laws. Letus look at for example the momentum u2 = ρv. In the absence of viscosity, wehave q2 = ρv2 + p(ρ, e). From this comes

∂tv + ∂y P(τ, e) = 0,

where P(τ, e) := p(τ−1, e). Similarly, for the energy, u3 = 12ρv2 + ρe and q3 =

(u3 + p)v :

∂t

(1

2v2 + e

)+ ∂y(P(τ, e)v) = 0.

The conservation of mass gives nothing new since it was already used to constructthe change of variables. With u1= ρ and q1= ρv, we only obtain the trivial equation1t + 0y = 0. To complete the system of equations for the unknowns (τ, v, e) wehave to involve a trivial conservation law. For example with u4 ≡ 1 and q4 ≡ 0,we obtain

∂tτ = ∂yv.

We note that in lagrangian variables the perfect gas law is written P = (γ − 1)e/τ .If we take into account the thermal and viscous effects, then q2 = ρv2+ p(ρ, e)−

ν(ρ, e)vx . As τvx = vy we obtain

∂tv + ∂y P(τ, e) = ∂y

(ν

τ∂yv

).

10 These assertions are correct even (ρ, ρv) are no better than locally integrable.

10 Some models

Similarly, q3 = (u3 + p)v − νvvx − kθx gives

∂t

(1

2v2 + e

)+ ∂y(Pv) = ∂y

(ν

τ(v∂yv)

)+ ∂y

(k

τ∂yθ

).

Criticism of the change of variables

Although this change of variable is perfectly justified, even if (e, v) is boundedwithout more regularity as well as v−1 (see D. Wagner [110]), it raises a majordifficulty if the vacuum is somewhere part of the space. In this case, the jacobianρ of (x, t) �→ (y, t) vanishes and it is no longer a change of variable. The specificvolume then reduces to a Dirac mass, with norm equal to the length of the intervalof the vacuum. It becomes critical to give sense to the equations (it is nothing otherthan the conservation law of a mathematical difficulty). The equations in euleriancoordinates are also ill-posed in the vacuum: the velocity cannot be defined and thefluxes q2 and q3 are singular. Indeed, returning to the variables u = (ρ, ρv, E), wehave q2 = u2

2/u1 + p, which makes no sense for ρ = 0.

1.3 The equation of road traffic

Let us consider a highway (a unique sense of circulation will be sufficient for ourpurpose), in which we take no account of entries or exits. We represent the vehicletraffic as the motion of a one-dimensional continuous medium, which is reasonableif the physical domain which we consider is very great in length in comparisonwith the length of the cars. In normal conditions, we have a conservation law of‘mass’

∂tρ + ∂xq = 0,

where q = ρv is the flux, or flow, and v is the mean velocity. Unlike the case of afluid there is no conservation law of momentum or of energy. The drivers choosetheir velocities according to the traffic conditions. It results in a relation v = V (ρ)where V is the speed limit if ρ is small. The function ρ �→ V is decreasing andvanishes for a saturation value ρm, for which neighbouring vehicles are bumper-to-bumper. The space of the states is therefore U = [0, qm].

This model is a typical example of a scalar conservation law. The state lawq(ρ)= ρV (ρ) has the form indicated in Fig. 1.1. We notice that each possible valueof the flow corresponds to two possible densities, of different velocities, with theexception of the maximal flow.

1.4 Electromagnetism 11

Fig. 1.1: Road traffic: flux vs density (in France).

A more precise model is obtained by taking the drivers’ anticipation into account.If they observe an upstream increase in the density (respectively a diminution),they show a tendency to brake (respectively to accelerate) slightly. In other terms,v − V (ρ) is of the opposite sign to that of ρx . The simplest state law which takesaccount of this phenomenon is v = V (ρ)− ερx , with 0 < ε� 1, which leads to theweakly parabolic equation

ρt + q(ρ)x = ε(ρρx )x .

1.4 Electromagnetism

Electromagnetism is a typically three-dimensional phenomenon (d = 3), whichbrings vector fields into play: the electric intensity E , the electric induction D, themagnetic intensity H , the magnetic induction B, the electric current j and the heatflux q. Denoting by e the internal energy per unit volume, the conservation lawsare

Faraday’s law

∂t B + curl E = 0,

with which is associated the compatibility condition div B = 0 (absence ofmagnetic charge),

Ampere’s law

∂t D − curl H + j = 0,

conservation of energy

∂t E + div(E ∧ H + q) = 0.

12 Some models

Maxwell’s equations

In the first instance let us neglect the current and the heat flux (which is correct forexample in the vacuum). Combining the three laws, we obtain

∂t e = H · ∂t B + E · ∂t D.

If the system formed by the laws of Faraday and Ampere is closed by the state laws

H = H (B, D), E = E (B, D),

from the conservation of energy it is then deduced that

H (B, D) · dB + E (B, D) · dD

is an exact differential. Following Coleman and Dill [9], we can then postulate theexistence of a function W : R

3 × R3 → R such that

Hj = ∂W

∂ Bj, E j = ∂W

∂ D j, j = 1, 2, 3.

We have e = W (B, D); the conservation laws are called Maxwell’s equations:

∂t B + curl∂W

∂ D= 0, ∂t D − curl

∂W

∂ B= 0.

These lead to Poynting’s formula

∂t W (B, D)+ div(E ∧ H ) = 0,

which shows that W is an entropy of the system, generally convex. Some otherentropies of the system, not convex, are the components of B ∧ D.

Now, taking into account the charge and the heat, the complete model is thefollowing:

∂t B + curl∂W

∂ D= 0, ∂t D − curl

∂W

∂ B= − j,

∂t (W (B, D)+ ε0)+ div(E ∧ H + q) = 0,

where ε0 is the purely calorific part of the internal energy.11 For a regular solutionwe have

∂tε0 + div q = E · j,

where the right-hand side represents the work done by the electromagnetic force

11 We have made the hypothesis that the underlying material is fixed in the reference frame. For a material inaccelerated motion, see for example the following section.

1.4 Electromagnetism 13

(the Joule effect). We notice that transfer between the two forms of energy ispossible. In the vacuum, the current is zero and there is neither temperature, norheat flux; next, following Feynman [21] (Chapter 12.7 of the first part of vol. II),the Maxwell equations are linear in a large range of the variables. The energy W isthus a quadratic form:

W (B, D) = 1

2

(1

µ0‖B‖2 + 1

ε0‖D‖2

).

The constants of electric and magnetic permittivity have the values (in S.I. units)ε0 = (36π · 109)−1 and µ0 = 4π · 10−7. Their product is c−2, the inverse of thesquare of the velocity of light.

In material medium, conducting and isotropic, the state law has the same formbut with constants ε > 0 and µ > 0 of greater value. The number (εµ)−1/2 is againequal to the velocity of propagation of plane waves in the medium. In media whichare poor conductors (dielectrics) the state law is no longer linear. The isotropymanifests itself by the condition

W (RB, RD) = W (B,D), ∀R ∈ O3(R).

This implies the existence of a function w of three variables, such that

W (B, D) = w(‖B‖, ‖D‖, B · D).

Finally, paramagnetic bodies present phenomena of memory (with hysteresis),which do not come into the body of systems with conservation laws.

Plane waves

Henceforth, let us neglect the thermodynamic effects as well as the electric current.For a plane wave which is propagating in the x1-direction we have ∂2 = ∂3 = 0,with the result that ∂t B1 = ∂t D1 = 0. There remain four equations, in which wewrite x = x1, the unique space variable:

∂t B2 − ∂x∂W

∂ D3= 0, ∂t B3 + ∂x

∂W

∂ D2= 0,

∂t D2 + ∂x∂W

∂ B3= 0, ∂t D3 − ∂x

∂W

∂ B2= 0.

Let us look at the simple case in which W is a function of ρ := (‖B‖2 + ‖D‖2)1/2

only. Introducing the functions y := B2 + D3 + i(B3 − D2), z := B2 − D3 +i(B3 + D2), we have yt − (ϕ(ρ)y)x = 0, zt + (ϕ(ρ)z)x = 0. The polar coordinates(r, s, α, β), defined by y = r exp iα and z = s exp iβ, enable us to simplify the

14 Some models

system into

αt − ϕ(ρ)αx = 0, rt − (ϕ(ρ)r )x = 0,

βt + ϕ(ρ)βx = 0, st + (ϕ(ρ)s)x = 0,

with the connection 2ρ2 = r2 + s2.

1.5 Magneto-hydrodynamics

Magneto-hydrodynamics (abbreviated as M.H.D.) studies the motion of a fluid inthe presence of an electromagnetic field. As it is a moving medium, the field actson the acceleration of the particles, while the motion of the charges contributes tothe evolution of the field. This coupling is negligible in a great number of situationsbut comes into action in a Tokamak, a furnace with induction, or in the interior ofa star.

The fluid is described by its density, its specific internal energy, its pressure,and its velocity. If no account is taken of the diffusion processes, we write theconservation laws of mass, of momentum, of energy and Faraday’s law asfollows:

ρt + div(ρv) = 0,

(ρvi )t + div(ρviv)+ ∂

∂xi

(p + 1

2‖B‖2

)− div(Bi · B) = 0, 1 ≤ i ≤ 3,

(ρ

(1

2‖v‖2 + ε

)+ 1

2‖B‖2

)t+ div

(ρ

(1

2‖v‖2 + ε

)+ pv + E ∧ B

)= 0,

Bt + curl E = 0.

We see from these equations that the magnetic field exerts a force on the fluidparticles and contributes to the internal energy of the system. The fact that theelectric field does not is the result of an approximation, the same as we made indisregarding Ampere’s law.

There are two state laws: on the one hand p= p(ρ, e), which always has the formP = (γ − 1)ρe for a perfect gas; on the other hand, E = B ∧ v. This expresses alocal equilibrium: the acceleration of the particles taken individually is of the formf + (E + v ∧ B)/m where m is the mass of a particle of unit charge and f is theforce due to the binary interactions. As m � 1 and since the velocity of the fluidremains moderate,12 E + v ∧ B is very small.

12 Under this hypothesis, the fluid is seen as a dielectric.

1.5 Magneto-hydrodynamics 15

For a sharper description, we take account of the processes of diffusion: theviscosity, Fourier’s law certainly, even Ohm’s law:13

E = B ∧ v + η j + χ ( j ∧ B).

Finally we take Ampere’s law into account, but we neglect in it the derivative ∂t Econsidering that E varies slowly in time:

j = curl B.

Each of the phenomena which we come to take into account is studied by addingone or several of the second order terms in the laws of conservation. Whether thefactors such as η, χ , k, α and β can be considered as small or not depends on thescale of the problems studied.

Plane waves in M.H.D.

Again, we consider the solutions for which ∂2 = ∂3 = 0 and β := B1 is constant.This behaviour is established when the initial condition satisfies it. In the sequelwe write

z := v1, w := (v2, v3), b := (B2, B3), x := x1.

In Faraday’s law Bt + curl E = 0, the component in the direction of x1 and thecompatibility condition div B = 0 are trivial. There remain seven equations in placeof eight, which is correct since B1 is no longer an unknown:

∂tρ + ∂x (ρz) = 0,

∂t (ρz)+ ∂x

(ρz2 + p(ρ, e)+ 1

2‖b‖2

)= 0,

∂t (ρw)+ ∂x (ρzw − ρb) = 0,

∂t

(ρ

(1

2z2 + 1

2‖w‖2 + e

)+ 1

2‖b‖2

)+ ∂x

(ρz

(1

2z2 + 1

2‖w‖2 + e

)

+ (p + ‖b‖2)z − βb · w)= 0,

∂t b + ∂x (zb − βw) = 0.

The system is simpler in lagrangian coordinates (y, t), defined by dy= ρ(dx−z dt) – see §1.2. Denoting by τ = 1/ρ the specific volume, these equations are

13 Which replaces the hypothesis E = B ∧ v.

16 Some models

transformed to

τt = zy,

zt +(

p(1/τ, e)+ 1

2‖b‖2

)y= 0,

wt − βby = 0,(1

2z2 + 1

2‖w‖2 + e + 1

2‖b‖2

)t+

((p + 1

2τ‖b‖2)z − βb · w

)y= 0,

(τb)t − βwy = 0.

A combination of these equations gives, for a regular solution, et + pzy = 0 oragain et + pτt = 0, that is to say

S(τ, e)t = 0,

S being the thermodynamic entropy (θ dS = de+ p dτ ). The analogous calculationin eulerian variables yields the transport equation

(∂t + z∂x )S = 0,

which shows that ρS is an entropy, in the mathematical sense, of the model.

A simplified model of waves

Let us consider the system of plane waves of M.H.D. in eulerian variables to fixthe ideas, with β �= 0. It admits in general seven distinct velocities of propagationλ1 < λ2 < · · · < λ7 among which λ4 = z, λ2 = z− βρ−1/2, and λ6 = z+ βρ−1/2

(λ2 and λ6 are the speeds of the Alfven waves). The four remaining speeds are theroots of the quartic equation

((λ− z)2 − c2)((λ− z)2 − β2/ρ) = (λ− z)2‖b‖2/ρ,

c= c(ρ, e) being the speed of sound in the absence of an electromagnetic field.However, when b vanishes, we have λ3 = λ2 and λ5 = λ6. This coincidence oftwo speeds and the non-linearity of the equations induce a resonance. For waves ofsmall amplitude, this phenomenon can be described by an asymptotic development.

First of all, a choice of a galilean frame of reference allows the assumptionthat the base state u0, constant, satisfies w0 = 0 (we already have b0 = 0) andz0√

ρ0 = β0. We thus have λ2(u0)= λ3(u0)= 0: the resonance occurs along curves(in the physical plane) with small velocities. If u − u0 is of the size ε � 1 thisvelocity is also of the order of ε, which leads to the change of the time variable

1.6 Hyperelastic materials 17

s := εt , so ∂t = ε∂s . The other hypotheses are

on the one hand ρ = ρ0 + ερ1 + · · ·, z = z0 + εz1 + · · ·, e = e0 + εe1 + · · ·,on the other hand w=√ε · (w1(s, x)+εw2(s, t, x)+· · ·), b = √ε · (b1(s, x)+

εb2(s, t, x)+ · · ·). We note that, although√

ε is great compared with ε, thesehypotheses ensure that λ2 and λ3 are of the order of ε.

The examination of the terms of order ε in the conservation laws shows that ρ1, z1,e1 and w1 are explicit functions of b1. Finally, the terms of order ε3/2 in Faraday’slaw, averaged with respect to the slow variable t to eliminate b2, furnish a systemwhich governs the evolution of U := b1:

∂tU + σ∂x (‖U‖2U ) = 0, (1.3)

where σ is a constant which depends only on (ρ0, e0). In this book, we shall copi-ously use the system (1.3) to illustrate the various theories, but we shall also makeappeal to a slightly more general one:

∂tU + ∂x (ϕ(‖U‖)U ) = 0

where ϕ: R+ → R is a given smooth function.

1.6 Hyperelastic materials

We shall consider a deformable solid body, which occupies, at rest, a referenceconfiguration which is an open set � ⊂ R

d . We describe its motion by a mapping(x, t) �→ (y, t), � → R

d , where y is the position at the instant t of the particlewhich was situated at rest at x in the reference configuration. We define the velocityv: � → R

d and the deformation tensor u: � → Md (R) by

v = ∂y

∂t, uα j = ∂yα

∂x j.

In the first instance we write the compatibility conditions

∂t uα j = ∂ jvα, ∂kuα j = ∂ j uαk, 1 ≤ α, j, k ≤ d.

A material is said to be hyperelastic if it admits an internal energy density of the formW (u) and if the forces due to the deformation derive from this energy (principle ofvirtual work):

f = ( f1, . . . , fd ), fα = − δE

δyα

.

Here δ/δy denotes the variational derivative of E [y] := ∫W (∇ y) dx :

fα =∑j=1

∂ j∂W

∂uα j.

18 Some models

The fundamental law of dynamics is written

∂tvα = fα + gα,

where g represents the other forces, due to gravity or to an electromagnetic field(but here we do not consider any coupling). Finally, U := (u, v) obeys a system ofconservation laws of first order (for which n = d(d + 1))

∂t uα j = ∂ jvα, 1 ≤ α, j ≤ d,

∂tvα =d∑

k=1

∂k∂W

∂uαk+ gα, 1 ≤ α ≤ d.

These equations can be linear, when W is a quadratic polynomial, but this type ofbehaviour is not realistic. In fact, the energy is defined only for u ∈GLd (R) withdet(u) > 0 (the material does not change orientation), and must tend to infinity whenthe material is compressed to a single point:

limu→0n

W (u) = +∞.

The models of elasticity are thus fundamentally non-linear. Other restrictions onthe form of W are due to the principle of frame indifference:

W (u) = W (Ru), R ∈ SOd (R), (1.4)

and, if the material is isotropic,

W (u) = W (uR), R ∈ SOd (R). (1.5)

From (1.4), there exists a function w: S+ → R, on the cone S+ of positive definitesymmetric matrices such that

W (u) = w(uTu).

If, in addition, (1.5) holds, then the function S �→w(S) depends only on the eigen-values of S.

We find an entropy of the system in writing the conservation of energy:

∂t

(1

2‖v‖2 +W (u)

)=

d∑α, j=1

∂ j

(vα

∂W

∂uα j

).

The total mechanical energy (v, u) �→ 12‖v‖2+W (u) is not always convex.14 How-

ever, it is in the ‘directions compatible’ with the constraint ∂kuα j = ∂ j uαk . In other

14 There are obstructions due to the invariances mentioned above and to the fact that W tends to infinity at 0 andat infinity. See [8] Theorem 4.8-1 for a discussion.

1.7 Singular limits of dispersive equations 19

words, W is convex on each straight line u + Rz where z is of rank one (theLegendre–Hadamard condition). This reduced concept of convexity is appropri-ate for problems with constraints. In particular, the mechanical energy furnishesan a priori estimate. A constitutive law currently used is that of St Venant andKirchhoff:

w(S) = 1

2λ(Tr E)2 + µ Tr(E2), E := 1

2(S − In).

On the other hand, other entropies do not have this convexity property; for allk ≤ d

∂t (v · uk) = ∂k

(1

2‖v‖2 −W (u)

)+

d∑j=1

∂ j

(uαk

∂W

∂uα j

).

Strings and membranes

More generally, we can consider a material for which x ∈ � (with � ⊂ Rd ) but with

(y, t) ∈ Rp with p ≥ d. The case p= d is that described above. When p = 3 and

d = 2 it is a mater of a membrane or a shell, while p ≥ 2 and d = 1 corresponds toa string. For a membrane or a string, the equations are the same as in the precedingparagraph, but the Greek suffixes go from 1 to p instead of from 1 to d. There arethen n = p(d + 1) unknowns and just as many equations of evolution.

Let us look at the case of string: u is a vector and W (u) = ϕ(‖u‖), because offrame indifference, ϕ being a state law. We have

∂W

∂uα

= 1

rϕ′(r )uα, r := ‖u‖,

with the result that dW is the product of ϕ′ (called the tension of the string) by theunit tangent vector to it: r−1u. There are four or six equations:

ut = vx , vt = (r−1ϕ′(r )u)x + g.

1.7 Singular limits of dispersive equations

The systems of conservation laws which are presented here proceed from com-pletely integrable dispersive partial differential equations. We take as an examplethe Korteweg–de Vries (KdV) equation

ut + 6uux = uxxx , (1.6)

but there are others, of which the best known is the cubic non-linear Schrodingerequation.

20 Some models

Certain solutions of (1.6) are progressive periodic waves: they have the formu = u(x − ct) with u′′′ = 6uu′ − cu′, with the result that

1

2u′2 = u3 − 1

2cu2 − au − b,

where a and b are constants of integration. The triplet (a, b, c) defines a uniqueperiodic solution (to within a translation) when the polynomial equation P(X ) :=X3− 1

2cX2− aX − b = 0 has real roots: u1 < u2 < u3. We then have min u(x) =u1 and max u(x) = u2.

What are of interest here are such periodic solutions of the KdV equation, whichare, in first approximation, modulated by the slow variables (s, y) := (εt, εx) with0 < ε � 1.

uε(x, t) = u0(a(s, y), b(s, y), c(s, y); x − c(s, y)t)+ εu1(s, y, x, t)+ O(ε2).

We require that u1 and u0 be smooth functions and that u1 be almost periodic withrespect to (x, t).

The choice of the parameters (a, b, c) is not the most practical from the pointof view of calculations. We proceed to construct another set, with the aid of theexpressions

i1 := u, i2 := 1

2u2, i3 := 1

2u2

x + u3.

These are invariants of the KdV equation in the sense that sufficiently smoothsolutions15 satisfy

∂t ik + ∂x jk = 0, 1 ≤ k ≤ 3,

with

j1 = 3u2 − uxx , j2 = 2u3 + 1

2u2

x − uuxx ,

j3 = 1

2u2

xx + 6uu2x − uxuxxx − 3u2uxx + 9

2u4.

Let (a1, a2, a3) ∈ R3 be a triplet such that there exists a function w ∈ H1(S1),

S1 = R/Z, with∫ 1

0w dξ = a1,

∫ 1

0

1

2w2 dξ = a2,

∫ 1

0w3 dξ < a3.

Then the set X (a1, a2, a3) of the couples (v, Y ) ∈ H1(S1)× (0,+∞) such that∫ 1

0v dξ = a1,

∫ 1

0

1

2v2 dξ = a2,

∫ 1

0

(v3 + 1

2Y 2v′2

)dξ = a3

15 There is no interest in the question of smoothness here; let us say that it is does not cause trouble.

1.7 Singular limits of dispersive equations 21

is not empty. It corresponds (via (v, Y ) �→ u(·/Y )) to the periodic functions ofH 1(R), the period not being fixed a priori, with the prescribed means

〈u〉 = a1,1

2〈u2〉 = a2,

⟨u3 + 1

2u′2

⟩= a3.

It can be shown without difficulty that the mapping

X (a1, a2, a3) → R, (v, Y ) �→ Y,

attains its lower bound (strictly positive), which is denoted by S(a1, a2, a3). Anoptimal pair (v, S) defines, via u(x) := v(x/S), a progressive periodic wave of theKdV equation. In general, (v, S) is unique to within a translation, with the resultthat if a differential polynomial P is given, then the mean 〈P(u, ux , . . . , ∂

mx u)〉 is

perfectly determined and depends only on (a1, a2, a3). Those which we shall needare the functions

Jk(�a) = 〈 jk(u)〉.For example,

J1(�a) = 〈3u2 − uxx 〉 = 〈3u2〉 = 6a2.

The two other functions have much less explicit expressions, which involve ellipticfunctions.

Let us denote by U (�a, x, t) ‘the’ periodic solution such that 〈ik〉 = ak , U being ofclass C

∞ with respect to each of its five variables. The modulated solutions whichwe consider are written

uε(x, t) = U (�a(εx, εt); x, t)+ εu1(εx, εt ; x, t)+ O(ε2).

Our purpose is to determine the evolution of �a as a function of (y, s). We write forthat the conservation laws

∂t ik[uε]+ ∂x jk[uε] = 0, 1 ≤ k ≤ 3.

In these, the terms of order ε0 are absent because (x, t) �→ U is an exact solutionof the KdV equation. There remain

∂sik[U ]+ ∂y jk[U ]+ ∂t · · · + ∂x · · · = O(ε),

where the imprecise expressions are smooth and almost periodic in (x, t). Weeliminate their derivatives in x or t by taking the mean in Bohr’s sense (withrespect to (x, t)) of this equality:

∂s〈ik[U ]〉 + ∂y〈 jk[U ]〉 = O(ε).

22 Some models

As the left-hand side does not depend on ε, there only remains

∂s �ak + ∂y Jk(�a) = 0, 1 ≤ k ≤ 3, (1.7)

which makes up a closed system of three conservation laws.

Remarks We do not have to use the solutions of (1.7) before the formation of shocks.In fact, if �a is discontinuous along a curve, the asymptotics cannot be justified andthe periodic solutions have to be replaced by more complicated, almost periodicsolutions. The equations of modulation are then made up of 2p + 1 conservationlaws in place of three (see [61]).

The validity of the asymptotics is closely linked to the hyperbolicity of (1.7),which allows it to have local smooth solutions. This property has been studied byLevermore [63].

The invariants i1, i2, i3 are only the first of a denumerable list (ik)k≥1, where ik isa polynomial in (u, ux , . . . , ∂

k−2x u). The expressions Ik(�a) := 〈ik(U, . . . ,U (k−1))〉

are thus entropies of the system (1.7): ∂s Ik + ∂y Jk = 0. Other entropies exist, inparticular

∂s S(a)− ∂y(cS) = 0,

where c = c(�a) is the speed of the progressive wave U (see the book by Whitham[112]).

1.8 Electrophoresis

Electrophoresis is a procedure of separating ions in an aqueous solution, by meansof an electromagnetic field. We refer the reader to the article by Fife and Geng [22]for more general models than that presented here.

The medium is one-dimensional (d = 1). The ions represent a negligible fractionof the total mass, with the result that we can suppose the solution to be at rest. Eachkind of ion (there are n + 1) has density ui (x, t) ≥ 0 for 0 ≤ i ≤ n. The unknownof the problem is U := (u0, . . . , un). The flux of mass of the ion of the ith kind is

fi := µi zi Eui − di∂xui

where µi > 0 is the mobility, zi the charge and di ≥ 0 the diffusivity (these numbersare constants). The electric current is thus

J = −z · f = −n∑

i=1

zi fi .

The electric field is given by Ampere’s law β∂x E = −z · u. When 0 < β � 1, we

1.8 Electrophoresis 23

are led to make the hypothesis of electric neutrality:

z · u =(

n∑i=1

ziui

)≡ 0. (1.8)

The conservation law of the i th type of ion is

∂t ui + ∂x fi = 0.

We therefore deduce from (1.8) that ∂x (z · f ) = 0, or ∂x J = 0. In this context thecurrent J is imposed by the experimentalist; it will, in general, be constant. It is adatum of the problem, which allows the expression of E as a function of U via

En∑

j=1

µ j z2j u j =

n∑j=1

z jd j∂xu j − J.

Finally the vector (u0, . . . , un) obeys the system of conservation laws

∂t ui − ∂xµi zi Jui∑nj=0 µ j z2

j u j= ∂x

(n∑

j=0

bi j∂xu j

), 1 ≤ i ≤ n, (1.9)

where

bi j = diδji −

µi zi z j d jui∑nk=0 µk z2

kuk.

We notice that the above system is not completely parabolic since zT B = 0; thiscomes from the electric neutrality, which renders the unknowns dependent on eachother. We obtain a system conforming more with the general body of this bookin eliminating one of the unknowns, for example u0, and writing the conservationlaws for u := (u1, . . . , un).

Let us look at the example where z0 = 1 and zi = −1 otherwise. Then u0 =∑ni≥1 ui . If we neglect the diffusion of the ions (di = 0), the system becomes

∂tvi + ∂xαivi∑nk=1 vk

= 0, 1 ≤ k ≤ n,

where αi =µi J > 0 and vk := (µk + µ0)uk . This system has very strong proper-ties, which render the study of the Cauchy problem easy (see Chapter 13). Whendiffusion is taken into account, the right-hand side has to be replaced by

∂x

( n∑j=1

βi j∂xu j

),

with βi j = bi j + bi0. In the equi-diffusive case in which di = Dµi , with D a

24 Some models

constant, we have

βi j = Dµi

(δ

ji +

µ0 − µ j

Sui

), S :=

n∑k=1

(µ0 + µk)uk .

It is easily shown (this is a variant of Gerschgorin’s theorem) that the eigenvaluesof β all lie in the right half-plane R z > 0, since D > 0 and ui > 0 for all i . Thesystem is then parabolic. It is one of the rare natural examples where the diffusionmatrix is invertible.

Let us mention another procedure for the separation of the constituents of amixture, which makes use of gravity and a temperature gradient in a column, whichfurnishes equations very close to those we have established here. This procedure,called chromatography, separates the solvents according to their molar masses.

2

Scalar equations in dimension d = 1

In this chapter we consider a scalar unknown function u(x, t). The equation gov-erning it is a conservation law, completed by an initial condition:

ut + f (u)x = 0, x ∈ R, t > 0,

u(x, 0) = u0(x), x ∈ R.

(2.1)

2.1 Classical solutions of the Cauchy problem

A classical solution of the Cauchy problem is a solution of class C1 for t > 0, conti-

nuous for t ≥ 0, which satisfies (2.1) pointwise. When u0 is also of class C1, a

classical solution is of class C1 for t ≥ 0. To avoid the related phenomena of

propagation with infinite speed, we suppose in addition that u0 is bounded on R.

The linear case

First of all let us examine the case in which f is given by the formula f (u) = cu,c being a constant. Then

d

dt(u(x + ct, t)) = (ut + cux )(x + ct, t) = 0.

Thus, t �→ u(x + ct, t) is a constant, with value u0(x). Replacing x by x − ct weobtain

u(x, t) = u0(x − ct)

for the unique solution of (2.1). For all initial data, there therefore exists one andonly one solution which has the same regularity.

25

26 Scalar equations in dimension d = 1

Non-linear case. The method of characteristics

We now abandon the linear hypothesis. The flux f is a given function of class C∞.

We write c(u) = f ′(u).Let u ∈ C

1 be a solution of the Cauchy problem. We define the characteristiccurves, or simply the characteristics, in the band R× [0, T ] as the integral curvest �→ (X (t), t) of the differential equation

dX

dt= c(u(X, t)).

In the linear case, c is constant with the result that the characteristics are a prioriknown straight lines. This is no longer true in the general case, where they dependon the solution itself. Let us calculate

d

dtu(X (t), t) = dX

dtux (X, t)+ ut (X, t) = (ut + c(u)ux )(X, t),

the last equality being the conservation law. Thus, u is constant along each charac-teristic, taking the value u0(y) where (y, 0) is the base of the latter. It follows thatthe slope of this curve has the constant value c(u0(y)). This is thus a straight line:

X (t) = y + tc(u0(y)).

The method of characteristics, which considers the solution of (2.1) by leading toan algebraic equation, is therefore the following.

Being given (x, t) ∈ R× [0, T ], find y, a solution of the equation

y + tc ◦ u0(y) = x .

Then put u(x, t) = u0(y).

Let Ft: R→R be the function defined by Ft (y) = y + tc ◦ u0(y). If u0 iscontinuous, so also is Ft (y). Since Ft (±∞) = ±∞, the mean value theoremensures the existence of a value y such that Ft (y) = x . But this non-linear equationcan have several solutions, thus preventing the construction of a classical solution.We make that precise in the statement of the proposition below.

Proposition 2.1.1 Let u0 ∈ C1(R) be, together with its derivative, bounded. We

define T ∗ = +∞ if c ◦ u0 is increasing,

T ∗ = −(

infd

dxc ◦ u0

)−1

otherwise. Then (2.1) possesses one and only one solution of class C1 in the band

R× [0, T ∗) and does not possess one in any greater band R× [0, T ].

2.2 Weak solutions, non-uniqueness 27

Proof Let p = c ◦u0. We have Ft = 1+ tp′ ≥ 1− t/T ∗ > 0, for t ∈ [0, T ∗). Thesolution of Ft (y) = x is then unique since Ft is strictly increasing. In addition,the implicit function theorem ensures that (x, t) �→ y(x, t) is of class C

1. Let usthen verify that u(x, t) defined by u = u0(y(x, t)) is a solution of (1.1). First of allwe have y(x, 0) = x so that u(x, 0) = u0(x). Then, in differentiating the equationFt (y) = x , we obtain Ft ′(y)yt = −p(y) and Ft ′(y)yx = 1. Hence

Ft ′(y(x, t))(ut + c(u)ux ) = u′0(y)Ft ′(y)(yt + c(u)yx )

= u′0(y)(−p(y)+ c(u0(y))) = 0.

Blow-up in finite time

When classical solutions are provided by the method of characteristics there is noother that can be constructed, at least for 0 ≤ t < T ∗. Let us now show that thatsolution cannot be prolonged beyond that.

Let T > 0 be such that there exists a regular solution on R× [0, T ] . We differ-entiate the quantity v = c′(u)ux along the characteristics. The following formulais obtained by differentiating the conservation law with respect to x .

d

dtv(X, t) = −c′(u)2u2

x = −v2.

This is an equation of Ricatti type. If T ∗ < ∞, there exist values of y for whichp′(y) = c′(u0(y))u′0(y) < 0. For these, the function v(X (t), t) blows up at the time−1/p′(y) because its initial value is p′(y). More precisely, for t < max(T, T ∗)

infx∈R

v(x, t) = (t + (inf p′)−1)−1.

Thus, we have T ≤ −(inf p′)−1 which proves the proposition.

Remark The phenomenon of blow-up which we have just described is of non-linearorigin since it does not occur in the linear case. Notice that the hypothesis T ∗ < ∞supposes that c is not constant. The exercises 2.1 and 2.2 also describe the effectsof the non-linearity.

2.2 Weak solutions, non-uniqueness

Classical solutions are not sufficient to resolve (1.1), so we turn to weak solutions,that is say solutions in the sense of distributions. This choice is consistent withthe underlying physics of this type of problem. In particular, the most interesting


solutions being piecewise continuous, we find indirectly from distributions thetransmission conditions which are introduced, for example, into fluid mechanics.

To give a meaning to the conservation law, it is enough that u and f (u) be distri-butions. Since f , in general, is not linear, we must suppose that u is a measurablefunction so that f (u) is defined pointwise. We then shall say that u is a weak so-lution of the equation ut + f (u)x = 0 in an open set ω of R

2 if u ∈ L1loc(ω),

f (u) ∈ L1loc(ω) and if for every test function ϕ ∈ D (ω), we have

∫∫ω

(u∂ϕ

∂t+ f (u)

∂ϕ

∂x

)dx dt = 0.

We shall say that u is a weak solution of the Cauchy problem (2.1) in the bandQ = R × [0, T ] if u ∈ L1

loc(Q), f (u) ∈ L1loc(Q), and if for all test functions

ϕ ∈ D (Q).

∫∫Q

(u∂ϕ

∂t+ f (u)

∂ϕ

∂x

)dx dt +

∫R

u0(x)ϕ(x, 0) dx = 0.

In the account given below, we consider the simple case in which u ∈ L∞(Q),which ensures that u ∈ L1

loc(Q), f (u) ∈ L1loc(Q), and which is consistent with

the maximum principle which we shall establish. This choice is nevertheless nota natural one once we consider systems of conservation laws since the max-imum principle is then an exception. In addition, the quantities which have aphysical meaning are those that are involved in Green’s formula, that is

∫u dx

(mass in a domain at a given instant) and∫

f (u) dt (flux of mass across a bound-ary during a given time). We thus see that a natural space for u is C ((0, T );L1

loc(R)).The reader should be able easily to verify that the notion of a weak solution

extends that of a classical solution: every classical solution of (1.1) is also a weaksolution.

The Rankine–Hugoniot condition

Let us consider a pair (u, q) of functions, piecewise continuous in the domain ω,whose line of discontinuity lies along a regular curve �, which separates ω into twoconnected components ω±. We assume that (u, q) is of class C

1 in ω− and in ω+.Finally, we denote by u+(x, t) the limit of u(y, s) when (y, s) tends to (x, t) ∈ �

and stays in ω+. In the same way we define q+(x, t) then u−(x, t) and q−(x, t)along �, and we write [h](x, t) = h+(x, t) − h−(x, t), the jump across � of eachpiecewise continuous function h.

We now wish to translate into simple terms the equation ut + qx = 0.


Fig. 2.1: Curves of discontinuity, unit normals.

Lemma 2.2.1 Under the above hypothese, the pair (u, q) satisfy the equation inthe distributional sense in ω if and only if

(1) On the one hand, u and q satisfy the equation pointwise in ω+ and ω−,(2) On the other hand, the jump condition [u]nt + [q]nx = 0 is satisfied along �,

where n is a unit normal vector to � in (x, t).

Proof Let us begin with the necessary condition. Let (u, q) be a solution of theequation in ω. We have ∫∫

ω

(u∂ϕ

∂t+ q

∂ϕ

∂x

)dx dt = 0.

First of all choosing test functions whose support is in ω−, we see that (u, q) is aweak solution in ω−. In the same way we have the result for ω+. With a generaltest function, we calculate the integral with the aid of Green’s formula:

0 =(∫∫

ω−+

∫∫ω+

)(u∂ϕ

∂t+ q

∂ϕ

∂x

)dx dt

=∫

∂ω−

(u−n−t + q−n−x

)ϕ ds −

∫ω−

ϕ(ut + qx ) dx dt

+∫

∂ω+

(u+n+t + q+n+x

)ϕ ds −

∫ω+

ϕ(ut + qx ) dx dt.

In the above formula, ∂ω± denote the boundaries of the domains ω±, and n± aretheir unit normal vectors, pointing outwards (see Fig. 2.1). The preceding argumentshows that the integrals over ω± are null. On the other hand, the border of ω− ismade up of part of the boundary of ω on which ϕ is zero and also of �. Theremaining part is therefore∫

�

(u−n−t + q−n−x

)ϕ ds +

∫�

(u+n+t + q+n+x

)ϕ ds = 0.

However, n− = −n+ along �, with the result that∫�

([u]nt + [q]nx )ϕ ds = 0.

This equality being true for every smooth function (say, of class C1) in ω it is true

when we replace ϕ by any smooth function, defined and with compact support on�. From this we deduce the jump condition (2).

Conversely, the same calculation, taken in the reverse sense, shows that theseconditions are sufficient.

When q = f (u), the jump condition is called the Rankine–Hugoniot condi-tion. Let M be a Lipschitz constant of f in the larger interval [a, b] in which utakes its values. We have |[ f (u)]| ≤ M |[u]| with the result that |nt | ≤ M |nx |.The curve of discontinuity can thus be parametrised by the variable t in theform

� = {(X (t), t): t ∈ ]t1, t2[}.The Rankine–Hugoniot condition can then be written in the definitive form

[ f (u)] = dX

dt[u].

We see again that the speed c plays a role like that in the method of character-istics, since provided that [u] is not null, the mean value theorem is given by

dX

dt= c(u(t))

where u(t) is a number between u−(X (t), t) and u+(X (t), t). In particular, whenthe amplitude [u] of the discontinuity is weak, its speed approaches that of theneighbouring characteristics.

Examples (1) The simplest discontinuous solutions are of the form

u ={

u−, x < σ t,

u+, x > σ t,

where σ = ( f (u+)− f (u−))/(u+ − u−). Indeed, u satisfies the equation triviallyoutside of the straight line x = σ t .

(2) For Burgers’ equation f (u) = 12u2, the speed of propagation of the disconti-

nuities is dX/dt = 12 (u+ + u−).

(3) For the model of road traffic, this speed is a new concept, distinct from thespeed of the vehicles. The discontinuities arise from the sudden variations in thedensity of the traffic, for example between traffic moving below a point and blocked

Fig. 2.2: Non-trival solution of a trivial Cauchy problem.

above. This point actually moves since it is nothing but X (t) except if the flow ofvehicles is the same below as above, the growth of the speed compensating exactlythe diminution in the density.

Non-uniqueness for the Cauchy problem

To construct a Cauchy problem which admits more than one weak solution, weclearly must choose a non-linear flux. We choose the simplest example, the Burgersequation. If u ≡ 0, we have a trivial classical solution u ≡ 0. Here is anothersolution, using the example treated above (see Fig. 2.2):

u(x, t) =

0,

−2p,

2p,

0,

x < −pt,

−pt < x < 0,

0 < x < pt,

pt <x .

This example gives, in fact, an infinity of solutions of the same Cauchy problem,parametrised by the positive real number p. We can, for the moment, conclude thatbetween the framework of the classical solution for which the existence is missing,and that of weak solutions whose uniqueness is not guaranteed, it is necessary tofind an intermediate theory for which the Cauchy problem is well-posed in the senseof Hadamard, that is to say satisfies the following three conditions.

(1) For each given initial datum in a function space Y , the Cauchy problem admitsa solution in a function space Z which is contained in L∞loc(R; Y ).

(2) That solution is unique.(3) The mapping Y → Z which with a given initial datum associates a solution is

continuous.

2.3 Entropy solutions, the Kruzkov existence theorem

Approximate solutions; entropy inequalities

The examination of various models has suggested that the conservation laws areonly a simplification of a more complex reality and, for example, should better bewritten

ut + f (u)x = εuxx . (2.2)

Here the positive number ε is a diffusion coefficient, ε << 1. The Cauchy problemfor (2.2) can be shown to have one and only one classical solution uε which satisfiesthe maximum principle. Let us assume also that the sequence {uε} converges almosteverywhere to a function u when ε → 0 (this is proved under sufficiently generalhypotheses [86], see also Theorem 5.4.1).

Lemma 2.3.1 Suppose that u0 ∈C b (R). If uε(x, t) → u(x, t) almost everywherein Q = R× (0, T ] then u is a weak solution of (2.1).

Proof As has been noted, uε is a classical solution, with the result that all theintegrations by parts are admissible. In fact, uε ∈C

∞ (Q) ∩ C (Q) and for everytest function ϕ with support in R× (−∞, T )

0 =∫∫

Qϕ(εuε

xx − uεt − f (uε)x

)dx dt

=∫∫

Q(uε(εϕxx + ϕt )+ f (uε)ϕx ) dx dt +

∫R

u0(x)ϕ(x, 0) dx .

From the maximum principle, and since u0 ∈ C b (R), the family {uε} is boundedby a constant. The theorem of dominated convergence thus ensures that, whenε → 0, ∫∫

Quε(εϕxx + ϕt ) dx dt →

∫∫Q

uϕt dx dt

and ∫∫Q

f (uε)ϕx dx dt →∫∫

Qf (u)ϕx dx dt.

Finally, we obtain the desired result∫∫Q

(uϕt + f (u)ϕx ) dx dt +∫

R

u0(x)ϕ(x, 0) dx = 0.

The following calculus leads to a radically new procedure whose importance is suchthat it dominates the whole theory of hyperbolic systems of conservation laws: the

2.3 Entropy solutions, the Kruzkov existence theorem 33

entropy inequality, which is one of the means of recognising, from among the weaksolutions, the solutions of physical origin. For a scalar equation, this criterion has theadvantage, beyond taking account of a residual diffusion, of resolving an essentialmathematical problem, the uniqueness of the Cauchy problem, while preservingthe existence.

The concept of entropy, or rather the notion of the entropy–entropy-flux pair,refers to the pair of regular functions (E, F) defined on the space of the states u, forwhich every classical solution of ut + f (u)x = 0 also satisfies E(u)t + F(u)x = 0.For the moment, let us say that in the scalar case, every regular function E is anentropy of which the corresponding flux is given to within a constant by F ′ = f ′E ′.If E is convex, then uε satisfies

E(uε)t + F(uε)x = E ′(uε)(uε

t + f (uε)x) = εE ′(uε)uε

xx

= εE(uε)xx − εE ′′(uε)(uε

x

)2 ≤ εE(uε)xx .

Integrating this inequality, multiplied by a test function with positive values,over Q :

0 ≤∫∫

Qϕ(εE(uε)xx − E(uε)t − F(uε)x ) dx dt

=∫∫

Q(E(uε)(εϕxx + ϕt )+ F(uε)ϕx ) dx dt +

∫R

E(u0(x))ϕ(x, 0) dx

→∫∫

Q(E(u)ϕt + F(u)ϕx ) dx dt +

∫R

E(u0(x))ϕ(x, 0) dx,

with the preceding hypotheses.

Proposition 2.3.2 Under the hypotheses of Lemma 2.3.1, the solution u of (2.1)also satisfies the following inequalities, for all pairs (entropies = E, flux = F)with E continuous and convex:∫∫


∫R

E(u0(x))ϕ(x, 0) dx ≥ 0,

for all ϕ ∈ D (R× (−∞, T )), and ϕ ≥ 0.

Proof It only remains to pass from convex entropies of class C2 to continuous

convex entropies, and then defining what an entropy flux is in this context.Let E be a convex function. Then E is locally a uniform limit of convex functions

of class C∞, for example functions En = E ∗ ρn , convolution products by ρn(s) =

nρ(ns) where ρ ∈ D (R). The convexity is assured by ρ ≥ 0. Let Fn be theassociated flux, fixed by

Fn(s) =∫ s

0f ′(y)E ′n(y) dy.

we have

Fn(s) = f ′(s)En(s)− f ′(0)En(0)−∫ s

0f ′′(y)En(y) dy,

which shows that Fn converges locally uniformly to the continuous function

F(s) = f ′(s)E(s)− f ′(0)E(0)−∫ s

0f ′′(y)E(y) dy.

The entropy inequality, stated in the proposition, is already true for the pairs(En, Fn). A new passage to the limit when n → ∞ shows that it is still true for(E, F).

Definition 2.3.3 We say that a weak solution of (2.1) is an entropy (or admissible)solution if it satisfies the entropy inequalities for every convex continuous entropyE of flux F :∫∫


∫R

E(u0(x))ϕ(x, 0) dx ≥ 0 (2.3)

for all ϕ ∈ D+(R× (−∞, T )).

We note that taking into account that test functions may not take the value zero onR× {0} is essential in the entropy inequalities. If, as in certain existing articles orbooks, we suppress the initial integral in (2.3) in restricting the entropy inequalitiesto test functions ϕ ∈ D

+(R × (0, T )), we lose the property of uniqueness (seeTheorem 2.3.5) in letting abnormal solutions continue to exist. A significant exam-ple is Exercise 2.12. As in the definition of a weak solution of the Cauchy problem,(2.3) expresses at the same time an initial condition, in the form of an inequality,and a partial differential relation in the open set R × (0, T ). In using the formal-ism of distributions and that of the traces of certain functional spaces, these twoconditions can be written

trt=0E(u) ≤ E(u0) = E(trt=0u), (2.4)

E(u)ϕt + F(u)ϕx ≤ 0. (2.5)

Let k ∈ R, the function u �→ |u − k| is convex and continuous, its flux beingequal (to within a constant) to ( f (u)− f (k)) sgn(u − k), where

sgn(s) =

1, s > 0,

0, s = 0,

−1, s < 0.


An entropic solution thus satisfies the inequalities∫∫Q

(ϕt |u − k| + ϕx ( f (u)− f (k)) sgn(u − k)) dx dt (2.6)

+∫

R

|u0(x)− k|ϕ(x, 0) dx ≥ 0.

Conversely, let us suppose that u ∈ L∞ satisfies (2.6). Let ϕ belong to D+(R×

(−∞, T )). The function u0 and the solution u are assumed to take values in abounded interval (a, b). For k = a, we have |u− k| = u−a and |u0− k| = u0−a,with the result that∫∫

Q(ϕt u + ϕx f (u)) dx dt +

∫R

u0(x)ϕ(x, 0) dx

≥ a

(∫∫Q

ϕt dx dt +∫

R

ϕ(x, 0) dx

)+ f (a)

∫∫Q

ϕx dx dt = 0

on integrating the second term by parts. Similarly, taking k = b, |u − k| = b − uand |u0−k| = b−u0, we obtain the inequality opposite to the preceding one. Thusfor ϕ ∈ D

+(R× (−∞, T ))∫∫Q

(ϕt u + ϕx f (u)) dx dt +∫

R

u0(x)ϕ(x, 0) dx = 0.

By linearity, this is again true without sign condition on ϕ: u is a weak solution of(2.1).

Let us now show that u is indeed an entropic solution. Being given a convexcontinuous entropy E , of flux F , there exists for all α > 0 an entropy Eα of fluxFα which satisfies

E(s) ≤ Eα(s) ≤ E(s)+ α for s ∈ [a, b],Eα is convex, piecewise affine: Eα(s) = b0 + b1s + j a j |s − k j |, with a j > 0.

For that it is enough to interpolate E linearly on a sufficiently fine grid. Certainly,Fα(s) = b1 f (s) + j a j ( f (s) − f (k j )) sgn(s − k j ). In making use of (2.6) and(2.1), we see that (2.3) is valid for Eα. As Eα and Fα converge uniformly to E andF on [a, b] we can pass to the limit in the integrals, with the result that (2.3) is validfor E . Finally, we have

Proposition 2.3.4 A bounded measurable function u on R × (0, T ) is an entropysolution of (2.1) if and only if it satisfies (2.6) for all k ∈ R and all ϕ ∈ D

+(R+ ×[0, T )).


Irreversibility

The definition 2.3.3 introduces the concept of irreversibility in the solution of (2.1).Previously, a weak solution u was reversible in the sense that the function v definedby v(x, t) = u(−x, s − t) was also a weak solution in the band R × (0, s) for thegiven initial function v0(x) =: u(−x, s).

Exercise Prove this result rigorously, that is to say with test functions.

On the other hand, the entropy inequalities change when we pass from u to v (verifythis likewise), with the result that an entropy solution of (2.1) is not reversible, thatis to say that v is an entropy solution only if the entropy inequalities are indeedequalities ∫∫


∫R

E(u0(x))ϕ(x, 0) dx = 0,

∀ϕ ∈ D+(R× (−∞, T )).

We presume that the reversible solutions are in fact a little more regular than theothers at least if the flux f is sufficiently non-linear, say if f ′′ is not identicallyzero. A weak form of this statement is found in §2.4.

However, in the linear case, we can show that there is an equivalence betweenthe notions of weak solution and of entropy solution. A good method is to make useof the theorem of existence and uniqueness, 2.3.5 below. This states that if u0 is abounded, measurable function, then the entropy solution exists (it is in fact unique,but that does not play a part in this argument). This is a weak solution, but we showeasily by duality (this is nothing but an application of the Hahn–Banach theorem)that the weak solution of (2.1) is unique in the linear case. Thus every weak solutionis an entropy solution. In particular, since it is reversible, it satisfies entirely theentropy equalities. By linearity, these equalities hold without a convexity conditionon the entropy and with no sign condition on the test function.

Existence and uniqueness for the Cauchy problem

The Cauchy problem is well-posed in the class of entropy solutions for the scalarconservation laws. Although Kruzkov’s result is valid for an equation with variablecoefficients and in several space dimensions, we begin by enunciating a versionwhich corresponds to the framework which we have adopted until now.

Theorem 2.3.5 (Kruzkov [79]) For every bounded measurable function u0 onR, there exists one and only one entropy solution of (2.1) in L∞(Q) ∩ C ([0, T );

L1loc(R)). It satisfies the maximum principle

‖u‖L∞(Q) = ‖u0‖L∞(R). (2.7)

The theorem is, in fact, more complete than that, but so as not to overload thestatement of the theorem, we have preferred to summarise below the principalproperties of the solution.

Proposition 2.3.6 Let u0 and v0 be two bounded measurable functions and u and v

the entropy solutions associated with them. Let M = sup{| f ′(s)|; s ∈ [inf(u0, v0),sup(u0, v0)]}. Then:

(P1) For all t > 0 and every interval [a, b], we have

∫ b

a|v(x, t)− u(x, t)| dx ≤

∫ b+Mt

a−Mt|v0(x)− u0(x, t)| dx .

(P2) In particular, if u0 and v0 coincide on {x : |x− x0| < d}, then u and v coincideon the triangle {(x, t): |x − x0| + Mt < d}.

(P3) If u0 − v0 ∈ L1(R), then u(t)− v(t) ∈ L1(R) (writing u(t) := u(·, t)) and

‖v(t)− u(t)‖L1(R) ≤ ‖v0 − u0‖L1(R),∫R

(v(x, t)− u(x, t)) dx =∫

R

(v0(x)− u0(x)) dx .

(P4) If u0 ∈ L1(R), then u(t) ∈ L1(R), for all t > 0, and

‖u(t)‖L1(R) ≤ ‖u0‖L1(R),

∫R

u(x, t) dx =∫

R

u0(x) dx .

(P5) If u0(x) ≤ v0(x) for almost all x ∈ R, then also u(x, t) ≤ v(x, t).(P6) If u0 ∈ BV(R), the space of functions of total bounded variation, then u(t) ∈

BV(R) and

T V (u(t)) ≤ T V (u0).

Comments (1) Theorem 2.3.5 allows us to construct an operator S(t) which with agiven initial value u0 associates at the instant t > 0 the entropy solution u(t). Thefamily (S(t))t ≥ 0 is a semi-group because

the conservation law does not involve the time explicitly (if u is a solution, thenus := u(·, · + s) is also a solution, for the given initial value u(s)).

we verify easily that if v is the entropy solution for the initial value u(s), thenthe function u defined by

u(x, t) ={

u(x, t), t < s,

v(x, t − s), t > s

is also an entropy solutionof (2.1),with the result that S(t + s)u0 = S(t)S(s)u0,thanks to uniqueness.

(2) The property (P3) expresses the fact that t �→ S(t) is a contraction semi-group in L1(R)∩ L∞(R), with respect to the L1-norm. However, the above resultsare much more general since S(t) is defined on L∞(R). We do not know of anon-decoupled system of at least two conservation laws which possess such a con-traction property, even for a metric structure. It is suspected that it does not exist,which renders difficult the question of uniqueness for systems (cf. [104]).

(3) The property (P1) obviously contains the uniqueness property of the entropysolution, but it contains a more precise fact, plain from (P2): the value of the en-tropy solution u at a point (x, t) depends only on the restriction of u0 to an interval[x−Mt, x+Mt]. A perturbation (sufficiently small) with compact support disjointfrom this interval does not modify the value u(x, t). We call the domain of depen-dence of (x, t) the smallest compact set K such that, for every bounded functiona with compact support disjoint from K , the solution of the Cauchy problem withinitial condition v0 =: u0 + εa coincides with u at (x, t) for ε sufficiently small.The domain of dependence of (x, t) is thus included in [x − Mt, x + Mt], butthis is not necessarily an interval once shocks are developed, for the latter createshadow zones. More generally we can define the domain of dependence of (x, t) atthe instant t0 < t by considering the Cauchy problem with a given initial conditionat the time t0. A symmetrical notion is that of the domain of influence of a point(x0, t0), made up of points (x, t) with t > t0 of which (x0, t0) belongs to the domainof dependence.

(4) The property (P4) is a trivial consequence of (P3), which leads also to (P5)by the following calculation:

‖v(t)− u(t)‖L1(R) ≤ ‖v0(t)− u0(t)‖L1(R) =∫

R

(v0(x)− u0(x)) dx

=∫

R

(v(x, t)− u(x, t)) dx,

that is to say v − u ≥ 0.(5) Similarly, (P3) implies (P6). If u0 ∈ BV(R), then u0(·+ h)− u0 is integrable

for all h ∈ R and

TV(u0) = limh→0

h−1∫

R

|u0(x + h)− u0(x)| dx .


Thus, u(· + h, t)− u(·, t), is integrable, and

TV(u(t)) = limh→0

h−1∫

R

|u(x + h, t)− u(x, t)| dx

≤ limh→0

h−1∫

R

|u0(x + h)− u0(x)| dx

= TV(u0).

(6) The monotonic property (P5) implies that to a monotonic given initial functionthere corresponds a solution having the same monotonic property with respectto x .

(7) Kruzkov’s theorem is in reality of much more general power. It applies toscalar equations with d ≥ 2 space variables when the fluxes depend explicitly onthe space and time variables and in the presence of a source term:

∂t u +∑

1≤α≤d

∂xαf α(x, t, u) = g(x, t, u), x ∈ R

d (2.8)

The sole hypothesis, except the regularity of f and g is that g− divx f is uniformlybounded with respect to x ∈ R

d . There exists one and only one entropy solutionu ∈ L∞(Q) ∩ C ([0, T ); L1(Rd )) of the Cauchy problem for (2.8), that is to saysatisfying for every positive test function ϕ

0 ≤∫∫

(0,T )×Rd(|u − k|ϕt + ∇xϕ · ( f (x, t, u)− f (x, t, k)) sgn(u − k)) dx dt

+∫

Rd|u0 − k|ϕ(x, 0) dx

+∫∫

(0,T )×Rd(g(x, t, u)− (divx f )(x, t, k))ϕ sgn(u − k) dx dt.

The properties (P1) to (P6) remain true in this context provided that g ≡ 0 and thatf depends only on u, with the following adaptation for (P1). If ω is a bounded openset of R

d

∫ω

|v(x, t)− u(x, t)| dx ≤∫

ω+B(0;Mt)|v0(x)− u0(x)| dx .

In addition, f being vector-valued, | f ′| denotes the norm of f ′ in the definitionof M . The statements remain valid whatever the norm chosen.

Application: admissible discontinuities

Piecewise smooth entropy solutions

Since the entropy inequalities are trivial for classical solutions, it is importantto understand them for less smooth solutions. The simplest, best understood andmost useful case is that of piecewise smooth solutions. This case is placed in thesame class of solutions as §2.2. We have seen that the curve of discontinuity isparametrised by the time: t �→ X (t). We fix naturally ω− and ω+ by x < X (t) andx > X (t) respectively. If ϕ ∈ D

+(ω) and if E is a convex entropy of class C1 of

flux F ,

0 ≤∫∫

ω

(ϕt E(u)+ ϕx F(u)) dx dt

=(∫∫

ω−+

∫∫ω+

)(ϕt E(u)+ ϕx F(u)) dx dt

= −(∫∫

ω−+

∫∫ω+

)(E(u)t + F(u)x )ϕ dx dt +

∫�

([E(u)]n−t + [F(u)]n−x

)ϕ ds

=∫

�

([E(u)]n−t + [F(u)]n−x

)ϕ ds.

As ϕ is an arbitrary positive function, we deduce that along �

[E(u)]n−t + [F(u)]n−x ≥ 0,

that is

[F(u)] ≤ dX

dt[E(u)]. (2.9)

By continuity, we deduce also the following inequality:

[( f (u)− f (k)) sgn(u − k)] ≤ dX

dt[|u − k|], k ∈ R. (2.10)

It is easy to make the reverse argument. First of all, (2.10) and the Rankine–Hugoniot condition imply (2.9) (indeed, we see that (2.10) leads to the Rankine–Hugoniot condition). Then we show that a piecewise smooth function, which isa classical solution outside of � and which satisfies (2.9) along �, is an entropysolution. Finally we obtain the following result.

Proposition 2.3.7 Let u be a piecewise C1 function in Q, whose discontinuities are

carried by the union � of Lipschitz curves, pairwise disjoint. Then u is an entropysolution of (2.1) if and only if u is a classical solution outside of � and satisfies(2.10) on �.

Oleınik’s condition

Now let us analyse the condition (2.10) in detail. We can fix a point of � and supposethat u+ �= u−. Let I be the interval with extremities u− and u+. In choosing koutside of I , we obtain successively two inequalities which, together, express theRankine–Hugoniot condition. We thus have

dX

dt= [ f (u)]

[u].

Finally we take k ∈ I , that is to say k = au− + (1− a)u+, a ∈ [0, 1]. Then(f (u+)+ f (u−)− 2 f (k)− [ f (u)]

[u](u+ + u− − 2k)

)sgn(u+ − u−) ≤ 0.

But u+ + u− − 2k = (2a − 1)(u+ − u−), with the result that

(a f (u−)+ (1− a) f (u+)− f (au− + (1− a)u+))

× sgn(u+ − u−) ≤ 0, a ∈ [0, 1]. (2.11)

There are therefore two cases, according to the sign of u+ − u−.

Case u−< u+ then a discontinuity is admissible if and only if the graph of f ,restricted to the interval [u−, u+], is situated above its chord.

Case u−> u+ then a discontinuity is admissible if and only if the graph of f ,restricted to the interval [u+, u−], is situated below its chord.

Examples 2.3.8 If f is convex, its graph is always below its chord; a discontinuityis thus admissible if and only if u+ < u−.

If f is concave, its graph is always above its chord; a discontinuity is thusadmissible if and only if u− < u+.

In the general case, an admissible discontinuity is reversible if and only if f isaffine between u− and u+ .

Shocks

An important consequence of (2.11) is Lax’s inequality (also called Lax’s shockcondition). On dividing (2.11) by a|u+−u−| (respectively (1−a)|u+−u−|), thenon making a tend to 0 (respectively to 1), we obtain

f ′(u+) ≤ dX

dt≤ f ′(u−). (2.12)


Fig. 2.3: Admissible discontinuities.

Fig. 2.4: Inadmissible discontinuities.

These inequalities express that the characteristics, which are straight lines in ω+or ω−, defined, if need be, up to � , usually cannot emerge from �. A borderline caseis the one of a tangentially emerging characteristic. In most cases, these inequalitiesare strict, for example if f is strictly convex or concave. In this favourable case,the characteristics can only penetrate into � coming from the past and not leavingtowards the future. More precisely, from a point of � two characteristics and onlytwo can be drawn, both directed towards the past. In addition, not being able toencounter another discontinuity by going back in time, they end up at a point (x0, 0).This reasoning is still correct if f ′′ is of constant sign, a characteristic then beingable to coincide with �. Finally:

Proposition 2.3.9 If f is concave or convex, the entropy solution of (2.1) can becalculated by the method of characteristics.

We shall see below that in this case, there exists an explicit formula, due to Lax([59]), giving the unique entropy solution of (2.1). For a more general flux, there

2.4 The Riemann problem 43

does not always exist such a formula, in particular the method of characteristicsdoes not work because these do not necessarily return to the initial point. In fact,there exist admissible discontinuities for which for example f ′(u+)= dX/dt whilef ′′(u+) �= 0. We can then construct a Cauchy problem and its entropy solution ofwhich a characteristic emerges from � towards the future (see Exercise 2.9).

Definition 2.3.10 We shall say that a discontinuity (u−, u+, σ = dX/dt) is a shockif the inequalities of (2.12) are strict. We shall say that it is a semi-characteristicshock (to the left or to the right according to the side on which the characteristicsare tangents) if one is strict and the other is an equality. Finally we shall say it isa characteristic shock if the two are equalities without f being affine from u− tou+. If f is affine between u− and u+, we shall speak of a contact discontinuity.

One of the principal properties of shocks is to induce a little more smoothness,because it propagates along the characteristics. For example if u is of class C

1 tothe left of �, with a continuous limit u− on this side, if � is of class C

1 and iff ′(u−) �= dX/dt , then this continuous extension is of class C

1 up to �. We shallsee in an exercise that this cannot be true in a semi-characteristic shock to the left.

2.4 The Riemann problem

Self-similar solutions. Rarefactions

The Riemann problem is the Cauchy problem in the particular case of a given initialcondition of the form

u0(x) ={

uL, x < 0,

uR, x > 0.

The role of the Riemann problem is to furnish all the solutions of the Cauchyproblem which are invariant under the group of homotheties (x, t) �→ (ax, at),a group which leaves invariant all the conservation laws of the first order. Moreprecisely, let a > 0. If u0 is as above, and if u is the corresponding entropy solution,then ua := u(ax, at) is the entropic solution for the given u0(ax), here u0(x). FromTheorem 2.3.5, this solution is unique: ua ≡ u. Choosing a = t−1, we obtainu(x, t) = u(x/t, 1). We must therefore seek the solution of the Riemann problemamong the self-similar solutions of the form u(x, t) = v(x/t) with v ∈ L∞. Thesystem (2.1) then reduces, in the sense of distributions, to

f (v)ξ = ξvξ ,

v(−∞) = uL,

v(+∞) = uR.

(2.13)

Since we seek the entropic solution, we have also

[( f (v)− f (k))sgn(v − k)]ξ ≤ −ξ |v − k|ξ .The sense to give to the conditions at infinity for v is trivial, for because of property(P1) of propagation with finite speed, we see that u ≡ uL for x < Mt . Here, M =sup{| f ′(s)|: s ∈ I (uL, uR)} where I (uL, uR) is the interval with ends uL and uR. Infact we therefore have v(ξ ) = uL for ξ < −M ; similarly v(ξ ) = uR for ξ > M .

As to the general Cauchy problem, we must first of all consider the smoothsolutions of (2.13). But we must take care that these lead to solutions of (2.1) thatare singular at the origin.

If v ∈ C (R) is piecewise of class C1, we develop (2.13): (c(v)−ξ ) (dv/dξ ) = 0.

This equality is trivial in certain zones, for example for |x |> M since v is constantthere. Where v′ �= 0, we have c(v(ξ )) = ξ , which leads to the following definition.

Definition 2.4.1 A rarefaction wave is a self-similar solution u(x, t) = v(x/t) ofclass C

1 in a wedge at < x < bt , which thus satisfies c(u(x, t)) = x/t .

In a rarefaction wave, v is injective, thus monotonic, and takes its values in aninterval in which c is monotonic, that is to say where f is either convex or concave.For example, if f is convex on R, the rarefaction waves are increasing.

Another type of self-similar wave has already been encountered in the preceding,it consists of shocks. These are defined by v(ξ )= v− for ξ < σ , v(ξ )= v+ for ξ > σ ,v−, v+, and σ being linked by the Rankine–Hugoniot condition and satisfyingOleınik’s condition.

Thus we have three kinds of self-similar solutions: the rarefaction waves, theshocks and the constants. The solution of the Riemann problem does not make useof any other.

The solution of the Riemann problem

In the Riemann problem, the initial datum is monotonic, with the result that thesolution u is also monotonic with respect to the space variable. The same is true forξ �→ v(ξ )= u(ξ, 1), which thus has, at the most, a denumerable number of disconti-nuities. These will be the shocks. The construction of the solution is the following.For uL= uR, the solution is constant. If not, we denote by χ the characteristicfunction of the interval I (uL, uR) with values 0 and∞.

Case uL < uR Let g= sup{h convex: h ≤ f + χ}. Then d := g′, defined on[uL, uR], is increasing and we put

d(v(ξ )) = ξ, ξ ∈ [d(uL), d(uR)].

2.5 The case of f convex. The Lax formula 45

This formula defines v in a unique manner except on the set of critical values of dwhich is denumerable. For ξ < d(uL), we put v= uL, while for ξ > d(uR) we putv= uR.

Case uL > uR Here, g is the smallest of the concave functions which bound f −χ

above. Its derivative d , defined on [uR, uL] is decreasing and therefore allows us todefine v on [d(uL), d(uR)] by d(v(ξ )) = ξ.

Let us show that this construction solves (2.13) as well as satisfying the entropyinequalities. By symmetry it is enough to consider the case uL < uR. For ξ = x/t ∈[d(uL), d(uR)], we have f (v) = g(v) almost everywhere since f and g coincideexcept on the critical values of d . For every s ∈ [uL, uR], we thus have

f (s) ≥ g(s) ≥ g(v)+ ξ (s − v) = f (v)+ ξ (s − v), (2.14)

the second inequality being due to the convexity of g.Choosing s = v(ξ + a), a �= 0 in (2.14), then dividing by |a| and letting a tend

to zero through positive and negative values respectively, we obtain (2.13). Theentropy inequality is thus satisfied for k ≤ uL, and for k ≥ uR uL ≤ v ≤ uR.

If uL < k < uR, the monotonicity of v ensures the existence of a real numberξ0 such that v(ξ ) ≤ k for ξ < ξ0 and v(ξ ) ≥ k for ξ > ξ0. Let us define w :=( f (v)− f (k))sgn(v − k)− ξ |v − k|. Owing to (2.13), we have wξ = −|v − k| onthe open set R − {ξ0}. To deduce the entropy inequality wξ + |v − k| ≤ 0, it isenough to establish Oleınik’s inequality [w] ≤ 0 at ξ0, which comes from (2.14)when we choose ξ = ξ0 + a, s = k with a > 0 and we make a tend to zero.

2.5 The case of f convex. The Lax formula

If theflux is strictly convex,Lax’s inequality for thediscontinuities is strict ( f ′(u+)<σ < f ′(u−)) and it reduces to u− > u+. As they cannot be tangent to a shock curve,the characteristics are not able to emerge. For a piecewise smooth solution, we seethat the characteristic passing through a point (x0, t0) is a straight line which ex-ists at least in the past on the time interval [0, t0]. In fact, it is limited only by itsencounter with a shock, which can only be produced in the future, owing to Lax’sinequality. If (x0, t0) is on a shock curve, there are two such characteristics, oneto each side of the shock; at one such point, the calculation of the characteristicsfurnishes two results, which correspond to the two degrees of freedom, the valueof u and the shock speed. Otherwise, there is only a single characteristics.

The Hamilton–Jacobi equation

Let u be the entropy solution of (2.1). This conservation law is the compatibilitycondition which ensures the existence of a functionv satisfyingvx = u,vt = − f (u).


Since u is measurable and bounded, v is Lipschitz and satisfies the above equationsalmost everywhere. We thus have vt + f (vx ) = 0 almost everywhere; this is aparticular case of the Hamilton–Jacobi equation.

Since f is convex, we have for all s ∈ R.

vt ≤ f ′(s)(s − vx )− f (s),

which has first integral

v(y + f ′(s)t, t) ≤ v0(y)+ t(s f ′(s)− f (s)), s, y ∈ R, t > 0. (2.15)

As s→ f ′(s) is strictly increasing, we can make the change of variables (y, s, t) �→(y, x, t) with x = y + t f ′(s). We derive s= b((x − y)/t) where b is the deriva-tive of the convex conjugate function g= f ∗ of f for g′ ◦ f ′ = id. We then haves f ′(s)− f (s)= g((x − y)/t). On minimising the right-hand side of (2.15) withrespect to y keeping (x, t) fixed we therefore obtain the inequality

v(x, t) ≤ V (x, t) =: infy∈R

(v0(y)+ tg((x − y)/t)). (2.16)

Let (x, t) ∈ R×R+ and let us choose a characteristic τ �→ X (τ ) which ends on

(x, t). We have seen that it is defined on [0, t]; we write z = X (0). The calculationbelow, made along the characteristic, shows that the inequality (2.16) is optimal.A rigorous justification of this calculation for every entropy solution will be foundin [60].

d

dτv(X, τ ) = vt + c(u)vx = c(u)u − f (u) = g(c(u)).

We have also dX/dτ = c(u), which is constant. Finally,

v(x, t) = v0(z)+ tg(c(u)) = v0(z)+ tg((x − z)/t).

Theorem 2.5.1 (Lax [59]) If f is strictly convex and u0 ∈ L∞(R), then theentropy solution of (2.1) is given by u = vx where

v(x, t) = supy∈R

(v0(y)+ t f ∗

(x − y

t

)),

v0 being a primitive of u0.

A dual formula to Lax’s

Lax’s formula is rendered possible because in the case in which f is convex, we haveseen that the characteristics all issue from the axis t = 0. For a general flux f , this

2.6 Proof of Theorem 2.3.5: existence 47

is no longer possible because characteristics can originate in a semi-characteristicshock. However, it will be seen in Exercise 2.9 that this cannot happen if the initialdatum is monotonic. We thus have in this case an explicit formula, which is thedual of that of Lax in the sense that one is of the form v(x, t) = infy∈R sups∈R(. . .)while the other is written v(x, t) = sups∈R infy∈R(. . .).

Proposition 2.5.2 (Kunik [57]) Let u0 be an increasing given initial function, theflux f being an arbitrary smooth function. Then u = vx where v is given by theformula

v(x, t) = sups∈R

(sx − t f (s)− v∗0(s)).

Here, v0 is a primitive (convex by hypothesis) of u0 and v∗0 is its convex conjugatefunction.

For a discussion of this formula, the reader should see [92].

2.6 Proof of Theorem 2.3.5: existence

The approach by semi-groups

An original proof of Kruzkov’s theorem, due to Crandall [14], is based on theparticular properties of the scalar case, notably the fact that the solution of theCauchy problem furnishes a semi-group of contractions in L1 ∩ L∞(R), which isdescribed by the properties (P3) and (2.7). In the one-dimensional case, which isours, we can again take advantage of the order structure on R to have the shortestof the proofs of existence of an entropy solution of (2.1).

We begin by considering a simplified case, that in which, as well as u0 beingmerely integrable on R, the flux f is uniformly monotonic (inf{ f ′(s): s ∈ R} > 0)and satisfies f ′′ = 0 outside a compact set. We define an unbounded operator A inthe following manner.

Domain of A: D(A) = W 1,1(R) = {v ∈ L1(R), vx ∈ L1(R)}.Graph of A: Av = f (v)x , ∀v ∈ D(A).

As f is Lipschitz, v �→ f (v) is a continuous operator of L1(R) into itself,with the result that the graph of A is closed (we say that A is closed). The con-struction of an entropy solution of (2.1) is made by approaching the conserva-tion law by a difference equation. We seek a solution uε ∈ C (R+; L1(R)) of the

following equation:

uε(t)− uε(t − ε)

ε+ Auε(t) = 0, t ≥ 0,

uε(t) = u0, t < 0.

(2.17)

We then make use of an abstract theorem from the theory of semi-groups [13].

Theorem 2.6.1 Let X be a Banach space and A a closed operator with domainD(A) dense in X, accretive and such that (idX + λA)(D(A)) = X for all λ > 0.Then, for all ε and every u0 ∈ X, the problem (2.17) possesses a unique solutionuε ∈ C (R+; X ). In addition, uε(t) converges in X, uniformly on every compactset of R

+ to a function t �→ S(t)u0. The family (S(t))t≥0 is a contraction semi-group in X, that is to say,

(1) S(t)S(s) = S(t + s), ∀t, s ≥ 0,(2) ‖S(t)v − S(t)w‖ ≤ ‖v − w‖, t ≥ 0, v, w ∈ X,(3) S(0) = idX ,(4) (t, v) �→ S(t)v is continuous on R

+ × X.

The accretivity of which mention is made in the hypotheses of the theorem is thefollowing property.

Definition 2.6.2 The operator A is said to be accretive if for all v, w ∈ D(A) andevery λ ≥ 0, we have

‖v − w‖ ≤ ‖v + λAv − w − λAw‖.

We shall make use of the Banach space X = L1(R).

Accretivity of A

The following lemma is classical, but we give a proof for the convenience of thereader. In all that follows sgn s vanishes if s = 0 .

Lemma 2.6.3 Let z ∈ W 1,1(R). Then zx sgn z = |z|x .

Proof Let j be a Lipschitz function and ( jn)n≥0 a sequence of Lipschitz functionsof class C

1, such that jn converge uniformly to j and j ′n converge pointwise to j ′

while staying uniformly bounded. For the lemma, we choose j = sgn and jn(s) =√(s2 + n−1). As W 1,1⊂C (R), we have jn(z) ∈ W 1,1(R) and jn(z)x = j ′n(z)zx .Let us pass to the limit in this equality. On the one hand jn(z) → j(z) uniformly,

so jn(z)x → j(z)x in D′(R). On the other hand, j ′n(z)zx → j ′(z)zx pointwise,

2.6 Proof of Theorem 2.3.5: existence 49

staying bounded in L1(R). We can therefore apply the theorem of dominated con-vergence. The convergence is valid in L1(R), so in the sense of distributions.

Now let v, w ∈ D(A). The lemma and the growth of f give | f (v) − f (w)|x =( f (v)− f (w))x sgn(v − w). Hence

‖v + λAv − w − λAw‖ ≥∫

R

(v + λ f (v)x − w − λ f (w)x ) sgn(v − w) dx

=∫

R

(|v − w| + λ| f (v)− f (w)|x ) dx

=∫

R

|v − w| dx,

since∫

Rzx dx = 0 for all z ∈ W 1,1(R). Finally, A is accretive.

The range of idX + λA

Let λ > 0. The equation (idX + λA)v = h, for h ∈ X = L1(R) and v ∈ D(A),leads to the ordinary differential equation

dv

dx= g(x, v) =:

h(x)− v

λ f ′(v). (2.18)

The function g being uniformly Lipschitz with respect to v and uniformly in-tegrable with respect to x , the Cauchy problem for (2.18) possesses one and onlyone solution defined on R. We denote by vn the solution of (2.18) on (−n,∞)which satisfies the initial condition vn (−n)= 0. Every solution of (2.18) satisfiesd|v|/dx ≤ �|h| − ω|v| where 0 < ω < 1/(λ f ′(z)) < � for all z. We deduce that,for x > 0,

|v(x)| ≤ e−ωx |v(0)| +�

∫ x

0eω(y−x)|h(y)| dy

and hence that limx→+∞ v(x) = 0 (by applying the theorem of dominated conver-gence to the integral). Making use of the same differential inequality, we derive now

ω

∫ +∞

−n|vn(x)| dx ≤ �‖h‖1

which shows that the sequence (vn)n∈N, continued by 0 for x < −n, is bounded inL1(R).

Similarly, we have d|v|/dx ≥ −ω|h| − �|v| which on integrating and takingaccount of the preceding estimate gives |vn(x)| ≤ (ω+�2/ω)‖h‖1. This sequenceis thus also bounded in L∞(R). Making use of the differential equation, we see


that the sequence is uniformly equi-continuous on every compact set. It thus admitsa cluster point when n → ∞, for the topology of uniform convergence on everycompact set. Denote this limit by v, which is continuous. We can pass to thelimit in the integrated form of the differential equation, which shows that v isa solution of (2.18). In addition, by Fatou’s lemma, v is integrable on R withω

∫R|v(x)| dx ≤ �‖h‖1. We thus have v ∈ X . Since in addition vx = g(·, v) ∈ X ,

indeed we have v ∈ D(A) and v + λAv = h. Thus, idX + λA is surjective. As Ais accretive, idX + λA is equally injective.

Passage to the limit

We can thus apply Theorem 2.6.1. The limiting solution uε therefore exists, isunique and converges uniformly on every compact set [0, T ] to a function t �→ u(t)with values in L1(R): u ∈ C (R+; L1(R)). Let k ∈ R; then applying Lemma 2.6.3to z = f (uε(t))− f (k), we have

|uε(t)− k| + ε| f (uε(t))− f (k)|x = (uε(t − ε)− k) sgn(uε(t)− k)

≤ |uε(t − ε)− k|.For ϕ ∈ D

+(R2), we therefore have

0 ≤∫∫

R×R+

ϕ(t)

( |uε(t − ε)− k| − |uε(t)− k|ε

− | f (uε(t))− f (k)|x)

dx dt

=∫∫

R×R+

(|uε(t)− k|ϕ(t + ε)− ϕ(t)

ε+ | f (uε(t))− f (k)|ϕx

)dx dt

+ 1

ε

∫ ε

0dt

∫R

ϕ(x, t)|u0(x)− k| dx

ε→0→∫∫

R×R+(|u − k|ϕt + | f (u)− f (k)|ϕx ) dx dt +

∫R

ϕ(x, 0)|u0(x)− k| dx,

because of the uniform convergence ofϕ(t+ε)−ϕ(t)/ε toϕt and of ε−1∫ ε

0 ϕ(x, t) dtto ϕ(x, 0). Finally, the existence of at least an entropy solution is demonstrated inthe case u0 ∈ L1(R) when f ′′ is null outside a compact set and inf f ′> 0.

Let us notice immediately an essential property of this solution – the maximumprinciple.

Proposition 2.6.4 Let u0 ∈ L1∩ L∞(R). The solution, limit of uε, of (2.1) satisfies‖u‖∞ ≤ ‖u0‖∞.

Proof As a result of Theorem 2.6.1 it is enough to show that if h ∈ L1 ∩ L∞

then the solution of v + εAv = h satisfies ‖v‖∞ ≤ ‖h‖∞. But v ∈ W 1,1(R) is

2.7 Proof of Theorem 2.3.5: uniqueness 51

continuous and tends to zero at infinity, so attains its bounds. There, vx is nulland v = h.

The general case

If u0 ∈ L1(R) and if f ′′ is null outside a compact set, then inf f ′ > −∞ and we areled to the preceding case by choosing r > − inf f ′. The function fr (s) =: f (s)+rssatisfies the hypotheses of the preceding paragraphs. The Cauchy problem for fr andu0 therefore possesses an entropy solution ur , which furnishes an entropy solutionof (2.1) by u(x, t) =: ur (x + r t, t). (Exercise: prove in detail this assertion.)

If u0 ∈ L1 ∩ L∞(R) and f is an arbitrary function of class C2, let us put

M = ‖u0‖∞. We choose a function g of class C2 such that g(s) = f (s) for |s|< M

and g′′(s) = 0 for |s| > M + 1. The solution already constructed of the Cauchyproblem for u0 and g is valid since ‖u0‖ ≤ M , with the result that f (u) ≡ g(u).To be precise, that shows that u is a weak solution of (2.1) and also that u satisfiesthe entropy inequalities for |k| ≤ M . But for |k| > M , these inequalities are trivialsince ‖u0‖ ≤ M and u is a weak solution.

It remains for us to prove the existence of a solution under the hypotheses ofTheorem 2.3.5, that is when, u0 ∈ L∞(R). For that, we make use of the propertiesof uniqueness and of propagation with finite speed which will be proved in thefollowing section.

Let M be as defined above. For T > 0 and y ∈ R, we denote by uy,T the solutiongiven by the method of semi-groups when we choose the prescribed initial conditionto be u0χ [y − MT, y + MT ] ∈ L1(R) (here χ takes values 0 and 1). Its restrictionto the triangle �y,T =: {(x, t):|x − y| + Mt < MT } is denoted by vy,T . From(P2) vy,T and vz,S and coincide on the intersection of �y,T and �z,S . We thereforeconstruct a function on R × R

+ by putting u(x, t) = uy,T (x, t) if (x, t) ∈ �y,T ,which is an entropy solution of (2.1). In fact, the localization by a test functionϕ makes the variational formulation for u equivalent to that for uy,T by choosing(y, T ) such that supp ϕ ∩ (R× R

+) is contained in �y,T .

2.7 Proof of Theorem 2.3.5: uniqueness

The essential idea of the proof of uniqueness is the inequality

|u − v|t + (( f (u)− f (v)) sgn(u − v))x ≤ 0,

which is satisfied by two entropic solutions of (2.1). The entropy inequalities arededuced in the following sub-section. On integrating over a domain of the form|x − x0| +Mt < b, where M bounds the speed of propagation of the waves above,we deduce the property (P1) of Proposition 2.3.6.


An inequality for two entropy solutions

Proposition 2.7.1 Let u and v be two entropy solutions of (2.1) of which the initialvalues are respectively u0 and v0. For all ϕ ∈ D

+(Q), where Q = R× [0, T ), wehave ∫∫

Q(|u − v|ϕt + ( f (u)− f (v)) sgn(u − v)ϕx ) dx dt (2.19)

+∫

R

|u0(x)− v0(x)|ϕ(x, 0) dx ≥ 0.

Proof Let � ∈ D+(Q × Q). We apply (2.6) with the solution u, with the constant

k = v(y, s) and with the test function �(·, ·, y, s), then we integrate the inequalityobtained with respect to (y, s) over Q. We make the same calculation replacing u byv and conversely, with the test function �(x, t, ·, ·). The sum of the two inequalitiesobtained is:

0 ≤∫∫∫∫

Q×Q|u(x, t)− v(y, s)|(�t +�s)(x, t, y, s) dt ds dx dy

+∫∫∫∫

Q×Qsgn(u(x, t)− v(y, s))( f (u(x, t))

− f (v(y, s)))(�x +�y) dt ds dx dy

+∫∫

R×Q|u0(x)− v(y, s)|�(x, 0, y, s) dx dy ds

+∫∫

Q×R

|u(x, t)− v0(y)|�(x, t, y, 0) dx dt dy. (2.20)

Let ϕ ∈ D+(Q). We apply (2.20) to the function

�ε = ϕ(x, t)χε(x − y, t − s) (2.21)

where χε(x, t) = ε−2χ (x/ε, t/ε) is a positive approximation to the Dirac mass atthe origin: χ ∈ D

+(R2),∫

R2 χ dx dt = 1. In fact we shall choose χ to be of the

form θ (x)η(t), the support of η being in [−2,−1] .We now apply a technical lemma.

Lemma 2.7.2 Let F be a locally Lipschitz function on R2. Then for �ε of the

form (2.21):

(1) When ε → 0, the integral∫∫∫∫Q×Q

F(u(x, t), v(y, s))�ε(x, t, y, s) dx dt dy ds (2.22)


converges to the integral∫∫Q

F(u(x, t), v(x, t))ϕ(x, t) dx dt. (2.23)

(2) When ε → 0, the integral∫∫∫R×Q

F(u0(x), v(y, s))�ε(x, 0, y, s) dx dy ds (2.24)

tends to the integral ∫R

F(u0(x), v0(x))�(x, 0) dx . (2.25)

In fact if � is of the form (2.21), then (�t+�s)(x, t, y, s) = ϕt (x, t)χε(x− y, t−s)and (�x+�y)(x, t, y, s) = ϕx (x, t)χε(x− y, t−s) are again of form (2.21). As thefunctions coming into play with u and v in the integrals are locally Lipschitz on R

2

we can pass to the limit in the first three terms of (2.20) making use of the lemma.Also, the last integral is zero because of the factor η(t/ε) = 0. The proposition isclearly proved.

Let us pass to the proof of the lemma which is only a result from measure theory.

Proof (1) First of all,∫

Qχε(x − y, t − s) dy ds = 1 because of the condition onthe support of η. Thus (2.23) has the value∫∫∫∫

Q×QF(u(x, t), v(x, t))�ε(x, t, y, s) dx dt dy ds.

The functions u and v are bounded and F is locally Lipschitz with the result that

|F(u(x, t), v(x, t))− F(u(x, t), v(y, s))| ≤ C |v(x, t)− v(y, s)|.It is enough therefore to show that

Iε(v) =:∫∫∫∫

Q×Q|v(x, t)− v(y, s)|�ε(x, t, y, s) dx dt dy ds

=∫∫∫∫

Q×R2|v(x, t)− v(x + εy, t + εs)|ϕ(x, t)χ (y, s) dx dt dy ds

converges to zero when ε → 0, the second equality occurring for ε small enough.Let U ∈ Q be a compact neighbourhood of the support of ϕ. For ε sufficiently small,the integral Iε is borne only by U and we have the upper bound

Iε(v) ≤ 2‖ϕ‖∞‖v‖L1(U ).

Similarly, the theorem of dominated convergence shows that if v is continuousIε → 0 as ε → 0. To conclude the proof in the general case we thus choose δ > 0arbitrarily small and w continuous and bounded such that ‖v − w‖ ≤ δ. SinceIε(v) ≤ Iε(v − w)+ Iε(w), we have that limsupε→0 Iε(v) ≤ δ, which implies thestated result.

(2) Similarly∫

Q χε(x− y,−s) dy ds = 1 and we are led back to the convergenceto 0 of the integral∫∫∫

R×Q|v0(x)− v(y, s)|ϕ(x, 0)χε(x − y,−s) dx dy ds.

The same method as above shows that∫∫∫R×Q

|v(x, s)− v(y, s)|ϕ(x, 0)χε(x − y,−s) dx dy ds.

tends to zero with ε. It therefore remains to show the convergence to zero of

Jε(v) =:∫∫

R×Q|v0(x)− v(x, s)|ϕ(x, 0)χε(x − y,−s) dx dy ds.

From the choice (2.21), the integral with respect to y is harmless:

Jε(v) =:∫∫

Q|v0(x)− v(x, s)|ϕ(x, 0)ε−1η(−s/ε) dx ds.

Next, since v is continuous on (0, T ) with values in L1loc(R), we may replace v0 by

v(·, 0) and we conclude by noting that

Jε(v) ≤ ‖ϕ‖∞∫ 2

1‖v(0)− v(s)‖L1(V )η(−s/ε) ds/ε

= ‖ϕ‖∞∫ ∞

0‖v(0)− v(εs)‖L1(V )η(−s) ds

where V is an interval containing the support of ϕ(·, 0).

Integration of the inequality (2.19)

Let (a, b) be an open interval of R and s > 0 a real number. We begin by constr-ucting a trapezium B of Q whose horizontal sections Bt are such that

(1) Bs = (a, b),(2) if t < τ , then Bt contains the domain of dependence of Bτ (see Fig. 2.5).

For that, we denote by I the smallest interval containing the essential values ofu and v. Since these solutions are supposed bounded, the same is true of I . Nextwe choose the number M which bounds above the absolute value of the velocity

Fig. 2.5: The domain of integration.

of wave propagation: M = : supr∈I | f ′(r )|. Finally we define B={(x, t)∈ Q:a − M(s − t) < x b and choose an even function θ ∈ D

+(R)decreasing on R

+ and such that θ(y) = 1 for |y| d.We also choose a function χ ∈ D

+((−∞, T )) such that χ (0) = 1 and χ (t) = 0 fort > s ′, where s < s ′< s + b/M . We apply (2.19) with the test function ϕ(x, t) =:χ (t)θ (|x | + Mt). Writing F(u, v) = ( f (u)− f (v)) sgn(u − v), we have

|u − v|ϕt + F(u, v)ϕx = χ ′(t)θ (|x | + Mt)|u − v|+χ (t)θ ′(|x | + Mt)(M |u − v| + F(u, v) sgn x).

Now |F(u, v)| ≤ M |u − v|, so the last bracket is positive. As χ ≥ 0 and θ ′ ≤ 0,we have

|u − v|ϕt + F(u, v)ϕx ≤ χ ′(t)θ (|x | + Mt)|u − v|.

Substituting this result in (2.19), we obtain∫∫Q

(|u − v|χ ′(t)θ (|x | + Mt)) dx dt +∫

R

|u0 − v0|θ (|x |) dx ≥ 0. (2.26)

Now, let us make d tend to b. The theorem of dominated convergence allowsus to pass to the limit in the two integrals which renders the formula correct whenθ (|x | + Mt) is simply the characteristic function of the set B. Let us write thenh(t) =: ‖u(t)− v(t)‖L1(Bt ). Since t �→ u(t) is continuous with values in L1

loc(R), h


is continuous and we deduce from (2.26) the inequality∫ s

0h(t)χ ′(t) dt + h(0) ≥ 0, (2.27)

for every function χ ∈ D+((−∞, s ′)) such that χ (0) = 1. This property implies

classically that h is decreasing on [0, s ′]. We thus have∫ b

−b|u(x, s)− v(x, s)| dx = h(s) ≤ h(0) =

∫ b+Ms

−b−Ms|u0(x)− v0(x)| dx,

which is exactly the inequality sought.

End of the proof of Proposition 2.3.6

It remains only to show that the property (P3) is valid. We thus assume that u0−v0 ∈L1(R) . Then the property (P1) implies that for every bounded interval I ,∫

I|u(x, t)− v(x, t)| dx ≤ ‖u0 − v0‖L1(R).

By means of Fatou’s lemma we then deduce that u(t)− v(t) ∈ L1(R) and that

‖u(t)− v(t)‖1 ≤ ‖u0 − v0‖1.

Next, we express that u and v are weak solutions of (2.1). Choosing a test functionof the form ϕε(x, t) = χ(t)θ (εx) where χ ∈ D ((−∞, T )) and θ (x) ∈ D (R) withθ ≡ 1 in a neighbourhood of the origin, we have∫∫

Q((u − v)ϕεt + ( f (u)− f (v))ϕεx ) dx dt (2.28)

+∫

R

(u0 − v0)(x)ϕε(x, 0) dx = 0,

that is to say,∫∫Q

(u − v)χ ′(t)θ (εx) dx dt + ε

∫∫Q

( f (u)− f (v))χ (t)θ ′(εx) dx dt (2.29)

+χ (0)∫

R

(u0 − v0)θ (εx) dx = 0. (2.30)

Each factor u(t)−v(t), u0−v0 and f (u(t))− f (v(t)) is integrable on R uniformlywith respect to t , the last because we have | f (u)− f (v)| ≤ M |u−v|. When ε → 0,the theorem of dominated convergence allows us to pass to the limit in each of thethree integrals, the second tending to zero. There remains∫∫

Q(u − v)χ ′(t) dx dt + χ (0)

∫R

(u0 − v0) dx = 0, (2.31)

2.8 Comments 57

which can be written in the form∫ T

0 hχ ′ dt + χ (0)h(0) = 0 with this time h(t) =:∫R

(u(t) − v(t)) dx which is a continuous function. We deduce that h is a constantfunction, that is to say∫

R

(u(x, t)− v(x, t)) dx =∫

R

(u0(x)− v0(x)) dx .

2.8 Comments

Oleınik’s inequality

We have seen in an exercise that if f is uniformly convex, that is to say, if there existsa number α > 0 such that f ′′ ≥ α, then the entropy solution satisfies Oleınik’sinequality

ux ≤ 1

αt. (2.32)

The positive distribution 1/αt − ux is thus a bounded measure for every com-pact interval and for t > 0 and the same is true of ux . Thus, u(t)∈BV(I ) for everyt > 0 and every compact interval I , even if u0 is only in L∞(R). There is there-fore a smoothing phenomenon which we did not observe in the linear case. Inparticular, the resolvent operator S(t) is compact as a mapping of L∞loc(R) intoitself, and we find again the irreversible character of the entropy formulationof (2.1).

Since f is Lipschitz, f ◦ u(t) is also of bounded variation on I ; we have in fact

TV( f ◦ u(t); I ) ≤ M TV(u(t)).

Thus f (u)x ∈ L∞loc((0, T ); Mb(I )) and, because of (2.1), the same is true of ut . Wededuce that t �→ u(t)|I is not only continuous, but even Lipschitz on (0, T ), withvalues in L1:

‖u(t)− u(s)‖L1(I ) ≤ M∫ t

sTV(u(τ )) dτ.

We can rewrite this result by decomposing ux into its positive and negative parts:ux = u+x − u−x . With I = (a, b) we have∫

Iu−x dx = u(a)− u(b)+

∫Iu+x dx ≤ 2‖u(t)‖ + |I |/αt.

From this, on using the maximum principle, we have

‖u(t)− u(s)‖L1(I ) ≤ M

(2|t − s|‖u0‖∞ + |I |

αlog

t

s

). (2.33)


Initial datum with bounded total variation

For a general flux, the property (P6) ensures that an initial datum in BV(R) leadsto a solution u whose total variation with respect to x at each instant is bounded byTV(u0). The preceding calculation remains valid in part but we find∫

R

|u(t)− u(s)| dx ≤ M |t − s|TV(u0).

Uniqueness: the duality method

In the method of Kruzkov, the uniqueness is the consequence of a monotonic prop-erty. When we consider systems rather than scalar equations, such a property isno longer available and the uniqueness question no longer has a general answer.An alternative argument is needed in approaching the problem. That which wasfirst presented in this spirit is the duality method of Holmgren. Although its gen-eralization to the case of systems is delicate and of limited range it has givenseveral interesting results. Let us look at its application to a scalar question due toOleınik [81].

Being given two entropy solutions u and v of the adjoint equation ut+ f (u)x = 0,their difference z = u − v satisfies the linear equation with variable coefficientszt + (hz)x = 0 with

h(x, t) = H (u, v) := f (v)− f (u)

v − u.

The following calculation uses the general solution of the adjoint equation pt +hpx = 0 with a given final condition P at a positive time T . In principle p(·, t)is obtained by forming the composite function P ◦ U where U (t ; T ) is the flowof the differential equation x = h(x, t). However, the discontinuities of h preventthe definition of U and deprive us of the solution of the adjoint equation. We getround this obstacle by smoothing h by hε = h∗ρε, where ∗ denotes the convolutionproduct with respect to the variable x alone while ρε = ρ(x/ε)/ε, with the classicalhypotheses onρ. Being given a smooth function P , we thus denote by pε the solutionof the Cauchy problem

pεt + hε pε

x = 0, pε(x, T ) = P(x).

We assume that the initial values u0 and v0 are bounded functions; we denote byJ a bounded interval of R which contains their values. The solutions consideredsatisfy the following properties:

u and v have values in J ,u(t)− u0 and v(t)− v0 are integrable,

2.8 Comments 59

as ε �→ 0+, ‖u(t)− u0‖L1(R) and ‖v(t)− v0‖L1(R) tend to zero,if f ′′ > α on J , then ux ≤ 1/(αt) and vx ≤ 1/(αt).

In particular, the calculations which follow apply to the solutions obtained as limit-ing values of the approximate solutions furnished by the parabolic equation ut +f (u)x = ηuxx as η tends to zero.

From now on, we assume that f is uniformly convex on J we denote by α (>0)and β the lower and upper bounds of f ′′ on J . On J × J , H is bounded andincreasing with respect to each argument. In particular h is bounded: |h(x, t)| ≤ Mand hence |hε| ≤ M . As hε is Lipschitz with respect to x (uniformly with respectto t but not with respect to ε), we have at our disposal a flow for the differentialequation x = hε(x, t), which enables us to solve the approximate adjoint equationpε

t + hεpεx = 0 in the class of functions Lip(R).

If v0−u0 is integrable so also is z(t) by hypothesis. Let us write Green’s formulaon R× (τ, T ), with 0 < τ < T :

0 =∫ T

τ

∫R

(pε(zt + (hz)x )+ z

(pε

t + hε pεx

))dx dt

=∫

R

P(x)z(x, T ) dx −∫

R

pε(x, τ )z(x, τ ) dx +∫ T

τ

∫R

z(hε − h)pεx dx dt.

As pε is constant along the integral curves of hε this becomes∣∣∣∣∫

R

P(x)z(x, T ) dx

∣∣∣∣ ≤ ‖P‖∞‖z(τ )‖1 +∫ T

τ

∫R

|z| |hε − h| |pεx | dx dt.

To exploit this inequality, we establish an estimate for pεx which does not depend

on ε; this is possible only with the hypothesis of genuine non-linearity made above.By Taylor’s formula

hx = Hu(u, v)ux + Hv(u, v)vx = 1

2f ′′(u1)ux + 1

2f ′′(u2)vx ≤ β/αt,

which implies that hεx = hx ∗ ρε ≤ β/(αt). Let us write qε := pε

x , which satisfies

(∂t + hε∂x ) log|qε| = −hεx ≥ −β/αt.

The function |qε|tβ/α increases along the characteristics of the modified adjointproblem. Therefore,

|pεx (x, t)| ≤

(T

t

)β/α

‖Px‖∞.

Finally,∣∣∣∣∫

R

P(x)z(x, T ) dx

∣∣∣∣ ≤ ‖P‖∞‖z(τ )‖1+∫ T

τ

‖Px‖∞(

T

t

)β/α

dt∫

R

|z| |hε−h| dx .

The function (T/t)β/α|z| |hε − h| is bounded above by the integrable function2M(T/τ )β/α|z| and converges almost everywhere to zero when ε tends to zero. Itsintegral also converges to zero as a result of the theorem of dominated convergence.There remains ∣∣∣∣

∫R

P(x)z(x, T ) dx

∣∣∣∣ ≤ ‖P‖∞‖z(τ )‖1.

Making τ tend to zero we obtain∣∣∣∣∫

R

P(x)z(x, T ) dx

∣∣∣∣ ≤ ‖P‖∞‖v0 − u0‖1,

for every bounded Lipschitz function P , which is equivalent to saying that ‖v(T )−u(T )‖1 ≤ ‖v0 − u0‖1. This implies uniqueness.

Remark For a system, deriving an estimate for pεx is the delicate point. In addition,

the adjoint problem not being a transport equation, the constant which we obtainin the eventual upper bound∣∣∣∣

∫R

pε(x, t)z(x, τ ) dx

∣∣∣∣ ≤ C‖P‖ · ‖z‖

is, in general, strictly greater than 1, with the result that ‖v(T )−u(T )‖1 ≤ C‖v0−u0‖1: the semi-group of a system is not contracting in L1(R). This point is madeprecise by Temple [104].

2.9 Exercises

2.1 We suppose that f is uniformly convex, that is that f ′′(s) ≥ α, where α is astrictly positive constant. Show that the classical solution satisfies ux < 1/αt .

2.2 In the case of the road traffic model, what comparison can we make betweenthe speed of the waves c(ρ) and that of vehicles V (ρ)?

2.3 We consider a scalar conservation law in several space dimensions:

ut +d∑

i=1

Ai (u)xi = 0.

We note that ut + divx A(u) = 0, and we suppose that the vector field u �→A(u) is smooth. We write a = A′.

(1) Let u be a classical solution of this conservation law in a domain Rd×

[0, T ) and u0 its initial value. Show that u is constant along the charac-teristics defined by dx/dt = a(u(X (t), t)) and these are straight lines.

2.9 Exercises 61

Fig. 2.6: Generic blow-up by a cusp.

(2) Let q = divx (a(u)). Show that along the characteristics q satisfies the dif-ferential equation dq/dt + q2 = 0. Deduce that if there exists a pointy ∈ R

d such that divx (a(u0))(y) < 0, then T is finite, more preciselyT ≤ T ∗ =: −(infx divx (a(u0)))−1.

(3) Conversely, show that, if 1 + T infx divx (a(u0)) ≥ 0, then there exists aclassical solution of the Cauchy problem on the domain R

d × [0, T ).

2.4 We consider the Burgers equation ( f (u) = 12u2) with given initial condition

u0 of class C1 and with non-empty compact support. In the formula T= −

(infR u0)−1, the lower bound is thus attained and T <∞. We suppose that itis attained at a single point y0 and that u0

′′′(y0) > 0 (we have u0′′(y0) = 0 and

u0′′′(y0) ≥ 0 a priori). We write x0 =: y0 + T u0(y0).

(1) Show that u(T ) is continuous on R, and of class C1 outside of x0. Prove

that limx→x0 ux (x, T ) = −∞.(2) Show that (u(T ) − u(x0, T ))3 is of class C

1 on R and that its derivativeat x0 has the value −6(T 4u0

′′′(y0))−1. Hint: (y, 0) being the base of acharacteristic which ends in (x, T ), find an equivalent of x − x0 as afunction y − y0. Figure 2.6 illustrates this generic behaviour.

2.5 Let f be a function which is not affine. To fix the ideas, there are given threenumbers v < w < z, w = av+(1−a)z, such that f (w) < a f (v)+(1−a) f (z).Using either one elementary discontinuity or two, construct two piecewiseconstant solutions for the Cauchy problem in which the given initial conditionis u0(x) = v if x < 0, and u0(x) = z if x > 0. Adapt the question and thesolution if f (w) < is replaced by f (w) >.

2.6 We assume that f is convex. Show that a discontinuity is admissible if andonly if it satisfies one entropy inequality

[F(u)] ≤ dx

dt[E(u)]

for at least one strictly convex entropy.

Fig. 2.7: Semi-characteristic shock.

2.7 Weak shocks. When [u] → 0, the speed σ = [ f (u)]/[u] of a shock is of theform σ = c(u±)+ O([u]) from Taylor’s formula.

(1) Show that in fact,

σ = c

(1

2(u+ + u−)

)+ O([u]2).

(2) Find an equivalent of the rate of dissipation of entropy [F − σ E] when[u] → 0 (F ′ = cE ′) by supposing that E ′′> 0 and f ′′> 0.Solution: [F − σ E] ∼ κ[u]3 where κ is calculable.

(3) Construct an example in which this equivalence is an equality.

2.8 Let a < b < c be three real numbers. We assume that (a, b) and (b, c) areentropic discontinuities of the equation ut + f (u)x = 0, with speeds σ1 andσ2. Using Lax’s inequality show that σ2 ≤ σ1. Deduce that (a, c) is an entropicdiscontinuity.

2.9 Let u be an entropic solution of (2.1) which is smooth except along a curve�: t �→ (X (t), t) of class C

1, along which there is a semi-characteristic shockto the right: f ′(u+) = dx/dt . To fix ideas, we assume that the shock isdecreasing: u+< u−. We also impose the generic condition f ′′(u+) �= 0.

(1) Show that � is the envelope of a family of straight line characteristics andthat its concavity is turned towards the left. Deduce that the continuousextension of u to the left side of � is not C

1 (see Fig. 2.7).(2) Show that f ′′(u+) < 0 and that there exists a local diffeomorphism G

which depends only on f such that along �, u− = G(u+) .(3) On the other hand, deduce that t �→ u+(t) is of class C

1.

2.9 Exercises 63

(4) We can then differentiate the Rankine–Hugoniot condition. Show that

( f ′(u+)− f ′(u−))2∂xu− = d2 X

dt2(u+ − u−) ≥ 0.

(5) Show that if u0 is monotonic, then the entropy solution, if it is piecewisesmooth, does not behave as a semi-characteristic shock (verify that wecan reduce this case to the one treated above).

2.10 Converse case. We consider an initial condition u(x, 0) = b(x) where b∈C

1(R∗), and b and b′ are bounded, having limits to the left at zero. Weassume that b≡ 0 for x > 0 and that b− = b(0−) > 0, b′(0−) > 0. Finally,we suppose that (b−, 0, σ0) is a semi-characteristic shock with σ0 = f ′(0)and f ′′(0) < 0. Show that there exists T > 0 such that the entropy solution issmooth off a curve �: t �→ (X (t), t) issuing from the origin, along which asemi-characteristic shock takes place. Using the method of characteristics tothe left side of �, derive an ordinary differential equation for X (t).

2.11 N-wave. We consider the Burgers equation ut + ( 12u2)x = 0.

(1) Let

u(x, t) ={

x/t, |x | < √t,

0, |x | > √t .

Show that u is a weak solution of (2.1) for the given initial conditionu0 ≡ 0.

(2) Show that u satisfies Oleınik’s condition along the two curves of discon-tinuity.

(3) Explain why u is not the entropy solution of a Cauchy problem.

2.12 Show that in the solution of the Riemann problem, we are necessarily in oneof the following cases.

There is no discontinuity, the solution is a rarefaction between the constantstates uL and uR.

The solution is a shock, possibly a (semi-)characteristic shock.The solution is a contact discontinuity.The solution involves one or several rarefactions and one or several discon-

tinuities. If there is a discontinuity at ξ = d(uL), it is semi-characteristicon the right, or characteristic, or is a contact discontinuity. If there isa discontinuity at ξ = d(uR), it is semi-characteristic on the left, orcharacteristic, or is a contact discontinuity. The other discontinuities arecharacteristics or are contact discontinuities.

2.13 We consider the equation ut + f (u)x = 0 with f (u)= (u2 − 1)2. Solve theCauchy problem for the following initial condition:

u0(x) =

−1, x <−1,

a, −1 < x <1,

1, x > 1,

where a ∈ ( 13 , 1).

2.14 (See [41]) Let E be a strictly convex entropy of flux F . We introduce theconvex conjugate function to E by

E∗(λ) = sups∈R

(sλ− E(s)).

Let u = v(x/t) be an entropic solution of a Riemann problem.

(1) Show that F(v)ξ ≤ ξ E(v)ξ .(2) Deduce that, for all real λ, the function ξ �→ F(v) − λ f (v) − ξ (E(v) +

E∗(λ)− λv) is decreasing.(3) Using the property, show anew that v satisfies Oleınik’s inequality.(4) We assume that lims→∞ |s|−1E(s) = +∞. We know then that E∗∗ =

E [19]. Deduce the inequality

E

(f (v(ξ ))− f (v(τ ))+ τv(τ )− ξv(ξ )

τ − ξ

)

≤ F(v(ξ ))− F(v(τ ))+ τ E(v(τ ))− ξ E(v(ξ ))

τ − ξ.

for all ξ < τ .(5) Deduce that, if v is differentiable,

E(v + ξvξ − f (v)ξ ) ≤ E(v)+ ξ E(v)ξ − F(v)ξ .

Show that this inequality also implies that ξvξ = f (v)ξ .

2.15 Let f and g be two regular functions. We denote the resolvent semi-group of(2.1) by S f (t), that is, the mapping u0 �→ u(·, t) which is defined from L∞

into itself. By replacing f by g, we also consider Sg.

(1) If f and g are convex, and if u0 is an initial condition in a Riemannproblem, show that S f (t) ◦ Sg(s)u0 = Sg(s) ◦ S f (t)u0, for all s, t > 0.

(2) If f is strictly convex and g strictly concave, show that to the contraryS f (t) ◦ Sg(s)u0 �= Sg(s) ◦ S f (t)u0.

2.9 Exercises 65

(3) More generally, show that if S f (t) and Sg(s) commute, then:(i) At each point f ′′ and g′′ are of the same sign.

(ii) The semi-characteristic shocks are the same for the two equationsut + f (u)x = 0 and ut + g(u)x = 0.

(4) If f ′′ > 0 on (0,+∞) and f ′′ < 0 on (−∞, 0), show that the fluxes g forwhich S f (t) and Sg(s) commute obey a second order linear differentialequation.

(5) Solve this equation when f (u) = u3.

2.16 Let f and g be two convex fluxes. Again taking the notation of the precedingexercise, show by using Lax’s formula that

S f (s) ◦ Sg(t) = Sg(t) ◦ S f (s).

2.17 We assume that f is strictly convex and that lims→∞( f (s)/|s|) = +∞. Weare given that u0 ∈ L∞(R).

(1) Show that, for all t > 0 and all x > 0, the lower bound of v0(y)+tg((x − y)/t) is attained at least one point y ∈ R.

(2) Let t > 0 and x1, x2 ∈ R with x1 < x2. We denote by yi a point at whichv0(y) + tg((xi − y)/t) attains its minimum. Using the convexity of g,show that y1 < y2.

(3) Deduce that, t being fixed, y is unique except for a set of values of x , atmost denumerable. To what do these exceptional values correspond?

2.18 Starting from Lax’s formula establish again Oleınik’s condition, if f is convex,or from the dual formula if u0 is increasing.

2.19 We consider a conservation law ut+ f (u)x = 0 where f ′′ does not vanish. Letu0 ∈ L1(R) ∩ L∞(R) and u be the entropy solution of the Cauchy problem.As a way of simplifying the calculations, we assume that u is of class C

1 offthe shock curves which are assumed to form a finite family of smooth curves.

(1) Let T > 0 and let x1 < x2 be two real numbers. Let (y j , 0) be the base ofa characteristic passing through (x j , T ). By integrating the conservationlaw over a suitable domain, show that

T (F ◦ u(x2, T )− F ◦ u(x1, T )) =∫ y2

y1

u0(x) dx −∫ x2

x1

u(x, T ) dx,

where F(s) := f (s)− s f ′(s).(2) Deduce that

TV(F ◦ u(·, T )) ≤ 2

T‖u0‖1.

(3) Assuming that f (0) = 0, show that

‖F ◦ u(·, T )‖∞ ≤ 2

T‖u0‖1.

2.20 Application: The Burgers equation. We assume that u0 is bounded and withcompact support: supp u0 ⊂ [a, b].

(1) Show that ‖u(t)‖∞ ≤ 2(t−1‖u0‖1)1/2. Deduce that

supp u(t) ⊂ [a − 2(t‖u0‖1)1/2, b + 2(t‖u0‖1)1/2].

(2) Using Oleınik’s inequality (x < y implies u(y, t)− u(x, t) ≤ (y − x)/t ,(2.32)), deduce that

TV(u(t)) ≤ 2b − a

t+ 8

(‖u0‖1

t

)1/2

.

2.21 Let f be an increasing function (with f ′> 0) of class C2. Define a concept

of an entropy solution of the mixed problem

ut + f (u)x = 0, (x, t) ∈ R+ × R

+,

u(x, 0) = u0(x), x ∈ R+,

u(0, t) = 0, t ∈ R+.

Show that this entropy solution exists for all u0 ∈ L∞(R+).2.22 We suppose that f is convex and that, more precisely, f ′′ ≥ α, where α

is a strictly positive constant. Show that each approximate solution uε inthe semi-group method satisfies (uε)x ≤ 1/αt . Deduce that the same istrue for the entropy solution in a distributional sense which should be madeprecise.

2.23 We are given u0 ∈ L∞(R) periodic, of period L .

(1) Show that the entropy solution of the Cauchy problem is equally periodicwith respect to x , with the same period (use uniqueness).

(2) Show that the average of u(t) over a period is constant:

L−1∫ L

0u(x, t) dx = L−1

∫ L

0u0(x) dx .

(3) Suppose that, in addition, α =: infx∈R f ′′ > 0. With the help of Oleınik’sinequality, show that

supx∈R

u(x, t)− infx∈R

u(x, t) ≤ L/tα.

Deduce that u(t) converges uniformly to the mean value of u0.

2.9 Exercises 67

(4) Example: Solve explicitly the Cauchy problem for the Burgers equationwith u0(x) = x − E[x], where E[x] denotes the integral part of x .

2.24 We consider a scalar conservation law in spatial dimension d = 2, but whoseflux has only a single component:

∂u

∂t+ ∂

∂x1f (u) = 0,

u(x1, x2, 0) = a(x1, x2).

We refer to the comment on Kruzkov’s theorem which concerns the scalarconservation laws in more than one space dimension. The initial function ais bounded and integrable, and u is the entropy solution.

(1) As u ∈ C (R+t ; L1(R2)), we can speak of the integrable function u(t) foreach value of t . Show that, for all t > 0 and almost all x2 ∈ R, we have∫

R

u(x1, x2, t) dx1 =∫

R

a(x1, x2) dx1.

Denote that value by m(x2) .(2) We recall that, for every measurable function F from R

p into R, the totalvariation of F is

TVp(F) = supξ∈S p−1

limh→0

1

|h|∫

Rp|F(y + hξ )− F(y)| dy.

show that, for all t > 0,

TV2(u(·, t)) ≥ TV1(m).

(3) Compare with Exercise 2.19.(4) Generalise to the spatial dimension d ≥ 2.(5) Construct an example, with d = 2, f (s) = 1

2s2 and a(·, x2) an odd func-tion for every x2 and such that m ≡ 0 (because of oddness); neverthelessu(t) should satisfy

limt→ inf+∞ TV2(u(t)) > 0.

Use could be made of N-waves.

3

Linear and quasi-linear systems

The object of this chapter is to derive the algebraic and geometrical propertieswhich ensure that the Cauchy problem for a first order system of conservationlaws is well-posed. In fact, we consider two classes. First of all we consider thequasi-linear systems of the first order, which are of the form

∂t u +d∑

α=1

Aα(u)∂xαu = b(u). (3.1)

The vector field b is defined and smooth on an open domain U ∈ Rn . Similarly

the mappings u �→ Aα(u) are defined and smooth on U, with values in the spaceof matrices Mn(R).

The second class, contained in the preceding, will be that of systems of conser-vation laws

∂t u +d∑

α=1

∂xαf α(u) = b(u). (3.2)

A system of the form (3.2) is clearly quasi-linear, with Aα(u) = du f α(u), wheredu denotes differentiation with respect to u.

Let us consider the Cauchy problem for one or other of these systems:

u(x, 0) = u0(x, 0). (3.3)

If u0 is a constant, the obvious solution is a function of t alone, which satisfies theordinary differential equation u′ = b(u). Let us look at the case of a given initialcondition of the form uε

0 = u0 + εv0(x). As we wish that the solution dependscontinuously on the given initial conditions, we pay attention to a solution of theform uε(x, t) = u(t)+ εv(x, t)+ O(ε2) on a bounded time interval. The corrector

68

3.1 Linear hyperbolic systems 69

v is a solution of the linear problem

∂tv +d∑

α=1

Aα(u(t))∂xαv = db(u(t)) · v, (3.4)

v(x, 0) = u0(x). (3.5)

A necessary condition for the asymptotic expansion uε to be correct is certainlythat v exists! In fact, as the remainder of order ε2 has to be determined amongother things with the help of v, it is important that v has sufficient smoothness, atleast that of v0(x). We begin, therefore, by considering the Cauchy problem for alinear system of the first order whose general coefficients depend on the time. Theright-hand side is the least important since we can make it as small as we please bychanging the time variable t �→ ηt . Thus we suppose that b ≡ 0.

3.1 Linear hyperbolic systems

We now therefore consider the following general system:

∂t u +d∑

α=1

Aα(t)∂xαu = 0, (3.6)

u(x, 0) = u0(x), (3.7)

where the matrices Aα depend on t in a smooth manner.Although it is possible to study the Cauchy problem for (3.6) in a space of smooth

functions, for which the partial derivatives have the usual sense, it is simpler andmore general to consider the weak solutions. A weak solution of (3.6) is a tempereddistribution u, that is to say an element of the dual of the Schwartz class (see below)which satisfies the conditions⟨

u, ∂tϕ +d∑

α=1

AαT∂xαϕ

⟩S ′,S

+∫

Rd

u0(x)ϕ(x, 0) dx = 0,

∀ϕ ∈ S (R× R). (3.8)

Since the Fourier transform F with respect to x of (3.6) leads to a linear ordinarydifferential equation, the natural body of a study of the Cauchy problem is the space(L2(Rd ))n or every other space which simply enables the definition of F and itsinverse, for example a Sobolev space Hs(Rd )=W s,2. We notice that the spacesW s,p(Rd ) are in general inappropriate for F as F sends L p(Rd ) into Lq (Rd ) if andonly if p−1 + q−1 = 1, and p ≤ 2, with the result that an isomorphism from L p

to its own dual is possible if and only if p= 2. In the case of constant coefficients,

70 Linear and quasi-linear systems

we know in fact [4] that the Cauchy problem is not well-posed in L p for p �= 2,when the matrices Aα do not commute with each other. Obviously in one spacedimension that obstruction does not take place.

Fourier analysis

The Fourier transform F is an isometry of L2(R) which is defined on the subspaceL1 ∩ L2 by the formula

F u(ξ ) := (2π )−n/2∫

Rd

e−iξ ·xu(x) dx .

Its inverse is the conjugate transformation

F u(ξ ) = (2π )−n/2∫

Rd

eiξ ·xu(x) dx .

We shall use only one of the properties of differentiation (F ∂xαu)(ξ ) = iξαF u(ξ ),

which also furnishes a formula for F . It follows that F is an isomorphism ofthe Schwartz class S (Rd ), the space of functions defined on R

d , of class C∞

and decreasing rapidly, along with all their derivatives (decreasing at infinity morerapidly than every non-zero rational fraction). By duality, the Fourier transfor-mation extends to an isomorphism of the dual space S

′(Rd ), via the Plancherelformula 〈F ϕ, u〉S ′,S := 〈ϕ, F u〉S . If t �→ u(·, t) is continuous in (0, T )with values in L2 (interpreting (3.6) in the sense of distributions) we can ap-ply the operator F . We then have the equivalent differential system, for v :=F u(ξ, t),

∂tv = −iA(ξ, t)v, (3.9)

v(ξ, 0) = v0(ξ ) := F u0(ξ ), (3.10)

where A(ξ, t) :=∑1≤α≤d ξα Aα(t).

We thus express v with the help of the resolvent: v(ξ, t) = R(t, 0, ξ )v0(ξ ) whereR(·, s; ξ ) is the matrix solution of the following Cauchy problem:

d

dtR(·, s; ξ ) = −iA(ξ, ·)R(·, s; ξ ),

R(s, s; ξ ) = Idn.

Since F is an isometry, the Cauchy problem (3.6) is well-posed in L2 if and onlyif there exists a constant CT independent of u0 and such that

supt∈(0,T )

‖v(t)‖L2 ≤ CT ‖v0‖L2 .


But since v0 �→ v(t) is just a multiplication operator v0(ξ ) �→ R(t, 0, ξ )v(ξ ),this is equivalent to saying that

supt∈(0,T )ξ∈R

d

‖R(t, 0, ξ )‖ ≤ CT . (3.11)

In this inequality, the constant CT depends on the matrix norm chosen, but the factthat it is finite is independent of this. Making the change of variables (t, ξ ) �→(t/a, aξ ), the system is transformed into

d

dτR = −iA(η, aτ )R,

which shows that R(aτ, 0, η/a)→ exp(−iτ A(η, 0))when a→ 0.Wededuce there-fore that a necessary condition for (3.11) is

supτ∈R

η∈Rd

‖exp(−iτ A(η))‖ ≤ CT , (3.12)

where A(η) stands for the restriction of A to the initial time or, similarly, to anyother instant.

Definition 3.1.1 We say that the linear system with constant coefficients

∂t u +∑

1≤i≤d

Aα∂xαu = 0 (3.13)

is hyperbolic if there exists C such that supξ∈R ‖exp(−iA(ξ ))‖ ≤ C .

The first important result is the following.

Theorem 3.1.2 For a linear system of the first order with constant coefficients, theCauchy problem is well-posed in L2 if and only if this system is hyperbolic. For ahyperbolic system, being given u0 ∈ L2(Rd ), there exists one and only one solutionof (3.6) in C (R; L2(Rd )).

Proof From the preceding analysis, we see that the hyperbolicity ensures that theCauchy problem is well-posed in L2 and more precisely that ‖u(t)‖L2 ≤ C‖u0‖L2

for all t ∈ R, for R(t, 0, ξ ) = exp(−iA(tξ )). Let us show that in fact t �→ u(t) is acontinuous map of R into R

d . Now |v(ξ, t)| ≤ C |v0(ξ )|, which bounds |v| aboveby a square-integrable function independent of t . As t �→ v(ξ, t) is continuous, thetheorem of dominated convergence assures the continuity demanded.

Conversely, suppose that the Cauchy problem is well-posed on (0, T ), the solutionbeing of class L2. Then for t fixed and non-zero, u0 �→ u(t) and thus v0 �→

v(t) are continuous endomorphisms of L2. As the second is only a multiplicationoperator by exp(−it A(ξ )), we easily calculate its norm, which takes the valuesupξ∈R

d ‖exp(−it A(ξ ))‖. It must therefore be bounded.

Geometric conditions of hyperbolicity

It is not simple, a priori, to verify the property of hyperbolicity for a given systemsince it demands the calculation of exponentials of matrices depending on d pa-rameters. This calculation can only be carried out by the diagonalisation of eachmatrix A(ξ ), by reason of the formula exp(PMP−1) = P(exp M)P−1. First of all,hyperbolicity implies

supξ∈R

d

ρ(exp(−iA(ξ ))) < ∞,

where ρ(M) denotes the spectral radius of a matrix M . But as

exp(−iA(mξ )) = (exp(−iA(ξ )))m

for every integer m, it is equivalent to saying that ρ(exp(−iA(ξ )))= 1 for all ξ ∈Rd ,

that is, since the eigenvalues of exp M are the exponentials of those of M , that thespectrum SpA(ξ ) of the matrices A(ξ ) is real. In addition, if one of these matricespossesses an eigenvalue λ of which the algebraic and geometrical multiplicitiesdiffer one from the other, then there exist two non-zero vectors w and z such thatAw = λw and Az = λz + w. We then deduce that

e−it A(ξ )z = e−itλ(z − itw),

which contradicts the boundedness condition (3.12). We have therefore

Lemma 3.1.3 If the system (3.13) is hyperbolic, then the matrix A(ξ ) is diagonal-isable with real eigenvalues, for all ξ in R

d .

Although the converse of this lemma turns out to be true (this is the so-calledKreiss matrix theorem), we shall give two proofs of hyperbolicity under more re-strictive (but rather natural) conditions. Let us look first of all at the case d = 1.Writing A := A(1), we have A(ξ )= ξ A. If A is diagonalisable with real eigenval-ues, A=PDP−1, we have exp(−iA(ξ ))= P(exp(−iξ D))P−1. Now exp(−iξ D) isa diagonal matrix whose diagonal terms are the complex numbers eiλ of modulusone where λ ∈ Sp(A), it is therefore bounded and the same is true of exp(−iA(ξ )).

In the case d ≥ 2, we proceed in the same way, but the matrices P and D dependon ξ : exp(−iA(ξ )) = P(ξ ) exp(−iD(ξ ))P−1(ξ ). The matrix D is homogeneous of

degree 1 with respect to ξ and we can choose P to be homogeneous of degree 0.Then

‖exp(−iA(ξ ))‖≤ K (ξ )‖exp(−iD(ξ ))‖,where K (ξ ) := ‖P(ξ )‖ · ‖P(ξ )−1‖. If the matrix A(ξ ) is diagonalisable with realeigenvalues, then again ‖exp(−iD(ξ ))‖ = 1 and so ‖exp(−iA(ξ ))‖ ≤ K (ξ ). Fromthis we have the sufficient condition

Proposition 3.1.4 We suppose that the matrices A(ξ ) are diagonalisable with realeigenvalues, uniformly with respect to ξ , that is that ξ �→ K (ξ )=‖P(ξ )‖·‖P(ξ )−1‖is bounded on R

d . Then the system (3.13) is hyperbolic.

An essential application of this proposition is the following.

Definition 3.1.5 The system (3.13) is symmetrisable hyperbolic if there exists apositive definite symmetric matrix S such that the matrices Sα := S Aα are sym-metric.

Theorem 3.1.6 If the system (3.13) is symmetrisable hyperbolic, then it is hyper-bolic.

Proof Let S(ξ ) :=∑1≤α≤d ξαSα. We have A(ξ ) = S−1S(ξ ). Let be the positive

symmetric square root of S−1. As S(ξ ) is symmetric, it is diagonalisable in anorthonormal basis, that is to say that there exists a matrix O(ξ ) ∈ On(R) such thatO(ξ )−1S(ξ )O(ξ ) is diagonal and real. Then we can choose P(ξ )=O(ξ ) andwe have K (ξ ) ≤ K0 := ‖‖‖−1‖.

Another favourable case, independent of the preceding one, is that of strictly hy-perbolic systems.

Definition 3.1.7 We say that the system (3.6) is strictly hyperbolic if the matricesA(ξ ) are diagonalisable with real eigenvalues, with constant multiplicities when ξ

ranges over Rd − {0}.

It comes to the same thing to say that the eigenvalues are continuous functions onR

d − {0}, ξ �→ λ j (ξ ), with

λ1(ξ ) < λ2(ξ ) < · · · < λr (ξ ).

Theorem 3.1.8 If the system (3.13) is strictly hyperbolic, then it is hyperbolic.


Proof The key point of the proof is a geometrical lemma, which uses only thecontinuity of the mapping ξ �→ A(ξ ).

Lemma 3.1.9 If the matrices A(ξ ) are diagonalisable with eigenvalues of multi-plicities independent of ξ for ξ �= 0, then the eigenspaces depend continuously(and even analytically, but that is indifferent to us) on ξ .

For all ξ0 �= 0, there thus exist a compact neighbourhood V (ξ0) and a continuousoption, hence bounded, of P(ξ ) on V (ξ0). As the sphere Sd−1 is compact, it iscovered by a finite number of such neighbourhoods, with the result that we canchoose a bounded option (but not necessarily continuous) of P on Sd−1, that is onR

d since P is homogeneous of degree zero with respect to ξ .It remains to prove the lemma. Let ξ0 �= 0 and ξm be a sequence tending to

ξ0. The eigenvalues λ j (ξ ) are continuous functions of the coefficients, thus ofξ . Let n j be the dimension of the sub-eigenspace associated with E j (ξ ). As thegrassmannian variety of the sub-spaces of dimension n j of R

n is compact, thesequence E j (ξm) takes a limiting value Fj which is included, by continuity, inE j (ξ0). Their dimensions being the same, these two sub-spaces are equal.

Example 3.1.10 Let us consider the example of the model below, for which n =d = 2:

ut +(

1 0

0 −1

)ux +

(0 1

1 0

)uy = 0. (3.14)

We have

A(ξ ) =(

ξ1 ξ2

ξ2 −ξ1

).

The eigenvalues are ±|ξ |, that is to say that the speeds of propagation (see below)take the values±1. They are independent of the direction, which is not the generalcase but corresponds to an invariance of the system under the group O2(R).

Plane waves

The normalised eigenvalues c j (ξ ) := λ j (ξ )/|ξ | must be seen as the speeds ofpropagation of plane waves in the direction ξ for ξ �= 0. There are two ways inwhich to see that.


First of all, if u0 ∈ L2(Rd ), the solution of the Cauchy problem is given formallyby

u(x, t) = (2π )−12 n

∫R

dP(ξ )ei(x ·ξ I2−t D(ξ ))w0(ξ ) dξ,

wherew0(ξ ) := P(ξ )−1v0(ξ ). Thus,u appears as an infinite sumofone-dimensionalsolutions

(x, t) �→ P(ξ )ei(x ·ξ I2−t D(ξ ))w0(ξ ).

These can be decomposed in their turn, using the column vectors r j (ξ ) of P(ξ ),which are the eigenvectors of A(ξ ), into plane waves of the form

(x, t) �→ a j (ξ )ei(x ·ξ−tλ j (ξ ))r j (ξ ).

The second approach, more elementary, consists of verifying that, for all ξ �= 0and every locally integrable function f , the plane wave

u(x, t) := f (x · ξ − tλ j (ξ ))r j (ξ )

is a weak solution of (3.13). Since here u(x, t) = u0(x − c j (ξ )tν) where ν = ξ/|ξ |is a unit vector, the number c j (ξ ) clearly plays the role of a speed of propagation.

Exercises

3.1 Show that every scalar equation (n = 1) is hyperbolic.3.2 Show that if d = 1, every hyperbolic system is symmetrisable.3.3 Assume that the matrices Aα commute with each other: Aα Aβ = Aβ Aα.

(1) Show that (3.6) is hyperbolic if and only if each one-dimensional systemvt + Aαvx = 0 is hyperbolic.

(2) By a linear change of variables u �→ v := Pu, show that the system isequivalent to n decoupled transport equations:

∂tvi + Vi · ∇xvi = 0.

3.4 Show that the system of Maxwell’s electromagnetic equations is hyperbolic.Here n = 6, d = 3 and u is composed of two vector fields B and E . Theequations are

Bt + curl E = 0, (3.15)

Et − c2 curl B = 0. (3.16)

Calculate the speeds of propagation and determine which correspond to planewaves of a physical nature, that is, which satisfy the constraint divB = 0.

3.5 We consider the linear system of isotropic elasticity of small deformations.The unknown is the displacement y(x, t) ∈ R

d . The equations of the secondorder are

∂2yi

∂t2= α�yi + β∂idiv y.

The parameters α > 0, β > 0 are the Young’s moduli of the material.

(1) Put the system into a first order form with a constraint,

∂t u +d∑

α=1

Aα∂αu = 0,

d∑α=1

Bα∂αu = 0.

(2) Calculate the plane waves. Deduce that the system is strictly hyperbolic.Calculate the speeds of propagation. Note: they are traditionally denotedby cS < cP and correspond respectively to shear waves, in which the ma-terial vibrates transversely to the direction of propagation, and to pressurewaves where the vibration is parallel to this propagation.

(3) Show that the system is symmetric hyperbolic in expressing the conser-vation of mechanical energy. This is the sum of a kinetic term and of anenergy of deformation.

3.6 We suppose that u0 ∈ Hs(Rd ), that it to say that ξ �→ (1 + |ξ 2|) 12 sv0(ξ ) is

square-integrable. Show that the weak solution of a hyperbolic problem (3.6)satisfies u ∈ C (R; Hs(Rd )) ∩ C

1(R; Hs−1(Rd )).3.7 (1) Let θ ∈ R. Find a matrix Mθ such that u �→ uθ where

uθ (x, y, t) := Mθu(x cos θ + y sin θ,−x sin θ + y cos θ, t)

preserves the set of solutions of (3.14).(2) Show that it is not possible to choose the matrix of the change of basis

P(ξ ) with the result that ξ �→ P(ξ ) is continuous on S1.3.8 We assume that u0 ∈ Hs(Rd ) for s > 0 sufficiently large (see Exercise 3.6)

and we consider a symmetrisable hyperbolic system.

(1) Show that there exists a number M > 0 such that, for all ξ ∈ Sd−1 and allw ∈ R

n , we have

|(S(ξ )w, w)| ≤ M(Sw, w),

where (·, ·) denotes the usual scalar product in Rn .

(2) Verify, for the solution of the Cauchy problem, the equality

∂t (Su, u)+∑

1≤α≤d

∂xα(S Aαu, u) = 0.

Fig. 3.1: The cone of dependence of a disk D(0, R).

(3) Deduce that, if u0 is zero for |x | < R, then u is zero in the interior of thecone defined by t > 0 and |x | +Mt < R (integrate the preceding formulaon the cone: see Fig. 3.1).

(4) Express this result in terms of a propagation phenomenon with finitespeed.

3.9 We consider a hyperbolic system for which n = d = 2.

(1) By a linear change of variable u �→ v := Pu, reduce to the case whereA1 is a diagonal matrix.

(2) Show that if A1 is of the form aI2, a ∈ R, the system (3.6) may bereduced to the one-dimensional case, with a given initial value dependingon a parameter.

(3) We suppose now that A1 is diagonal but is not of the form aI2. In calcu-lating the characteristic polynomial of A2 + x A1, show that either A2 isdiagonal or a2

12a221 > 0.

(4) Show then that the system is symmetrisable.

3.10 We consider the system (3.6), where the matrices Aα depend on the time t .We suppose that at each instant, the system is symmetrisable by a matrix S0(t)which is of class C

1 with respect to t :

S0(t) is symmetric and positive definite,S(ξ, t) := S0(t)A(ξ, t) is symmetric.

(1) Show that (R∗S0 R)t = R∗(dS0/dt)R, where R(t, 0, ξ ) is the resolventconsidered above.

(2) Show that there exists a number cT > 0 such that, for all t ∈ [0, T ] andall ξ ∈ R

d , Tr(R∗(dS0/dt)R) ≤ cT Tr(R∗S0 R).(3) Deduce that there exists a number CT > 0 such that, for all t ∈ [0, T ] and

all ξ ∈ Rd , ‖R(ξ, t)‖ ≤ CT .

The Cauchy problem for (3.6) is therefore well-posed in L2(Rd ).


3.11 We suppose that (3.6) is symmetrisable hyperbolic. Let B ∈ Mn(R). We wishto show that the Cauchy problem for the system

∂t u +d∑

α=1

Aα(t)∂xαu = Bu (3.17)

is well-posed in L2(Rd ). We denote by S(t) the solution operator of (3.6),u0 �→ S(t)u0 = u(t) (see the preceding exercise for the construction of thisoperator, an endomorphism of Hs(Rd ) (for s ≥ 0).

(1) Let u0 ∈ L2(Rd ) and f ∈ L1(R; L2(Rd )). Show that the inhomogeneousCauchy problem

∂t u +∑

1≤α≤d

Aα(t)∂xαu = B f, (3.18)

u(x, 0) = u0(x) (3.19)

possesses a unique solution in C (R; L2(Rd )), given by Duhamel’s formula

u(t) = S(t)u0 +∫ t

0S(t − s)B f (s) ds.

We write u = T f .(2) We construct a sequence (um)m∈N by u0(x, t) ≡ u0(x), and um+1 = T um .

Show that T restricted to the space L1(0, T ; L2(R)) is a contractionmapping provided that T > 0 is sufficiently small.

(3) Deduce that the sequence (um)m∈N converges in C (0, T ; L2(Rd )) and thatits limit is a weak solution of (3.17) in the band R

d × (0, T ).(4) Making use of the fact that the number T does not depend on u0 show

that (3.17) possesses a weak solution on Rd × R.

(5) Show that themapping u0 �→ u is continuous in L2(Rd ) in C (0, T ; L2(Rd )),u being the solution of (3.17).

(6) Show that there exists a constant C , depending only on S0 and B, suchthat

d

dt

∫R

d(S0u, u) dx ≤ ‖B‖

∫R

d|u|2 dx ≤ C

∫R

d(S0u, u) dx .

(Do it first of all for u0 ∈ Hs(Rd ) for s sufficiently large, then deduce thegeneral case with the help of the preceding question).

(7) Deduce that the solution of (3.17) is unique (reduce to the case u0 ≡ 0,then apply Gronwall’s lemma).

3.2 Quasi-linear hyperbolic systems 79

3.2 Quasi-linear hyperbolic systems

We return to the case of quasi-linear systems, that is to say to systems of the form(3.1). We have seen that a formal analysis of the stability of perturbations of smallamplitude, via an asymptotic development with respect to this amplitude, requiresthe hyperbolicity of the linearised system. That condition is in fact far from beingsufficient, but as we have not found one which is truly satisfying, mathematicianshave for a long time adopted the following definition. We shall use the notation

A(ξ ; u) =∑

1≤α≤d

ξα Aα(u)

and more generally P(ξ ; u), . . ., for the matrices having been defined in the studyof the linear case but which now depend on the state u of the system.

Definition 3.2.1 The quasi-linear system (3.1) is said to be hyperbolic if for all u,the linear system

∂t u +∑

1≤α≤d

Aα(u)∂xαu = 0

is hyperbolic, the matrices P(ξ ; u) and their inverses being bounded on every com-pact set of Sd ×U.

This definition does not assure us that the Cauchy problem for (3.1) is well-posed(one may no longer apply Kreiss’ matrix theorem; much more, the well-posednessis no longer a matter of matrices only), even in a space of smooth functions andlocally in time. Its popularity comes from the fact that it is invariant under the changeof unknown u �→ v = ϕ(u). If ϕ is a diffeomorphism of U into V , this change ofvariable transforms a quasi-linear system into another quasi-linear system

∂tv +∑

1≤α≤d

Bα(v)∂xαv = 0,

where the matrices Bα are conjugate to the matrices Aα:

Bα(ϕ(u)) = dϕ(u) Aα(u) (dϕ(u))−1.

In fact, if A(ξ ; u)= P(ξ, u)D(ξ, u)(P(ξ, u))−1, we diagonalise B(ξ, u) by the ma-trix dϕ(u)P(ξ, u) which is bounded on every compact set when P is.

However, other changes of the unknown field (the dependent variables) can trans-form a hyperbolic quasi-linear system into a non-hyperbolic quasi-linear system.The basic example is the following, which is clearly hyperbolic in the sense of the


definition given above. We have n = 2, d = 1:

∂t u +(

u1 0

0 u1

)∂xu = 0. (3.20)

We transform this system by v1 = u1, v2 = ∂xu2 and we obtain, on differentiatingthe second equation of (3.20) with respect to x ,

∂tv +(

v1 0

v2 v1

)∂xv = 0, (3.21)

whose matrix is no longer diagonalisable, except for v2 = 0.

3.3 Conservative systems

The systems originating in physics or mechanics form in fact a much more restrictedclass than those of the first order quasi-linear systems. These are the systems ofconservation laws which can be written in the form

∂t u + ∂xαf α(u) = 0, (3.22)

where f α is a smooth vector field defined on U. We can re-write such a systemunder the quasi-linear form. We shall then have Aα(u) = d f α(u) and A(ξ, u) =d(ξ · f )(u).

The conservation principle which is essential to define the weak solutions beyondthe formation of the discontinuities is not preserved by the diffeomorphisms u �→v = ϕ(u) in general. And even when a diffeomorphism ϕ transforms (3.22) intoanother conservative system vt + div g(v) = 0, the notion of a weak solution willbe in general modified, which is unacceptable because, even in dimension d = 1,the Rankine–Hugoniot conditions [ f (u)] = σ [u] and [g(ϕ(u))] = σ [ϕ(u)] are notequivalent. There are notable exceptions. For example when f is linear, and moregenerally if all the characteristic fields (see below) are linearly degenerate (idem).For a general system, the only transformations which preserve the notion of a weaksolution are the affine functions u �→ Au+b, because then g(v)= A f (A−1(v−b))and [g(v)] = A[ f (u)] = σ A[u] = σ [v]. From which comes the importance of anaffine theory of the system of laws of conservation. One should consult the thesisof B. Sevennec [93] on this subject. The simplest result in this direction is that ofG. Boillat. For this statement we first of all need a new concept.

Definition 3.3.1 A characteristic field of a quasi-linear system of the form (3.1) isa mapping (ξ, u) �→ (λ, E) defined and smooth on an open set O of U, where λ is

3.3 Conservative systems 81

an eigenvalue of A(ξ, u), of constant multiplicity, and E the associated eigenspace,whose dimension is the multiplicity of λ.

We have thus excluded the case where λ is associated with a non-trivial Jordan formin the decomposition into characteristic sub-spaces of the matrix A(ξ, u). Clearlyξ �→ λ(ξ, u) is homogeneous of degree 1. When there is a single spatial dimensionwe set ξ = 1 and a characteristic field is merely a mapping u �→ (λ, E). Anotheressential notion is that of differential eigenform, that is to say of a left eigenvectorfield (ξ, u) �→ l:

l(ξ, u)(A(ξ, u)− λ(ξ, u)In) = 0.

Definition 3.3.2 A characteristic field is said to be linearly degenerate on O if thedifferential of λ is zero on E = Ker(A − λIn) when u ranges over O :

duλ · r ≡ 0, ∀r ∈ Ker(A − λIn), ∀u ∈ O .

Theorem 3.3.3 (Boillat [3]) Let us consider a system of conservation laws (3.22).We suppose that A(ξ, u) has an eigenvalueλ(ξ, u) whose multiplicity m is a constantgreater than or equal to 2. Then the characteristic field (λ, Ker (A− λIn)) is linearlydegenerate. In addition, ξ �= 0 being given, the affine sub-spaces u+Ker (A(ξ, u)−λ(ξ, u)In) are the tangent spaces to a family of sub-manifolds of dimension m.

Obviously, the integral sub-manifolds mentioned in the theorem form a foliationof the open set O called the characteristic foliation associated with λ. The charac-teristic foliation depends only on the direction of ξ but on neither its sense nor itsnorm. If the multiplicity of λ has the value 1, then Ker (A − λIn) is generated bya vector r (ξ, u), with the result that the foliation still exists, formed by the integralcurves of the vector field u �→ r .

Proof Let us fix ξ �= 0. Let u �→ r be a smooth field of eigenvectors on O . Sincem > 1, we can choose a second field of eigenvectors u �→ s, smooth and linearlyindependent of the first at every point. Differentiating the relation (du(ξ · f ) −λ(u))r (u) ≡ 0 in the direction s we obtain

D2(ξ · f )(r, s)− (duλ · s)r = (d f − λ)dur · s.Interchanging the roles of r and s, we have also

D2(ξ · f )(s, r )− (duλ · r )s = (d f − λ)dus · r.As D2 f is a symmetric bilinear form, we can eliminate the term D2(ξ · f )(r, s)between these two equalities. This gives

(d f − λ){r, s} = (duλ · s)r − (duλ · r )s, (3.23)

where {r, s} denotes the Poisson bracket of the field vectors r and s.


The right-hand side of equation (3.23) is a vector of Ker(dξ · f −λ) with the resultthat {r, s} ∈Ker(dξ · f −λ)2=Ker(dξ · f −λ), the second equality being due to theequality of the algebraic and geometric multiplicities of λ. The set of eigenvectorfields associated with the eigenvalue λ is thus a Lie algebra. This property, calledthe Frobenius integrability condition, ensures the existence of the foliation whoseaffine spaces u + Ker(A(ξ, u)− λ(ξ, u)In) are the tangent spaces.

Finally, again using (3.23), we find the relation of linear dependence (duλ · s)r −(duλ · r )s = 0, which implies the nullity of the coefficients. For example, we haveduλ · r = 0. The characteristic field is degenerate.

In the light of this theorem, we are led to a finer definition of hyperbolicity in thecase of a quasi-linear system. In addition to imposing that the matrices A(ξ, u) arediagonalisable on R with eigenvalues of constant multiplicities m j we shall demand,when m j ≥ 2, that the corresponding characteristic field be linearly degenerate andthat the eigenvector fields form a Lie algebra. This now excludes the pathologicalexample (3.20) of the preceding section.

3.4 Entropies, convexity and hyperbolicity

Physical systems

Most of the systems arising in physics or mechanics are conservative systems andadmit a supplementary conservation law of the form E(u)t + divx F(u) = 0 wherethe time component E is a strictly convex function on U. The convexity makesgood sense here since it is an affine notion and the only group of transformationsof dependent variables u which we accept is the affine group. On the other hand,care must be taken not to apply a non-linear change of variables, even in preservingthe conservative nature of the system, for the function v �→ E(ϕ−1(v)) need not beconvex if E is.

Definition 3.4.1 We say that a real function u �→ E is an entropy of the system (3.1)if there exists a mapping u �→ F(u) with values in R

d , called the entropy flux, suchthat every classical solution of (3.1) satisfies the equality E(u)t + divx F(u) = 0.

The entropy–entropy-flux pairs are thus the solutions of the linear first order equa-tions in U,

∂Fα

∂u j=

∑1≤i≤n

aαi j

∂E

∂ui, ∀ 1 ≤ j ≤ n, 1 ≤ α ≤ d. (3.24)

The entropies and their convexity have an essential role in the theory of hyper-bolic systems of conservation laws. In particular, the mere convexity assures thehyperbolicity.

3.4 Entropies, convexity and hyperbolicity 83

Theorem 3.4.2 (26, 35) If a conservative system (3.22), whose state u(x, t) takes itsvalues in a convex domain U, possesses a strictly convex entropy in the sense thatD2

u E is positive definite at every point, then matrices A(ξ, u) are symmetrisable:there exist positive definite symmetric matrices S(u) (in fact S(u) = D2

u E) suchthat the matrices S A(ξ, u) are symmetric.

From Theorem 3.1.6, such a system is thus hyperbolic, strictly hyperbolic wherethe eigenvalues are of constant multiplicities. We thus shall adopt the followingdefinition.

Definition 3.4.3 A physical system is a system of conservation laws whose stateu(x, t) takes its values in a convex domain U of R

n and which possesses a stronglyconvex entropy on U.

Proof of theorem

We introduce the conjugate convex function E∗(q) := supu∈U (q ·u−E(u)), definedon the range of du E , and we make the change of variables q := du E(u) whoseinverse is u = dq E∗(q). We have E∗(q) = q ·u−E(u). The matrix S(u) = D2E(u)is positive definite symmetric and we have Sut = qt .

We rewrite the equations (3.24):

∂Fα

∂u j=

∑1≤i≤n

qi∂ f α

i

∂u j.

Writing g(q) = f (u(q)), we have also

∂ f αi

∂u j=

∑1≤k≤n

∂gαi

∂qk

∂qk

∂u j=

∑1≤k≤n

∂gαi

∂qk

∂2E

∂u j∂uk,

and, similarly, with H (q) := F(u(q)),

∂Fα

∂u j=

∑1≤k≤n

∂ Hα

∂qk

∂qk

∂u j=

∑1≤k≤n

∂ Hα

∂qk

∂2E

∂u j∂uk.

The equation which governs the entropies and their fluxes can thus be written

Sdq H = S∑

1≤i≤n

qidq gi .

Since S is invertible, this becomesdq H =∑1≤i≤n qidq gi = dq (

∑1≤i≤n qi gi )−g.

Finally, we have

gak =

∂hα

∂qk,


where

hα =( ∑

1≤i≤n

∂E

∂uif αi − Fα

)◦ u.

The system (3.22) thus has the equivalent form (even when this concerns the weaksolutions)

∂t (dq E∗)+ divx (dqh) = 0. (3.25)

Finally, in quasi-linear form, for the classical solutions, it is symmetric:

S−1∂t q +∑

1≤α≤d

D2hα∂xαq = 0.

In particular, S Aα = S(D2hα)S is symmetric.

Exercises

3.12 For a hyperbolic linear system ut+ Aux = 0, find all the entropies. Determinethose which are convex.

3.13 For a decoupled system of scalar conservation laws ∂t ui + ∂x fi (ui ) = 0,1 ≤ i ≤ n, find all the entropies and determine those which are convex.

3.14 Let H : U → R be a smooth function. Find a non-trivial entropy (i.e., non-affine) for the system

∂t ui + ∂x∂ H

∂ui= 0, 1 ≤ i ≤ n.

3.15 Let U = (0,+∞) × Rn−1 and H : U → R be a smooth function. Find a

list of entropies of the form Eg(u) = E0(u)g(q1, . . . , qn−1), parametrised bysmooth functions g of n − 1 variables, for the Keyfitz and Kranzer system:

∂t ui + ∂x (H (u)ui ) = 0.

3.16 (Converse to Theorem 3.3.3) Let ut + f (u)x = 0 be a strictly hyperbolicsystem whose one eigenvalueλ is linearly degenerate. Letv := (u, z) ∈ U×R

be new dependent variables. We consider the augmented system vt +g(v)x =0, defined by

ut + f (u)x = 0,

zt + (λ(u)z)x = 0.

Show that this system is hyperbolic and has the same propagation speeds asthe preceding system, the multiplicity of λ being augmented by unity.

3.4 Entropies, convexity and hyperbolicity 85

3.17 This problem occurs in spatial dimension d = 1. We suppose that the eigen-values of d f (u) are real and strictly positive for all u ∈ U and that f isproper, that is to say that limd(u;∂U)→0 | f (u)| = ∞.

(1) Show that the mapping u �→ v := f (u) is invertible from U into Rn .

We write g = f −1. (Optional question, alternatively pass directly to thefollowing question.)

(2) Using this change of dependent variables, we write G(v) := F(u) andH (v) := E(u). Show that dvG(v) = du E(u).

(3) Show that G is an entropy of flux H , of the system vs + (g(v))y = 0, andthat G is strictly convex if and only if E is strictly convex. Hint: Verifythat D2

vG = D2u E · dvg and deduce that the list of signs of eigenvalues of

dg is equal to the signature of D2vG if D2

u E > 0.(4) Deduce that the system is hyperbolic in the spatial direction x , that the no-

tions of weak and entropy solutions are the same, that they respectively oc-cur in R×R

+with t for time variable, or in R+×R with x for time variable.

3.18 We consider gas dynamics in dimension 1 in its lagrangian representation

vt = ux ,

ut + (p(v, e))x = 0,(e + 1

2u2

)t+ (up)x = 0.

We denote by S(v, e) a smooth function such that Sv = pSe and Se > 0. Weput T = S−1

e .

(1) Show that the system is hyperbolic if and only if ppe − pv > 0. We shallthen write c(v, e) = √(ppe − pv).

(2) Show that for every numerical function g, E = g ◦ S is an entropy.(3) We define the differential form α = p dv + de = T dS. Show that

T 2D2E = (g′′ − Teg′)α2 − T g′(c2dv2 − 2peα dv + du2),

where D2E is the hessian form of E in the variables (v, u, ε := e+ 12u2).

(4) Deduce that E is a convex entropy (with respect to (v, u, e+ 12u2)) if and

only if

i. g′ ≤ 0,ii. c2g′′ ≥ T 2(pvSee − peSve)g′.

(5) For a perfect gas (P(v, e) = (γ − 1)e/v where γ > 1 is a constant), wecan choose T = e and S = (γ − 1) log v+ log e. Show that E is convexif and only if g′ ≤ 0 and γ g′′ + g′ ≥ 0. In particular,−S is itself a convexentropy.


3.19 We keep the notation of the preceding exercise. The aim of the present oneis to determine all the entropies of gas dynamics in lagrangian coordinates indimension 1. Let E be a general entropy and F its flux.

(1) Show that p and S are independent functions. Henceforth, we shall expressE and F as functions of (u, p, S).

(2) Write down the equations satisfied by the pair (E, F). Show that E de-composes in the form E = ε(p, S)+ a(p, u).

(3) We suppose that c and p are two independent functions, that is to saythat cS �= 0. Show that εp + cεpS/(2cS) = −ap and deduce that we canchoose to decompose E into the sum E = ε(p, S) + a(u) and that thenF is of the form pa′(u)+ h(p).

(4) Show that a′′ is a constant. Deduce that E is a linear combination of twoentropies, one which we knew already and an entropy which depends onlyon (p, S).

(5) We are therefore led to set up the list of entropies of the form E(p, S).Show that then F depends only on u and is affine. Again, using an entropyalready known, this leads to the case F ≡ 0.

(6) Show that then E is of the form g ◦ S.(7) In the (non-realistic) case where c is a function of p alone, show that

the system decouples, at least for smooth solutions, into two independentsystems, one governing S, the other governing the pair (u, p). This latter,consisting of two equations only, possesses many entropies as we shallsee in Chapter 9, those evidently not being of the form g ◦ S.

3.5 Weak solutions and entropy solutions

The hyperbolic quasi-linear systems having a complexity at least as large (it is infact greater) as that of scalar equations, classical solutions only exist in generalduring a finite time, after which they give place to less smooth solutions, typicallybounded measurable ones. Those satisfy partial differential equations in the senseof distributions, which can only be written properly for systems of conservationlaws (in fact a product Aα∂xα

u does not have a sense for u ∈ L∞(ωx,t )).Without repeating the analysis made in the preceding chapter, we define a notion

of weak solution which translates well that which a conservation law wants tocommunicate to a physical level when u is piecewise continuous.

Definition 3.5.1 Let ω be an open set of Rd+1 and u ∈ L∞(ω)n . We say that u is a

weak solution of (3.2) in ω, if for all � ∈ D (ω)n , we have∫ ∫ω

(u · ∂t�+

∑1≤α≤d

f α(u) · ∂xα�+ b(u) ·�

)dx dt = 0.

3.5 Weak solutions and entropy solutions 87

Wenote thatwehave takenvector-valued test functions.Bychoosing� = (0, . . . , 0,

ϕ j , 0, . . . , 0)T where ϕ j ranges over D (ω) and j over {1, . . . , n}, this reduces towriting that (u j , f 1

j (u), . . . , f dj (u)) satisfies the j th conservation law in the sense

of distributions. For the Cauchy problem, we have

Definition 3.5.2 Let T be a positive real number, u0 ∈ L∞(Rd )n and u ∈ L∞(Rd ×(0, T ))n . We say that u is a weak solution of the Cauchy problem

∂t u +∑

1≤α≤d

∂xαf α(u) = b(u), (x, t) ∈ R

d × (0, T ),

u0(x, 0) = u0(x), x ∈ Rd ,

(3.26)

if, for all � ∈ D (Rd × (−∞,T ))n , we have∫ ∫R

d×(0,T )

(u · ∂t�+

∑1≤α≤d

f α(u) · ∂xα�+ b(u) ·�

)dx dt

+∫

Rdu0(x) ·�(x, 0) dx = 0.

However, as in the scalar case, the class of weak solutions is not appropriate becausethe solution of the Cauchy problem is not in general unique, whereas the modelsconsidered are conceived in a deterministic setting. We thus must introduce a newadmissibility criterion to select, from all weak solutions, that which is stable from thephysical or mathematical point of view, hoping that there exists only one. The onlycriterion of general power, whose application is not restricted to piecewise smoothsolutions, is Lax’s entropy condition, which, for physical systems, can be written:

Definition 3.5.3 We consider a physical system whose strongly convex entropy andits flux are denoted by E and F . Being given u and u0 as above, u being a weaksolution of the Cauchy problem, we say that u satisfies Lax’s entropy condition if,for all ϕ ∈ D

+(Rd × (−∞, T )), we have∫ ∫R

d×(0,T )(E(u)∂tϕ + F(u) · ∇xϕ + dE(u) · b(u)ϕ) dx dt

+∫

Rd

E(u0(x))ϕ(x, 0)dx ≥ 0.

The entropy condition implies, when we take only test functions with compactsupport in R

d × (0, T ), the inequality (in the sense of distributions)

∂t E(u)+ divx F(u) ≤ 0.


As we have seen in the scalar case (see the exercise on the N-wave), the entropycondition is, in general, strictly more precise than this single inequality.

The above definitions are written in the most general framework, but the hypoth-esis u ∈ L∞ can often be weaker. For example, in the linear case, we know that forthe majority of systems, the Cauchy problem is not well-posed in L∞, but that it isin L2

loc. Natural conditions are to suppose that u and E(u) are in C ((0, T ); L1loc(R

d ))and f (u) ∈ L1

loc(Rd × (0, T )). However, without the growth of u at infinity being

controlled, there is a risk of an unavoidable blow-up in finite time of the entropysolution if, because of the non-linearity, a propagation speed is unbounded. It isthus prudent to restrict the study of the Cauchy problem to solutions satisfyingu ∈ C ((0, T ); L1

loc(Rd )) ∩ L∞(Rd+1).

The Rankine–Hugoniot condition

We are now going to interpret the weak formulation for a discontinuous but piece-wise smooth solution. To be precise, we consider an open set ω of R

d+1 which aregular hypersurface separates into two connected components ω+ and ω−. Thefield of unit vectors normal to and directed towards ω+ is denoted by ν. The fieldof unit vectors normal to ∂ω± and pointing outwards is denoted by ν±. Along ,we have ν− = ν = −ν+ (see Fig. 3.2). We assume that u is a function defined onω with values in U whose restrictions to ω+ and ω− are of class C

1 and can beextended by continuity to . Their traces on are denoted by u+ and u− ; u isgenuinely discontinuous as long as u+ �= u−.

Fig. 3.2: Surface of discontinuity and unit normal.

3.5 Weak solutions and entropy solutions 89

Let us suppose that u is a weak solution of (3.2) in ω. Then for every test function� ∈ D (ω)n , using Green’s formula we have that

0 =∫ ∫

ω

(u · ∂t�+

∑1≤α≤d

f α(u) · ∂xα�+ b(u) ·�

)dx dt

=(∫ ∫

ω−+

∫ ∫ω+

)� ·

(b(u)− ut −

∑1≤α≤d

∂xαf α(u)

)dx dt

+∫

∂ω−

(ν−0 u +

∑1≤α≤d

ν−α f α(u)

)·� ds(x, t)

+∫

∂ω+

(ν+0 u +

∑1≤α≤d

ν+α f α(u)

)·� ds(x, t).

When � ranges over D+(ω+), we deduce from

0 =∫ ∫

ω+� ·

(b(u)− u −

∑1≤α≤d

∂xαf α(u)

)dx dt,

that the conservation law holds at each point of ω+, because b(u) − ut −∑1≤α≤d ∂xα

f α(u) is continuous on ω+. Similarly in ω−. Finally the integrals onω+ and on ω− are zero for every test function in the above formula. It only remainsto show that the boundary integrals on ∂ω+ and ∂ω− are zero. These simplify fortwo reasons. First of all because � is zero on ∂ω, with the result that these integralsreduce to the domain . Then, because on , ν+ = −ν−. Finally, with the usualnotation [u] = u+ − u−,

0 =∫

(ν0[u]+

∑1≤α≤d

να[ f α(u)]

)·� ds(x, t).

When � ranges over D (ω)n , the traces of � on run through a sub-space densein C

1(). Since ν0u +∑1≤α≤d να f α(u) is continuous on , we therefore obtain

at each point the Rankine–Hugoniot condition:

ν0[u]+∑

1≤α≤d

να[ f α(u)] = 0. (3.27)

Conversely, if u is a classical solution of (3.2) in ω+ and ω− and if u satisfiesthe Rankine–Hugoniot condition along , the same calculation, carried out in thereverse order, shows that u is a weak solution of (3.2) in ω.


Let Mα be the Lipschitz constant of f α on the interval with extremities u− andu+. We deduce from (3.27) the inequality

|ν0| ‖[u]‖ ≤∑

1≤α≤d

|Mανα| ‖[u]‖.

This shows that, if u is genuinely discontinuous, then |ν0| ≤∑

1≤α≤d |Mανα|. Asν is a unit vector, it follows that (ν1, . . . , νd ) does not vanish: the hypersurface

is never tangent to the horizontal space {0} × Rd , it is a space-like hypersurface.

As in the one-dimensional case, we introduce a unit vector

n = (ν1, . . . , νd )

|ν| , |ν| =√(

ν21 + · · · + ν2

d

),

and a number

V = − ν0

|ν| .

The field n is a field of unit vectors normal to the section t = ∩ ({t} ×Rd ) and

V is the normal speed of the displacement of t with respect to the time.Rewriting the Rankine–Hugoniot condition with the help of V and of n, we have

n · [ f (u)] = V [u]. (3.28)

The above formula is vectorial. There is no ambiguity in the scalar productbetween the vector n and the tensor [ f (u)] as the dimensions of these, d andp, are in general distinct. Since n is a unit vector, we observe that the speed ofpropagation of a discontinuity is dominated by the Lipschitz constant of f on theinterval [u−, u+], which confirms the general idea that the hyperbolic systems areassociated with phenomena of propagation with finite speeds.

In what concerns Lax’s entropy condition, the same calculation as above, carriedout with positive test functions, shows that a weak solution u of (3.2) is an entropysolution if and only if

n · [F(u)] ≤ V [E(u)]. (3.29)

Note that the significance of this inequality does not depend on the direction ofn. In fact, if we replace n by −n this results in interchanging ω+ and ω−, hencereplacing [E(u)] by −[E(u)] and [F(u)] by −[F(u)].

Reversibility

If u is a weak solution of (3.2), then v(x, t) := u(x,−t) is also a weak solution ofthe system vt −

∑1≤α≤d ∂xα

f α(v) = 0. On the other hand, in conserving the samefield of unit vectors n, the speed V and the flux F change sign. It follows that u

3.6 Local existence of smooth solutions 91

and v cannot satisfy their entropy conditions simultaneously unless the inequality(3.29) is an equality. Thus as long as (3.29) is strict, the solution u is irreversible.

3.6 Local existence of smooth solutions

The method of characteristics allowed us to show the existence of a smooth solutionin a band (0, T )× R for a scalar equation when the given initial condition is itselfsmooth. This result remains true for sufficiently general systems, but as the methodof characteristics is not transposable to this case,1 we must turn back to the energyestimates in the Sobolev spaces Hs . By this method, we can treat only symmetrisablesystems. The regularity demanded for the given initial condition grows with thedimension. Typically, we need to show that if u ∈ Hs(Rd ), then the non-linearterms of the form g(u)∇xu are bounded. From the Sobolev injection theorems andthe Gagliardo–Nirenberg inequality, this reduces to requiring that s > 1+ 1

2d. Thefollowing theorem expresses that the Cauchy problem for a symmetrisable systemis locally well-posed for s > 1+ 1

2d [28].

Theorem 3.6.1 (L. Garding, J. Leray) Let s be a real number, s > 1 + 12d.

Let U1 be an open set relatively compact in U and u0 ∈ Hs(Rd ) with valuesin U1.

Then there exists a time T > 0 such that the symmetric hyperbolic system

S0(u)∂t u +∑

1≤α≤d

Sα(u)∂xαu = b(u)

has a classical solution u ∈ C1([0, T ]× R

d ) satisfying the initial condition

u(x, 0) = u0(x).

In addition, u ∈ C ([0, T ]; Hs(Rd ))∩C1([0, T ]; Hs−1(Rd )) and this solution is

unique.

Remark The time T obtained in the proof of the theorem depends a priori on‖u0‖s , on U1 and on the smooth functions S0, Sα and b defined on U. When s isan integer, the norm on Hs(Rd ) is defined classically by

‖v‖s :=( ∑|γ |≤s

∫R

d|Dγ v|2 dx

) 12

,

1 In fact, it remains effective in dimension d = 1, even for systems. See a proof in Courant and Hilbert [12],pp. 476–8.


where γ = (γ1, . . . , γd ) denotes a positive multi-integer of length |γ | = γ1+· · ·+γd and Dγ is the differential operator(

∂

∂x1

)γ1

· · ·(

∂

∂xd

)γd

.

Indications about the proof

The proof of Theorem 3.6.1 makes use of an iterative scheme in which each iterationconsists of solving a linear system with variable coefficients but of class C

∞.We choose for this a sequence (uk

0)k≥0 of functions of class C∞ such that the

series∑

k ‖uk+10 − uk

0‖s converges, the limit of the sequence being u0. Since, byhypothesis, Hs(Rd ) is included in C

1(Rd ), we may suppose that each element ofthis sequence has values in U2, a neighbourhood of U1 relatively compact in U.If uk ∈ C

∞([0, Tk]× Rd ) takes its values in U, the linear system

S0(uk)∂t uk+1 +∑

1≤α≤d

Sα(uk)∂xαuk+1 = b(uk),

uk+1(x, 0) = uk0(x)

(3.30)

makes sense and possesses a solution of class C∞ in the band [0, Tk]×R

d . This is aconsequence of the linear theory (which we shall not develop in this work becauseof space constraints). We call Tk+1 ≤ Tk , the maximal time in which uk+1 takes itsvalues in U.

The aim of these estimates is two-fold: on the one hand to control the distanceof uk(x, t) from the boundary of U, since the coefficients of the system can reachsingular values at the boundary of this domain (leading for example to an infinitepropagation speed), on the other hand to control the norm of uk in C

1 (which canonly be done by passing through Hs) so as to be able to pass to the limit in theproducts S0(uk)∂t uk+1, etc. In both cases, it is obviously essential to show that thesequence of times of existence Tk is bounded below by a number T > 0, which isthe number T stated in the theorem.

As frequently happens in the study of non-linear (and also linear) partial differ-ential equations, the estimates for the scheme are adapted from an a priori estimateobtained on the equation we seek to solve, when we assume that it possesses asolution which is sufficiently smooth. It is there that the deep idea remains, the restis essentially a matter of technique. For simplicity, we shall suppose that U = R

d ,which reduces the first estimate to a control of |uk |∞, the norm of uk in L∞, whichitself is bounded by ‖uk‖s,T := sup0≤t≤T ‖uk(t)‖s . Finally, there is only a single im-portant estimate, that of ‖uk‖s,T . We shall assume also that s is an integer, so that weshall avoid having to manipulate with fractional derivatives in the energy estimates.


In fact, in the case of the iterative scheme, a supplementary difficulty appears, itmust be shown that the whole sequence (uk)k≥0 converges. For that, we shall firstestablish a bound on ‖uk+1 − uk‖0,T by using that on ‖uk‖s,T .

A priori estimate of ‖u‖s,T

As we have said above, we work directly on a smooth solution u of the problem tobe solved. The calculations carried out below do not constitute a proof of Theorem3.6.1. In addition, the right-hand side b(u) plays a minor role in the theory and weshall suppose it to be identically zero.

For an integer k ≥ 0, let us denote by vk the list of derivatives of u of orderk:

vk = (Dγ u)|γ |=k .

For a monomial

M(v1, . . . , vk) =k∏

j=1

vβ j

j ,

we denote its weight by

p(M) =k∑

j=1

j |β j |.

Differentiating the system k times with respect to the spatial variables, we obtaina system, linear with respect to derivatives of higher order:

A0(u)∂tvk +∑

1≤α≤d

Aα(u)∂xαvk = P(u; v1, . . . , vk), (3.31)

where P(u; v1, . . . , vk) is a polynomial homogeneous in weight, of weight k + 1.In the energy method we use as a matter of fact a norm equivalent to the usual normon (L2)n , which depends on the solution and on the time, namely

[w(t)] :=(∫

Rdw∗A0(u(t))w dx

) 12

.

So as not to overburden the notation, we have chosen not to mention the depen-dence of this norm on u(t), but we hope that the context will recall it clearly.The equivalence of [·] and the usual norm ‖ · ‖0 of (L2)n is not in general uni-form with respect to u, in any case if A0 or A−1

0 is not bounded as a func-tion of u. Precisely, there exists an increasing numerical function C ≥ 1, such


that

C(|u(t)|∞)−1‖w‖0 ≤ [w] ≤ C(|u(t)|∞)‖w‖0.

Taking the scalar product of (3.31) with vk , we obtain

∂t

(1

2v∗k A0(u)vk

)+

∑1≤α≤d

∂xα

(1

2v∗k Aα(u)vk

)

= vk · P + 1

2v∗k

(∂t A0(u)+

∑1≤α≤d

∂xαAα(u)

)vk

= Q(u; v1, . . . , vk) (3.32)

where Q(u; . . .) is a homogeneous polynomial in weight, of weight 2k + 1.Integrating (3.32) over a ball of radius R and admitting that vk tends to zero at

infinity sufficiently fast that the boundary integrals of v∗k Aα(u)vk tend to zero whenR tends to infinity,2 we deduce that

d

dt

(1

2[vk]2

)=

∫R

dQ(u; v1, . . . , vk) dx . (3.33)

Let us for the moment accept the following lemma.

Lemma 3.6.2 If Q(u; v1, . . . , vk) is a polynomial, homogeneous in weight withrespect to v, of weight 2k + 1, then there exists a numerical function Ck such that,for all u ∈ Hk(Rd )n we have∫

Rd

Q(u; dxu, . . . , Dk

xu)

dx ≤ Ck(|u|∞, |dxu|∞)∥∥Dk

xu∥∥2

0.

Applying the lemma to the formula (3.33), then summing from k = 0 to k = s, weobtain

d

dt

(1

2[u]2

s

)=

(max

0≤k≤sCk

)‖u‖2

s , (3.34)

where we have defined

[w]s :=( ∑

0≤k≤s

[Dk

xw]2

) 12

.

2 Each approximation uk0 to the given initial function u0 is with compact support, with the result that the approx-

imate solution uk is with compact support with respect to x . In practice, the functions vk are therefore withcompact support.


The expression [·]s is a norm on Hs(Rd )n , equivalent to the norm ‖ · ‖s but thisequivalence depends on u(t) in the same manner as for [·]:

C(|u(t)|∞)−1‖w‖ ≤ [w]s ≤ C(|u(t)|∞)‖w‖.We can thus simplify the inequality (3.34), to reduce it to

d

dt[u] ≤ c1(|u|∞, |dxu|∞)‖u‖s, (3.35)

where c1 is an explicit numerical function.Integrating (3.35) from 0 to t , we find that

[u(t)]s ≤∫ t

0c1(|u(τ )|∞, |dxu(τ )|∞)‖u(τ )‖sdτ + [u0]s,

that is to say,

‖u(t)‖s ≤ C(|u(t)|∞)

×{∫ t

0c1(|u(τ )|∞, |dxu(τ )|∞)‖u(τ )‖sdτ + [u0]s

}. (3.36)

This inequality is more precise than we need immediately, but it will serve us inthe sequel to establish a characterisation of the maximal time of existence of theclassical solution.

For the moment, we make use of a rough upper bound, which expresses thatHs(Rd ) is included in C

1(Rd ); let |u|∞+|dxu|∞ ≤ c2‖u‖s , where c2 is a constant.From (3.36) we then derive

‖u(t)‖s ≤ C(c2‖u(t)‖s)

{∫ t

0c3(‖u(τ )‖s)‖u(τ )‖sdτ + [u0]s

}, (3.37)

where c3 is an explicit numerical function.Let us introduce the numbers R := C(c2‖u0‖s)[u0]s and c4, the supremum of

the expression C(c2z)c3(z)z when z ranges over the interval [0, R + 1]. We have‖u0‖s ≤ R. The inequality (3.37) then ensures that ‖u(t)‖ ≤ R + 1 provided that0 ≤ t ≤ T where T := R/c4.

Lemma 3.6.3 There exist a time T > 0 and a real number L > 0 such that everysmooth solution of the Cauchy problem satisfies

sup0≤t≤T

‖u(t)‖s ≤ L .

The numbers L and T depend both on the system considered and on the initialnorm ‖u0‖s , s > 1+ 1

2d.


Corollary 3.6.4 With the notation of the preceding lemma, there exists a numberL1 such that every regular solution of the Cauchy problem satisfies

sup0≤t≤T

∥∥∥∥∂u

∂t

∥∥∥∥s−1

≤ L1.

Proof We recall the formula (3.31) for k ≤ s − 1, which shows that

Dkx (ut ) = A0(u)−1

(Pk −

∑1≤α≤d

Aα∂xαvk

).

The samecalculationwhichpreviously showed that A−10 Pk is bounded in L∞(0,T;

L2(Rd )) is still valid and even trivial for A−10 Aα∂xα

vk .

Proof of Lemma 3.6.2

The proof of this lemma is based on the Gagliardo–Nirenberg inequality. If r is apositive real number, i an integer between 0 and r and if z belongs to L∞ ∩ Hr ,then Di

x z ∈ L2r/ i with

∣∣Dix z

∣∣2r/ i ≤ Ci,r |z|1−i/r

∞ ‖z‖i/rr , (3.38)

where |·|q denotes the usual norm of Lq . Applying this inequality to v1 and i = j−1for j ≤ r + 1, we have

|v j |2r/( j−1) ≤ C j−1,r |v1|1−( j−1)/r∞ ‖u‖( j−1)/r

r+1 .

Thus, for u ∈ W 1,∞ ∩ Hk, k ≥ j , we have v j ∈ L p j for all p j comprisedbetween 2 and 2(k − 1)/( j − 1). A monomial of the form

Q :=∏

1≤ j≤k

vγ j

j ,

of weight 2k + 1, thus belongs to L p for p satisfying a ≤ p ≤ b, where

1

a= 1

2

∑j

|γ j |, 1

b= 1

2

∑j

j − 1

k − 1|γ j |.

In addition, we have

|v j |p j ≤ C |v1|1−2/p j∞ ‖u‖2/p j

k .

We thus have |Q|p ≤ C |v1|σ∞‖u‖θk , where

θ :=∑

j

2

p j|γ j | = 2

p, σ :=

∑j

|γ j | − θ.

The lemma then comes from the inequalities a ≤ 1 ≤ b, which we prove now.We have ∑

1≤ j≤k

|γ j | ≥∑

1≤ j≤k

j

k|γ j | = 2+ 1

k,

which indicates that∑

1≤ j≤k |γ j | ≥ 3 (because this is an integer). We deduce thata ≤ 2/3, as well as b ≥ 1 (which achieves the proof of the lemma) by reason ofthe formula

1

b= 1+ 1

2(k − 1)

(3−

∑|γ j |

).

Convergence of the iterative scheme

Returning to the iterative scheme, we shall accept that the a priori estimates es-tablished in §3.6 remain valid for the approximate solutions introduced there, evenif it entails that the time T of existence common to all the solutions um is a littlesmaller (but, however, strictly positive). There are thus a time T1 and a number R1

such that, for all m ≥ 1 with s > 1+ 12d, we have

sup1≤t≤T1

‖um(t)‖s ≤ R1, (3.39)

sup0≤t≤T1

‖∂t um(t)‖s−1 ≤ R1. (3.40)

We are going first to show the convergence of the iterative scheme in L2(Rd ) ona time interval eventually smaller, then we conclude in Hr (Rd ) for all 0 ≤ r < sby interpolation. For a start, let us define the difference zm := um+1− um and formthe difference of two successive equations of the scheme:

A0(um)∂t zm +

∑1≤α≤d

Aα(um)∂xαzm = Fm, (3.41)

where

Fm = (A0(um−1)− A0(um))∂t um +

∑α

((Aα(um−1)− Aα(um))∂xαum)

= R(um−1, um, dt,xum, zm−1),

R being linear with respect to its last argument because of Taylor’s formula (meanvalue theorem). Thus

|Fm |2 ≤ C(R1)|zm−1|2.Multiplying (3.41) by zm , it becomes

∂t

(1

2zm A0(um)zm

)+

∑α

∂xα

(1

2zm Aα(um)zm

)

= zm · Fm + 1

2zm(∂t A0(um)+

∑α

∂xαAα(um))zm .

Integrating again over Rd and assuming that zm tends to zero at infinity rapidly

enough for the integrals of ∂xα(zm Aα(u)zm) to be null (same remark as previously),

it becomes, for 0 ≤ t ≤ T1,

d

dt

∫R

dzm A0(u)zm dx ≤ C(R1)|zm |2|zm−1|2 + |∂t A0(um)+

∑α

∂xαAα(um)|∞|zm |22

≤ C1(R1)[zm][zm−1]+ C2(R1)[zm]2,

which immediately reduces to

2d

dt[zm] ≤ C1(R1)[zm−1]+ C2(R1)[zm].

Integrating from 0 to T ∗ ≤ T1 and writing ym := sup0≤t≤T ∗[zm(t)], we have

ym ≤ δym−1 + βm, (3.42)

where δ = C2(R1)T ∗eC1(R1)T ∗ and βm = eC1T ∗[zm(0)]. We then choose T ∗ to besuch that 0 < δ < 1; this is possible and we obtain

∑m

ym ≤ 1

1− δ

∑m

βm . (3.43)

Lemma 3.6.5 The sequence (βm)m≥0 has a finite sum.

Proof Since u0 ∈ Hs with s > 1+ 12d, we have u0 ∈ C

1. We choose um0 = u0 ∗ jm ,

the convolution product of u0 with a smoothing function jm(x) := 2md j(2mx) wherej ∈ D (Rd ) and

∫R

d j dx = 1. We have∫

Rd ( j1 − j0) dx = 0 with the result that

we can write j1 − j0 in the form divp, p being of class C∞ and with compact

support. Let pm(x) = 2(m−1)(d−1) p(2m−1x). We have jm − jm−1 = div pm and

|pm |1 = 2−mC . Thus

zm(0) = −pm ∗ dxu0,

|zm(0)|2 ≤ |pm |1|dxu0|2 ≤ |pm |1‖u0‖1

≤ 2−m‖u0‖1 ≤ 2−m‖u0‖s,

which shows clearly that the sequence has a finite sum.

We deduce from the lemma and (3.43) that the sequence (ym)m≥0 equally has afinite sum, that is to say that um converges at least in L∞(0, T ∗; L2(Rd )) since thenorms [·] and | · |2 are equivalent on L2(Rd ) uniformly for t varying from 0 to T ∗.If u is the limit of this sequence, then u ∈ L∞(0, T ∗; L2(Rd )) by Fatou’s lemma. Inaddition by an interpolation lemma between L2 = H0 and Hs (see [1]), we havefor all 0 ≤ r ≤ s

‖u − um‖r ≤ |u − um |1−r/s2 ‖u − um‖r/s,

the right-hand side of which tends to zero from the preceding argument and Lemma3.6.2. Thus, the sequence (um)m≥0 tends to u in L∞(0, T ∗; Hr (Rd )) for all r < s.

Finally, the convergence of the equations of the iterative scheme takes place, ina uniform manner, with the result that u is clearly a regular solution of the Cauchyproblem.

Remarks (1) In fact, the approximate solutions are continuous with respect to thetime with values in Hs−1, hence with values in Hr for all r < s (again the argumentby interpolation). In the above results we can thus replace L∞(0, T ∗; Hr (Rd )) byC (0, T ∗; Hr (Rd )).

(2) As in addition u ∈ L∞(0, T ∗; Hs(Rd )), we deduce that in fact, u is continuouswith respect to the time, with values in Hs equipped with its weak topology. Showingfinally, by the same type of estimates as those already used, that t �→ [u(t)]s iscontinuous, we deduce the result stated, that is to say that u ∈ C (0, T ∗; Hs(Rd ))(use the fact that a weakly convergent sequence in a Hilbert space, of which thelimit of the norms is equal to the norm of the limit, converges strongly).

(3) The smooth solution is in fact unique, as we can convince ourselves byrecalling the inequality (3.42) either for two approximate solutions or for twosmooth solutions of the same Cauchy problem; then we have β = 0 and so y ≤ δywhich gives y = 0, that is to say that the difference between the two solutions isnull in L∞(0, T ∗; L2(Rd )), and therefore null.

More generally, this calculation can be carried out with two solutions u and v

corresponding to two distinct initial conditions. If z := v−u, we obtain a Gronwall


inequality

d

dt

∫R

dz A0(u)z dx ≤ C(R)[z]2,

which produces

sup0≤t≤T ∗

|v − u|2 ≤ CeCt |v0 − u0|2,

where the constants C and T ∗ > 0 depend only on ‖u0‖s and ‖v0‖s .(4) An important question concerns the manner in which the solution ceases to

be smooth, when that is the case. The answer, necessarily partial, is furnished bythe inequality (3.35). Let T be the maximal time of existence of the solution ofclass Hs with s > 1+ 1

2d, and suppose T to be finite. Then the Gronwall inequalityshows us that

∫ T

0c1(|u|∞, |dxu|∞) dt = +∞.

We deduce easily (again using the estimate of the local time of existence of thesmooth solution) that

limt→T

max(|u(t)|∞, |dxu(t)|∞) = +∞. (3.44)

If there is a blow-up in finite time, it thus is produced in the same manner as in thescalar case, by the blow-up of the first derivatives, unless of course u itself becomesunbounded. Returning to a domain U of admissible states, this should signify thatu(x, t) approaches the boundary of U at a certain point when t → T .

(5) However, and contrary to what occurs in dimension d = 1, it is possiblethat the solution remains smooth for all time even for very non-linear systemsprovided that the dimension is high enough and that u0 is sufficiently small andwith compact support. The first observation in this direction seems to be due, fora non-linear wave equation, to Klainerman [53, 54] and a comprehensive study ofthe subject is to be found in the work of Li Ta-Tsien [64]. In this will be foundnumerous references to the works of other authors, among them L. Hormander andF. John.

In the special case of the full gas dynamics with the perfect gas law (p =(γ − 1)ρe), the author and Magali Grassin have recently obtained global exis-tence theorems for non-small data. These are chosen with a small density, an en-tropy close to a constant, and an initial velocity field which makes the particlesspread.

3.7 The wave equation 101

3.7 The wave equation

The wave equation, through second order in the time and in space, comes into thecategory studied in §3.1 if we put u = (∂tv, −∇xv)T. In fact,

∂2t v = �v, v(x, 0) = v0(x), ∂tv(x, 0) = v1(x)

is equivalent to

∂t u +(

0 div

∇ 0d

)u = 0, u(x, 0) = (v1(x), −∇v0(x)).

It is clearly a linear first order system with n = d + 1 and

A(ξ ) =(

0 ξT

ξ 0d

).

This system is symmetric and therefore hyperbolic. Its propagation velocities areclearly ±1 and 0 in each direction ξ but as we are interested only in solutionssatisfying ∂αuβ+1 = ∂βuα+1, only ±1 remain.

The wave equation arises in many physical problems, specially in relativity andelectromagnetism, often coupled with other equations. For that reason, it is impor-tant to have qualitative results concerning the behavior of solutions. These are basedon an explicit formula, much easier to use than that obtained by Fourier analysis.It is extracted from three facts:

the invariance of the equation under the action of isometries of Rd ,

the easy solvability in the case d = 1,the possibility of passing from a dimension d to a dimension d + 2.

Huygens’ principle

We introduce the spherical mean, for x ∈ Rd , t, r > 0:

I (x, t, r ) := 1

ωd−1rd−1

∫S(x,r )

v(y, t) ds(y),

where S(x, r ) denotes the sphere with centre x and radius r in Rd ,ωd−1 is the (d−1)-

dimensional measure of the unit sphere Sd−1 of Rd and ds(y) is the usual measure

on S(x, r ). If x ∈ Rd and if v is a solution, we verify that (z, t) �→ I (x, t, ‖z‖) is

also a solution of the wave equation3 which is written

∂2t I = ∂2

r I + d − 1

r∂r I. (3.45)

3 As far as here, the method is valid for every linear partial differential equation invariant under the action ofisometries of R

d ; for example the heat equation ∂tv = �v.

This is called the Euler–Poisson–Darboux equation. We easily show I is a solutionif and only if J := r∂r I + (d − 2)I is a solution of the analogous equation whered is replaced by4 d − 2 .When d is odd, that allows us to reduce easily to the trivialcase d = 1. For example, if d = 3, K := r I is a solution of ∂2

t K = ∂2r K . We thus

have K (x, t, r ) = p(x, t + r ) + q(x, r − t). Here p(x, ·) is defined on R+, while

q(x, ·) is defined on the whole of R.We determine p and q with the help of the initial conditions and of the limits on

K :

K (x, 0, r ) = r I0(x, r ), ∂t K (x, 0, r ) = r I1(x, r ), K (x, t, 0) = 0,

where I j (x, r ) denotes the mean of v j on the sphere S(x, r ). We thus have, forr > 0,

∂r p(x, r ) = 1

2(∂r (r I0)+ r I1), ∂rq(x, r ) = 1

2(∂r (r I0)− r I1).

Finally, for r < 0, we have q(x, r ) = −p(x,−r ).We recover the solution v by the relation

v(x, t) = I (x, t, 0) = limr→0+

K (x, t, r )

r

= limr→0+

p(x, r + t)− p(x, t − r )

r

= 2∂r p(x, t) = ∂t (t I0(x, t))+ t I1(x, t).

Finally, the solution of the Cauchy problem in dimension d = 3 is given by theformula

v(x, t) = ∂

∂t

(1

4π t

∫S(x ;t)

v0(y) ds(y)

)+ 1

4π t

∫S(x,t)

v1(y) ds(y).

Of course, the above calculation proceeds by necessary conditions only, but aswe end up with an explicit formula, it is easy to verify that this actually definesa solution of the Cauchy problem, for example, when v0 and v1 are sufficientlysmooth.

The case of the dimension d = 2, which is likewise interesting, is solved byassociating an extra spatial variable. If the functions v j (x1, x2) are the Cauchy dataand v(x1, x2, t) is the solution then V (x1, x2, x3, t) := v(x1, x2, t) is the solutionof the wave equation in dimension 3 for the data Vj (x1, x2, x3) := v j (x1, x2). We

4 On the other hand, there is no simple way to pass from d to d − 1; we shall see that that has importantconsequences.

thus have

v(x, t) = ∂

∂t

1

4π t

(∫S2(x,0;t)

v0(y1, y2) ds(y)

)+ 1

4π t

∫S2(x,0;t)

v1(y1, y2) ds(y).

Parametrising each hemisphere of S2(x, t) by (y1, y2) we obtain the formula

v(x, t) = ∂

∂t

1

2π

(∫D(x ;t)

v0(y)√t2 − ‖x − y‖2

dy

)+ 1

2π

∫D(x ;t)

v1(y)√t2 − ‖x − y‖2

dy,

less elegant than that in dimension 3 (D(x ; r ) denotes the disk with center x andradius r ).

The two formulae derived above show a very different qualitative behaviouraccording to the dimension: if d = 3, the value of v at (x, t) depends only onthe Cauchy data by the restriction of v1, of v0 and of ∇v0 on the light cone {y ∈R

3; ‖y − x‖ = t} (Huygens’ principle). On the other hand, if d = 2, the value of v

at (x, t) depends on the restriction of the data v j to the whole disk D(x ; t).

Conservation and decay

As the system associated with the wave equation is symmetrical, the energy, whichis the norm of u(t) in (L2(Rd ))d+1, remains constant in the course of the time. Itis not necessary to show this my making use of the Fourier transformation. It isenough to integrate the conservation law

1

2∂t (|∂t u|2 + |∇u|2) = div(∂t u∇u)

over the frustum {(x, t); 0 < t < T, ‖x‖+ t ≤ R}. By Green’s formula this becomes

1

2

∫B(x ;R−T )

(|∂tv|2 + |∇v|2)(x, T ) dx

= 1

2

∫B(x,R)

(|v1|2 + |∇v0|2) dx +∫

�

(∂tv∂rv − 1

2(|∂tv|2 + |∇v|2)

)ds

where � denotes the lateral boundary of the domain, ds a conveniently normalisedmeasure of area and ∂r the radial derivative. The boundary integral is negative bythe Cauchy–Schwarz inequality, with the result that

1

2

∫B(x ;R−T )

(|∂tv|2 + |∇v|2)(x, T ) dx ≤ E0 := 1

2

∫R

d(|v1|2 + |∇v0|2) dx .

Thus, the energy E(T ) := ∫R

d (|∂tv|2 + |∇v|2)(x, T ) dx is bounded above by E0.But reversing the direction of the time, we therefore have E0 ≤ E(T ) and finallyE(T ) ≡ E0.


However, this result is mediocre when compared with what can be obtained bymaking use of other conservation laws. These are consequences of Emmy Noether’stheorem and the invariance of the wave equation under the action of the Lorentzgroup. The most important conservation law is

∂t e3 = div q3,

where5

e3(x, t) := (r2+ t2)

(|∂tv|2+

d∑j=1

λ2j

)+ 4t∂tv

d∑j=1

x jλ j+(d−1)(d−3)r2 + t2

r2|v|2,

with

λ j := ∂v

∂x j+ d − 1

2

x j

r2v.

The same method as was used for the energy shows that E3(T ) := ∫R

d e3(x, T ) dxis constant if E3(0) is finite, that is if

∫R

d

((r2 + t2)

(|v1|2 +

d∑j=1

�2j

)

+ 4tv1

d∑j=1

x j� j + (d − 1)(d − 3)(1+ t2/r2)|v0|2)

dx,

is finite, where we have written � j := ∂ jv0 + (d − 1)x jv0/(2r2). We remark thatthe integrand Q is a positive semi-definite quadratic form of (v1, �, v0) providedthat d is different from 2:

Q = (r2 + t2)|�T|2+ 1

2(r − t)2(v1 −�R)2+ 1

2(r + t)2(v1+�R)2

+(d − 1)(d − 3)(1+ t2/r2)|v0|2,where we have decomposed � into its radial and tangential components

�R := x

t; � = ∂rv0 + 1

2(d − 1)v0/r, �T := �−�R

x

r= (∇v0)T.

For t > 0, we have

Q ≥ t2(|�T|2 + 1

2(v1 +�R)2

).

5 Exercise: calculate the expression for q3.


We deduce∫R

d|∇Tv|2dx + 1

2

∫R

d

(∂tv + ∂rv + d − 1

2rv

)2

dx ≤ (2/t2)E3(0).(3.46)

Similarly, if we restrict ourselves to the complement of a conical neighbourhoodof the light cone we have∫

|r−t |>θ t

(∂tv − ∂rv − d − 1

2rv

)2

dx ≤ 4/(θ t)2E3(0). (3.47)

The upper bound (3.46) shows that the solution behaves asymptotically, for t →+∞, as a function V which satisfies

∇TV = 0, ∂t V + ∂r V + d − 1

2rV = 0,

that is to say V = V (t, r ) and V = r12 (1−d)W (t−r ). Finally (3.47) reduces to saying

that ∫|s|>η

(W ′(s))2 ds ≤ const

η2.

The reader wishing to go further on the dispersion properties of the wave equation,for example in the presence of a potential or of an obstacle, should consult thememoir of Cathleen Morawetz [79].

4

Dimension d = 1, the Riemann problem

4.1 Generalities on the Riemann problem

We work in a space of dimension d = 1. The systems studied have the conservativeform

ut + f (u)x = 0, (4.1)

f being a smooth vector field, defined on a convex set U of Rn , with a non-empty

interior. We assume that A(u) = d f (u) is diagonalisable over R with eigenvalues ofconstant multiplicities, which to fix ideas we arrange in increasing order: λ1(u) <

λ2(u) < · · · < λp(u).As in the scalar case, the study of the Riemann problem is essential, both for

numerical methods and for the understanding of the Cauchy problem for (4.1). Itallows us, for example, to define numerical schemes which are sufficiently pre-cise. With one among them Glimm has been able to prove [32] the sole globalexistence theorem in time of weak solutions which is of some significance (seeChapter 5).

The Riemann problem consists of solving the Cauchy problem for (4.1) whenthe initial condition takes the form

u0(x) ={

uL, x < 0,

uR, x > 0.

}(4.2)

If u is a weak solution (respectively an entropy solution in the case where (4.1)has a strictly convex entropy) of (4.1), (4.2) and if a > 0, then ua(x, t) := u(ax, at)defines another solution. We hope that the solution, at least one that makes sensephysically, is unique, without which the system is worthless. If such is the case,we have ua ≡ u for all a > 0, which reduces to saying that u depends only on thevariable ξ := x/t . We shall denote by v: R → U the function u(·, 1), with theresult that u(x, t) = v(x/t). The Riemann problem thus reduces to the ordinary

106

4.2 The Hugoniot locus 107

differential equation

( f (v))′(ξ ) = ξv′(ξ ), ξ ∈ R, with g′ = dg/dξ, (4.3)v(−∞) = uL, (4.4)v(+∞) = uR. (4.5)

As in the scalar case the solution of the Riemann problem will be a juxtapositionof constant states, of rarefaction waves and of discontinuities. These last could beshock waves or contact discontinuities. The case of the (semi-)characteristic shockswill be ignored a priori although it presents an interest for applications. We havepreferred to put the stress on the most fundamental questions.

An essential difference from the scalar case occurs as long as f (u) = Au withA diagonalisable on R (the linear case). In this case, let us decompose the vectoruR− uL into a series of eigenvectors. We have uR− uL =

∑1≤ j≤p v j with Av j =

λ jv j . The solution of the Riemann problem is given by

u(x, t) = uL +∑

j :x>λ j t

v j .

In the non-linear case, the solution will be equally composed of p waves, clearlydifferentiated by their physical meanings, separated by p + 1 constant states u0 =uL, u1, . . . , u p = uR.

The great variety of the class of strictly hyperbolic conservative systems hindersa truly general study of the Riemann problem, with the notable exception of thecase where the initial data satisfy |uR − uL| � 1, which is the object of Theorem4.6.1 below. In particular, we shall be led to make a hypothesis of a geometricalnature which ensures that each of the p waves mentioned above is simple, that isto say that it consists of a shock, a contact discontinuity or a rarefaction wave, butnot of several of these waves. In two examples, the p-system and gas dynamics,we shall give the complete solution of the Riemann problem without a hypothesisconcerning the smallness of uR − uL.

4.2 The Hugoniot locus

Local description of the Hugoniot locus

We begin by describing the possible discontinuities (a, b, σ ) with respect to theRankine–Hugoniot condition, which is written here:

f (b)− f (a) = σ (b − a). (4.6)

In the first instance, we are interested in the possible pairs (a, b), reducing thusby projection the trival triplets to the single point (a, a). Fixing the left state (or the

108 Dimension d = 1, the Riemann problem

right as for the moment we have perfect symmetry), a ∈ U, we define the Hugoniotlocus of a by

H (a) := {b ∈ U: ∃ σ ∈ R, f (b)− f (a) = σ (b − a)}.The theorem below describes the structure of H (a) in the neighbourhood of a.

Theorem 4.2.1 We suppose that the eigenvalues of d f are simple (and hencep = n). In the neighbourhood of a, the Hugoniot locus of a is the union of nsmooth curves Hk(a), 1 ≤ k ≤ n. The k-th curve is tangent at a to the eigenvectorrk(a) of d f(a); it is in fact second order tangent at a to the integral curve of theeigenfield rk.

Exercises

4.1 In the case of eigenvalues λ j with constant multiplicities n j , 1 ≤ j ≤ p, showthat H (a) is locally the union of p sub-manifolds Hj (a) of respective dimen-sions n j , the j th being tangent to the eigenspace E j (a) := ker(A(a)− λ j In)(we still suppose that A is diagonalisable in R). If n j ≥ 2 show that Hj (a) is infact an integral manifold of the associated eigenvector field, that is that Hj (a)is tangent to E j (b) at each of its points b (Hint: make use of Theorem 3.3.3.)

4.2 Describe H (a) in the linear case.4.3 Describe H (a) for a system of two decoupled equations

vt + g(v)x = 0,

wt + h(w)x = 0.

4.4 Describe H (a) for the p-system.

Fig. 4.1: The Hugoniot locus H (a). For simplicity, we have supposed that the eigenvaluesare simple.

4.2 The Hugoniot locus 109

4.5 Let s �→ u(s) be the solution of the differential equation

du

ds= r j (u), u(0) = a.

Let s �→ v(s) be a parametrisation of Hj (a) (which is not necessarily the sameas that previously introduced, in the case where Hj (a) was the integral curveof r j ).

(1) Let G(s) := ( f (u(s))− f (a))∧ (u(s)− a) which is an element of �2(Rn)(an element of degree 2 in the exterior algebra of R

n). Calculate G ′, G ′′, G′′′

and verify that G(0) = G ′(0) = G ′′(0) = G′′′(0) = 0.

(2) Show (without calculating Giv completely) that

Giv(0) = 2(dλ j · r j )(dr j · r j ) ∧ r j .

(3) Withoutmakinguseof thepreceding calculations, show that if u(s)−v(s)=O(s4), then G(s) = O(s5).

(4) We suppose that at every point a ∈ U, Hj (a) is tangent of the third orderto the integral curve of r j . Show that either the j th characteristic field islinearly degenerate, or the integral curves of r j are straight lines in U (seeTemple [103]).

(5) In both cases, show that Hj (a) is the integral curve of r j passing through a.

Some symmetric functions

Since a and b play symmetric roles in the Rankine–Hugoniot condition, it is con-venient to use symmetrical functions in a and b which generalise objects alreadydefined on U. For example, writing

A(u, v) :=∫ 1

0d f ((1− t)u + tv) dt,

we have A(u, v) = A(v, u) and Taylor’s formula gives f (v)− f (u) = A(u, v)(v−u). The Rankine–Hugoniot condition is thus written

(A(a, b)− σ In)(b − a) = 0. (4.7)

When v− u is small, a symmetric function in u and v possesses a precise equiv-alent to the second order:

Lemma 4.2.2 Let (u, v) �→ M(u, v) be a symmetric function of class C2 defined

in U and let m(u) = M(u, u). Then, when v → u, we have

M(u, v) = m

(u + v

2

)+ O(|v − u|2).


Proof From Taylor’s formula,

M(u, v)−m

(u + v

2

)= (dv M−du M)

(u + v

2,

u + v

2

)·(

v − u

2

)+O(|v−u|2).

But the symmetry implies (differentiate the equality M(a, b) = M(b, a) withrespect to one of the vectors, then put a = b = u)

dv M(u, u) = du M(u, u) = 1

2dm(u). (4.8)

Since the eigenvalues of A(u) are real and simple, every real matrix close to A(u)has its eigenvalues real and simple. This is the case of A(a, b) with b a neighbourof a since A(a, a) = A(a). We shall denote by µ j (a, b) these eigenvalues, andby R j (a, b) some associated eigenvector fields, chosen in a smooth manner, thatis of class C

p as functions of a and b if f is of class Cp+1. These functions are

symmetric and we have µ j (a, a) = λ j (a).

Proof of Theorem 4.2.1

The formulation (4.7) of the Rankine–Hugoniot condition shows that if b ∈ H (a)is in the neighbourhood of a but is distinct from it, then there exist an integerj, 1 ≤ j ≤ n, and a small real number s �= 0 such that

σ = µ j (a, b), b = a + s R j (a, b).

To the integer j corresponds the sub-set Hj (a) of the Hugoniot curve of a. Thediscontinuities (a, b, σ ) where σ = µ j (a, b) are called the j-discontinuities.

Let us define, j being fixed, a smooth function N from R×U into Rn by

N (s, u) := u − a − s R j (a, u).

As N (0, a) = 0 and du N (0, a) = In is invertible, the implicit function theoremshows that Hj (a) is, in the neighbourhood of a, a smooth curve parametrised bys ∈ (−s0, s0), s �→ ϕ j (s; a), of which we are going to study a Taylor expansion atthe origin.

To the first order, since ϕ j (0; a) = a,

ϕ j (s; a) = a + s R j (a, a + sr j (a)+ O(s2))

= a + sr j (a)+ s2dv R j (a, a) · r j (a)+ O(s3)

= a + sr (a)+ 1

2s2(dr j · r j )(a)+ O(s3)

which shows that Hj (a) is second order tangent to the integral curve of the vectorfield r j and the theorem is proved.

4.3 Shock waves 111

We notice that these two curves are not in general third order tangents (seeExercise 4.5 above).

4.3 Shock waves

Entropy balance

Theorem 4.2.1 does not indicate, among the discontinuities, those which have aphysical sense. Let us suppose that (4.1) is of a physical nature, its entropy, strictlyconvex, being denoted by E (with D2E > 0) and its flux by F . Lax’s entropy condi-tion expresses that the rate of dissipation of [F] − σ [E] is non-positive. The fol-lowing theorem provides an equivalent rate when b is close to a.

Theorem 4.3.1 Let E be an entropy of class C3 of the system (4.1) and F its flux.

Then, for b ∈ Hj (a) a neighbour of a (b = ϕ j (s, a), that is b − a ∼ sr j (a)), wehave

[F]− σ [E] = s3

12(dλ j · r j )D

2E(r j , r j )+ O(s4),

the values of dλ j , r j , and D2E being calculated at a.

Proof Since dF = dE d f , we have

[F]− σ [E] =∫ 1

0(dF − σdE)(a + t(b − a)) · (b − a) dt

=∫ 1

0dE(a + t(b − a))(d f (a + t(b − a))− σ ) · (b − a) dt. (4.9)

Similarly and with the Rankine–Hugoniot condition we have

0 = [ f ]− σ [u] =∫ 1

0(d f (a + t(b − a))− σ ) · (b − a) dt. (4.10)

Taking the scalar product of (4.10) by dE((a + b)/2) and subtracting from (4.9)the result is

[F]− σ [E]

=∫ 1

0

(dE(a+ t(b− a))− dE

(a + b

2

))(A(a+ t(b− a))− σ ) · (b − a) dt.


But with (4.7)

[F]− σ [E] =∫ 1

0

(dE(a + t(b − a))− dE

(a + b

2

))(A(a + t(b − a))

− A(a, b)) · (b − a) dt.

We now make use of Lemma 4.2.2:

A(a + t(b − a))− A(a, b) = A(a + t(b − a))− A

(a + b

2

)+ O(|b − a|2)

=(

t − 1

2

)D2 f (a) · (b − a)+ O(|b − a|2).

Similarly

dE(a + t(b − a))− dE

(a + b

2

)=

(t − 1

2

)D2E(a) · (b − a)+ O(|b − a|2).

Thus,

[F]− σ [E] = C D2E(b − a, D2 f (b − a, b − a))+ O(|b − a|4)

= Cs3D2E(r j , D2 f (r j , r j ))+ O(|b − a|4)

where

C :=∫ 1

0

(t − 1

2

)2

dt = 1

12.

The theorem thus results from the following two important lemmas.

Lemma 4.3.2 For 1 ≤ j ≤ n, we have

D2 f (r j , r j ) = (dλ j · r j )r j +∑k �= j

c jkrk,

where the c jk depend on the normalisation of the eigenbasis.

Lemma 4.3.3 The basis (r j )1≤ j≤n is orthogonal for the symmetric bilinear formD2E.

Actually, we have from these lemmas

D2E(r j , D2 f (r j , r j )) = (dλ j · r j )D2E(r j , r j )+

∑k �= j

c jkD2E(r j , rk)

where the terms of the last sum are zero by orthogonality.

4.3 Shock waves 113

Proof of Lemmas 4.3.2 and 4.3.3

Let us begin with Lemma 4.3.3.

Proof Differentiating the equality dF · h = dE · (d f · h), we have

D2F(h, k) = D2E(d f · h, k)+ dE · (D2 f (h, k)).

By symmetry we derive

D2E(d f · h, k) = D2E(d f · k, h).

Putting h = r j and k = rk in the above equality we obtain

(λ j − λk)D2E(r j , rk) = 0,

which proves the lemma.

Now, let us look at Lemma 4.3.2.

Proof We differentiate the relation (d f − λ j )r j = 0 in the direction h:

D2 f (r j , h)+ (d f − λ j )(dr j · h) = (dλ j · h)r j .

We put h = r j in this formula, then we remark that Im(d f − λ j ) is spanned by thevectors rk for k �= j since d f is diagonalisable.

Genuinely non-linear characteristic fields

Of course, when D2E > 0, that is for what concerns us, we have D2E(r j , r j ) > 0and the sign of [F] − σ [E] is uniquely determined by those of s and of dλ j · r j

provided that this latter number is not zero. This justifies the following definition.

Definition 4.3.4 We say that the j th characteristic field is genuinely non-linear ata if dλ j · r j is non-zero at a. We say that it is genuinely non-linear if it is genuinelynon-linear at every point of U.

The notion of a genuinely non-linear field means that λ j is monotonic along theintegral curves of r j and thus also in the neighbourhood of a along Hj (a). Thisis the antithesis of a linearly degenerate field, which does not mean a field is oneor the other: the rate of variation dλ j · r j of the eigenvalue along the eigenfieldcan be zero on a closed set of U with empty interior, for example a hypersurfacetransverse to r j . In this case, λ j is not monotonic along the integral curves of r j , oralong the curves of the Hugoniot locus Hj .

Fig. 4.2: Lax shocks. Here n = 3: in full line (resp. in dotted line) the characteristicsincoming (resp. outgoing).

For a genuinely non-linear field, there is canonical choice of right or left eigen-fields (note that because of Theorem 3.3.3, a genuinely non-linear field correspondsto a simple eigenvalue) by the normalisation

dλ j · r j ≡ 1, l j · r j ≡ 1.

We shall take care not to confuse the differential forms dλ j and l j , as l j · rk ≡ 0for k �= j, while this is not the case in general for dλ j (this causes the coupling ofthe equations of the system).

We now have

Proposition 4.3.5 We suppose that the j-th characteristic field is genuinely non-linear and that we have adopted the above normalisation.

If b ∈ Hj (a) is in the neighbourhood of a, the discontinuity (a, b, σ = µ j (a, b))satisfies Lax’s entropy condition if and only if s ≤ 0.

We have seen in the scalar case an inequality comparing the speed of the disconti-nuity with those of the waves to the right and to the left of a shock. Lax [59] hasintroduced for systems the following definition.

Definition 4.3.6 We say that the discontinuity (a, b, σ ) is a j-shock in the sense ofLax if it satisfies the inequalities

λ j (b) ≤ σ ≤ λ j (a), λ j−1(a) < σ < λ j+1(b).

Lax’s shock condition is one of numerous conditions of admissibility of dis-continuities, in fact the simplest. It has the great merit of having a geometricalinterpretation in terms of the stability of a discontinuity subject to a perburbationof small amplitude [64]. It expresses that at a point of discontinuity there are n+ 1incoming characteristics, of which the speeds are the eigenvalues λ1(b), . . . , λ j (b),λ j (a), . . . , λn(a), leading to n+1 scalar data (instead of n at a point of continuity),

4.3 Shock waves 115

which shows up the fact that the speed of the discontinuity is itself an unknown(the shock curve is a free boundary).

The major inconvenience of Lax’s shock condition is that it is unable to beexpressed for a weak solution, but only for a piecewise smooth solution, contrary tothe entropy condition. On the other hand, it keeps its meaning for piecewise smoothsolutions of (4.1) even when the system (4.1) does not possess a non-trivial convexentropy. Finally these two entropy conditions are equivalent for discontinuities ofsmall amplitude.

Theorem 4.3.7 We suppose that the j-th characteristic field is genuinely non-linear.If b ∈ Hj (a) in the neighbourhood of a, the discontinuity (a, b, σ = µ j (a, b))satisfies the Lax entropy condition if and only if it is a j-shock.

In fact, σ �= λ j (a), λ j (b) for b �= a, the two inequalities σ < λ j (a) and λ j (b) <

σ are equivalent while λ j−1(a) < σ < λ j+1(b) is trivial.

Proof From Lemma 4.2.2,

σ = λ j

(a + b

2

)+ O(|b − a|2) = λ j (a)+ 1

2sdλ j · r j + O(s2),

with the result that σ �= λ j (a) if s �= 0, that is, if b �= a. Similarly σ �= λ j (b). Infact σ − λ j (a) is of the same sign as s, as is λ j (b)− σ because

σ = λ j (b)− 1

2sdλ j · r j + O(s2).

Finally,λ j−1(a) < λ j (a) ∼ σ andσ ∼ λ j (b) < λ j+1(b) complete the proof.

Exercise

4.6 We consider the p-system

ut + vx = 0,

vt + p(u)x = 0,

}

where p′ > 0.

(1) Calculate the eigenvalues λ1 < λ2 of the system and the associated vec-tors. Show that each field is genuinely non-linear in (u, v) if and only ifp′′(u) �= 0.

(2) Let (a, b) ∈ R2 and 1 ≤ i ≤ 2. Describe Hi (a, b) as a curve parametrised

by v = b + εϕ(u, a) where ε = (−1)i .(3) Show that E(u, v) := 1

2v2+e(u) where e′ = p is a strictly convex entropy.Calculate its flux.


(4) Let (u, v) ∈ H (a, b). Calculate the rate [F]− σ [E] of production of en-tropy as a function of u and a only. Show that its sign is equal to that ofε[u] (respectively of−ε[u]) if p is convex (respectively concave) betweena and u.

(5) We suppose that (u − a)p′′(u) > 0 for u �= a. Show that ((a, b), (u, v), σ )with (u, v) ∈ H1(a, b) satisfies Lax’s entropy condition, and Lax’s shockcondition, but that those for which (u, v) ∈ H2(a, b) satisfy neither the onenor the other. Compare with Proposition 4.3.5.

4.4 Contact discontinuities

If the j th characteristic curve is linearly degenerate, Theorem 4.3.1 does not allowus to determine from among the j-discontinuities those which satisfy the entropycondition. In fact all satisfy it, for these are contact discontinuities. They are thusreversible: (a, b, σ ) and (b, a, σ ) are admissible. More precisely:

Theorem 4.4.1 We suppose that the j-th characteristic field is linearly degeneratewith one simple eigenvalue.

Then Hj (a) coincides with the integral curve of r j and the rate of production ofentropy is zero for every j-discontinuity. Finally λ j (a) = σ = λ j (b).

Proof First of all, it suffices to show that if b is on the integral curve γ j (a) of r j

through a, then b ∈ Hj (a). Let us notice, first of all, that, since dλ j · r j ≡ 0, λ j isconstant on γ j (a). Thus

f (b)− f (a)− λ j |γ j (a)(b − a) =∫ s

0

d

dt( f (u)− f (a)− λ j |γ j (a)(u − a)) dt

=∫ s

0(d f (u)− λ j (u))r j (u) dt

= 0.

Hence γ j (a) ⊂ Hj (a), that is to say that these curves are identical. Then

F(b)− F(a)− λ j |γ j (a)(E(b)− E(a))

=∫ s

0

d

dt(F(u)− F(a)− λ j |γ j (a)(E(u)− E(a))) dt

=∫ s

0dE(u)(d f (u)− λ j )r j dt

= 0.

4.4 Contact discontinuities 117

When λ j is of constant multiplicity m > 1 (and therefore is linearly degenerate)the algebraic properties described above are still valid for the integral manifold � j

of ker(d f − λ j ): if b ∈ � j (a), then b ∈ H (a) and the discontinuities (a, b, λ j (a))and (b, a, λ j (a)) are both admissible. We shall again therefore denote by Hj (a) thisintegral manifold.

Most physical systems, for which n ≥ 2, possess simultaneously genuinely non-linear fields and one (or several) linearly degenerate fields. The two concepts aretherefore equally important. The presence of a linearly degenerate field is oftenassociated with an invariance group of the system, for example a rotation group[24]. It would be erroneous to believe that the linearly degenerate fields are simplerin their structure, easier to understand, or to treat, under the pretext that the linearsystems are less complicated. It is rather the contrary that occurs. For example, onaccount of their dissipative aspect, the genuinely non-linear fields lead to stablestructures (the shocks) which are less perturbed, even by the addition of a parabolicterm (a viscosity) into the system [70] (see Chapter 7). On the other hand the con-tact discontinuities have a marginal stability in dimension d = 1 and can even beplainly unstable in higher spatial dimensions (Kelvin–Helmholtz or Richtmyer–Meshkov instability for gas dynamics). Even in dimension 1, their behaviour withrespect to a parabolic perturbation of the system is extremely complex and can-not in general be described with the help of conservative integrals. Finally, lineardegenerate fields can lead to solutions which display large amplitude oscillationseven if of high frequency, for example sequences of entropy solutions of the formuε(x, t) = v(ε−1ϕ(x−ct)), (see Chapter 10). The persistence of structures of largeamplitude and arbitrarily high frequency renders null and void the linearisationby which we have justified hyperbolicity as a geometrical condition allowing theCauchy problem to be well-posed. In Chapter 10, we shall see therefore an exten-sion of the notion of hyperbolicity which takes into account the large amplitudesall over a linearly degenerate field and which reduces to the actual notion in thecase without linearly degenerate fields and in the case of linear fields.

Riemann invariants

The integral curves of a vector field can be described as level sets of a list of n − 1independent functions defined on U. Let us consider a simple eigenvalue λ j of d fand its field of eigenvectors r j . Let us choose arbitrarily a hypersurface transverseto r j and a regular function v0: → R. Under sufficiently general hypotheses,

meets each integral curve in one point and one only. The Cauchy problem

dv · r j = 0, u ∈ U,

v(u) = v0(u), u ∈ ,

}(4.11)


has a unique global solution. Let us choose on independent functions vα0 , 1 ≤

α ≤ n−1, that is to say such that the differential form ω0 := dv10∧· · ·∧dvn−1

0 doesnot vanish on the space tangent to . We verify easily that for the correspondingsolutions vα of (4.11), the form ω := dv1 ∧ · · · ∧ dvn−1 satisfies a differentialequation of the form dω · r j = L(ω, dr j ) where L is bilinear. From the Cauchy–Lipschitz theorem, ω is zero at a point of U only if it is identically zero on theintegral curve passing through that point. By hypothesis, ω is thus not zero on anypart: the functions vα remain independent at every point. In particular, the level setsof (v1, . . . , vn−1) are smooth curves, which are the integral curves of r j since eachvα is constant along these.

A method of describing the integral curves of r j is thus to find n−1 independentsolutions of the linear differential equation dv · r j = 0. Each non-trivial solution(that is to say of which the differential does not vanish) is called a weak Riemanninvariant. There is no general method of solving that equation as that comes downto knowing how to integrate all the differential equations. But in most applications,separation of variables, homogeneity properties or considerations of symmetryenable us to set up an explicit list. In the following sections, we shall see how toproceed for the p-system and for gas dynamics, where we shall solve globally theRiemann problem.

If the field is linearly degenerate of multiplicity m j , (4.11) must be replaced by

dv|ker(d f−λ j ) = 0, u ∈U. (4.12)

The initial data are specified over a sub-manifold of codimension m j , transverseto ker(d f − λ j ). From Theorem 3.3.3, the system (4.12) has n − m j independentsolutions vα. The level sets of (v1, . . . , vn−m j ) are again integral manifolds of theeigenfield. We notice that if dλ j does not vanish, the condition of linear degeneracymakes λ j be a Riemann invariant.

Exercises

4.7 The following theorem, due to B. Sevennec [93], is a difficult geometrical prob-lem. Its physical interpretation is still not clear, at the moment of publicationof this work.

Theorem 4.4.2 Let λ be an eigenvalue of constant multiplicity m of a linearlydegenerate field for a physical system. let � be an integral manifold of the fieldof the corresponding eigenspaces (see Theorem 3.3.3). Using the Legendretransformation q := du E, the set

�∗ := {(q, E∗(q)) : u ∈ �}is included in an affine subspace of dimension m + 1.

4.5 Rarefaction waves. Wave curves 119

Verify this statement for the following examples (in each case, first of allidentify the genuine non-linearity or the linear degeneracy of the fields).

(1) The system of Keyfitz and Kranzer ut + (ϕ(r )u)x = 0, r := ‖u‖.(2) Gas dynamics in eulerian variables:

ρt + (ρv)x = 0,

(ρv)t + (ρv2 + p(ρ, e))x = 0,(1

2ρv2 + ρe

)t+

((1

2ρv2 + ρe + p

)v

)x= 0.

(3) Gas dynamics in lagrangian variables:

vt = zx ,

zt + q(v, e)x = 0,(e + 1

2z2)

t + (qz)x = 0.

(4) The dynamics of an elastic string:

vt = wx ,

wt = (T (r )v/r )x , r = ‖v‖.

}

4.8 Let ut + f (u)x = 0 be a strictly hyperbolic system, of which we choose acharacteristic field, of multiplicity p. We also choose independent Riemanninvariants w1, . . . , wn−p for this field.

(1) Show that there exists a mapping u �→ B(u) with values in Mn−p(R) suchthat, for every classical solution of the system, we have

w(u)t + B(u)w(u)x = 0,

with w = (w1, . . . , wn−p)T.(2) What are the eigenvalues of B(u)?(3) How is B transformed when we change the choice of Riemann invariants?

4.5 Rarefaction waves. Wave curves

Rarefaction waves are, as in the scalar case, the solutions of (4.3) which are of classC

1 in an interval (ξ1, ξ2), with v �= 0. Expanding the differential equation, we arriveat (d f (v)−ξ )v′ = 0 which shows thatv′(ξ ) is an eigenvector of d f (v(ξ )), associatedwith the eigenvalue ξ . Thus there exists an integer j, 1 ≤ j ≤ n, such that

ξ = λ j (v(ξ )), ξ ∈ (ξ1, ξ2), (4.13)v′(ξ ) ‖ r j (v(ξ )), ξ ∈ (ξ1, ξ2). (4.14)

Differentiating (4.13) with respect to ξ we deduce 1 = dλ j (v(ξ )) · v′(ξ ), whichby (4.14) implies that dλ j · r j �= 0. Hence the j th characteristic field is genuinely

non-linear. Conversely, if a characteristic field, let us say the j th, is genuinelynon-linear in an open set V of U, let us consider a curve γ j , connected in V , beingan integral curve of the vector field r j . Then λ j is monotonic increasing along γ j ,varying from ξ− to ξ+. If ξ−< ξ1 < ξ < ξ2 < ξ+, then the equality (4.13) determinesa unique point v(ξ ) of γ j , and the mapping ξ �→ v(ξ ) clearly defines a rarefaction,which we call a j-rarefaction.

Let us now consider the possibility of having a j-wave, that is a j-rarefaction, aj-shock or a j-contact-discontinuity, which passes from a fixed state a to a neigh-bouring state b in going from left to right, that is, in the direction of ξ increasing. Wesuppose that the j th field is linearly degenerate or else that it is genuinely non-linearat a.

If it is linearly degenerate, then we go from a to b by a j-contact-discontinuitywhenever b ∈ γ j (a), the integral curve of r j passing through a (to be replaced bythe integral manifold of the eigenfield if λ j is of multiplicity m ≥ 2).

If the field is genuinely non-linear at a there are two possibilities. Either b ∈γ j (a), but then b − a ∼ sr j (a) with s > 0 and the wave is a rarefaction (we havenormalised r j by dλ j · r j (a) = 1), or b∈ Hj (a), but now b − a ∼ sr j (a) withs < 0, and the wave is a shock. In each of these two situations, b describes a curveparametrised by s and of which a is one extremity. The union of these two curvesand of a forms a curve of class C

2 (from Theorem 4.2.1), tangent at a to the vectorr j (a), the j-wave curve originating at a and which we denote by O j (a). In thelinearly degenerate case, the curve (or the variety of dimension m) of the j-waveis γ j (a), which we still denote by O j (a).

Finally, to each field, genuinely non-linear at a or linearly degenerate, therecorresponds a manifold of j-wave indexed by a, defined in the neighbourhood ofa and denoted by O j (a), such that for b = O j (a), there exists a j th simple wavepassing from a to b. To be perfectly clear, it is even necessary to speak of the directwave curve O

dj (a) (we fix the left state of the wave) in contrast to the reverse wave

curve Orj (a) (where we fix the right state of the wave). The relation between the

two families of curve is given by the equivalence

b ∈ Odj (a) = O j (a) ⇔ a ∈ O

rj (b).

The extension of the wave curves O j (a) beyond a neighbourhood of a is animportant question when we have in mind realistic problems where the variation ofthe solution is not small. The procedure is clear in the case of a linearly degeneratefield: as long as the field is linearly degenerate extend O j (a) by an integral curveof r j , that is as a Hugoniot curve. For genuinely non-linear field at a, we canalso extend O j (a) on the side of s > 0 by an integral curve of r j , as long as λ j

is strictly increasing along it. On the other hand, on the side of s < 0 (shocks),the monotonicity of λ j is neither a necessary condition (as can be seen in the

4.5 Rarefaction waves. Wave curves 121

scalar case) nor a sufficient condition, since it does not imply in an obvious wayLax’s entropy condition or Lax’s shock condition. The extension must follow theHugoniot curve (in so far as it is a curve) until a suitable entropy condition leadsto the exclusion of certain discontinuities. When a field is genuinely non-linear,except on a hypersurface, transverse to r j , there will correspond to it compositewaves, in which (semi-)characteristic shocks are combined with expansion waves,as in the scalar case. The description of these curves is much more complicatedthan any we have seen up until now and can only be made by taking a well-definedexample and treating it thoroughly.

The most satisfactory entropy condition for a characteristic field of which theexpression dλ j ·r j changes sign is that of Liu [66, 67] which generalises to systemsOleınik’s criterion. First of all, let us denote by σ (a, b) the speed of the discontinuitybetween a and b, when f (b)− f (a) is parallel to b − a and a �= b.

Definition 4.5.1 Let (uL, uR; σ (uL, uR)) be a discontinuity of the system ut +f (u)x = 0. We say that it is admissible (in the sense of Liu) if the followingconditions are fulfilled.

(1) There exists an index j such that λ j is simple, such that Hj (uL) extends to acurve of class C

1 as far as uR and that Hj (uR) extends to a curve of class C1

as far as uL.(2) For all u ∈ Hj (uL), located between uL and uR, we have

σ (uL, uR) ≤ σ (uL, u).

(3) For all u ∈ Hj (uR), located between uL and uR, we have

σ (u, uR) ≤ σ (uL, uR).

Let (uL, uR, s) be a discontinuity satisfying Liu’s criterion. When u tends to uL whileremaining in Hj (uL), σ (uL, u) tends to λ j (uL). We thus have σ (uL, uR) ≤ λ j (uL).Similarly, letting u tend to uR along Hj (uR), we obtain λ j (uR) ≤ σ (uL, uR). Liu’scriterion is therefore more precise than Lax’s criterion.

Exercise: Prove that properties (2) and (3) in Definition 4.5.1 are equivalent toeach other.

Parametrisation of wave curves

We have seen that a canonical choice of r j is possible for a non-linear field. Similarly,there exists a canonical parametrisation of the wave curve O j (a), by b = ϕ j (s; a)where s := λ j (b) − λ j (a). We recall the inequality s > 0 on the side of the rar-efactions and the inequality s < 0 on the side of the shocks. We notice that thisparametrisation is of class C

2 but not any better in general. On the side of the

rarefactions, s is exactly the time it takes to pass from a to b in solving the differ-ential equation u′ = r j (u) where r j is normalised by dλ j · r j = 1.

On the other hand, there is not one favoured parametrisation of a wave curve ofa linearly degenerate field and anyone must find that which is most suitable for thecalculations of this or that example.

Finally, let us note, what will serve in the proof of Theorem 4.6.1, that eachfunction ϕ j is of class C

2 with respect to its arguments (ε, a). In fact, the integralcurves of a vector field, being the solutions of a differential equation, depend in aC∞ way (if the field itself is C

∞) on the ‘time’ ε and the initial point a. Similarly,the Hugoniot curves are projections on R

n of a manifold (that of the pairs (a, b) forwhich ( f (b) − f (a)) ∧ (b − a) = 0, which is of class C

∞ if f is itself C∞) and

if this projection is made transversely to the tangent space at (a, a); these curvesare thus regular. Finally, one glues together the relevant pieces of the Hugoniotand integral curves in a C

2 way with the following Taylor expansion at a point ofcoincidence:

ϕ j (ε, b)− ϕ j (0, a) = b − a + εr j (a)+ εdr j (a) · (b − a) (4.15)

+ 1

2ε2(dr j · r j )+ O(ε3 + ‖b − a‖3).

4.6 Lax’s theorem

The form of the solution of the Riemann problem

If i < j , an i-wave which joins a to b and a j-wave that joins b to c have differentvelocities, that is to say that, if they are centred at the origin, with the view ofsolving a Riemann problem, the zones where they do vary can be disjoint. Moreprecisely, let x/t ∈ [s1, s2] and x/t ∈ [s3, s4] be these zones. The i-wave has valueb for x > s2t whereas the j-wave has value b for x < s3t . We can construct aself-similar solution, which will be a solution of the Riemann problem betweena and c, by gluing these two waves provided that s2 < s3. We shall see that thiscondition is realised except perhaps if the two waves are shock waves of sufficientlylarge amplitudes.

If one of the waves is a shock wave, for example the i-wave, then s1 = s2 =: σi ,the velocity of the shock wave, and this satisfies the Lax inequalities λi (b) ≤ σi ≤λi (a) and λi−1(a) < σi < λi+1(b). If the i-wave is a rarefaction wave or a contactdiscontinuity, then λi (a) = s1 and λi (b) = s2, with s1 = s2 in the latter case.

If both waves are rarefaction waves or contact discontinuity, then s2 = λi (b) <

λ j (b) = s3.If the i-wave is a rarefaction wave or a contact discontinuity and the j-wave is a

shock wave, then s2 = λi (b) ≤ λ j−1(b) < σi = s3.

4.6 Lax’s theorem 123

Fig. 4.3: Solution of the Riemann problem. Here n = 2.

The case of an i-shock-wave and a j-rarefaction-wave or contact-discontinuityis symmetric with the preceding case.

If both waves are shock waves and j = i + 1, we are unable to deduce the orderof s2 and s3 from the inequalities s2 < λi+1(b), s2 ≤ λi (a), s3 > λ j−1(b) ands3 ≥ λ j (c), unless at least one of the shock waves is weak. For example if thei-shock-wave is of small amplitude, then s2 ∼ λi (b) ≤ λ j−1(b) < s3, fromwhich we deduce that s2 < s3.

Of course, the case of shock waves of large amplitude remains to be examinedcase by case. For a great number of physical systems, the gluing of an i-shock-waveand a j-shock-wave is always possible for i < j .

Generalising the idea developed above, we therefore seek the solution of theRiemann problem between two states uL and uR under a succession of constantstates u0 = uL, u1, . . . , un−1, un = uR, separated by simple waves. For 1 ≤p ≤ n a p-wave localised in a sector of the form x/t ∈ [s2p−1, s2p] takes u p−1

to u p and the sequence (sk)1≤k≤n is increasing. We have u p+1 ∈ Op(u p), that isu p+1 = ϕp(εp; u p) where εp ∈R has the value λp(u p+1)− λp(u p) if the p-th fieldis genuinely non-linear.

Now, let us define a mapping �(·, a), from a neighbourhood of the origin in Rn

into a neighbourhood of a ∈ U , by

�(ε; a) := ϕn(εn; ϕn−1(εn−1; . . . ; ϕ1(ε1; a) . . .)).

The preceding construction relies on the solution of the equation

�(ε; uL) = uR, (4.16)

where ε ∈ Rn is the unknown vector. In fact, we have seen that a solution of the

Riemann problem obtained by the gluing of a simple wave of each family (nwaves in all) corresponds to a solution of (4.16). Conversely if ε is a solution

of (4.16), if we define u0 = uL, then u p+1 = ϕp(ε; u p) by induction on p, wehave un = uR and we can join u p to u p+1 by a p-wave, these waves gluingexcept perhaps in the case where a p-wave and a (p+ 1)-wave are shock waves oflarge amplitude, that is if εp < 0 and εp+1 < 0 are both large. Let us note that thesolution of (4.16), if it exists, can have one (or several) vanishing component(s);for example εp = 0 means that there is not a p-wave, that is that u p+1 = u p. Inthis case, because of Lax’s inequalities, the (p− 1)-wave and the (p+ 1)-wave arealways gluable.

Local existence of the solution of the Riemann problem

The main theorem, due to Lax [59], is the following.

Theorem 4.6.1 (Lax) Let ut + f (u)x = 0 be a strictly hyperbolic system of con-servation laws in an open set of U. We suppose that each characteristic field iseither genuinely non-linear or linearly degenerate in the neighbourhood of a.

For every neighbourhood W ⊂U of a, there exists a neighbourhood V ⊂Wof a such that, for uL, uR ∈ V , the Riemann problem has a solution of the formdescribed above and with values in W. In addition such a solution is unique.

Because we do not know of a uniqueness theorem suitable for the Cauchy problemin the case of systems, we cannot guarantee that the solution constructed in this wayis the only one, although it is difficult to imagine a solution which does not have thestructure imposed here: self-similar with simple waves separating n + 1 constantstates. For a physical system, Heibig [39, 40] shows that a self-similar solution ofthe Riemann problem which satisfies the entropy inequality necessarily possessesthis structure. This shows that if the non-linearity of the fields is well-defined, asole strictly convex entropy might be sufficient to characterise the mathematicallyreasonable solutions.

Proof Let us carry out the proof in the case of simple eigenvalues.Since each function ϕ j is of class C

2, � is C2 with respect to (ε, u) throughout

a neighbourhood of (0, a) in Rn × U and similarly for the partial functions �k

defined by

�k(ε1, . . . , εk ; a) := ϕk(εk ; ϕk−1(. . . ; ϕ2(ε2; ϕ1(ε1; u)) . . .)).

Let us calculate the differential of � with respect to ε at ε = 0, by induction on k.

4.6 Lax’s theorem 125

We have ϕ1(ε1; a) = a + ε1r1(a)+ O(ε21). If

�k(ε1, . . . , εk ; a) = a +∑

1≤ j≤k

ε j r j (a)+ O(‖ε‖2),

then

�k+1(ε1, . . . , εk+1; a) = �k(ε1, . . . , εk ; a)+ εk+1rk+1(�k(ε1, . . . , εk ; a))

+O(‖ε‖2)

= a +∑

1≤ j≤k

ε j r j (a)+ εk+1(rk+1(a)+ O(‖ε‖))+ O(‖ε‖2)

= a +∑

1≤ j≤k+1

ε j r j (a)+ O(‖ε‖2),

which justifies the induction hypothesis. For k = n, we find that dε�(0; a) is thematrix whose column vectors are the eigenvectors r j of d f (a). These forming abasis in R

n , this matrix is invertible. We shall now make use of the implicit functiontheorem in the following quantitative form.

Let G be a function of class C2 defined on a ball B(x0; ρ) of R

n , with values inR

n . We suppose that dG(x0) is invertible. Then there exist two numbers η > 0 andL > 0, which depend only on ρ, on ‖dG(x0))‖ and on supB(x0,ρ) ‖D2G(x)‖, suchthat

G is injective on B(x0; η),the image under G of the ball B(x0; α) contains the ball B(G(x0); α/L) for all

α < η.

Now, let K be a compact neighbourhood of a, contained in W , and ρ > 0 suchthat B(a; 2ρ) is contained in K . The implicit function theorem gives two constantsL > 0 and η > 0 corresponding to ρ, to infU∈K ‖dε�(0; u)‖ and to sup(ε,u)∈B1×K

‖D2�(ε; u)‖. If uL ∈ B(a; ρ) then B(uL; ρ)⊂ K and the preceding argument ap-plies: for all uR ∈ U satisfying ‖uR − uL‖ < η/L , there exists one and only oneε ∈R

n such that ‖ε‖< Lη and uR ∈�(ε; uL). In particular, the Riemann prob-lem clearly has a solution when uL, uR ∈ B(a; 1

2η/L), and we have ‖ε‖ <

L‖uR − uL‖. Making use of the fact that each �k is locally Lipschitz, we see thatall the values taken by the solution of the Riemann problem, even those which areto be found in a k-rarefaction-wave (corresponding to �k(ε1, . . . , εk−1, α; u) with0 < α < εk), are in W provided that max(‖uL − a‖, ‖uR − a‖) is sufficientlysmall.


The proof gives an equivalent of ε and of the constant intermediate states when uL

and uR are close. In fact 0 = �(ε; uL)− uR = −[u]+∑1≤ j≤n ε j r j (uL)+O(‖ε‖2).

We thus have ε j ∼ l j (uL) · [u] when [u] = uR − uL is small, with the usualnormalisation l j · r j = 1, l j being a left eigenvector associated with the eigenvalueλ j . In particular, the intermediate states are given by

uk = uL +∑

1≤ j≤k

(l j (uL) · [u])r j (uL)+ O(‖[u]‖2).

In the strictly hyperbolic case where the eigenvalues λ1, . . . , λp are not necessarilysimple, the solution of the Riemann problem contains only p distinct waves, but eachset Hj (a), of dimension m j equal to the multiplicity of λ j , can be parametrised bya vector ε j running over a neighbourhood of the origin in R

m j , this parametrisationbeing smooth. The preceding proof carries over without change because the tangentspaces to Hj (uL) are linearly independent and their direct sum is R

n . The calculationof the intermediate states up to ‖[u]‖2 is still easy: we decompose [u] into a sumof eigenvectors of d f (uL),

[u] =∑

1≤ j≤p

v j .

We then have

uk = uL +∑

1≤ j≤p

v j + O(‖[u]‖2).

Comments B. Riemann, in his memoir to the Royal Academy of Sciences ofGottingen (1860), introduced most of the essential ideas for 2 × 2 systems inrestricting himself to the study of the system of isentropic gas dynamics. He cal-culates the expressions r and s which we today call the Riemann invariants. Heshows that (x, t), considered as a function of r and s, satisfies a linear hyperbolicsystem with variable coefficients: this is the hodograph method. For such a sys-tem, written in the form of a single equation of the second order, he describesthe method of duality (which bears his name) which reduces the solution of thenon-characteristic Cauchy problem to that of Goursat problems for the adjointequation. Noting that the velocity of propagation depends on the state, he showsthat the first derivatives rx and sx blow up in a finite time for general data. Riemanndeduces the appearance of shock waves, writes the Rankine–Hugoniot condition(of which he seems to have no previous knowledge). In the fifth section there ap-pears explicitly ‘Lax’s shock condition’. Finally, Riemann solves the ‘Riemannproblem’ for isothermal gas dynamics (p = ρ), thus avoiding a discussion of thepossible vacuum.

4.7 The solution of the Riemann problem for the p-system 127

4.7 The solution of the Riemann problem for the p-system

Hypotheses

Let us consider the p-system, which is equivalent to the non-linear wave equationytt = (p(yx ))x :

ut + vx = 0,

vt + p(u)x = 0.

}(4.17)

Since f (u, v) = (v, p(u))T the matrix d f has the value

(0 1

p′(u) 0

).

Its eigenvalues are the roots of x2 = p′(u). If p′(u) = 0, the matrix d f is not diago-nalisable. The system is hyperbolic if and only if p′(u) > 0, which we suppose fromnow. The eigenvalues are λ1 = −√p′(u) and λ2 =

√p′(u). The corresponding

eigenvectors are

r1 =(

−1√p′(u)

), r2 =

(1√p′(u)

).

We therefore have dλ j · r j = p′′(u)/2√

p′(u): the characteristic fields are si-multaneously genuinely non-linear or linearly degenerate. To solve the Riemannproblem we assume the genuinely non-linear case. At the expense of a changeof variables (x, u) = (−x,−u), we can suppose that p′′(u) > 0, that is that pis strictly convex. This hypothesis ensures that limu→∞

√p′(u) > 0, but not

that limu→−∞√

p′(u) > 0, that is that the system might not be uniformly hy-perbolic when u→−∞. We are therefore driven by subsequent needs to makea slightly stronger hypothesis concerning the hyperbolicity: we suppose that∫−∞

√p′(u) du = +∞.

Rarefaction waves

A 1-Riemann-invariant is a non-trivial solution of the equation dw · r j = 0, thatis of

dw

du=

√p′(u)

dw

dv.

The simplest solutionof this equation isw := v+ g(u)where g(u) := ∫ u0

√p′(s) ds.

The integral curves r1 are thus parametrised by u and have the form

v = v0 − g(u)+ g(u0) = v0 −∫ u

u0

√p′(s) ds.

Similarly, the parametrisation by u of the integral curves of r2 is

v = v0 + g(u)− g(u0) = v0 +∫ u

u0

√p′(s) ds.

Two points (u−, v−) and (u+, v+) of the same integral curve of r j are linked in thisorder by a j-rarefaction-wave if and only if λ j is strictly increasing along the lengthof this curve from (u−, v−) to (u+, v+). As λ2 increasing with u, a 2-rarefaction-wave is characterised by

v+ = v− +∫ u+

u−

√p′(s) ds, u− < u+. (4.18)

Similarly, a 1-rarefaction-wave is characterised by

v+ = v− −∫ u+

u−

√p′(s) ds, u− > u+. (4.19)

We notice that in both cases, v−< v+. In addition, if instead of ordering thestates following the xs increasing we order them following the times increasing,as this is possible since the waves have non-zero speeds, then we see that, in everyexpansion wave, u decays with time.

Shocks

The Rankine–Hugoniot condition between two states [u−, v−] and [u+, v+] iswritten

[v] = σ [u], [p(u)] = σ [v].

We derive [p(u)] = σ 2[u], which gives σ = ±S where S = √[p(u)]/[u]. Since

p is strictly convex, we have (S − λ+2 )(S − λ−2 ) < 0 and (S + λ+1 )(S + λ−1 ) < 0,where we have written λ+j = λ j (u+, v+), etc. In the 1-shock-wave, the conditionσ < λ+2 thus implies that σ = − S. Lax’s inequality is then written

√p′(u−) <√

[p(u)]/[u] <√

p′(u+), which because of the convexity of p is equivalent tou− < u+. Finally, a 1-shock-wave is characterised by

v+ = v− − [u]√

[p(u)/[u] = v− −√

[p(u)][u], u− < u+. (4.20)

Similarly, in a 2-shock-wave, σ = S while the Lax inequality√

p′(u+) <√[p(u)]/[u] <

√p′(u−) is equivalent to u+< u−. A 2-shock is thus characterised

by

v+ = v− + [u]√

[p(u)/[u] = v− −√

[p(u)][u], u− > u+. (4.21)

Now, let us show that the shock condition which we are about to treat is hereequivalent to Lax’s entropy condition, with the result that our analysis does notdepend on the admissibility criterion adopted.

The natural entropy for this problem is a total energy, sum of the kinetic energyand of a potential energy e(u) defined within a constant e′ = p:

E(u, v) = 1

2v2 + e(u), F(u, v) = vp(u).

The rate of entropy production is

[F]− σ [E] = [vp(u)]− σ

[1

2v2 + e(u)

]

= [v]p− + v+[p]− σ

2[v](v− + v+)− σ [e]

= σ [u]p− + 1

2[p][v]− σ [e]

= σ

(1

2[u](p− + p+)− [e]

),

where we have used the Rankine–Hugoniot condition to eliminate v+. In addition

[u]p− + p+

2− [e] = (u+ − u−)

∫ 1

0(sp+ + (1− s)p− − p(su+ + (1− s)u−)) ds

is of the same sign as [u] since p is strictly convex. Thus [F]− σ [E] has the samesign as σ [u]. For σ < 0, as discontinuity is thus entropic if and only if u− < u+,while for σ > 0, it is so if and only if u+< u−. This confirms the criterion obtainedvia Lax’s shock inequalities.

Wave curves

To resume the two preceding sections, each wave curve is parametrised by u. Asingle function of two variables is enough to make this point obvious. Let us put

θ (u, u1) =

∫ u1

u

√p′(s) ds, u < u1,

−√(p(u)− p(u1))(u − u1), u > u1.

Fig. 4.4: Two 2-wave curves for the p-system. The relation P ∈ O2(Q) is not transitive;the curves do not permit the definition of a coordinate system.

Being given two states (a, b) and (u, v), we have

(u, v) ∈ O1(a, b) ⇔ v = b + θ (u, a),

just as

(u, v) ∈ O2(a, b) ⇔ v = b + θ (a, u).

In the plane R2, the 1-wave curves are strictly decreasing and we infer one from

the other by vertical translation. The 2-wave curves are strictly increasing and weinfer one from the other by vertical translation. These curves are unbounded in thevertical direction. In fact we have made the hypothesis that θ (−∞, u1) = +∞. Onthe other hand, if u → +∞, then θ(u, a) < −√(p(a + 1)− p(a))(u − a) givesθ (+∞, a) = −∞, and so on.


Let (uL, vL) and (uR, vR) be the two initial states of the Riemann problem. Thesolution is a priori made up of a 1-wave and a 2-wave which separate two initialstates and an intermediate state (u0, v0). Making use of the parametrising of thewave curves, the solution of the Riemann problem comes down to finding thesolution (u0, v0) of the system

v0 = vL + θ (u0, uL),

vR = v0 + θ (u0, uR)

}(4.22)

and verifying that the two waves are gluable, that is if they are two shock waves,then σ1 < σ2. This last point is trivial since we always have σ1 < 0 < σ2.

Eliminating v0 from the equations (4.22), we are led to the scalar equation in thesingle unknown u0,

G(u0)= 0, (4.23)

where G(u) := θ (u, uL)+θ (u, uR)− [v]. The function G is continuous (it is in factof class C

2) and satisfies G(±∞) = ∓∞. Also G ′< 0 as for u ≤ a, θu(u, a) =−√p′(u) < 0 and on the other hand, for u > a, 2θθu(u, a) = p′(u)(u − a)+p(u)− p(a) > 0 (as the sum of two positive terms), while θ ≥ 0.

The equation (4.23) thus has one and only one solution u0. The pair (u0, vL +θ(u0, uL)) is then the unique solution of (4.22). Finally, the Riemann problem forthe p-system has one and only one solution. We can even make precise the natureof the waves produced as a function of the values of uL, uR and [v] by consideringthe signs of G(uL) = θ (uL, uR)− [v] and of G(uR) = θ (uR, uL)− [v], since G isdecreasing and G(u0) = 0.

Case uL ≤ uR. Then we have G(uR) ≤ G(uL).

If [v] < θ (uR, uL), then u0 > uR. There are two shock waves.If θ(uR, uL) < [v] < θ (uL, uR), then uL < u0 < uR. There are one 1-shock-

wave and one 2-rarefaction wave.If θ (uL, uR) < [v], then uL > u0. There are two rarefaction waves.

Case uR ≤ uL. Then G(uL) ≤ G(uR).

If [v] < θ (uL, uR), then u0 > uL. There are two shocks.If θ (uL, uR) < [v] < θ (uR, uL), then uR < u0 < uL. There are one 1-rarefaction

and one 2-shock.If θ (uR, uL) < [v], then u0 < uR. There are two rarefactions.

Comments (1) We note that in these criteria, the two values of θ (uL, uR) andθ(uR, uL) are of opposite signs. In particular, if vR = vL then one of the wavesis a shock wave while the other is a rarefaction wave, this remaining true if vR− vL

is small with respect to |uR − uL|.(2) When [v] is equal to one of the values θ(uL, uR) and θ (uR, uL), one of

the waves disappears, that is the median state (u0, v0) is equal to (uL, vL) or to(uR, vR).


4.8 The solution of the Riemann problem for gas dynamics

Hypotheses

We consider the system of gas dynamics in one spatial dimension with euleriancoordinates. The choice of lagrangian coordinates would make the solution ofthe Riemann problem more difficult as the eventual appearance of the vacuum isrepresented by a Dirac measure for the specific volume. Also, when we approachmulti-dimensional configurations only the eulerian coordinates have a practicalinterest. The system is thus

ρt + (ρv)x = 0,

(ρv)t + (ρv2 + p(ρ, e))x = 0,(ρ

(e + 1

2v2

))t+

(ρ

(e + 1

2v2

)v + pv

)x= 0.

(4.24)

The velocity v takes its values in R while ρ and e take positive values or zero.Many of the calculations are simpler if we use the state variables (ρ, e) and v, be-cause of the simplicity of the non-conservative form of the system in these variables,when ρ > 0:

(∂t + v∂x )ρ + ρvx = 0,

(∂t + v∂x )v + ρ−1 px = 0,

(∂t + v∂x )e + ρ−1 pvx = 0.

(4.25)

The matrix of this system is

A = v I3 +

0 ρ 0

ρ−1 pρ 0 ρ−1 pe

0 ρ−1 p 0

.

The eigenvalues are the solutions of (λ − v)3 = (λ − v)(pρ + ρ−2 ppe). In theform (4.25) we see that the system has a singularity all over the plane ρ = 0. Thiscorresponds to the fact that, when ρ is identically zero, the conservative variablesρ, q = ρv and ε = ρ(e+ 1

2v2)) are not independent of each other since they are allzero together (with the result that for (4.24) the singularity reduces to a single point(0, 0, 0)). The density being zero on an interval expresses the fact that this interval isfree of gas. We cannot exclude this situation in the solution of the Riemann problem,which introduces an indeterminacy in the variables which describe the flow. Indeed,it is clear that the vacuum has zero energy, momentum and pressure, on the otherhand the velocity is not defined (this is the quotient q/ρ), which prevents giving asense to the energy fluxρ(e+ 1

2v2)v+ pv. We admit that generally, the speed havingto be a bounded variable, the energy flux is also identically zero in the vacuum.

4.8 The solution of the Riemann problem for gas dynamics 133

The system is therefore hyperbolic for ρ > 0 if, and only if pρ + ρ−2 ppe > 0,which we shall henceforth assume to be the case (the matrix is not diagonalisableif pρ + ρ−2 ppe = 0).

We express the eigenvectors and the eigenvalues of A as a function c := (pρ +ρ−2 ppe)1/2, the speed of sound in the gas:

λ1 = v − c, λ2 = v, λ3 = v + c,

r1 =

−ρ

c

−ρ−1 p

, r2 =

pe

0

−pρ

, r3 =

ρ

c

ρ−1 p

.

We have dλ j · r j = (∂ρ + ρ−2 p∂e)(ρc) for j = 1, 3. The first and the third fieldsare of the same nature, in general genuinely non-linear, as for a perfect gas (statelaw p = (γ − 1)ρe where γ > 1 is a constant, c2 = γ (γ − 1)e = γ p/ρ anddλ j · r j = 1

2 (1 + γ )c > 0). On the other hand, the second field is always linearlydegenerate, independent of the state law chosen.

Let us note finally that the same uncertainty as formerly occurs for the speed ofsound in the vacuum. For example for a perfect gas, c2 = γ (γ − 1)e which doesnot make sense.

The rarefaction waves

Let us examine the 1-rarefaction-waves by first calculating the correspondingRiemann invariants. A 1-Riemann-invariant is a solution w of dw · r1 = 0, thatis of ρwρ − cwv + ρ−1 pwe = 0. The simplest solutions are v + g(ρ, e) and Swhere g is a solution of the linear first order equation.

ρgρ + ρ−1 pge = c,

and S, the specific entropy considered by the physicists (but which is not an entropyin the mathematical sense) satisfies the homogeneous equation ρSρ+ρ−1 pSe = 0.

Let us remark that the non-linearity condition of the first and third fields can beinterpreted by saying that ρc and S are two independent functions.

In a symmetric way, the Riemann invariants for the third field are S andv− g(ρ, e). A translation parallel to the velocity axis thus preserves the familyof integral curves of r j for j = 1, 3. Also, the symmetry (ρ, v, e) �→ (ρ,−v, e)exchanges the two families of curves. More precisely, if there exists a 1-rarefactionto pass from a state u− = (ρ−, v−, e−) to a state u+ = (ρ+, v+, e+), then v−c(ρ, e)is monotonic increasing in going from u− to u+ following the integral curve of r1.It follows that v + c(ρ, e) is monotonic decreasing from U− = (ρ−,−v−, e−) toU+ = (ρ+,−v+, e+) along the integral curve of r3, that is we can pass from U− to

U+ by a 3-rarefaction-wave. The symmetry thus as a matter of fact exchanges thecurves of the rarefaction waves relative to the first and third characteristic fields.

The curve of the 1-rarefaction-wave issuing from u− can be parametrised bythe pressure p+. Indeed, dp · r1 = −ρc2 < 0 shows that p is strictly monotonicalong the integral curve of r1. If we suppose in fact that dλ1 · r1 > 0 (which is truefor known gases and in particular for perfect gases, knowing that γ > 1), then thepressure decreases along the 1-rarefactions (this is the origin of the name rarefactionwave). Writing that S and v + g(ρ, e) are conserved, we obtain a parametrisationof the form

ρ+ = ϕ(p+; ρ−, p−),

v+ = v− − θ (p+; ρ−, p−)

}(p+ ≤ p−).

By symmetry, the 3-rarefaction-waves have the parametrised form

ρ− = ϕ(p−; ρ+, p+),

v− = v+ + θ (p−; ρ+, p+)

(p− ≤ p+).

Since the first and the third components of r1 are negative, ρ and e also decreasealong the 1-rarefaction-wave: ϕ is increasing with respect to its first argument.Additionally, θ (p; ρ−, p−) = g(p, S−)− g(p−, S−) where

ρc = (ρ2∂ρ + p∂e)g = gp(ρ2∂ρ + p∂e)p + gS(ρ2∂ρ + p∂e)S = ρ2c2gp.

Hence, θ is strictly increasingly with respect to p. As p diminishes, the velocitygrows in a 1-rarefaction-wave. We can write

v+ = v− +∫ p+

p−

dp

ρc

where the integral is evaluated along the rarefaction curve. By symmetry, we obtainthe following result.

Proposition 4.8.1 With respect to x, the variations of ρ, p, e and v across ararefaction wave are as follows, provided dλ1 · r1 > 0.

In a 1-rarefaction-wave, the density, the pressure and the specific energy diminishwhile the speed grows.

In a 3-rarefaction-wave, the density, the pressure, the specific energy and thespeed grow.

On the other hand, when we follow the particles paths, the density, the pressureand the specific energy decrease across a rarefaction wave.

Remark As we are now going to see, in the 1-shock-waves and the 3-shock-waves,the variations of the pressure and of the speed are opposite to those stated above.


The shocks

Let us write the Rankine–Hugoniot condition for a discontinuity of velocity s:

[ρv] = s[ρ],

[ρv2 + p] = s[ρv],[(1

2ρv2 + ρe + p

)v

]= s

[1

2ρv2 + ρe

].

By defining z := v−s, we obtain the reduced conditions which amount to sayingthat (ρ, v − s, e)± satisfy the Rankine–Hugoniot condition for a zero velocity:

[ρz] = 0,

[ρz2 + p] = 0,[(1

2ρz2 + ρe + p

)z

]= 0.

Then, let us denote by m the common value of ρ−z− and ρ+z+. There follow

m[z]+ [p] = 0,

m

[1

2z2 + e

]+ [pz] = 0.

By combining these two equalities, we obtain

m

[1

2z2 + e

]= −[p]z+ − p−[z] = (mz+ − p−)[z] = (mz− − p+)[z]

= 1

2(m(z− + z+)− (p− + p+))[z]

= m

[1

2z2

]− 1

2(p− + p+)[z];

this is to say

m[e] = −1

2(p− + p+)[z] = −1

2m(p− + p+)[ρ−1].

There are now two cases, according as m is zero or not. We shall see the casem = 0 later since it corresponds to contact discontinuities. We therefore assumethat m �= 0 which implies the fundamental relation across the discontinuities fromwhich the velocities of the gas and of the discontinuity itself have been eliminated:

[e]+ 1

2(p+ + p−)

[1

ρ

]= 0. (4.26)

This relation, which appears to be a necessary condition to satisfy the Rankine–Hugoniot condition, is merely sufficient: if ρ−, ρ+, e−, e+ satisfy it we choose form one of the roots, which we hope are real, of m2[ρ−1] + [p] = 0, and then wedefine z+ = m/ρ+. The (arbitrary) choice of s brings to an end the construction ofv± with the definitions v± = z± + s.

Now, let us look at the admissibility of the shock waves. The convex entropy isof the form E = ρh(S) where h is a suitable numerical function. Its flux is F = vE ,whence the rate of entropy production is P = m[h(S)]. Its sign depends on theone hand on that of z± and on the other on that of [S], provided that h is strictlymonotonic. Again, the symmetry (x, ρ, v, e) �→ (−x, ρ,−v, e) exchanges the ad-missible discontinuities of one family (s < v) with the admissible discontinuitiesof the other family (s > v): if (U−,U+, s) is admissible then (U−,U+;−s) is also,in the notation adopted above.

Lemma 4.8.2 Let S = S(ρ, e) be an entropy density in the sense of thermody-namics, that is to say a solution of ρ2Sρ + pSe = 0 satisfying Se > 0 (the inverseT = 1/Se is called the absolute temperature).

For every numerical function h, the function E := ρh ◦ S an entropy in themathematical sense, of flux F = vE. If (ρ, q, ε) := (ρ, ρv, 1

2ρv2 + ρe) �→ E isconvex, then h is decreasing.

Proof Because of (4.25), we verify immediately that (∂t + v∂x )S = 0 for thesmooth solutions: the specific entropy is constant along the trajectories of the gasparticles (this is no longer true when a particle goes through a shock wave). Itis the same for h(S). Combining with the conservation law for mass, it becomesEt + (vE)x = 0: E is an entropy of flux vE .

If E is convex, then the mapping a �→ k(a) := E(ρ, a, ε + v(a − q)) is convexfor every choice of (ρ, q, ε), that is, writing S1 := h ◦ S,

0 ≤ k ′′(a) = ρ−1(−S1e + (v − ρ−1a)2S1

ee

).

In particular,

0 ≤ k ′′(q) = −ρ−1S1e .

We thus find that S1e ≤ 0, that is, h′ ≤ 0. If the entropy is not trivial, h is strictly

decreasing.

Let us note finally that if m �= 0, a part of the Lax inequalities is satisfied. Forexample, if m > 0, then s < min(λ+2 , λ−2 ) = min(v+, v−) which leads us to associatethe discontinuity with the first characteristic field: it will be admissible if and onlyif this is a 1-shock-wave. Similarly, if m < 0, it will be admissible if and only if itis a 3-shock-wave.

The 1-shock-waves

Taking advantage of all the admissibility conditions introduced already, we re-quire that the shock waves satisfy simultaneously the entropy condition, that is,

m[S] ≥ 0 (since h is decreasing), and the Lax shock condition which for a 1-shock iswritten

v+ − c+ < s < min(v− − c−, v+).

We thus have m > 0 and the entropy condition reduces to [S] > 0. Observing thatthe gas particles traverse the 1-shock-wave from the left towards the right, afterm > 0, we reformulate this condition by saying that the entropy density, which isconstant along the trajectories in the absence of a shock, grows along these acrossa shock wave.

The 3-shock-waves

In a symmetrical manner, a 3-shock-wave satisfies the Lax inequalities

max(v−, v+ + c+) < s < v− + c−.

The entropy condition becomes [S] < 0. Since m < 0, the trajectories traverse the3-shock-waves from right to left, with the result that the entropy again grows alongthe trajectories. Finally:

Theorem 4.8.3 Let (u−, u+; s) be a discontinuity which is not of contact (i.e.m �= 0). Then it satisfies Lax’s entropy criterion if and only if the entropy densityS grows across the discontinuity when we follow the motion of the particles.

This statement is nothing but the second law of thermodynamics, due to Carnot.An important consequence of this criterion is the maximum principle for S.

Corollary 4.8.4 For a bounded and piecewise smooth entropy solution of theCauchy problem, we have the maximum principle

infx∈R

S(x, t) ≥ infx∈R

S(x, 0).

Proof For t > 0 and x ∈ R, we consider the particle path which has arrived at x atthe time t . It clearly started out at the initial instant t0 = 0 since the trajectories onlytraverse the 1-shock-waves and 3-shock-waves and never meet with the 2-waveswhich are contact discontinuities of the same speed as the flow (s = λ−2 = λ+2 ).The value of S along the length of the trajectory only varies when crossing a shockwave and it increases in doing so. We therefore have

S(x, t) ≥ S(x0, 0) ≥ infy∈R

S(y, 0),

which proves the corollary.

Parametrisation of shock curves

Let us consider for example the 1-shock-waves, because of the symmetry evokedabove with the 3-shock-waves. Because of the omnipresence of the operator L :=∂ρ + ρ−2 p∂e and the condition of hyperbolicity, it is more convenient to use thevariables (p, S) than (ρ, e). In fact, Lp = c2 > 0 and LS = 0 show that (ρ, e) �→(p, S) is a change of variables. We then have the formulae

c2∂p = L ,

c2∂S = T (pρ∂e − pe∂ρ).

The genuinely non-linear hypothesis (adapted from the case of perfect gases) is∂p(ρc) > 0.

The function g used to define the Riemann invariants of the 1- and 3-characteri-stic-fields is a solution of the equation ∂pg = ρ−1c−1. Finally we have the formulae

ρ2c2∂pe = p > 0,

ρ2c2∂p1

ρ= −1 < 0,

ρ2c2∂Se = ρ2T pρ,

ρ2c2∂S1

ρ= T pe.

Since S is a Riemann invariant associated with the 1- and 3-waves, we knowthat, for a weak shock, the jump [S] is of the order of the cube of the amplitudeof the shock wave (C 2 matching of shock and rarefaction curves). As p is not aRiemann invariant for these fields (we have seen that dp · r1 < 0), we can measurethis amplitude by the jump [p] of p. Hence, we have [S] = O([p]3). Now, usingthe relation (4.26), we write, with the now classical notation (〈k〉 := 1

2 (k+ + k−)),

0 =∫ A+

A−

(de + 〈p〉 d

1

ρ

)

=∫ A+

A−

(T dS + (〈p〉 − p) d

1

ρ

)

= 〈T 〉[S]+∫ A+

A−

((T − 〈T 〉) dS + (〈p〉 − p) d

1

ρ

)

= O([p4]+ 〈T 〉[S]+∫ A+

A−(〈p〉 − p) d

1

ρ

= O([p]4)+ 〈T 〉[S]+∫ A+

A−(〈p〉 − p)

((∂p

1

ρ

)dp +

(∂S

1

ρ

)dS

)

= O([p]4)+ 〈T 〉[S]+∫ p+

p−(p − 〈p〉)ρ−2c−2dp.

Now,

ρ−2c−2 = 〈ρ−2c−2〉 − a(p − 〈p〉)+ O([p]3),

where a = −∂p(ρ−2c−2) > 0 from the non-linearity hypothesis we have made.Hence, we have

0 = O([p]4)+ 〈T 〉[S]− a∫ p+

p−(p − 〈p〉)2dp

= O([p]4)+ 〈T 〉[S]− a

12[p]3.

Finally,

[S] ∼ a

12〈T 〉 [p]3.

The sign of variation of S is thus the same as that of p across a discontinuity.From the argument developed above, p thus grows across the 1-shock-waves anddecays across the 3-shock-waves with respect to x , which is the opposite situation tothat of the rarefaction waves (just as we have already observed for the p-system).Of course, if instead of making x vary while keeping t constant, we follow theparticles, we find that p grows along the trajectories, that is to say, that the shockwaves are compression waves.

In the favourable cases (those of the perfect gases for example as we shallsee later) equation (4.26) defines globally a curve p+ �→ (p+, S+) where S+ =(p+; p−, S−). On the other hand from the first two Rankine–Hugoniot relationswe derive the formula

[v]2 = [p]

〈p〉 [e] = −[p]

[1

ρ

]. (4.27)

Also, we have seen that m[z] + [p] = 0, that is m[v] + [p] = 0. For all theshock waves (at least for weak shocks, but we shall admit, this being true for aperfect gas, that it is so for shock waves of arbitrary amplitude), we have [v] < 0,then either m < 0 and [p] < 0 or m > 0 and [p] > 0. Again, this is the oppositeto the case of rarefaction waves. The curves of the 1-shock-waves are thus of theform

S+ = (p+; p−, S−),

v+ = v− − ψ(p+; p−, S−),

p+ > p−,

where ψ := √[p]/〈p〉[e] is a function defined for p+> p−. To avoid hav-

ing to extend ψ to other values of the variables, we make use of the symmetry


(x, t, p, v, S) �→ (−x, t, p,−v, S) to write the 3-shock-wave in the form

S− = (p−; p+, S+),

v− = v+ + ψ(p−; p+, S+),

p− > p+.

Wave curves

Finally, the 1-wave curves are described in the form

S+ = σ (p+; p−, S−),

v+ = v− − τ (p+; p−, S−),

by defining on the one hand τ := θ when p+ ≤ p−, and τ := ψ when p+> p−,

and on the other hand σ := when p+ > p− and σ := S− when p+ ≤ p−. Bysymmetry, the 3-wave curves are described by

S− = σ (p−; p+, S+),

v− = v+ + τ (p−; p+, S+).

Concerning the 2-waves, which are contact discontinuities because the secondfield is linearly degenerate, their curves are the integrals of the vector field r2, whichare given by

v+ = v−,

p+ = p−.


Being given a state to the left (pL, vL, SL) and a state to the right (pR, vR, SR) theusual procedure is to seek two intermediate states, of indices 1 and 2, and threewaves linking these four states. The central wave being a contact discontinuity, wehave p1= p2 and v1= v2. We shall denote these common values by p and v. Thereare thus a 1-wave from (pL, vL, SL) to (p, v, S1) and a 3-wave from (p, v, S2) to(pR, vR, SR). This results in the equations

S1 = σ (p; pL, SL), (4.28)

v = vL − τ (p; pL, SL), (4.29)

S2 = σ (p; pR, SR), (4.30)

v = vR + τ (p; pR, SR). (4.31)

The cancellation of v from equations (4.29) and (4.31) reduces the Riemannproblem to that of solving a single scalar equation in the unknown p:

τ (p; pL, SL)+ τ (p; pR, SR) = vL − vR. (4.32)

Once this equation has been solved, (4.28) and (4.30) yield the values of S1

and S2. Finally v is given by (4.29). However, the equation (4.32) does not al-ways have a solution. We shall see for example that for a perfect gas, τ is anincreasing function of p (this is true generally at least as long as p < p−, sincethen τ = g(p, S−)− g(p−, S−) and ∂pg > 0 is our hypothesis of non-linearity).If we admit this property, then the left-hand side of (4.32) is minimal for p= 0,taking a value V (pL, SL, pR, SR)= τ (0; pL, SL)+ τ (0; pR, SR), finite in general. IfvL− vR< V (pL, SL, pR, SR), equation (4.32) does not have a solution and we donot find a solution of the Riemann problem by the classical method.

This difficulty is removed by observing that a vacuum can occur when the pressureis zero. In this case, there is no contact discontinuity and the vacuum is foundbetween the straight lines of slopes v1 and v2. We have necessarily v1 < v2 andp1 = p2 = 0. Finally there are two cases:

EithervL−vR≥ V (pL, SL, pR, SR), and then the solution of the Riemann problemis made up of a wave of each family and there is no vacuum. In this casethe fact that the 1- and 3-waves are rarefaction waves or shock waves can bedetermined by considering the position of vL−vR with respect to τ (pR; pL, SL)and τ (pL; pR, SR), as for the p-system.

or vL − vR < V (pL, SL, pR, SR), and then the solution of the Riemann prob-lem is made up of a 1-rarefaction-wave (leading to a state of zero pres-sure), followed by a vacuum, followed by a 3-rarefaction-wave starting fromzero pressure. We verify clearly that in this case v1 < v2, since v2− v1 =vR− vL+ V (pL, SL, pR, SR).

The case of a perfect gas

We now take up the case where p := (γ − 1)ρe, where γ > 1 is a constant. Thefirst calculations are immediate and give

c =√

γ (γ − 1)e,

g = 2√

γ e

γ − 1,

S = (1− γ ) log ρ + log e,

T = e.

The functions of the form ρh ◦ S are mathematical entropies of flux ρvh ◦ S. Wehave seen in Exercise (3.7) that the mapping (1/ρ, v, e + 1

2v2) �→ S is concave. Itis written then as the infimum of a family A of affine functions:

S = inf

{l

(1

ρ, v, e + 1

2v2

): l ∈ A

}.

To every affine function l(τ, v, E) = α0 + α1τ + α2v + α3E , we make a corre-sponding affine function m(ρ, q, e) := α0ρ+α1+α2q+α3ε. This correspondencesends the set A onto the set B. Then

ρS = inf

{m

(ρ, ρv, ρ

(1

2v2 + e

)): m ∈ B

},

which shows that ρS is a concave entropy of the conservative variables

u :=(

ρ, ρv, ρ

(1

2v2 + e

)).

Let us look at the shock curves, in the light of equations (4.27) and (4.26). Thislatter is written

(γ − 1)(ρ+ p+ − ρ− p−)+ (γ + 1)(ρ+ p− − ρ− p+) = 0

which reduces to

[v]2 = (p+ − p−)2

ρ−(

γ+12 p+ + γ−1

2 p−) .

In a 1-shock-wave, [p] > 0, and hence we have (since [v] < 0)

v+ − v− = − p+ − p−√ρ−

(γ+1

2 p+ + γ−12 p−

) .

The parametrisation by p of the wave curves is done by means of the functionτ (p; p−, S−) defined here by

τ (p; p−, S−) =

(p − p−)

√ρ−

(γ + 1

2p + γ − 1

2p−

), p > p−,

2c−

((p

p−

)(γ−1)/2γ

− 1

)/(γ − 1), p ≤ p−.

The second line of the above formula is obtained by writing that, in a 1-rarefaction-wave, on the one hand τ = g− − g+, on the other hand S+ = S−, that is to sayp1−γ+ eγ

+ = p1−γ− eγ

−. We find that the minimal value V (pL, ρL, pR, ρR) below which

4.9 Exercises 143

the value of vL−vR leads to the creation (if we venture to so express it) of a vacuum,

V = − 2

γ − 1(cL + cR).

Finally, noticing that τ (+∞; p−, S−) = +∞, we discover that there is no upperlimit imposed on vL − vR. From the intermediate value theorem (τ is certainlycontinuous) equation (4.32) possesses at least one solution as long as vL−vR ≥ V .But as τ (·; p−, S−) is obviously increasing, this solution is unique.

Theorem 4.8.5 The Riemann problem for the dynamics of perfect gases has aunique solution. That is of classical form (a 1-wave followed by a contact disconti-nuity followed by a 3-wave) if vL− vR ≥ V (pL, SL, pR, SR). Otherwise, it is madeup of a 1-rarefaction-wave and of a 3-rarefaction-wave which join respectively thestates (pL, vL, SL) and (pR, vR, SR) to a vacuum.

4.9 Exercises

4.9 For gas dynamics, we consider one of the Hugoniot curves Hj (u−) withj = 1, 3.

(1) Express the differential of the restriction of p to Hj (u−) as a function ofthe differential of that of 1/ρ.

(2) Calculate the differential of the restriction of S to Hj (u−). Deduce that Sis monotonic along Hj (u−) so long as ρ2c2[1/ρ]+ [p] is not zero.

4.10 We consider only classical solutions of gas dynamics.

(1) Let s(x, t) be a quantity satisfying the transport equation (∂t +v∂x )s = 0.Show that ρ−1∂x s does too.

(2) Let g be a numerical function. Show that E := ρg(ρ−1Sx ) is an ‘entropy’with ‘flux’ F = vE .

(3) We construct by induction S0 := S, . . . , Sn := ρ−1∂x Sn−1. Let g be a realfunction of m+1 real variables. Show that ρg(S0, . . . , Sm) is an ‘entropy’of ‘flux’ F = vE .

4.11 We choose to express every thermodynamic quantity as a function of theentropy–pressure pair (p, S).

(1) Show that

dρ = c−2(dp − peT dS),

de = c−2(ρ−2 pdp + pρT dS).

(2) We consider the (unrealistic) case where all the fields are linearly degener-ate. Show that there exists a numerical function h such that ρ2c2h(S) ≡ 1.

(3) Deduce that e and ρ are expressed in the form

e = 1

2h(S)p2 + k(S),

1

ρ= −h(S)p + l(S).

(4) Show that [S] = 0 implies the relation (4.26). Deduce that in the neigh-bourhood of u−, S ≡ S− along the curves Hj (u−) for j = 1, 3.

4.12 We consider the interaction of two shock waves of the same family, (u−, u0;s−), (u0, u+; s+). More precisely, the initial condition is

u(x, 0) =

u−, x < −1,

u0, −1 < x < 1,

u+, x > 1.

We denote by z := 1/ρ the specific volume

(1) Calculate the solution of the Cauchy problem up to the time T of inter-action. Show that T is finite.

(2) We suppose that the by-product of the interaction, that is the solution ofthe Riemann problem with uL= u−, and uR= u+, reduces to a singleshock wave. Derive the formula

p−(z+ − z0)+ p+(z0 − z−)+ p0(z− − z+) = 0.

(3) In a shock wave with velocity s, show that

[p]

[z]= −(ρ(v − s))2,

(ρ, v) taking either of the values, downstream or upstream.(4) Deduce from what has come before that the result of the interaction of

two shock waves of the same family cannot reduce to a single shock wave(show that s− = s+ before finishing).

4.13 Solve the Riemann problem for the dynamics of barotropic gases:

ρt + (ρv)x = 0,

(ρv)t + (ρv2 + p(ρ))x = 0,

}(4.33)

where ρ(≥ 0) �→ p(ρ) ≥ 0 is the equation of state of the gas. We shall makethe appropriate hypotheses on p for the system to be hyperbolic and to havetwo genuinely non-linear characteristic fields (for the sense of variation of λ j

along the integral curve of r j , we shall adopt the same assumptions as for apolytropic gas). We shall take account of the fact that, as for a non-isothermalgas (i.e. a general gas), it must sometimes admit a vacuum. Following the

4.9 Exercises 145

position of vR − vL with respect to the appropriate values of H (ρL, ρR) andH (ρR, ρL) we shall determine the nature (shock or rarefaction) of the waves.

4.14 (See [5], [77], [78], [52], [94])We consider the system which describes the dynamics of an elastic string inR

2. We have v = (v1, v2) and w = (w1, w2) :

vt = wx ,

wt = (T (r )v/r )x , r := ‖v‖.We write also q := v/r , which is a unit vector.

We suppose that T (1) = 0 (in the reference frame, the string is at rest),T ′> 0 and T ′′ > 0.

(1) Show that the system is hyperbolic if and only if r > 1. We shall denotethis zone by H , which describes the string in extension.

(2) Show that the 1- and 4-characteristic-fields are genuinely non-linear, whilethe 2- and 3-fields are linearly degenerate. Establish a symmetry relationbetween the j-waves and the (5− j)-waves.

(3) Show that in parametric form each wave curve can be written in the form

w+ = w− + ϕ j (v+, v−).

What can we deduce about the ϕ j , from the symmetry relations?(4) We wish to solve the Riemann problem in the usual form. We suppose that

rL, rR > 1 and we denote the intermediate states by (u j , v j ) for 1 ≤ j ≤ 3.Show that q1 = qL, q2 = qR, and r1 = r2 = r3. We denote this commonvalue by r .

(5) Eliminating the vectors w j (and using (3) above) reduce the solution ofthe Riemann problem to that of a single vector equation with values in R

2,whose unknowns are q2 and r .

(6) Writing that q2 is a unit vector leads to the solution of a scalar equation ofthe form

J (r ; wR − wL, vL, vR) = 0.

(7) Study the variations of J with respect to its first variable on the interval(1,∞). Deduce that the Riemann problem has one and only one solutionwith values in H .

(8) Characterise the nature of the 1-wave and of the 4-wave as a function of thevalues, which should be calculated explicitly, of J (rL; wR − wL, vL, vR)and of J (rR; wR − wL, vL, vR).

5

The Glimm scheme

It is not the purpose of this work to introduce the schemes for the numerical simula-tion of the systems of conservation laws, which is very well done in other works [62,34]. But it is impossible to study the theory of systems without describing Glimm’sscheme, which gives the sole result of any generality concerning Cauchy’s problem.Curiously, this scheme, the only one for which we have at our disposal a conver-gence theorem for systems in one space dimension, is rarely used, no doubt byreason of its random aspect (which prevents it attaining a high precision) and alsobecause its extension to several space dimensions is disappointing.

We therefore restrict ourselves to systems of n conservation laws in one spacedimension,

ut + f (u)x = 0, x ∈ R, t > 0, (5.1)

u(x, 0) = u(x), x ∈ R, (5.2)

for which we know a priori the Riemann problem has a solution. For example, sincethe principal result concerns an initial datum u near to a constant, we can supposethat the system is strictly hyperbolic and that each of its characteristic fields is eithergenuinely non-linear or linearly degenerate, as we can use Lax’s theorem for thelocal solution of the Riemann problem.

5.1 Functions of bounded variation

Glimm’s theory makes use of the class of functions of bounded total variation onR. We recall that if E is a normed vector space and if I is an interval of R (whichcan be either open or closed), the total variation of a function v: R → E on I ,denoted by TV(v; I ), is the upper bound of

r∑j=1

‖v(x j )− v(x j−1)‖,

146

5.1 Functions of bounded variation 147

when x = (x0, . . . , xr ) runs through the set of finite increasing sequences withvalues in I . We say that v is of bounded total variation on I if TV(v; I ) < +∞.The set BV(I ; E) of these functions is a Banach space when we equip it with thenorm TV(v; I ) + ‖v(y)‖, y being a point chosen in I . The main property of thespace BV is Helly’s theorem.

Theorem 5.1.1 We suppose that E is of finite dimension and that I is bounded.Then the canonical mapping of BV(I ; E) into L1(I ; E) is compact.

We take heed of the fact that this mapping is not injective since the functions in L1

are defined only almost everywhere. We could make it injective by replacing BVby its quotent modulo the null functions almost everywhere. The norm of a classof functions v would then be the lower bound of the norms of its elements. Thistheorem can be seen as a variant of the Rellich–Kondrachov theorem for Sobolevspaces since the image of BV is also the space of locally integrable functions whosedistributional derivative is a bounded measure. This space is particularly appropriate(at least in one space dimension) in the study of weak solutions involving shockwaves and contact discontinuities and which are smooth elsewhere.

In several space dimensions, the situation is clearly less favourable sincewe know [4, 85] that the space BV is not suitable for the linear systemsut +

∑1≤ j≤d A jux j = 0, except when the matrices A j commute pairwise.

Although it is not excluded that the non-linearity of certain characteristic fieldscontributes to partially regularise the solution, the physical systems also havelinearly degenerate fields and it is improbable that functions of bounded vari-ation settle the question. However, no other satisfactory function space hasbeen suggested until now to study weak solutions.

Now, let us introduce some rules of calculation. When a function v: I → E ispiecewise continuous, discontinuous only at the points a1, . . . , as and piecewiseC

1 between these, the total variation of v is calculated simply by the formula

TV(v; I ) =s∑

j=1

(‖v(a j +0)−v(a j )‖+‖v(a j )−v(a j −0)‖)+∫

I\{a1,...,as}

∥∥∥∥dv

dx

∥∥∥∥dx .

If I = (a, b) and if c ∈ (a, b), then

TV(v; I ) = TV(v; (a, c))+ TV(v; (c, b))+ |v(c)− v(c − 0)| + |v(c)− v(c + 0)|.

The following inequality is a consequence of the definition.

148 The Glimm scheme

Proposition 5.1.2 If v ∈ BV(R) and if h > 0, we have∫R

|v(x + h)− v(x)| dx ≤ h TV(v; R).

In fact the number of jumps of a function of bounded variation is at most denumer-able and, free to modify v at these points with the result that v(x) is on the segmentof extremities v(x − 0) and v(x + 0), we have

TV(v; R) = limh→0

1

h

∫R

|v(x + h)− v(x)| dx .

Proof This uses only the triangle inequality and the relation of Chasles:∫R

|v(x + h)− v(x)| dx =∑k∈Z

∫ (k+1)h

kh|v(x + h)− v(x)| dx

=∑k∈Z

∫ h

0|v(y + (k + 1)h)− v(y + kh)| dy

=∫ h

0

∑k∈Z

|v(y + (k + 1)h)− v(y + kh)| dy

≤∫ h

0TV(v; R) dy = h TV(v; R).

It is easy now to prove Helly’s theorem. In fact if F is a bounded family of realfunctions defined on a bounded interval I we go on to consider the family F oftheir extensions by 0 to the whole of R. This family is again bounded in BV(R)since each extension satisfies TV( f ; R) ≤ TV( f ; I )+2‖ f ‖∞ ≤ 3‖ f ‖BV(I ). Fromthe above proposition, F is uniformly equi-integrable. In addition, the elements ofF are with support in I , which is a fixed compact set, and are uniformly boundedon R, since ‖ f ‖∞ ≤ ‖ f ‖BV(I ). Thus, the set F satisfies the hypotheses of therelative compactness theorem of Kolmogorov in L1(R) and we conclude that F isrelatively compact in L1(I ).

The last result which we need concerns functions which also depend on a timevariable.

Theorem 5.1.3 We suppose that E is finite-dimensional and we consider a sequence(vm)m∈N of functions defined on [0, T ]×R, with values in E satisfying the followingthree hypotheses:

(1) there exists a positive real number M such that TV(vm(t); R) ≤ M for allm ∈ N,

5.2 Description of the scheme 149

Fig. 5.1: The grid with sampling points.

(2) ‖vm(t, 0)‖ ≤ M for all m ∈ N,(3) there exists a sequence (εm)m∈N, which converges to 0+, such that∫

R

‖vm(t, x)− vm(s, x)‖ dx ≤ εm + M |t − s|

for all m ∈ N and all s, t ∈ [0, T ].

Then this sequence is relatively compact in L1loc((0, T )×R): we can extract a sub-

sequence whose restriction to every bounded open set � ⊂ (0, T ) × R convergesin L1(�). Also the limit v belongs to C ([0, T ]; L1(R)).

In fact the limit also satisfies∫R

‖v(t, x)− v(s, x)‖ dx ≤ M |t − s|,

and v(t, ·) (which is well-defined as an element of L1(R) for all values of t) admitsfor t ∈ (0, T ) a representation with bounded variation satisfying TV(v(t, ·); R) ≤M and ‖v(t, 0)‖ ≤ M .

The proof of this theorem will be given in §5.8.

5.2 Description of the scheme

The approximate solution of the Cauchy problem given by Glimm’s scheme dependson the choice of the space step �x and of the time step �t and on that of a sequencea= (a0, a1, . . .) of which each term is an element of (−1, 1). In general, the ratioρ := �x/�t is chosen a priori to ensure a sufficient speed of propagation (theCourant–Friedrichs–Lewy condition) and remains constant when we make h = �x


tend to zero. We then denote by uha the approximate solution, although it depends

also on ρ, as we only vary h and a in the sequel.This is defined by induction on n ∈N in each strip [n�t, (n + 1)�t]× R. Pos-

sibly this induction may fail at a stage N and the solution will be defined only on(0, N�t)×R. At the instants of the form n�t , uh

a is piecewise constant, constant onthe meshes I j := (( j − 1)�x, ( j + 1)�x) for j ∈ n+ 2Z. The meshes are thus ar-ranged in alternate rows. At the initial instant, we define uh

a(0, I j ) = u(( j+a0)�x),which is a sampling of the initial condition. Similarly, for n ≥ 1, we pass fromn�t − 0 to n�t + 0 through a sampling value

uha(n�t, I j ) := uh

a(n�t − 0, ( j + an)�x − 0), j ∈ n + 2Z.

This, the choice of the value to the left, is conventional and is intended to removethe ambiguity when we must sample a discontinuity in the approximate solution(which happens only exceptionally).

It remains to define uha in the strip (n�t, (n + 1)�t) × R as the exact solution

of the Cauchy problem in this strip, of which the initial condition is the piecewiseconstant function uh

a(n�t, ·). This exact solution is known to us during a certaintime interval δtn: it is the gluing of solutions of the Riemann problem. Let us writeun, j := uh

a(n�t, I j ). The Riemann problem centred in t = n�t and x = k�x(for k ∈ n + 1 + 2Z), of which the left and right states are respectively un,k−1

and un,k+1, admits a solution vn,k by hypothesis. The function v(·, ·) defined fort ≥ n�t by

vn(t, x) = vn,k(t, x), ∀(x, t) : x ∈ Ik,

is a weak entropy solution of the Cauchy problem considered while vn is continuousat the interfaces between the meshes, that is to say for x = j�x , j ∈ n + 2Z. Itis enough for this that the waves of the Riemann problem issuing from the nodek�x do not reach the boundaries of the mesh Ik , this for all k. Denoting by Vn theupper bound of the speeds of the waves of all the Riemann problems solved at theinstant n�t , we observe that vn is the exact solution sought in the time intervalδtn = V−1

n �x . The calculation of uha is thus effective until the following instant

(n + 1)�t provided the Courant–Friedrichs–Lewy (CFL) condition is satisfied:

ρVn ≤ 1. (5.3)

We note that Vn depends only on the (ordered) list of the states un, j , j ∈ n+ 2Z.In particular, suppose that u takes its values in a domain K ⊂ U, compact andinvariant for the Riemann problem, that is to say satisfying the following property:

For all vL, vR ∈ K , the solution of the Riemann problem between vL and vR

has its values in K .

5.2 Description of the scheme 151

Under this hypothesis, it is immediate that uha takes its value in K while the ap-

proximate solution is defined, that is, while the condition (5.3) is satisfied. But asK is compact, there exists a bound V K of the speed of the waves in the Riemannproblems whose initial data are in K . And as Vn ≤ V K , it is sufficient to choose apriori ρ = (V K )−1 for (5.3) to hold and for the approximate solution to be definedfor all time. This remark is due to D. Hoff [42].

In the general case, grosso modo for systems of at least three equations, there doesnot exist an invariant compact domain for the Riemann problem (see Chapter 8).We therefore define the approximate solution by induction on the strips (tn, tn+1)×R with tn+1 − tn = V−1

n �x , hoping that∑

n V−1n diverges. However, Glimm’s

theorem, which we are going to prove, states that for a small enough initial datumthe approximate solution remains in a fixed compact set, independent of ρ, withthe result that there again exists a value of ρ which satisfies the CFL condition atall stages of the calculation.

Theorem 5.2.1 (Glimm) Let u be an interior point of U. We suppose that eachcharacteristic field is either genuinely non-linear or linearly degenerate in theneighbourhood of u. Then, there exist two numbers δ > 0 and C > 0 such that forevery given initial function satisfying the hypothesis

‖u − u‖L∞ + TV(u) ≤ 0, (5.4)

the Cauchy problem (5.1), (5.2) possesses a weak solution in R+t × Rx satisfying

in addition

‖u(t, ·)− u‖L∞ ≤ C(‖u − u‖L∞ + TV(u)),TV(u(t, ·)) ≤ CTV(u),‖u(t, ·)− u(s, ·)‖L1 ≤ C |t − s|TV(u),u satisfies Lax’s entropy inequalities Et + Fx ≤ 0 for every convex entropy E of

flux F:∫∫R+×R

(E(u)ϕt + F(u)ϕx ) dx dt +∫

R

E(u)ϕ(x, 0) dx ≥ 0, ∀ϕ ∈ D+(R2).

Of course, this theorem is tied to Glimm’s scheme by a statement of convergence.However, as we shall see in the following section, the approximate solution, thoughit converges in general, does not converge to a weak solution of the Cauchy prob-lem when we choose certain sequences a. The convergence is linked to a randomproperty of the sequence a, which explains that we have recourse to a parameter ascomplex as a generic element of A = (−1, 1)N.

First of all, let us provide A with the uniform probability measured dν, productof the uniform measures dm j = 1

2da j on each factor (−1, 1). This is the unique

measure defined on the class of Borel sets of A which satisfies the identities∫A

g(a) dν(a) = 2−1−N∫ 1

−1. . .

∫ 1

−1G(a0, . . . , aN ) da0 . . . daN

when g(a) := G(a0, . . . , aN ) has only a finite number of arguments. We see fromthe Stone–Weierstrass theorem that these functions form a dense sub-space of C ( A)so the above formula allows us to define in a unique manner the integral of a function,continuous on A, with respect to dν.

Here is the convergence result.

Theorem 5.2.2 Under the hypotheses of Theorem 5.2.1, there exist two numbersh0 > 0 and ρ0 > 0 such that, for all a ∈ A and every step h = �x, 0 < h < h0,�t = ρ�x (0 < ρ < ρ0), the approximate solution is defined for all time andsatisfies at each instant∥∥uh

a − u∥∥

L∞ ≤ C(‖u − u‖L∞ + TV(u)),

TV(uh

a

) ≤ CTV(u).

In addition, there exists a subset N of measure zero in A such that, for all a∈ A\N ,the sequence (uh

a)h→0+ is relatively compact in L1loc(R

+ × R), its limits being theweak solutions of the Cauchy problem such as are described in Theorem 5.2.1.

Of course, the uniqueness of the entropy solution in the class where we show theexistence not being known,1 it is not possible to write a simpler statement. Thescalar case is the most favourable because of Theorem 2.3.5 and in this case it isthe whole sequence (uh

a)h which converges, this for almost all a. Also, the proofof the stability of the scheme (obtaining a priori estimates) is much more simpleand general in the scalar case (we have for example C = 1): we no longer supposethat the given initial function is small. It is the same for the systems said to be of B.Temple (see Chapter 13) which extend in a natural manner the scalar conservationlaws. This remark is also of value for a class of systems which we shall study in §5.6and which contains the isothermal model of gas dynamics. In the general case, wecan ask if the hypothesis of a small datum is essential for the existence of a weaksolution to the Cauchy problem, since the real world does not consist of such data.We do not have the means to answer this question, but it has been observed that theestimate TV(u(t, ·)) ≤ CTV(u) cannot be true if the constant C depends solely on‖u − u‖∞ for rather general systems of at least three conservation laws [47]. The

1 Actually, a recent result of A. Bressan states that the limit is unique whenever Glimm’s estimate and consis-tency hold.

5.3 Consistency 153

number 3 is here optimal since in the case of 2×2 systems (n = 2 equations), onlythe norm of u − u in L∞ must be supposed small (see §5.9).

As often in numerical analysis, the stability of the scheme implies its convergence(Theorem 5.4.1). It is the possibility of showing the stability which distinguishesthe general case from particular cases. In fact, only the convergence is the objectof a probabilistic analysis, while that of the stability, which leads to the estimate(5.4), proceeds from physical ideas and makes use of the notion of interactionpotential.

Before dealing with the proof of Theorem 5.2.2 we shall explain in the nextsection why certain sequences a ∈ A are not suitable.

5.3 Consistency

For most of the systems of physical interest, one at least of the characteristic fieldsis genuinely non-linear and the work of Lax shows that there exist pairs of states(uL, uR) linked by a shock wave. We denote by c the speed of this shock waveand we calculate the approximate solution provided by Glimm’s scheme when thegiven initial condition is

u(x) ={

uL, x < 0,

uR, x > 0.

Since the solution of the Riemann problem between two equal states is constantand since the scheme proceeds by sampling and by solutions of Riemann prob-lems, we see immediately that the states un, j are all with values in {uL, uR}. Moreprecisely, there exists a number jn ∈ n + 1 + 2Z such that un, j = uL if j < jnand un, j = uR if j > jn . The approximate solution is thus completely known ifwe realise the recurrence jn �→ jn+1. In the strip (tn, tn+1) × R, the approximatesolution takes the values

uha(t, x) =

{uL, x − jnh < c(t − tn),

uR, x − jnh > c(t − tn).

The CFL condition is written ρ|c| ≤ 1, and this does not depend on n. The approx-imate solution is thus defined for all time. Also, uh

a(tn+1 − 0, x) takes the value uL

on I j for all j < jn and the value uR for j > jn . Thus, un+1, j takes the valueuL for j < jn and the value uR for j > jn . As jn+1 − jn is odd, we thus havejn+1 = jn ± 1 and in fact

jn+1 = jn +{

1, an+1 < ρc,

−1, an+1 ≥ ρc.


Finally

jn = n + 1− 2 card{m ∈ N: 0 ≤ m ≤ n, am ≥ ρc}.The approximate solution takes the values uL and uR at one side and the other ofthe broken line which passes through the nodes Pn = (n�t, jnh). The slope of thestraight line O Pn has the value

p(a; h, n�t) = pn = jnh

n�t= jn

ρn∼ 1

ρ− 2

ρncard{m ∈ N: 0 ≤ m ≤ n, am ≥ ρc}.

If the approximate solution converges to the exact solution of the Cauchy problem,which is the shock wave of speed c, we must have limh→0 p(a; h, t) = c, that is tosay,

2

ncard{m ∈ N: 0 ≤ m ≤ n, am ≥ ρc} ∼ 1− ρc. (5.5)

If we wish to maintain that this convergence takes place for all the possible systems,that is for shock waves of arbitrary speed and numbers ρ compatible with the CFLcondition, it is necessary that

2

ncard{m ∈ N: 0 ≤ m ≤ n, am ≥ θ} ∼ 1− θ, (5.6)

when θ ∈ (−1, 1) and n → ∞. A sequence a which satisfies this property iscalled equi-distributed in (−1, 1). An equivalent condition is the convergence ofquadrature formulae:

limn→+∞

1

n + 1

n∑k=0

f (ak) = 1

2

∫ 1

−1f (x) dx, ∀ f ∈ C ([0, 1]). (5.7)

A classic example of an equi-distributed sequence is an = nξ (mod 2) where ξ isan irrational number. But to put Glimm’s scheme into operation, Collela [10] hasproposed the sequence of Van der Corput [38] in which the filling of the interval(−1, 1) is mades at an optimal speed (this refers to the notion of discrepancy ofa sequence which goes beyond the objective of this section). This sequence isobtained by writing each integer n as a number to the base 2, n = αr · · ·α1 withα j = 0 or 1 and αr = 1, then calculating the number bn written in binary mode asbn = α1, · · ·αr . Finally we put bn = 1−an . The first elements of this sequence are

−1 0

−1/2 1/2

−3/4 −1/4 1/4 3/4...

...

5.3 Consistency 155

Thus, only the equi-distributed sequences let us hope for convergence of uha to a

weak solution of the Cauchy problem. Happily we have

Proposition 5.3.1 Almost every (with respect to dν) sequence a ∈ A is equi-dis-tributed.

This proposition explains that the convergence of the scheme can take placefor almost every sequence a except for those that are badly distributed. In fact,T.-P. Liu has improved Glimm’s theorem in proving that convergence takesplace for every equi-distributed sequence [69].

Proof Let F be a dense denumerable subset of C ([−1, 1]). For f ∈ C ([−1, 1])and a ∈ A, we write

I ( f ) = 1

2

∫ 1

−1f (x) dx , In(a; f ) = 1

n + 1

n∑k=0

f (ak).

If a sequence a satisfies limn→+∞ In(a; f ) = I ( f ) for all f in F , then it isequi-distributed for if g ∈ C ([−1, 1]) and if ε > 0, there exists f in F such thatsupx | f (x) − g(x)| ≤ ε and there exists N ∈ N such that, for n > N , we have|In(a; f )− I ( f )| ≤ ε. But then, for n ≥ N , we have

|In(a; g)− I (g)| ≤ |In(a; g − f )| + |In(a; f )− I ( f )| + |I ( f − g)| ≤ 3ε.

We thus have limn→+∞ In(a; g) = I (g) for every continuous function g and this isequivalent to the equi-distribution of a. The set of sequences badly distributed isthus the (denumerable) union of sets N f defined by

J (a; f ) := limsupm→+∞

|Im(a; f )− I ( f )|2 > 0.

Now making use of Fatou’s lemma,∫A

J (a; f ) dν(a) ≤ limsupm→+∞

∫A|Im(a; f )− I ( f )|2 dν(a),

where the integral on the right-hand side has value

1

(m + 1)2

m∑j,k=0

∫A

f (a j ) f (ak) dν(a)− 2

m + 1I ( f )

m∑k=0

∫A

f (ak) dν(a)+ I ( f )2

= I ( f )2(

m

m + 1− 2+ 1

)+ 1

m + 1I ( f 2)

= 1

m + 1(I ( f 2)− I ( f )2)

m→+∞−→ 0.

Thus∫

A J (a; f ) dν(a) is null, with the result that N f is of null measure. Finally,the (denumerable) union of the N f for f ranging over F is of null measure.

5.4 Convergence

We show in this section that the stability of Glimm’s scheme in BV implies theconvergence to a weak entropy solution for almost every choice of the sequence a.

Theorem 5.4.1 Let u ∈ BV(R) and ρ > 0. We suppose that for h ∈ (0, h0)(with h0 > 0 suitably chosen) and for all a ∈ A, Glimm’s scheme defines a globalapproximate solution uh

a and that there exists a constant M > 0 such that we havefor all time ∥∥uh

a − u∥∥

L∞ + TV(uh

a

) ≤ M.

Then the sequences (uha)h>0 are relatively compact in L1

loc(R+ ×R) and their limit

are, for dν-almost all a ∈ A, weak solutions of the Cauchy problem (5.1), (5.2).

Let us note that we do not assume that the given initial condition is small in BV(R),with the result that, if we know how to prove the stability of the scheme for arbitrarilylarge given initial data, we immediately deduce an existence theorem for such data.

Compactness

The compactness of the sequences (uha)h>0 will follow from Theorem 5.1.3, once

we have verified the third hypothesis. Let us first suppose that s = n�t + 0 andn�t < t < (n + 1)�t − 0. In these calculations let us write u = uh

a . Then u(s, x)has value un, j on I j ( j+n even), in the same manner as u(t, jh). We therefore have∫

R

|u(t)− u(s)| dx =∑

j∈n+2Z

∫I j

|u(t)− u(s)| dx

=∑

j∈n+2Z

∫I j

|u(t, x)− u(t, jh)| dx

≤∑

j∈n+2Z

2h TV(u(t); I j ) = 2h TV(u(t)) ≤ 2Mh.

Then, if s = n�t − 0, and t = n�t + 0,∫R

|u(t)− u(s)| dx =∑

j∈n+2Z

∫I j

|u(s, ( j + an)h)− u(s, x)| dx

≤∑

j∈n+2Z

2h TV(u(s); I j ) ≤ 2Mh.

5.4 Convergence 157

Combining these two inequalities we bound the integral for any s and t by 2(N+1)Mh where N is the number of integers n such that s ≤ n�t ≤ t . Finally,∫

R

|u(t)− u(s)| dx ≤ 2M(|t − s| + h).

We can therefore apply Theorem 5.1.3.We note that we have implicity assumed that the CFL condition is satisfied,

by supposing that uha is globally defined. We make use of this when we write

u(t, jh) = un, j for n�t < t < (n + 1)�t .

Estimate of the error

The error due to the scheme is the value of the expression which should benull for a weak solution in the variational formulation of the Cauchy problem,namely

e(a; ϕ, h) :=∫∫

R+ ×R

(uh

a · ϕt + f(uh

a

) · ϕx)

dx dt +∫

R

u(x) · ϕ(x, 0) dx .

We prove here the following lemma.

Lemma 5.4.2 Under the hypotheses of Theorem 5.4.1 we have

limh→0+

∫A|e(a; ϕ, h)|2 dν(a) = 0

for all ϕ ∈ D (R2)n.

Proof Let ϕ ∈ D (R2)n . In each strip Bn = (n�t, (n + 1)�t) × R, uha is a weak

solution, piecewise smooth, of a Cauchy problem; that is we have∫∫Bn

(uh

a · ϕt + f(uh

a

) · ϕx)

dx dt =∫

R

uha((n + 1)�t − 0, x) · ϕ((n + 1)�t, x) dx

−∫

R

uha(n�t + 0, x) · ϕ(n�t, x) dx .

Summing these equalities (all these integrals except a finite number among themare null) we obtain e(a; ϕ, h) =∑

n≥0 en(a; ϕ, h) where

en(a; ϕ, h) :=∫

R

(uh

a(n�t − 0, x)− uha(n�t + 0, x)

) · ϕ(n�t, x) dx,

these quantities being all null but a finite number. It is easy to see that each en(a; ϕ, h)is O(h), for by writing wn := uh

a(n�t − 0) we have

en(a; ϕ, h) =∑

j∈n+2Z

∫I j

(wn(x)− wn(( j + an)h)) · ϕ(n�t, x) dx,

|en(a; ϕ, h)| ≤∑

j∈n+2Z

2h TV(wn; I j )‖ϕ‖∞ (5.8)

≤ 2h TV(wn)‖ϕ‖∞ ≤ 2hM‖ϕ‖∞.

Besides, en(a; ϕ, h) in fact depends only on (a0, . . . , an) and not on the entiresequence a and behaves in mean as O(h2). In fact

1

2

∫ 1

−1en(a; ϕ, h) dan =

∑j∈n+2Z

1

2h

∫∫I j×I j

(wn(x)− wn(y)) · ϕ(n�t, x) dx dy

= 1

4h

∑j∈n+2Z

∫∫I j×I j

(wn(x)− wn(y)) · (ϕ(n�t, x)

−ϕ(n�t, y)) dx dy.

Thus∣∣∣∣12∫ 1

−1en(a; ϕ, h) dan

∣∣∣∣ ≤ h∑

j∈n+2Z

TV(wn; I j )TV(ϕ(n�t); I j )

≤ 2h2∑

j∈n+2Z

TV(wn; I j )‖ϕx‖∞ ≤ 2h2 TV(wn)‖ϕx‖∞

≤ 2h2M‖ϕx‖∞. (5.9)

We develop now the integral over A.∫A|e(a; ϕ, h)|2 dν(a) =

∑n,m≥0

∫A

enem dν(a)

=∑n≥0

∫A|en|2 dν(a)+ 2

∑0≤m<n

∫A

enem dν(a).

Let N be the number of strips Bn whose intersection with the support of ϕ is notempty. As this support is contained in a half-plane t ≤ T , we have

N ≤ 1+ T

�t= 1+ T

ρh.

The first sum is bounded above, from (5.8), by N (2hM‖ϕx‖∞)2 = O(h). Also,

5.4 Convergence 159

each integral∫

A emen dν(a) is O(h3), as by (5.9) and (5.8),∣∣∣∣∫

Aemen dν(a)

∣∣∣∣ =∣∣∣∣∫ 1

−1. . .

∫ 1

−1em

(1

2

∫ 1

−1en dan

)2−n da0 · · · dan−1

∣∣∣∣≤

∣∣∣∣∫ 1

−1· · ·

∫ 1

−12hM‖ϕ‖∞ · 2h2M‖ϕx‖∞2−n da0 · · · dan−1

∣∣∣∣= 4h3M2‖ϕ‖∞‖ϕx‖∞.

The second sum is thus bounded above by N (N − 1)4h3M2‖ϕ‖∞‖ϕx‖∞ = O(h).Finally ∫

A|e(a; ϕ, h)|2 dν(a) = O(h),

which proves the lemma.

Conclusion

From the lemma, when ϕ ∈ D (R2)n , there exists a negligible part Nϕ of A suchthat for a ∈ A \ Nϕ , we have limh→0 e(a; ϕ, h) = 0. Let us choose a denumerabledense subset F of D (R2)n . The set N , the union of the Nϕ when ϕ ranges overF , is negligible in A. Being given a ∈ A \ N , let us consider a limiting value ofthe sequence (uh

a)h in L1loc(R

+ × R) (we have seen that there exist such limits).We know that u(x, t) is the pointwise limit, almost everywhere in R

+ × R, of asub-sequence (u

h pa )p∈N where h p → 0+ is a suitable sequence of mesh sizes. From

the theorem of dominated convergence and since f is continuous, we have

limp→+∞

∫∫R+×R

f(u

h pa

) · ϕ dx dt =∫∫

R+×R

f (u) · ϕ dx dt,

and therefore

limp→+∞ e(a; ϕ, h p) =

∫∫R+×R

(u · ϕt + f (u) · ϕx ) dx dt +∫

R

u(x) · ϕ(x, 0) dx .

Finally, since a ∈ A \N , we deduce∫∫R+×R

(u · ϕt + f (u) · ϕx ) dx dt +∫

R

u(x) · ϕ(0, x) dx = 0

for all ϕ ∈ F and hence for all ϕ ∈ D (R2)n since F is dense and since the left-handside is a continuous linear form on D (R2)n . This completes the proof of Theorem5.4.1.


Entropy inequalities

When E: U → R is a convex entropy of flux F , we prove the entropy inequalityin the same manner provided that the shock waves used in the solution of theRiemann problem satisfy this inequality (this is the least that we can ask of them).The approximate solutions satisfy in each band the inequality, for ϕ ∈ D

+(R2),∫ (n+1)�t

n�t

∫R

(E(uh

a

) · ϕt + F(uh

a

) · ϕx)

dx dt +∫

R

E(uh

a(n�t + 0)) · ϕ(n�t) dx

≥∫

R

E(uh

a((n + 1)�t − 0)) · ϕ((n + 1)�t) dx .

Denoting still the error due to the scheme by e(a; ϕ, h), that is to say

e(a; ϕ, h) :=∫

R+×R

(E(uh

a

) · ϕt + F(uh

a

) · ϕx)

dx dt +∫

R

E(u) · ϕ(x, 0) dx,

we thus have e(a; ϕ, h) ≥∑n≥0 en(a; ϕ, h) where

en(a; ϕ, h) :=∫

R

(E(uh

a(n�t − 0))− E

(uh

a(n�t + 0))) · ϕ(n�t) dx .

On the compact set B(u; M), E is Lipschitz with a constant denoted by K . We thenhave, as in (5.8),

|en(a; ϕ, h)| ≤ 2hKM‖ϕ‖∞and similarly ∣∣∣∣12

∫ 1

−1en(a; ϕ, h) dan

∣∣∣∣ ≤ 2h2KM‖ϕx‖∞.

Writing e(a; ϕ, h) :=∑n≥0 en(a; ϕ, h), we deduce again that

∫A |e(a; ϕ, h)|2 dan =

O(h) and hence that limh→0 e(a; ϕ, h) = 0 for almost all a ∈ A, that is

liminfh→0

e(a; ϕ, h) ≥ 0.

The same arguments as were used in the preceding section then show the existenceof NE , a negligible part of A, containing N , such that for a ∈ A \NE , the limitingvalues of (uh

a)h>0 satisfy Lax’s inequality∫∫R+×R

(E(u) · ϕt + F(u) · ϕx ) dx dt +∫

R

E(u) · ϕ(0) dx ≥ 0

for all ϕ ∈ D+(R2).

5.5 Stability 161

5.5 Stability

Supplements apropos of the local Riemann problem

Since the characteristic fields are each either genuinely non-linear or linearly de-generate, Theorem 4.6.1 ensures, for every neighbourhood ω of u, the existenceof a neighbourhood ω1 ⊂ ω such that, for all (uL, uR) ∈ ω1 × ω1, the Riemannproblem between uL and uR admits a unique solution with values in ω. This so-lution is made up of the succession of waves of each family, a contact disconti-nuity if the field is linearly degenerate, a shock wave or a rarefaction wave oth-erwise. The k-wave links the constant states uk−1 and uk (u0 = uL, un = uR).Using the parametrisation s �→ ϕk(s, v) of the wave curve issuing from a pointv (with s = λk(ϕk(s, v)) − λk(v) if the kth field is genuinely non-linear), definedin Chapter 4, we construct the solution of the Riemann problem by solving theequation �(ε, uL) = uR where

�(ε, v) := ϕn(εn, ϕn−1(εn−1, . . . , ϕ1(ε1, v) · · ·)),The mapping � is of class C

2 on V × ω, V being a neighbourhood of the originwhich depends only on the choice of ω (which we take compact) and satisfies

∂�

∂εk(0, v) = rk(v),

rk(v) being an eigenvector of d f (v) associated with the eigenvalue λk and nor-malised by dλk · rk ≡ 1 when the field is genuinely non-linear. We thus haveεk = lk(uL) · (uR − uL) + O(‖uR − uL‖2) denoting by (lk)1≤k≤n the dual basisof (rk)1≤k≤n . The intermediate constant states are given by induction on k: uk =ϕk(εk ; uk−1).

The essential idea of the stability analysis made by Glimm [32] concerning hisscheme is that of the interaction of successive Riemann problems. If three states uL,um, uR are given in ω1, we have um = �(δ, uL), uR = �(ε; um) and uR=�(γ ; uL).What can we say about the vector γ when we express it as a function of theparameters (δ, ε, um) ? The cases uL = um and uR = um give us the formulaeγ (0, ε, um) = ε and γ (δ, 0, um) = δ respectively. As γ is of class C

2 (from theimplicit function theorem and since � is of class C

2), we deduce that γ (δ, ε, um)−δ − ε = O(‖ε‖2 + ‖δ‖2), which we are now going to improve.

In fact, we shall have γ = δ + ε when um is a value taken by the solution ofthe Riemann problem between uL and uR, that is to say when there exists an indexp ∈ {1, . . . , n} such that

for k > p, δk = 0,for k < p, εk = 0,the pth field is linearly degenerate, or εp = 0, or δp = 0 or (εp > 0 and δp > 0).

The last case cited is that of a p-rarefaction-wave that passes through the valueum. If the pth field is genuinely non-linear, the third condition is thus writtenε−p δ−p = 0 by writing z− = max(0,−z). The set of the above condition is thuswritten �(δ, ε, um) = 0, where � is the quadratic interaction term

�(δ, ε, um) =∑

1≤p<q≤n

|δqεp| +∑

p∈GNL

ε−p δ−p ,

the symbol∑

p∈GNL meaning that the summation is over the indices of the genuinelynon-linear fields only. Finally:

Lemma 5.5.1 The function (δ, ε, um) �→ γ − δ− ε is of class C2 in a neighbour-

hood of (0, 0, u). It satisfies

(1) γ − δ − ε = O(‖ε‖2 + ‖δ‖2),

(2) � = 0 =⇒ γ − δ − ε = 0.

Owing to a geometrical lemma of which we shall give the statement and the proofin §5.8, we deduce the result which will enable us to establish the stability.

Lemma 5.5.2 There exist a neighbourhood � of (0, 0, u) and a real number c0 > 0such that in � we have

‖γ (δ, ε, um)− δ − ε‖ ≤ c0 �(δ, ε, um).

In the sequel we shall take � of the form O × ω2, small enough for us to have�(ε, �(δ, um)) ∈ ω1 when (δ, ε, um) ∈ �.

A linear functional

If ‖u − u‖∞ is small enough, u has values in ω2 and we are able to start to putGlimm’s scheme into operation. The ratio ρ = �t/�x is fixed so that ρV ω1 < 1and the scheme stops if the approximate solution leaves ω2. One of our aims isto show that it does not leave if the given initial state is sufficiently close to u inBV(R). At the first iteration, we still have u1,k ∈ ω1 since u0,k ∈ ω2. As longas uh

a(n�t) is defined with values in ω2, we denote by ε(n, k), δ(n, k), θ (n, k)and �(n, k) = θ (n + 1, k) the respective solutions of �(ε, un+1,k) = un,k+1,�(δ, un,k−1) = un+1,k , �(θ, un,k−1) = un,k+1, and �(�, un+1,k−1) = un+1,k+1.Since un+1,k is an intermediate state of the Riemann problem between un,k−1 andun,k+1 (this is the sampling principle) we have θ (n, k) = δ(n, k) + ε(n, k) and

5.5 Stability 163

likewise

�(δ(n, k), ε(n, k)) = 0,

|θp| = |δp| + |εp|, 1 ≤ p ≤ n,

θ−p = δ−p + ε−p , p ∈ GNL.

(5.10)

In addition �(n, k) = γ (ε(n, k − 1), δ(n, k + 1), un,k). In future, we shall omitthe last argument which is always easy to identify. We define a functional L(n) ofwhich we shall show the equivalence with the total variation of uh

a(n�t):

L(n) :=∑

k∈n+1+2Z

‖θ(n, k)‖.

In fact, when uR and uL are in ω1 with uR = �(ε; uL), we have ε ∼ P(u) · (uR −uL) where P(v)−1 is the matrix whose columns are the eigenvectors of d f (v),normalised by dλp · rp ≡ 1 if p ∈ GNL. There thus exists a constant C > 1 suchthat

C−1‖uR − uL‖ ≤ ‖ε‖ ≤ C‖uR − uL‖, ∀uR, uL ∈ ω1.

Since un,k−1 and un,k+1 have values in ω2, the Riemann problem between thesestates has a solution with values in ω1 and the above inequality applies to all thequantities (δ, ε, θ, �)(n, k). The total variation of uh

a(n�t), by breaking it up ontransverse waves (Chasles’ relation), is bounded above by

Cn∑

p=1

|θp(n, k)| = C‖θ (n, k)‖

by choosing the norm l1 on R2. We deduce that

C−1TV(uh

a(t)) ≤ L(n) ≤ CTV

(uh

a(t))

for n�t ≤ t ≤ (n + 1)�t , and similarly

C−1TV(uh

a((n + 1)�t)) ≤ L(n + 1) ≤ CTV

(uh

a((n + 1)�t)).

In particular, L(0) ≤ CTV(u).From now on, looking to effect an induction from n to n+ 1, we omit the argument

n in (ε, δ, θ, �) and we suppose the sequence (un,k)k∈n+2Z takes values in ω2. We

shall bound L(n + 1) above.

L(n + 1) =∑

k∈n+2Z

‖�(k)‖ =∑

k∈n+2Z

‖γ (ε(k − 1), δ(k + 1))‖

≤∑

k∈n+2Z

(‖ε(k − 1)‖ + ‖δ(k + 1)‖ + c0�(ε(k − 1), δ(k + 1))).

Reordering the terms and with (5.10), we find

L(n + 1) ≤ L(n)+ c0�(n), (5.11)

where we have defined

�(n) :=∑

k∈n+2Z

�(ε(k − 1), δ(k + 1)).

Obviously, the presence of the positive term c0�(n) on the right-hand side does notallow us to come to a conclusion.

A quadratic functional

We now introduce the interaction potential

Q(n) :=∑

j,k∈n+1+2Z;j<k

�(θ ( j), θ (k)).

In particular

�(n) ≤ Q(n). (5.12)

Because of the formulae (5.10) and although � is not bilinear we can in any casedevelop

�(θ ( j), θ (k)) = �(ε( j), ε(k))+�(ε( j), δ(k))+�(δ( j), ε(k))+�(δ( j), δ(k)). (5.13)

Since � is sub-additive, we have also

�(�( j), �(k)) ≤ �(δ( j + 1), δ(k + 1))+�(δ( j + 1), ε(k − 1))+�(δ( j + 1), ρ(k))+�(ε( j − 1), δ(k + 1))+�(ε( j − 1), ε(k − 1))+�(ε( j − 1), ρ(k))+�(ρ( j), �(k)),

where we have denoted by ρ( j) the difference �( j)−ε( j−1)−δ( j+1). Reordering

5.5 Stability 165

the terms, we thus have

Q(n + 1) ≤∑

j,k∈n+1+2Z;j<k

{�(δ( j), δ(k))+�(δ( j), ε(k))+�(δ( j), ρ(k − 1))

+�(ε( j), δ(k))+�(ε( j), ε(k))+�(ε( j), ρ(k + 1))

+�(ρ( j + 1), �(k + 1))}+

∑j∈n+1+2Z

{�(δ( j), ε( j))−�(ε( j − 2), δ( j))}

= Q(n)−�(n)

+∑j<k

{�(δ( j), ρ(k − 1))+�(ε( j), ρ(k + 1))

+�(ρ( j + 1), �(k + 1))},as �(δ( j), ε( j)) = 0. The following three lines are elementary applications ofLemma 5.5.2.

�(δ( j), ρ(k − 1)) ≤ ‖δ( j)‖ ‖ρ(k − 1)‖ ≤ c0‖δ( j)‖�(δ(k), ε(k − 2)),

�(ε( j), ρ(k + 1)) ≤ c0‖ε( j)‖�(δ(k + 2), ε(k)),

�(ρ( j + 1), �(k + 1)) ≤ c0‖�(k + 1)‖�(δ( j + 2), ε( j)).

Bringing all these together, we arrive at

Q(n + 1) ≤ Q(n)+�(n)(2c0L(n)+ c2

0�(n)− 1). (5.14)

The ‘good’ functional, which permits us to estimate a priori the approximatesolution if TV(u)+ ‖u − u‖∞ is sufficiently small, is L := L + 2c0Q. We recallthat, u being fixed, ω, ω1 and ω2 have been chosen a priori, but compact, and c0

depends only on this construction. Taking (5.11), (5.14) and (5.12) into account weobtain the upper bound

L (n + 1) ≤ L (n)+ c0�(n){4c0L (n)− 1}, (5.15)

provided that (un,k)k∈n+2Z has values in ω2. It is important to note that we do notsuppose that the states un+1, j are in ω2.

The induction

We now choose a number α which satisfies the following two conditions:

α < 18c0C

the ball B(u; (1+ 5C2/4)α) is included in ω2.

Fig. 5.2: Wave curves of a system ‘a la Nishida’.

Finally, we suppose that the initial condition is close to the constant state u, in thesense that

‖u − u‖∞ + TV(u) ≤ α. (5.16)

We have L(0) ≤ Cα and hence L (0) ≤ L(0)(1 + 2c0L(0)) ≤ 54CTV(u) ≤ 5

4Cα

because Q(n) ≤ L(n)2 since �(δ, ε) ≤ ‖δ‖ ‖ε‖. In particular L (0) ≤ 1/4c0.Let us make the induction hypothesis L (n) ≤ L (0). Then L (n) ≤ 1/4c0 and

therefore ∑k∈n+1+2Z

‖un,k+1 − un,k−1‖ ≤ CL(n) ≤ CL (n) ≤ CL (0) ≤ 5C2α

4.

As limk→−∞ un,k = limx→−∞ u(x) and ‖u(−∞)− u‖ ≤ α, we have ‖un,k − u‖ ≤(1+ 5C2/4)α and hence un,k ∈ ω2. The approximate solution is thus again definedup to (n + 1)�t and the upper bound (5.15) is valid. As L (n) ≤ 1/4c0, we haveL (n + 1) ≤ L (0), which proves the induction.

Under the hypothesis (5.16), we thus have L (n) ≤ L (0) for all n ≥ 0, fromwhich we derive

TV(uh

a(n�t)) ≤ 5C2

4TV(u),

∥∥uha(n�t)− u

∥∥∞ ≤

(1+ 5C2

4

)(‖u − u‖∞ + TV(u)).

The analogous estimates for n�t < t < (n + 1)�t are elementary and are left to

5.6 The example of Nishida 167

the reader. This completes the proof of Theorem 5.2.2 and hence that of Theorem5.2.1.

5.6 The example of Nishida

This section is devoted to the study of certain systems for which we can showthe stability of Glimm’s scheme without supposing that the initial datum is small.Since the stability involves the convergence for almost all a ∈ A, we deduce aglobal existence theorem for a weak solution for the Cauchy problem without otherhypothesis than TV(u) < +∞.

Hypotheses and theorem

Since the initial datum is of arbitrarily large variation, the wave curves must bedefined globally in order that we can solve all the Riemann problems.

We only consider 2× 2 strictly hyperbolic systems (n = 2), with characteristicvelocities u �→ λ j (u), j = 1, 2, λ1 < λ2. We introduce the Riemann invariantsu �→ r (u) and u �→ s(u) which are the solutions without critical points of thedifferential equations

dr (d f − λ1) = 0,

ds (d f − λ2) = 0.

We suppose that u �→ (r, s) is a diffeomorphism of U on R2. The class of systems

which we consider is such that the wave curves O j (v), j = 1, 2, are connected,allowing the unique solvability of the Riemann problem and, last but not least,deducing the one from the other by the translations (r, s) �→ (r + r0, s+ s0) andthe symmetries (r, s) �→ (s + r0, r + s0).

To ensure the connectedness of the wave curves, we are led to suppose thecharacteristic fields are genuinely non-linear or else linearly degenerate (the caseof a scalar equation is sufficiently instructive). Each wave curve O j (v) contains thusa half straight-line ending in v, parallel to the axis s = 0 if j = 1, r = 0 otherwise.From the uniqueness of the solution of the Riemann problem, we deduce that O1(v)is transversal to the straight lines r = const. In fact, being given a and b in O1(v) withra = rb, these two states would be linked by a 2-rarefaction-wave, for example inthe direction a �→ b (or by a 2-contact-discontinuity-wave). The Riemann problembetween v and b would therefore have two distinct solutions

v1-S�−→ b,

and

v1-S�−→ a

2-R�−→ b,

which contradicts the uniqueness (here S stands for shock, and R for rarefactionwaves).

The curves O1 are thus parametrised by r and the property of translation showsthat there exists a smooth function f : R → R with for example f (R−) = {0} suchthat O1(v) is given by the equation

s − sv = f (r − rv).

By symmetry the 2-wave curves O2(v) are given by the equation

r − rv = f (s − sv).

Again, the uniqueness of the solution of the Riemann problem implies | f (τ )−f (σ )| �= |τ−σ | for τ �= σ . In fact, if f (τ )− f (σ ) = τ−σ , then the Riemann prob-lem between uL and uR (rL = sR = 0, sL = rR = τ − f (τ )) admits two solutions

uL1-S�−→ um(r = s = τ )

2-S�−→ uR,

uL1-S�−→ um(r = s = σ )

2-S�−→ uR.

Similarly, if f (τ ) − f (σ ) = σ − τ , the Riemann problem between uL and uR

(rL = sR = 0, sL = rR = τ − f (σ )) admits two solutions

uL1-S�−→ um(r = τ, s = σ )

2-S�−→ uR,

uL1-S�−→ um(r = σ, s = τ )

2-S�−→ uR.

Since f (R−) = {0} we deduce by continuity that | f (τ ) − f (σ )| < |τ − σ | forτ �= σ . Putting w = 1

2 (r + s) and z = 12 (r − s), we can rewrite the equations of

1-wave curves in the form

w − w− = F(z − z−)

and similarly the 2-wave curves as

w − w− = F(z− − z).

We have F ′ = (1+ f ′)/(1− f ′), with the result that F is strictly increasing. Thesolution of the Riemann problem leads to eliminating the component wm of themedian state um, and to searching for the unique root zm of the equation

F(z − zR)+ F(z − zL) = wR − wL.

The last hypothesis is the inequality

F(z1 + z2) ≥ F(z1)+ F(z2), ∀z1, z2 ≥ 0. (5.17)


Remarking that F(z) = z for z ≤ 0, we have also

F(z1 + z2) ≤ F(z1)+ F(z2), ∀z1, z2 ≤ 0. (5.18)

We are now able to state the result which Nishida [80] has obtained for the isother-mal model of gas dynamics. This example will be the object of §5.6.

Theorem 5.6.1 For a system of two conservation laws such as are described above,the scheme of Glimm is stable in BV(R) for every given initial condition u ∈BV(R)2. The families (uh

a)h>0 of approximate solutions are relatively compact inL1

loc(R+ × R) and their limiting values are for almost all a ∈ A weak entropy

solutions of the Cauchy problem.

Of course, only the stability has to be proved.

A distance in U

Being given two states a, b ∈ U, we define a ‘distance’ d(a, b) := |zm−za|+|zm−zb| where um = (wm, zm) is the median state in the Riemann problem between aand b:

F(zm − za)+ F(zm − zb) = wb − wa. (5.19)

In fact, d is not symmetric. Of the two properties of d of which we shall haveneed, the first is the triangular inequality.

Lemma 5.6.2 If a j ∈ U, j = 1, 2, 3, we have

d(a1, a3) ≤ d(a1, a2)+ d(a2, a3).

The proof of this lemma has remained complex for a long time, necessitating thestudy of fifteen or so cases, rarely presented in an exhaustive way, until done soelegantly by F. Poupaud [84].

Proof Let ai j be the median state between ai and a j . Eliminating w from theequations (5.19), we obtain

F(z13− z1)+F(z13− z3) = F(z12− z1)+F(z12− z2)+F(z23− z2)+F(z23− z3).

We let x = z13 − z1, y = z13 − z3, α = z12 − z1, β = z12 − z2, γ = z23 − z2,


δ = z23 − z3. The following equalities arise:

x − y = α − β + γ − δ, (5.20)

F(x)+ F(y) = F(α)+ F(β)+ F(γ )+ F(δ). (5.21)

If xy ≤ 0, then

d(a1, a3) = |x | + |y| = |x − y| = |α − β + γ − δ|≤ |α| + |β| + |γ | + |δ| = d(a1, a2)+ d(a2, a3).

If x and y are positive, we write that F(α+) + F(α−) = F(|α|) + F(0) ≥ F(α)with α+ = max(α, 0), and α− = max(−α, 0). We deduce from (5.21) that one ofthe two following inequalities is true:

F(x) ≤ F(α+)+ F(β−)+ F(γ+)+ F(δ−)

or

F(y) ≤ F(α−)+ F(β+)+ F(γ−)+ F(δ+).

If the first holds (the two cases are similar) then F(x) ≤ F(α+ + β− + γ+ + δ−)because of (5.17). Since F is strictly increasing, we have x ≤ α++β−+γ++ δ−.Finally,

d(a1, a3) = x + y = 2x − (x − y)

≤ 2(α+ + β− + γ+ + δ−)− α + β − γ + δ

= |α| + |β| + |γ | + |δ| = d(a1, a2)+ d(a2, a3).

The case x ≤ 0, y ≤ 0 is treated in the same way using (5.18).Simpler is the following case of an equality.

Lemma 5.6.3 If a2 is an intermediate value in the solution of the Riemann problembetween a1 and a3, then d(a1, a3) = d(a1, a2)+ d(a2, a3).

In fact, d(a, b) is nothing but the total variation of x/t �→ z in the solution of theRiemann problem between a and b and we can express it by Chasles’ relation.

Finally, d is equivalent to the usual distance of U.

Lemma 5.6.4 Let K be a compact set of R2 and C its image by (w, z) �→ u. There

exists a number cK ≥ 1 such that for all a, b ∈ C, we have

c−1K ‖b − a‖ ≤ d(a, b) ≤ cK‖b − a‖.

Proof First of all, C is a compact set and

0 < m = min F ′(z) ≤ M = max F ′(z) < +∞for the zs of the form z+ − z− with u± ∈ C . In each wave, we thus have

N−1|z+ − z−| ≤ |w+ − w−| ≤ N |z+ − z−|,with N = max(M, m−1). We therefore have

d(a, b) ≥ 1

1+ N1(|zm − za| + |wm − wa| + |zm − zb| + |wm − wb|)

where N1 is the same number but associated with a compact set C1 which containsall the median states of the Riemann problems between the points a and b of C .Thus

d(a, b) ≥ 1

1+ N1(|zb − za| + |wb − wa|) ≥ const.‖b − a‖

since (w, z) �→ u is Lipschitz of constant K in C .Conversely, the mapping (ra, sa, rb, sb) �→ (rm, sm) is of class C

2 with (rm, sm) =(rb, sa) + O(‖b − a‖2) when b tends to a, from Lax’s theorem. We thus havezm = 1

2 (rb − sa)+ O(‖b − a‖2) and

d(a, b) = 1

2(|rb − ra| + |sb − sa|)+ O(‖b − a‖2) ≤ const.‖b − a‖2

on the compact set C .

Stability

We make use of a functional slightly different from that of the general case:

M(n) :=∑

k∈n+1+2Z

d(un,k−1, un,k+1).

By the triangle inequality, we have

M(n + 1) ≤∑

k∈n+2Z

(d(un+1,k−1, un,k)+ d(un,k, un+1,k+1)).

Let us reorder the terms of this sum:

M(n + 1) ≤∑

k∈n+1+2Z

(d(un,k−1, un+1,k)+ d(un+1,k, un,k+1)).

But as un+1,k is an intermediate value in the solution of the Riemann problembetween un,k−1 and un,k+1, the right-hand side takes the value

∑k d(un,k−1, un,k+1)


from Lemma 5.6.3. We therefore have

M(n + 1) ≤ M(n).

By induction, while the approximate solution is defined, we have M(n) ≤ M(0),that is TV(zh

a(·, n�t + 0)) ≤ TV(z). As zn,−∞ = z(−∞), we deduce

|zn,k | ≤ |z(−∞)| + TV(z) := z.

Let us write c = max{|F ′(z)|; |z| ≤ z}. In each j-wave, we have TV(w) ≤ c TV(z)and hence TV(wh

a (·, n�t + 0)) ≤ c TV(z). Thus

|wn,k | ≤ |w(−∞)| + c TV(z),

which shows that uha takes its values in a compact set C which does not depend on h

or on a or on n or on ρ. We can thus choose ρ a priori for the approximate solutionto be defined for all time. In addition, the mapping u �→ (w, z) being bi-Lipschitzon the compact set in question, we have

TV(uh

a(·, n�t + 0)) ≤ cC

(TV

(wh

a

)+ TV(zh

a

))≤ (1+ c)cCTV(z) ≤ cTV(u),

which completes the proof of Nishida’s theorem.

The isothermal model of gas dynamics

An isothermal gas has for equation of state, with the standard notation, p(ρ) = c2ρ

where c is the constant speed of sound. By a change of scale, we can suppose thatc = 1. In lagrangian coordinates, the one-dimensional motion is governed by theequations

vt = qx ,

qt +(

1

v

)x= 0,

(5.22)

v being the specific volume and q the velocity of the gas. The characteristic speedsare λ = ±v−1 while the Riemann invariants are

r = q + log v, s = q − log v.

Here, w = q and z = log v. In a shock the Rankine–Hugoniot condition is written

[q]+ σ [v] = 0,

[1

v

]= σ [q]

and hence v−v+σ 2= 1. In a 1-shock-wave, we have σ = − (v−v+)−12 and therefore

w+ − w− =√

v+v−−

√v−v+

with v+ < v− as the entropy condition. On the other hand, in a 1-rarefaction-wave,s is constant, that is to say that w+−w− = log(v+/v−) and we have v2 > v1. Thusthe 1-wave curve is given by w+ − w− = F(z+ − z−) with

F(z) =

z, z ≥ 0,

2 sinhz

2, z ≤ 0.

In a 2-shock-wave, we have σ = (v−v+)−12 ,

w+ − w− =√

v−v+−

√v+v−

and v+ > v− as the entropy condition. Similarly, in a 2-rarefaction-wave, r isconstant, that is to say w+−w− = log(v−/v+) and we have v+ < v− and thus againw+−w− = F(z−−z+). The wave curves have thus the form demanded and we haveclearly F ′ > 0. In fact f ′ = (F ′ − 1)/(F ′ + 1) ∈ (−1, 1). It remains to verify thatF(a+b) ≥ F(a)+ F(b) for a, b ≥ 0 which is trivial, and F(a+b) ≤ F(a)+ F(b)for a, b ≤ 0, which is classic:

F(a + b) = 2

(sinh

a

2cosh

b

2+ sinh

b

2cosh

a

2

)

≤ 2

(sinh

a

2+ sinh

b

2

)= F(a)+ F(b),

where we have used sinh ≤ 0, cosh ≥ 1.As an application of Theorem 5.6.1, we therefore have

Theorem 5.6.5 (Nishida) Let v ∈ BV(R) and q ∈ BV(R) be such that infx v(x) >

0. Then the Cauchy problem for the system (5.22) possesses a weak entropy solutionwhich satisfies v(x, t) ≥ v∗ where v∗ is an explicitly calculable constant.

In fact, v∗ = exp(−z) with the notation of the preceding section. More precisely,

v∗ = v(−∞) exp(−TV(log v)).

Of course, this value is not, in general, an optimal bound.

5.7 2× 2 Systems with diminishing total variation

Description

We shall consider strictly hyperbolic systems, whose wave curves have for equationsr = const. (respectively s = const.), r and s being the Riemann invariants. Thesesystems were studied for the first time by B. Temple [103]. This property is satisfiedat least by the wave curves which correspond to a linearly degenerate field. As theappropriate Riemann invariant is likewise constant across a rarefaction wave, wesee that the hypothesis concerns only shock waves.

The solution of the Riemann problem is particularly simple in this case. Let usconsider a characteristic quadrilateral K ⊂ U defined by

K = {u ∈ U: r− ≤ r (u) ≤ r+, s− ≤ s(u) ≤ s+}and which is complete, that is to say that

K → [r−, r+]× [s−, s+], u �→ (r (u), s(u)),

is a diffeomorphism. Then for uL and uR in K , the Riemann problem has a uniquesolution with values in K , for which the median state um is determined by rm =rR, sm = sL.

In particular, the remark of Hoff (see §5.2) shows that if the Cauchy datum u hasvalues in K , there exists a value ρ of the ratio ρ = �t/�x such that uh

a is definedfor all time and has values in K .

Stability

To study the stability of Glimm’s scheme, we consider the functionals

V1(t) = TV(r ◦ uh

a(t); R),

V2(t) = TV(s ◦ uh

a(t); R).

If n�t < s, t < (n + 1)�t , uha(s) and uh

a(t) differ only by a diffeomorphism of R,

uha(s, x) = uh

a(t, ψs,t (x))

with the result that Vj (s) = Vj (t). In addition, sampling is an operation whichdiminishes the total variation:

Vj (n�t) ≤ Vj (n�t − 0).

After these two remarks which do not make use of the particular structure of thesystem we calculate Vj (n�t + 0). For j = 1, this is the sum of the variations of

5.7 2× 2 Systems with diminishing total variation 175

Fig. 5.3: The Riemann problem for a system ‘of B. Temple’.

r ◦ uha in the Riemann problems solved at the instant n�t . As r is constant in the

2-waves, V1(n�t + 0) is the sum of the variations of r in the 1-waves. Finally ris monotonic in the 1-waves (this is true for very general systems), for example insupposing that a 1-wave is either a shock wave, or a rarefaction wave, or a contactdiscontinuity, as r is monotonic in a rarefaction, Finally, the variation of r in aRiemann problem is |rR − rL| and we deduce that V1(n�t + 0) = V1(n�t) andsimilarly for V2. Thus t �→ Vj (t) is decreasing:

Vj (t) ≤ Vj (0) ∈ {TV(r ◦ u), TV(s ◦ u)}.

Finally, C being a Lipschitz constant on K of the diffeomorphism u �→ (r, s) andits inverse, we have the estimate

TV(uh

a(t)) ≤ CTV

((r, s) ◦ uh

a(t))

≤ CTV((r, s) ◦ u) ≤ C2TV(u),

which shows the stability of Glimm’s scheme in BV(R). We have seen that thisentails the convergence. Let us state the result.

Theorem 5.7.1 (Leveque and Temple, Serre) We suppose that the integral curvesof the eigenvector fields of d f are the wave curves of the system (5.1). Let K be acomplete characteristic quadrilateral in U.

For all u ∈ BV(R)2 with values in K , the Cauchy problem has a weak entropysolution with values in K and which satisfies

TV(r ◦ u(t)) ≤ TV(r ◦ u),

TV(s ◦ u(t)) ≤ TV(s ◦ u).

Example 5.7.2 The following system has been considered by numerous authors,for example [52]:

ut + (ϕ(u)u)x = 0, (5.23)

where u = (u1, u2) and ϕ ∈ C2(U = R

+×R; R). We write r = ‖u‖ (the Riemanninvariants will be denoted differently) and we suppose that ϕr > 0 (but ϕr < 0 couldalso arise) with r∂r := u1∂1+u2∂2. Finally, we suppose that limr→+∞ ϕ(u) = +∞.Thus, (θ, ϕ): U → [− 1

2π, 12π ]× (ϕ(0),+∞) realises a diffeomorphism.

The system (5.23) is strictly hyperbolic, with characteristic speeds λ1=ϕ, λ2 =ϕ + rϕr . The first characteristic field is linearly degenerate while the second isgenuinely non-linear when (rϕ)rr �= 0.

The Riemann invariants of this system are the angle θ = arctan(u2/u1) and thefunction ϕ. We have seen that only the shock waves have to be considered. Theseare relative to the second characteristic field and satisfy the Rankine–Hugoniotcondition

[(ϕ(u)− σ )u] = 0,

from which we deduce either uL ∧ uR = 0, or ϕL = ϕR = σ . But this lattercase is that of a contact discontinuity. The shocks therefore satisfy uL ∧ uR = 0,that is to say θL = θR. The system thus satisfies the hypotheses of the precedingsection. If the given initial condition is of bounded variation and with values in U

∗,

Fig. 5.4: Wave curves and invariant domain for the system (5.23).

5.8 Technical lemmas 177

if besides infx r (x) > 0, there exists a complete characteristic quadrilateral Kdefined by

|θ | ≤ 1

2π, inf

xϕ(u) ≤ ϕ ≤ sup

xϕ(u),

which contains the values of u (cf. Fig. 5.4). The compact set K is invariant forthe Riemann problem and thus for Glimm’s scheme. By Theorem 5.7.1, this oneconverges. The Cauchy problem therefore has a weak entropy solution with valuesin K for almost all (t, x) ∈ R

+ × R.In infx r (x) = 0, the situation is more delicate as the hypothesis does not ensure

that θ is of bounded variation, u �→ θ not being Lipschitz. On the other hand, as(θ, ϕ) �→ u is Lipschitz, it is sufficient to consider a given initial function u forwhich θ and ϕ are of bounded variation to obtain the convergence of Glimm’sscheme and the existence of a solution of the Cauchy problem.

5.8 Technical lemmas

Proof of Lemma 5.5.2

We begin by stating an intermediary result, the inspiration for which comes from acourse at Rennes given by G. Metivier.

Lemma 5.8.1 Let I be a part of {1, . . . , m}×{1, . . . , n} and f ∈ C2

b ([0,+∞)m+n).If f is identically zero when �I (x, y) := ∑

(i, j)∈I xi y j is identically zero, then wehave the inequality

| f (x ; y)| ≤ Cm,n�I (x, y) supa,b,i, j

∣∣∣∣ ∂2 f

∂xi∂y j(a; b)

∣∣∣∣, ∀x, y > 0.

The constant Cm,n depends only on m and n.

Proof By linearity, we can suppose that

supa,b,i, j

∣∣∣∣ ∂2 f

∂xi∂y j(a; b)

∣∣∣∣ ≤ 1.

We proceed by induction on (m, n). If m = 0 or n = 0, I is empty, hence f ≡ 0,and the result is obvious. Similarly if m, n ≥ 1 when I is empty. We supposetherefore that m ≥ 1, n ≥ 1, that I is not empty and that the lemma is true for thepairs (m − 1, n) and (m, n − 1). We can suppose that (1, 1) ∈ I .


We apply the lemma (as an induction hypothesis) to the functions

(x ′, y)f0�→ f (0, x ′; y),

(x, y′)f1�→ f (x ; 0; y′),

(R+)m+n−1 → R.

For example, f0 is identically zero on the zeros of �J where J = I ∩({2, . . . , m}×{1, . . . , n}). We thus have

| f0| ≤ Cm−1,n�J ≤ Cm−1,n�I .

Now let us fix a point (x ′, y′) of [0,+∞)m+n−2 and let us define

g(x1, y1) := f (x1, x ′; y1; y′)− f (0, x ′; y1, y′)− f (x1, x ′; 0, y′)+ f (0, x ′; 0, y′).

We have g(0, y) = g(x, 0) = 0 and so

g(x1, y1) =∫ x1

0

∫ y1

0

∂2g

∂a∂b(a, b) da db,

from where we derive |g(x1, y1)| ≤ x1y1. Finally,

| f (x, y)| ≤ x1y1 + (Cm−1,n + Cm−1,n−1 + Cm,n−1)�I (x, y)

≤ Cm,n�I (x, y).

The constant Cm,n is calculated by the recurrence relation

Cm,n : = 1+ Cm−1,n + Cm−1,n−1 + Cm,n−1.

We now apply Lemma 5.8.1 to the function (δ, ε) �→ f (δ, ε) := γ (δ, ε, um)−δ−ε

considering separately the 22n sectors of the form J1 × · · · × J2n with J j = R±.

Clearly, f must be defined over the whole of R2n by truncation and extension by

zero, or else quite simply remark that Lemma 5.8.1 remains true when we replace[0,∞] by [0, A] where A > 0 is an arbitrary finite number. Noting that �(δ, ε, um)is of the form �I (δ, ε) in each of the sectors, the inequality |γ (δ, ε, um) − δ − ε|≤ c0�(δ, ε, um) is thus established, this with the constant c0 depending on thebase point um. But from Lemma 5.8.1 it depends only on the upper bound, on aneighbourhood of (0, 0) of the second derivatives of γ (·, ·, um). This bound is itselfbounded above by a constant when um remains in a neighbourhood of u and thiscompletes the proof of Lemma 5.5.2.


Let us fix a compact set K = [0, T ]×[−L , L] of R+t ×Rx . The sequence (am(t))m∈N

of the restrictions of vm(t) to (−L , L) is relatively compact in L1(−L , L) for all

5.8 Technical lemmas 179

t ∈ [0, T ] by Helly’s theorem. Let Q be a dense subset of [0, T ] (for example therational numbers). We can extract, making use of the diagonal procedure, a sequence(am(k)(t))k∈N such that m(k) →+∞ and (am(k)(t))k∈N converges in L1(−L , L) forall t belonging to Q. We denote this limit by a(t).

Passing to the limit in the inequality∫ L

−L|am(k)(t, x)− am(k)(s, x)| dx ≤ εm(k) + M |t − s|, ∀t, s ∈ Q,

it becomes ∫ L

−L|a(t, x)− a(s, x)| dx ≤ M |t − s|, ∀t, s ∈ Q.

This shows that a is the restriction to Q of a Lipschitz function defined on [0, T ]with values in L1(−L , L). We again denote this function by a(t), which is clearlyunique.

Being given ε > 0 and t ∈ [0, T ], there exists s ∈ Q such that 2M |t − s| < ε.Then there exists l ∈ N such that εm(k) < ε and

∫ L−L |am(k)(s)− a(s)| dx < ε for all

k > l. For these indices we therefore have∫ L

−L|am(k)(t)− a(t)| dx ≤

∫ L

−L|am(k)(t)− um(k)(s)| dx

+∫ L

−L|am(k)(s)− a(s)| dx +

∫ L

−L|a(t)− a(s)| dx

≤ 2M |t − s| + εm(k) +∫ L

−L|am(k)(s)− a(s)| dx

≤ 3 ε,

that is to say that

limk→+∞

am(k)(t) = a(t), ∀t ∈ [0, T ],

for the norm of L1((−L , L)).Now let us define Hk(t) = ∫ L

−L |am(k)(t)− a(t)| dx . We have

Hk(t) ≤ 2M L + supt∈[0,T ]

∫ L

−L|a| dx ≤ 4M L, ∀t ∈ [0, T ].

We have seen also that (Hk(t))k∈N tends to zero for all t ∈ [0, T ]. The theorem ofdominated convergence thus ensures that

limk→+∞

∫ T

0Hk(t) dt = 0,


that is to say that (am(k)(t))k∈N tends to a in L1(K ). The sequence (am)m∈N is thusrelatively compact in L1(K ).

Finally, as R+ ×R is the denumerable union of such blocks, the diagonal proce-

dure allows us to find a sequence again denoted by (m(k))k∈N such that (am(k))k∈N

converges in L1(ω) for every bounded open set ω of R+ × R.

Remark The proof of Theorem 5.1.3 is simpler when εm = 0, for all m. This isthen a consequence of the theorems of Helly and of Ascoli and Arzela.

5.9 Supplementary remarks

Other numerical schemes

Several other numerical schemes are very close in conception to that of Glimm,especially that of Lax and Friedrichs. In the latter, only the sampling disappears, tobe replaced by an averaging:

un,k := 1

2h

∫ (k+1)h

(k−1)huh

a(n�t − 0, y) dy.

The scheme of Godunov differs from that of Lax and Friedrichs only by the positionof the meshes. In place of being in alternate rows, they are aligned according tothe rectangular network Z× 2Z. These three schemes are particular cases of thosedefined and studied by H. Gilquin [29], where the definition of un depends on aprobability measure dνn:

un,k :=∫ 1

−1uh

a(n�t − 0, (k + s)�x)dνn(s).

For all these schemes, which are monotonic (that is to say preserve the order)in the scalar case, convergence takes place provided that the sequence (dνn)n∈N isequi-distributed in (−1, 1) and that the approximation is stable in BV (exercise).Unfortunately, we do not in general know how to prove this stability, except forthe scalar conservation laws and their natural generalisations, the Temple systems(see Chapter 13). In the case of certain systems of two conservation laws, called2 × 2 systems, the method of compensated compactness has enabled us to obtaintheorems of convergence to weak solutions which we do not know to be of boundedvariation [10, 14].

The rich case

The systems of conservation laws called rich (or semi-hamiltonian according to theterminology of Tsarev [106, 107]) will be studied in greater detail in Chapter 12. The

5.9 Supplementary remarks 181

system (5.1) is called rich if it is strictly hyperbolic and if there exists a completesystem of Riemann invariants, that is to say a system of curvilinear coordinatesw1(u), . . . , wn(u) satisfying

dw j (u)(d f (u)− λ j (u)) ≡ 0, j = 1, . . . , n.

The 2 × 2 systems are rich as long as they are strictly hyperbolic. In addition, westill suppose that each characteristic field is genuinely non-linear or else linearlydegenerate in order to be able to use Lax’s theorem.

Lemma 5.9.1 If the system (5.1) is rich in ω, then there exists a constant c0 suchthat for all uL, uR, and um in ω1, we have

‖γ − ε − δ‖ ≤ c0(‖ε‖ + ‖δ‖)�(ε, δ),

with the notation um = ϕ(ε; uL) and uR = ϕ(δ; um) = ϕ(γ ; uL).

In making use of this estimate, Glimm [32] improved the stability result in weak-ening the condition of smallness on the given Cauchy condition: for every numberV0 > 0, there exists a number α such that if TV(u) < V0 and ‖u − u‖ ≤ α, thenGlimm’s scheme is stable in BV(R) (hence the Cauchy problem admits a weakentropy solution).

Proof We use the expansion (4.15) for three Riemann problems.We obtain

um − uL =∑

j

ε j rLj +

1

2

∑j

ε2j (dr j · r j )L +

∑j<k

ε jεk(drk · r j )L + O(‖ε‖3),

uR − um =∑

j

δ j rmj +

1

2

∑j

δ2j (dr j · r j )m +

∑j<k

δ jδk(drk · r j )m + O(‖δ‖3)

=∑

j

δ j rLj +

1

2

∑j

δ2j (dr j · r j )L +

∑j<k

δ jδk(drk · r j )L

+∑j,k

δkε j (drk · r j )L + O(‖δ‖3 + ‖ε‖3),

uR − uL =∑

j

γ j rLj +

1

2

∑j

γ 2j (dr j · r j )L +

∑j<k

γ jγk(drk · r j )L + O(‖γ ‖3).

We know that γ = ε + δ + O(‖ε‖2 + ‖δ‖2). Combining the above equalities andmaking use of the dual basis (l p(uL))1≤p≤n to that of the eigenvectors r j , we havetherefore

γp − εp − δp = l p ·∑j<k

δkε j (drk · r j − dr j · rk)L + O(‖ε‖3 + ‖δ‖3).

But the existence of the pth Riemann invariant wp is equivalent to the geometriccondition of Frobenius:

l p · (drk · r j − dr j · rk) = 0, ∀l ≤ j, k ≤ n.

Thus γp − εp − δp = O(‖ε‖3 + ‖δ‖3). Finally, the fact that the right-hand side iszero when �(ε, δ) = 0 leads by an analogous calculation to that in the general caseto the upper bound

|γp − εp − δp| ≤ const.(‖ε‖ + ‖δ‖)�(ε, δ).

‘Continuous’ Glimm functional

M. Schatzman [89, 90] has adapted the calculation of Glimm for an exact solutionof a system of conservation laws, when it is of class C

1 outside a denumerablefamily of shocks which statisfy Lax’s condition. Without going into details, let ussay that in the absence of shocks, the linear term is replaced by

V (u(t)) =∑

i

∫R

|li (u) · ux |dx,

the li being the linear eigenforms of d f . The quadratic form becomes then

Q(u(t)) =∫ ∫

x<yD(Z (x), Z (y)) dxdy, Z := (l1(u) · ux , . . . , ln(u) · ux ).

In the general case, these functionals contain supplementary terms to take intoaccount shock waves and can even be defined for a general spatial curve σ ; wewrite then V (u; σ ) and Q(u; σ ). For a given initial condition of small total variation,there exists a number K > 0 such that σ �→ (V + KQ)(u; σ ) is decreasing, whichfurnishes an a priori estimate of u(t) in BV(R), uniform in time.

5.10 Exercises

5.1 (1) Let θ be a real number and a = nθ − E(nθ ) the fractional part of nθ .Show that the sequence (an)n∈N is equi-distributed in (0, 1) if and only if θ

is irrational. When θ ∈ R\Qwe shall apply the criterion (5.7) to functionsjudiciously chosen, then we shall proceed by a density argument.

(2) Find a real number θ > 1 for which the sequence an = θn − E(θn) is notequi-distributed in (0, 1).

5.10 Exercises 183

(3) For n ∈ N, let an = 10−k2n , where k = E(log10 2n). It is thus an elementof the interval ( 1

10 , 1). Show that (an)n∈N is not equi-distributed in ( 110 , 1).

5.2 (1) Show that if an entropy E (not necessarily convex) of flux F satisfiesEt + Fx ≤ 0, for all simple admissible waves, then the weak solutionsobtained by Glimm’s scheme also satisfy this inequality.

(2) For the system (5.23), show that these solutions satisfy

(rg(θ))t + (rϕ(u)g(θ ))x = 0

for all g ∈ C [− 12π, 1

2π ]. Verify that these equations contain that of thesystem (5.23). Finally show the inequality

rt + (rϕ(u))x ≤ 0.

5.3 The dynamics of an isothermal gas in one dimension and in eulerian variablesand governed by the system

ρt + (ρu)x = 0,

(ρu)t + (ρu2 + c2ρ)x = 0,

}(5.24)

where U = R+∗ × R and c > 0 is the (constant) speed of sound. Show that

this system belongs to the class described in §5.6. The Cauchy problem thushas a weak solution.

5.4 We consider the so-called ‘Leroux’ system

ρt + (ρu)x = 0,

ut + (u2 + ρ)x = 0,

}

with U = R+ × R.

(1) Show that it is strictly hyperbolic in U∗. Show that the Riemann invariants

are the slopes of the straight lines which pass through (ρ, u) and whichare tangents to the parabola � of equation u2 + 4ρ = 0.

(2) Show that these straight lines are the wave curves of the system and thatU is an invariant domain for the Riemann problem.

(3) Deduce that if the given initial condition (ρ, u) is of bounded variationand if infx (ρ + u) > 0, then the Cauchy problem has a weak solution.

(4) Let � be one of the tangents of �, with equation αρ + βu = γ . Showthat the weak solution which we have constructed satisfies

((αρ + βu − γ )+)t +((

u + β

α

)(αρ + βu − γ )+

)x≤ 0.

5.5 We consider a strictly hyperbolic 2× 2 system of ‘Temple’ type. Let K ⊂ U

be a complete characteristic quadrilateral. We denote the Riemann invariantsby r and s.

(1) Show that if K is ‘small enough’, then

supu∈K

λ1(u) < infu∈K

λ2(u).

(2) We suppose from now on that

supu∈K

λ1(u) := c1 < c2 := infu∈K

λ2(u).

We consider a given initial datum u, such that

u(x) ={

uL, ∀x ≤ α,

uR, ∀x ≥ β.

Show that s(un, j ) = s(uL) for all j ≤ Jn , where

Jn = 2E

(α

2h

)− n + 2 card{m: 1 ≤ m ≤ n, am < ρc2}.

(3) Deduce that the weak solutions of the Cauchy problem, obtained byGlimm’s scheme, satisfy

(x < α + c2t) ⇒ (s(u(x, t)) = s(uL))

and similarly

(x > β + c1t) ⇒ (r (u(x, t)) = r (uR)).

(4) Show that there exists a time T of uncoupling, beyond which the evolu-tion proceeds according to independent scalar conservation laws. Moreprecisely, there exists a point X such that we have three zones for t > T .A zone is defined by c1(t − T ) < x − X < c2(t − T ) where u(t, x),constant, is determined by r (u) = r (uR) and s(u) = s(uL). A zone isdefined by x < X + c1(t − T ) where u(t, x) = ϕ1(ε(t, x); uL) and ε

is a solution of a scalar conservation law. Finally a zone is defined byx > X + c2(t − T ) where u(t, x) = ϕ2(δ(t, x); uR) and δ is the solutionof another scalar conservation law.

5.6 We consider a strictly hyperbolic 2× 2 system whose characteristic fields aregenuinely non-linear. To fix our ideas, the speeds, expressed as functions ofthe Riemann invariants, satisfy

∂λ1

∂r> 0,

∂λ2

∂s> 0.

5.10 Exercises 185

We suppose that the Riemann problem has a unique solution. We are given abounded initial condition u such that r ◦ u and s ◦ u are increasing and havevalues in a characteristic quadrilateral K ⊂ U.

(1) Show that u ∈ BV(R)2.

(2) Let uL and uR be the data of a Riemann problem. Show that if r (uL) ≤r (uR) and s(uL) ≤ s(uR) the waves of the Riemann problem are rarefac-tion waves. We might begin by studying the case of equality.

(3) Show by induction on n that the sequences (r (un, j )) j∈n+2Z and(s(un, j )) j∈n+2Z are increasing and that the Riemann problems which aresolved by putting into effect Glimm’s scheme only make use of rarefactionwaves.

(4) Deduce that uha is defined for all time t ≥ 0 and that

TV(r ◦ uh

a(t)) = TV(r ◦ u),

TV(s ◦ uh

a(t)) = TV(s ◦ u).

Conclude that the Cauchy problem possesses a weak entropy solution onR+ × R.

6

Second order perturbations

We are interested in this chapter in perturbations of hyperbolic systems of conser-vation laws

∂t u +d∑

α=1

∂α f α(u) = 0 (6.1)

by diffusion terms of second order. To adhere closely to physical examples, werestrict ourselves to perturbations of conservative form:

∂t u +d∑

α=1

∂α f α(u) = ε

d∑α,β=1

∂α(Bαβ(u)∂βu). (6.2)

In these models, Bαβ(u), for 1 ≤ α, β ≤ d, denotes matrices whose coefficientsare smooth functions defined on the state space U. The coefficient ε > 0 is ultimatelyto tend to zero. It may come from a change of scale (x, t) �→ (ηx, ηt) in which theprocess of diffusion seems secondary with respect to the transport phenomena. Forexample, for a gas, we generally admit that the thermal conduction and, especially,the viscosity have rather weak effects.

For a given initial state u0(x), the solution of the Cauchy problem for (6.2), whenit is well-posed, will be denoted by uε(t, x). The natural question is to know ifuε converges, in a sense to be made precise, when ε tends to zero, to an entropysolution u of the Cauchy problem for (6.1). We are also interested in the behaviourof uε in the neighbourhood of a shock wave of u. Let us say immediately that theconvergence is proved only in very few cases.

To be complete, we should hope equally to know that uε converges when thegiven initial state uε

0 also depends on ε and converges to a u0. This will lead us toconsider particular solutions in the form of a progressive wave U ((ν · x − ct)/ε).Their analysis, relatively simpler than that of the Cauchy problem, allows anotherapproach to the stability of shocks.

186

6.1 Dissipation by viscosity 187

6.1 Dissipation by viscosity

Not every perturbation of the form (6.2) leads to a well-posed Cauchy problem.A natural requirement is that it is linearly well-posed in the neighbourhood of aconstant state. By the change of variables (x, t) �→ (εx, εt), we are led to the caseε = 1. The linearised system

∂tv +d∑

α=1

Aα(u)∂αv =d∑

α,β=1

Bαβ(u)∂α∂βv (6.3)

possesses particular solutions of the form v(x, t) = exp(ωt + iξ · x)V , V ∈ Rn , if

and only if V is an eigenvector of the matrix

M(ξ ) :=d∑

α,β=1

ξαξβ Bαβ(u)+ id∑

α=1

ξα Aα(u)

associated with the eigenvalue −ω. The dispersion relation, which links ξ and ω,is thus

det(ωIn + M(ξ )) = 0. (6.4)

A necessary condition for the Cauchy problem to be well-posed for (6.3) is thatthe real part of the solutions ω of the equation (6.4) retains an upper bound whenξ ranges over R

d . In particular, making ‖ξ‖ tend to +∞, we see that the eigen-values of the matrices B(u; ξ ) := ∑

α,β ξαξβ B(u) must all have their real partsnon-negative, for ξ ∈ Sd−1. Quite evidently, none of these conditions is sufficient.Not only are they not sufficient for the Cauchy problem for (6.3) to be well-posedfor each ε > 0 (after all, we have not excluded that the matrices Bαβ are singu-lar, even zero), but they ensure still less the convergence of uε when ε tends tozero.

Non-dissipative case

To understand why the convergence of uε demands stronger hypotheses than theexistence, let us look at the case of a physical system, where (6.1) is compatiblewith a strongly convex entropy E (D2

u E > 0), of flux F . Let us suppose that thetensor B(u) satisfies the following condition:

∑α,β,i, j,k

∂2E

∂ui∂ukBαβ

k j (u)mα jmβi = 0, ∀m ∈ Md×n(R). (6.5)

188 Second order perturbations

This condition excludes neither the Cauchy problem being well-posed nor that itproduces a smoothing effect. For example, the linear system

∂t

(v

w

)= ε

(0 1

−1 0

)∂2

x

(v

w

)

(take E = v2 +w2 to satisfy (6.5)) leads to two uncoupled Schrodinger equationsvia the change of unknowns (v, w) �→ (v + iw, v − iw), an equation whose semi-group is smoothing for initial data with rapid decay.

For such systems, we have the identity

∂t E(uε)+ div F(uε) = ε∑α,β

∂α((du E · Bαβ)(uε)∂βuε), (6.6)

valid at least for smooth solutions, let us say of class C2. For these, when they

decay quickly enough at infinity, we deduce from (6.6) the conservation of energy(or of entropy, according to the context)∫

Rd

E(uε(t, x)) dx =∫

Rd

E(u0(x)) dx, ∀t > 0.

This shows that the sequence (uε)ε>0 is bounded in a certain Lebesgue–Orlicz spaceassociated with E . If in addition uε and f (uε) converge simultaneously to u andf (u) in the sense of distributions,1 then this convergence will occur in general forthe strong topology of a Lebesgue space because of the non-linearity of f (seeExercise 6.1). Free to extract a sub-sequence, pointwise convergence will occuralmost everywhere. Finally, we can think that (uε)ε remains localized (the speed ofpropagation is finite when ε is zero), at least enough to be able to apply the theoremof dominated convergence. We then obtain∫

Rd

E(u(t, x)) dx =∫

Rd

E(u0(x)) dx, ∀t > 0.

The conservation of energy when ε= 0 contradicts the development of shock wavesfor which we hope that the non-positive measure ∂t E(u)+ div F(u) is not identicallyzero.

Dissipation or production of entropy

Another way to see that the condition (6.5) is incompatible with the convergence ofuε to an entropy solution of (6.1) when that contains a shock wave is an asymptoticanalysis in a neighbourhood of a point of discontinuity (t0, x0) of this wave. If the

1 with the result that u is a weak solution of (6.1).


wave front is tangent to the hyperplane with equation y := ν · (x−x0)−c(t−t0) = 0,a change of scale suggests that uε has an asymptotic expansion of the form

uε(t, x) = U( y

ε, t, x

)+ O(ε).

The function z �→ U (z, t, x) is called a profile of the shock wave. With values inU, it is smooth and tends when z→±∞ to the states u±(t, x) situated on one sideor other of the shock. We shall see in Chapter 7 that U is the heteroclinic orbit ofa very simple vector field. On the right-hand side of (6.6), in the form ∂αeα, theterms eα obey the following asymptotic expansion:

eα =∑β

(du E · Bαβ)(U )U ′ + O(ε),

where the first term tends to zero in L ploc for all finite p since it is bounded and

localized in a zone of size ε round about the shock wave. Therefore eα ⇀ 0in D

′(Rd+1) and a passage to the limit in (6.6) must give the entropy equality∂t E(u)+ div F(u) = 0 instead of the entropy inequality expected.

We remark that the situation remains the same when we replace the condition(6.5) by the weaker hypothesis

∑α,β,i, j,k

ξαξβ

∂2E

∂ui∂ukBαβ

k j (u)ηiη j = 0, ∀ξ ∈ Rd , ∀η ∈ R

n.

The calculation is the same but the right-hand side of the balance of the entropy isnow written ∑

α

∂αeα + ε∑

α,β,i, j

Qαβ

i j (uε)(∂αuε

i ∂βuεj − ∂αuε

j∂βuεi

).

We must verify that the second sum tends to zero in the sense of distributionsby making use of the asymptotic expansion of uε. Now the coefficient of ε−2 in∂αuε

i ∂βuεj−∂αuε

j∂βuεi is identically zero (this is due to the structure of codimension

1 of the shock waves), with the result that the terms of this sum are of the formL(U )U ′ +O(ε) where L(U ) is bounded. This term thus tends to zero in D

′(Rd+1).This discussion shows that it is essential that the perturbation is strictly dissipative

for the entropy of the system. However, if we demand that it satisfies the Legendre–Hadamard condition∑

α,β,i, j,k

ξαξβ

∂2E

∂ui∂ukBαβ

k j (u)ηiη j ≥ c(u)‖ξ‖2‖η‖2, ∀ξ ∈ Rd ,∀η ∈ R

n, (6.7)

we shall miss most of the perturbations of physical origin. In fact, each time thatthe system (6.1) contains a conservation law such as that of mass, ρt + div(ρv)= 0,


this will not be perturbed, that is to say that the matrices B(u; ξ ) := ∑α,β ξαξβ

× Bαβ(u) will be singular, having a line of zeros. The left-hand side of (6.7) willthus be zero for η ∈ ker B(u; ξ ) and also for D2

u E · η ∈ ker B(u; ξ )T. Since thequadratic form η �→ (B(u; ξ )η |D2

u E · η) must be positive semi-definite, a naturalhypothesis is that there exists a continuous function c(u) > 0 such that(

B(u; ξ )η |D2u Eη

) ≥ c(u)‖B(u; ξ )η‖2, ∀ξ ∈ Sd−1, ∀η ∈ Rn. (6.8)

We say then that the tensor B is dissipative with respect to the entropy E .Now let us see a formal consequence of (6.8). The entropy balance for the

perturbed problem is now

∂t E(uε)+div F(uε)+ε∑

α,β,i, j,k

∂2E

∂ui∂uk(uε)Bαβ

k j (uε)∂αuεi ∂βuε

j =∑

α

∂αeα. (6.9)

If d = 1 (the multi-dimensional case is not as clear but the reader will treat it withoutdifficulty where E = ‖u‖2 when B is constant), we deduce

∂t E(uε)+ ∂x F(uε)+ εc(uε)‖B(uε)∂xuε‖2 ≤ ∂xeε.

Supposing that the solution is sufficiently smooth (so that the above inequalityis correct) and that it decays rapidly at infinity, the integration with respect to xyields

d

dt

∫R

E(uε) dx + ε

∫R

c(uε)‖B(uε)∂xuε‖2 dx ≤ 0.

Finally

∫R

E(uε(T, x)) dx + ε

∫ T

0dt

∫R

c(uε)‖B(uε)∂xuε‖2 dx ≤∫

R

E(u0(x)) dx,

which furnishes a priori estimates. If now the sequence (uε)ε is bounded inL∞(ω) and converges almost everywhere to u(t, x) in an open set ω of R

d+1, thenuε and f (uε) converge to u and f (u) in L1

loc(ω), hence in D′(ω), and uε

t + f (uε)x

tends to ut + f (u)x in the sense of distributions. In addition, the hypothesisabove shows that ε1/2∂x (ε1/2 B(uε)∂xuε) tends to zero in the sense of distribu-tions. Passing to the limit in (6.2), we obtain ut + f (u)x = 0. Passing also to thelimit in

E(uε)t + F(uε)x ≤ ε(dE(uε) · B(uε)uε

x

)x ,

we obtain this time the desired entropy inequality E(u)t + F(u)x ≤ 0.


Let us note again that the asymptotic analysis of a shock wave no longer contra-dicts the production of entropy. In fact, the dominant term of

∑α

∂αeα − ε∑

α,β,i, j,k

∂2E

∂ui∂uk(uε)Bαβ

k j (uε)∂αuεi ∂βuε

j

is −ε−1(B(U ; ν)U ′ |D2u E(U )U ′) which is of the form −ε−1V (ε−1y) where V is

positive, zero at infinity. This term has a mass independent of ε, strictly positive.The set of the terms considered therefore tends to a negative singular measurecarried by the front of the shock wave, which is nothing but the measure of entropydissipation (the opposite of the measure of entropy production).

Example 6.1.1 Let us illustrate the criterion (6.8) by the Navier–Stokes equationsof the dynamics of a compressible, viscous, heat-conducting fluid. The case d = 1is as usual the easiest to treat since we can use lagrangian coordinates. Denoting byτ , v, p(τ, e), T (τ, e), e the specific volume, the velocity, the pressure, the temper-ature, the specific internal energy, we express the relative viscosity and the thermalconduction as functions of τ and of e. The Navier–Stokes equations are written as

τt − vx = 0,

vt + px = ε(bvx )x ,(e + 1

2v2

)t+ (pv)x = ε(bvvx + kTx )x .

The coefficients b and k are positive. There are two points of view, according aswe neglect or not the effects due to the viscosity compared with those due to thethermal transfers (which is realistic in the case of a gas). In one case, we shall haveb ≡ 0 and k > 0, in the other b, k > 0. The diffusion tensor takes the value

B =

0

b dv

bv dv + k dT

.

Its kernel is the plane dT = 0 in the case without viscosity, the straight line dv =dT = 0 in the viscous case.

The mathematical entropy is the opposite E = −S(τ, e) of the physical en-tropy. This satisfies the relation T dS= de+ p dτ . We have T > 0. For the smoothsolutions of the Navier–Stokes equations, we find

St = ε

(kTx

T

)x+ ε

b

Tv2

x + εkT 2

x

T 2,

that is to say that

(Bη |D2Eη) = b

T(dv · η)2 + k

T 2(dT · η)2.

In both cases, the condition (6.8) is satisfied, with c = 1/(kT 2) in the inviscid caseand a more complicated expression in the viscous case.

Partially hyperbolic systems

The Navier–Stokes equations give the occasion to remark that the diffusion effectmay or may not have a smoothing effect.

In the inviscid case, with only thermal conduction, Ling Hsiao and Dafermos[16] have shown that shock waves can develop from very smooth initial data.For such solutions, τ , v and Tx are discontinuous while T is continuous (evenif its initial value T0 is not). The discontinuities satisfy the Rankine–Hugoniotrelations:

[v]+ s[τ ] = 0, [p] = s[v],

[T ] = 0, [pv] = s

[e + 1

2v2

]+ ε[kTx ].

The shock waves are therefore similar to those of an isothermal gas, which explainsthe importance given in the literature to this model.

We note however that to a sufficiently small and smooth initial condition therecan correspond a global smooth solution as has been shown by Slemrod [95]. On theother hand, in the viscous case, the diffusion is powerful enough for the solution ofthe Cauchy problem to be smooth for all time if the given initial condition is (see forexample [51]). In fact a discontinuity should have to satisfy the Rankine–Hugoniotconditions

[v]+ s[τ ] = 0, [v] = 0, [T ] = 0.

We should have s = 0. We show easily (see [43]) that d[τ ]/dt = O([τ ]) with theresult that no discontinuity can appear if it did not exist before the initial instant (andsimilarly no discontinuity can disappear). In fact, as the tensor B is not invertible,the Navier–Stokes system is not parabolic and its semi-group is not smoothing. Fora measurable bounded given initial condition, the velocity v and the temperatureT are a little smoothed in the sense that vx , Tx ∈ L2

loc(R+ × R), but the smooth-

ness of the specific volume τ is simply propagated. For example, if 0≤ s < 1 and1≤ p <∞,

τ ( · , t1) ∈ W s,ploc ⇐⇒ τ ( · , t2) ∈ W s,p

loc , ∀t1, t2 ≥ 0.

6.2 Global existence in the strictly dissipative case 193

The viscous isentropic case gives way to an analogous phenomenon. The Navier–Stokes equations are reduced to the conservation of mass and to Newton’s law:

τt − vx = 0,

vt + p(τ )x = ε(b(τ )vx )x .

A discontinuity satisfies the conditions [v] = 0 and [v]+ s[τ ] = 0, hence the condi-tions s = 0. Again, there is a discontinuity in τ at (t0, x0) if τ ( 0 , ·) is discontinuousat x0. The smoothness of τ is propagated without it improving.

We see the fact that the diffusion tensors are not invertible allow that discon-tinuous solutions exist for the perturbed problems. However, the diffusion is oftensufficient for the discontinuities not to appear spontaneously. When they do so eventhen, we must see there a genuinely non-linear hyperbolic behaviour and anticipatethat the equation (6.9) will not be satisfied. The solution of the Cauchy problem willnot be unique. We must then select the physically admissible solution by imposingthe entropy inequality

∂t E(u)+ div F(u)+ ε∑

α,β,i, j,k

Bαβ

k j

∂2E

∂ui∂uk∂αuε

i ∂βuεj ≤

∑α

∂αeα (6.10)

while giving a convenient sense to the third and fourth terms.

6.2 Global existence in the strictly dissipative case

The parabolic case is that where the Legendre–Hadamard condition (6.7) is satisfied.Following Hoff and Smoller [43, 44], we restrict ourselves to the case of a ‘diagonal’perturbation and with constant coefficients

Bαβ

i j = δji bαβ

i , ∀u ∈U.

The hypothesis (6.7) amounts to saying that the differential operators Qi :=−∑

α,β bαβ

i ∂α∂β are elliptic. The equation of the heat type vt + Qiv= 0 possessesa fundamental solution Ki of the form

Ki (t, x) = t−d/2ki

(x√t

)

where ki belongs to S(Rd ) (the Schwarz class), ki > 0 and∫

ki (y) dy= 1. Thesolution of the Cauchy problem for vt + Qiv = g is given by the Duhamel formula

v(t, · ) = Ki (t) ∗ v0 +∫ t

0Ki (t − s) ∗ g(s) ds

where ∗ denotes the convolution product in Rd and v0 the initial condition.


Local existence in L∞

Let us return to the non-linear system that we have to solve. In view of later appli-cations, we also take into account external forces g(x, t) = (g1, . . . , gn) which area priori given:

∂t ui + div f i (u)+ Qiui = gi , 1 ≤ i ≤ n. (6.11)

Being given an initial condition u0 :=Rd �→U, a solution of the Cauchy problem

is a solution of the non-linear integral equation

ui (t) = Ki (t) ∗ u0i −d∑

α=1

∫ t

0∂α Ki (t − s) ∗ f i

α(u(s)) ds

+∫ t

0Ki (t − s) ∗ gi (s) ds, 1 ≤ i ≤ n, (6.12)

when the terms have a meaning. We shall suppose that there exists a point u suchthat u0 − u is bounded and square-integrable.

The existence and uniqueness (local in time) of a solution of (6.12) are obtainedby writing it as a fixed point of the mapping

u �→ Lu,

(Lu)i (t) := Ki (t) ∗ u0i −d∑

α= 1

∫ t

0∂α Ki (t − s) ∗ f i

α(u(s)) ds

+∫ t

0Ki (t − s) ∗ gi (s) ds,

and by showing that this is a contraction in a suitable complete metric space.

Norms

For v ∈Rn we write ‖v‖ = max1≤i≤n |vi |. We have chosen this norm because it

defines invariant balls B(a; s) for the semi-group associated with the system ofnon-coupled linear equations ∂tvi + Qivi = 0, 1 ≤ i ≤ n (when Q1 = · · · = Qn ,then any norm of R

n can be used), thanks to the maximum principle. For m ∈ N∗

and 1 ≤ p ≤ ∞ we write

‖v‖∞p = ess sup0≤t≤T

‖v(t)‖p

the usual norm of L∞(0, T ; (L p(Rd ))m) and

‖v‖1p =∫ T

0‖v(t)‖p dt

that of L1(0, T ; (L p(Rd ))m). We shorten ‖ · ‖∞∞ to ‖ · ‖∞ (it is the norm of(L∞((0, T )×R

d ))m). We remark that these norms depend on the choice of T > 0 ifv is defined in a time interval containing (0, T ). For example, limT→0+ ‖v‖1p = 0when v belongs to L1(0, T1; (L p(Rd ))m). Finally we shall define, similarly, thenorms in Lq (0, T ; X ) where X is a Banach space, for example a Sobolev space.

Hypotheses

We are given two numbers r0 and r such that 0 < r0 < r and a point u such thatB(u; r ) ⊂U. We suppose that the initial datum has values in B(u; r0) (which isrestrictive only ifU �= R

n) and that u0− u is square-integrable. As far as the forcesgi are concerned their smoothness will be made precise in each statement. We shalldenote by GT the ball of (L∞(0, T ) × R

d )n defined by ‖u − u‖∞ ≤ r . It is acomplete metric space. We are going to consider L as a mapping defined on GT .The essential result concerning local existence in time is the following.

Lemma 6.2.1 Let g ∈ L1(0, S; (L∞(Rd ))n) with S > 0. There exists a time T > 0such that L is a contracting mapping of GT into itself for the norm ‖ · ‖∞. Ifin addition g ∈ L1(0, T ; (L2(Rd ))n), then L is equally contracting for the norm‖ · ‖∞ 2.

We deduce then from Picard’s theorem the statement of existence (in which we donot suppose that f is the flux of a hyperbolic system).

Corollary 6.2.2 Let g and T be as above. The mapping L possesses a unique fixedpoint u ∈ GT . In addition u ∈ L∞(0, T ; (L2(Rd ))n). In fact, there exists a numberC1 = C1(r0, r, T ) such that we have

‖u − u‖∞ ≤ C1(‖u0 − u‖∞ + ‖g‖1∞), (6.13)

‖u − u‖∞2 ≤ C1(‖u0 − u‖∞2 + ‖g‖12). (6.14)

Proof Let us first show that we can choose T with the result that L(GT ) ⊂ GT .Using the fact that

∫R

d Ki (t, x) dx ≡ 1, we have

Lu(t)− u = K (t) ∗ (u0 − u)−∫ t

0∇x K (t − s) ∗ ( f (u(s))− f (u)) ds

+∫ t

0K (t − s) ∗ g(s) ds.

We have used a vector notation to simplify the equations. Let us denote by M(r ) aLipschitz constant for the function f in B(u; r ). As Ki ≥ 0, we have ‖Ki (t)‖1 = 1,


therefore (and it is there that we use the fact that B(u; r0) has its edges parallel tothe axes)

‖Lu(t)− u‖∞ ≤ ‖u0 − u‖∞ + M(r )‖u − u‖∞∫ t

0‖∇x K (t − s)‖1 ds

+∫ t

0‖g(s)‖∞ ds.

However, since ∇x Ki (t) = t−12 (d+1)li (t−

12 x) where li ∈ S (Rd ), we have ‖∇x K (t)‖

= Ct−12 . Thus, for u ∈ GT .

‖Lu(t)− u‖∞ ≤ r0 + Cr M(r )∫ t

0

ds√t − s

+ ‖g‖1∞,

hence

‖Lu(t)− u‖∞ ≤ r0 + 2Cr M(r )√

T + ‖g‖1∞.

Choosing T small enough that

r0 + 2Cr M(r )√

T + ‖g‖1∞ ≤ r, (6.15)

we have that Lu ∈ GT . For u, v ∈ GT , q = 2 and q = ∞, we then have

‖Lv(t)− Lu(t)‖q ≤ M(r )∫ t

0‖∇x K (t − s)‖1‖v(s)− u(s)‖q ds

and therefore

‖Lv − Lu‖∞q ≤ 2CM(r )√

T ‖v − u‖∞q .

From (6.15) the ratio k = 2CM(r )T 1/2 is strictly less than 1. The mapping L isthus contracting in GT for the norm ‖ · ‖∞ 2.

To show that the fixed point u of L is in L∞(0, T ; (L2(Rd ))n), we note that thesequence of iterates (we take um+1= Lum and u0≡ u) is Cauchy in this space,hence converges for the norm ‖ · ‖∞2 to the limit u. But there exists a sub-sequencewhich converges almost everywhere and hence u= u is simultaneously in L∞(0, T ;(L2(Rd ))n) and in (L∞((0, T ) × R

d )n). Finally, the constant C1 has the value1/(1− k).

Estimate of the derivatives

To show that u(t) has partial derivatives of order p which are in L2 ∩ L∞, we showthat each iterate um has and that the corresponding norms remain bounded whenm tends to infinity. For that, we shall suppose that in the first instance u0 − u ∈


H p−1∩W p−1,∞. Using the smoothing property of the semi-group we shall removethis hypothesis later. Let us begin with the case2 p = 1.

Lemma 6.2.3 Let g ∈ L1(0, T ; (H1∩W 1,∞(Rd ))n) and T be as above. There existT0 ∈ (0, T ] and C2 > 1 such that the solution of (6.12) satisfies

u − u ∈ L∞(0, T0; L2),

t1/2∇xu ∈ L∞(0, T0; L2 ∩ L∞)

with the upper bounds

‖u − u‖∞ 2 ≤ C2(‖u0 − u‖2 + ‖g‖12),

‖t1/2∇xu‖∞ 2 ≤ C2(‖u0 − u‖2 + ‖∇x g‖12),

‖t1/2∇xu‖∞ ≤ C2(‖u0 − u‖∞ + ‖∇x g‖1∞).

Remark In this statement as in those that follow, the time of existence T0 dependsonly on r0, r and the norm of g in L1(0, S; X ) where X is an appropriate Banachspace (here, X = H 1 ∩ W 1,∞). The constants C1, C2 depend only on r , while T0

is bounded below by a number (2C M(r ))−2 which depends only on r .

Proof of lemma

The first inequality has already been proved. It is sufficient then to show that Lpreserves the above inequalities, that is to say that if v (given in GT ) satisfies them,then Lv satisfies them. Let v ∈ GT0 ∩ L∞(0, T ; L2) where T0 has still to be madeprecise. We have

∇x Lv(t) = ∇x K (t) ∗ (u0 − u)−∫ t

0∇x K (t − s) ∗ ∇x f (v(s)) ds

+∫ t

0K (t − s) ∗ ∇x g(s) ds.

Hence

‖∇x Lv(t)‖2 ≤ Ct−1/2‖u0 − u‖2 + C∫ t

0‖∇x f (v(s))‖2

ds√t − s

+ ‖∇x g‖12.

But

‖∇x f (v(s))‖q ≤ M(r )‖∇xv(s)‖q

2 From here on, we differ from the analysis of Hoff and Smoller, who do not estimate the L∞-norms of thederivatives. It does not seem possible to perform an induction argument using only the L2-norms and theLemma 2.1 of [44] appears to be false.

for all 1 ≤ q ≤ ∞. The above equation therefore becomes

‖∇x Lv(t)‖2 ≤ Ct−1/2‖u0 − u‖2 + CM(r )∫ t

0‖∇xv(s)‖2

ds√t − s

+ ‖∇x g‖12.

Let us choose T0 ∈ (0, T ] with the result that

l :=CM(r )√

T0

∫ 1

0

dσ√σ (1− σ )

< 1. (6.16)

Then

‖∇x Lv(t)‖2 ≤ Ct−1/2‖u0 − u‖2 + lT−1/20 ‖t1/2∇xv‖∞ 2 + ‖∇x g‖12,

from which

‖t1/2∇x Lv‖∞ 2 ≤ C‖u0 − u‖2 + l‖t1/2∇xv‖∞ 2 + T 1/20 ‖∇x g‖12.

As u0 ≡ u, the repeated use of the preceding argument implies

‖t1/2∇xum‖∞ 2 ≤ 1

1− l

(C‖u0 − u‖2 + T 1/2

0 ‖∇x g‖12).

Finally

‖∇x Lv(t)‖∞ ≤ Ct−1/2‖u0 − u‖∞ + C∫ t

0‖∇x f (v(s))‖∞ ds√

t − s+ ‖∇x g‖1∞,

which leads in a similar way to

‖t1/2∇x Lv‖∞ ≤ C‖u0 − u‖∞ + l‖t1/2∇xv‖∞ + T 1/20 ‖∇x g‖1∞

and to

‖t1/2∇xum‖∞ ≤ 1

1− l

(C‖u0 − u‖∞ + T 1/2

0 ‖∇x g‖1∞).

We continue with the derivatives of higher order.

Lemma 6.2.4 Let p≥ 2, g ∈ L1(0, S; (H p ∩ W p,∞(Rd ))n) and T0 be as above.There exists a polynomial P depending on p, with positive coefficients dependingon r, s, and on the norm of D2 f in C

p−2(B(u; r )), such that, if also u0− u ∈(H p−1 ∩W p−1,∞(Rd ))n, then the solution u of (6.12) satisfies

u − u ∈ L∞(0, T0; (H p−1 ∩W p−1,∞(Rd ))n),

t1/2(u − u) ∈ L∞(0, T0; (H p ∩W p,∞(Rd ))n),


with ∥∥∇ p−1x u

∥∥∞ 2 +∥∥∇ p−1

x u∥∥∞ ≤ P(‖u0 − u‖ + G p−1),∥∥t1/2∇ p

x u∥∥∞ 2 +

∥∥t1/2∇ px u

∥∥∞ ≤ P(‖u0 − u‖ + G p),

where ‖u0− u‖ is the norm in H p−1∩W p−1,∞ and G p that of g in L1(0, T ; (H p∩W p,∞(Rd ))n).

Proof We proceed by induction on p. We show that L preserves the inequalitiesthat u0 ≡ u satisfies trivially. We deduce that um , then u also, satisfies them, whichis the substance of the lemma. In the first place we have

∇ p−1x Lv(t) = K (t) ∗ ∇ p−1

x u0 −∫ t

0∇x K (t − s) ∗ ∇ p−1

x f (v(s)) ds

+∫ t

0K (t − s) ∗ ∇ p−1

x g(s) ds.

Hence, if q = 2 or q = ∞,∥∥∇ p−1x Lv(t)

∥∥q ≤

∥∥∇ p−1x u0

∥∥q + C

∫ t

0

∥∥∇ p−1x f (v(s))

∥∥q

ds√t − s

+∫ t

0

∥∥∇ p−1x g(s)

∥∥q ds.

However, ∇ p−1x f (v) − d f (v)∇ p−1

x v is a polynomial without constant term in thevariables ∇xv, . . . ,∇ p−2

x v, whose coefficients are the derivatives of f of ordersbetween 2 and p− 1, and hence are bounded on B(u; r ). From the induction hy-pothesis, the iterates um therefore satisfy∥∥∇ p−1

x f (v)− d f (v)∇ p−1x v

∥∥q ≤ Q(‖u0 − u‖ + G p−2)

where in fact Q depends only on the norm of u0− u in H p−2 ∩W p−2,∞, on G p−2

and on the norm of D2 f in C p−3(B(u; r )). Thus

∥∥∇ p−1x Lum(t)

∥∥q ≤

∥∥∇ p−1x u0

∥∥q + G p−1 + CM(r )

∫ t

0

∥∥∇ p−1x um(s)

∥∥q

ds√t − s

+C∫ t

0Q(‖u0 − u‖ + G p−2)

ds√t − s

and hence∥∥∇ p−1x Lum(t)

∥∥∞q ≤ Q1(‖u0 − u‖ + G p−1)+ k∥∥∇ p−1

x um∥∥∞q ,

from which ∥∥∇ p−1x um(t)

∥∥∞q ≤

1

1− kQ1(‖u0 − u‖ + G p−1). (6.17)

Similarly, applying the induction hypothesis and using (6.17) to bound the term‖∇ p

x f (um)− d f (um)∇ px um‖q , we obtain∥∥∇ p

x Lum(t)∥∥

q ≤ Ct−1/2∥∥∇ p−1

x u0∥∥

q + C∫ t

0

∥∥∇ px f (um(s))

∥∥q

ds√t − s

+∫ t

0

∥∥∇ px g(s)

∥∥qds

≤ Ct−1/2∥∥∇ p−1

x u0∥∥

q + G p

+C∫ t

0

(M(r )

∥∥∇ px um(s)

∥∥q + Q2(‖u0 − u‖ + G p−1)

) ds√t − s

.

Thus ∥∥t1/2∇ px Lum

∥∥∞q ≤ Q3(‖u0 − u‖ + G p)+ l‖t1/2∇ px um‖∞q ,

from which it follows that∥∥t1/2∇ px um

∥∥∞q ≤1

1− lQ3(‖u0 − u‖ + G p).

Now let us show that the solution is smooth when t > 0, this being true even fornon-smooth data. If u0− u belongs only to L2(Rd ) (with still ‖u0− u‖∞ ≤ r0), thenthe iteration converges to the unique solution u in L∞(0, T ; (L2 ∩ L∞(Rd ))n). Inaddition t1/2um ∈ L∞(0, T0; H1∩W 1,∞). Let t0 > 0. Making use of the semi-groupproperty of K , we have

Lv(t) = K (t− t0)∗ (Lv)(t0)−∫ t

t0

∇x K (t− s)∗ f (v(s)) ds+∫ t

t0

K (t− s)∗g(s) ds.

Applying this to v = um−1 and using the fact that Lv(t0) ∈ H1∩W 1,∞, we deducethat if also g ∈ L1(0, S; (H 2 ∩W 2,∞(Rd ))n), we have for all t0 < t < T0 and q = 2or∞,

∥∥(t − t0)1/2∇2x um

∥∥∞q ≤ P

(‖u0 − u‖2, ‖u0 − u‖∞, G1, G2,

1√t0

)

for a suitable polynomial P . In proceeding by induction on the order of the deriva-tives, we state that if g is still more smooth, there exists a polynomial Pp of p + 3variables such that if t∗ ∈ (0, T0], then

∥∥(t−t∗)1/2∇ px um

∥∥∞q ≤ Pp

(‖u0− u‖2, ‖u0− u‖∞, G1, . . . , G p,

1√t∗

)(6.18)

for q = 2, q = ∞, p ≥ 2 and for all m. The convergence towards the solution of(6.12) thus confirms that u ∈ L∞loc(0, T0; (H p ∩W p,∞)n) when g ∈ L1(0, S; (H p ∩W p,∞(Rd ))n). Let us sum up this in the following theorem.


Theorem 6.2.5 Let u0 ∈ (L2 ∩ L∞(Rd ))n be such that u0 takes its values ina block

∏ni=1[ai , bi ] strictly included inU. Let g ∈ L1(0, S; (H p ∩ W p,∞(Rd ))n).

Then there exists T0 ∈ (0, S] such that the system (6.11) possesses a unique solutionin C ([0, T0]; (L2(Rd ))n) ∩ L∞((0,∞)× R

d ) with u(0, · ) = u0. In addition thereexists C > 1 such that this solution satisfies

u ∈ L∞loc(0, T0; (H p ∩ W p,∞)n),

‖u − u‖∞ 2 + ‖t1/2∇xu‖∞ 2 ≤ C(‖u0 − u‖2 + ‖g‖12),

‖u − u‖∞ + ‖t1/2∇xu‖∞ ≤ C(‖u0 − u‖∞ + ‖g‖1∞).

Finally, for all t∗ ∈ (0, T0], there exists a polynomial Pp such that for all t ∈ (t∗, T0],we have∥∥(t − t∗)1/2∇ p

x u∥∥∞ 2 ≤ Pp(‖u0 − u‖2, ‖u0 − u‖∞, G1, . . . , G p),∥∥(t − t∗)1/2∇ p

x u∥∥∞ ≤ Pp(‖u0 − u‖2, ‖u0 − u‖∞, G1, . . . , G p).

Proof If u ∈ C ([0, T ]; (L2(Rd ))n) ∩ (L∞((0, T ) × Rd ))n is the solution of (6.2)

and satisfies u(0, · ) = u0, then this is a solution of (6.12) and we have seen thatthis exists (continuity with values in L2 comes from the fact that the um have thisproperty since K (t) ∗ u0 has it) and is unique. We have already shown all the otherstated properties.

Remark This theorem is not optimal. We can for example weaken the hypothesesconcerning g. If, in addition, g is somewhat smooth with respect to the time (forexample ∂t g ∈ (L2 ∩ L∞)n), the solution itself is also somewhat smooth whent > 0. That is shown as previously, by differentiating the integral equation as oftenas is necessary with respect to the time. As an example, we can state

Theorem 6.2.6 Let g ∈ (D (Rd+1))n and u0 ∈ (L2 ∩ L∞(Rd ))n. Then there existsT > 0 such that the (unique) solution of the Cauchy problem (6.11) is of class C

∞

on (0, T ]×Rd . If moreover u0 is of class C

∞, then the solution is of class C∞ on

[0, T ]× Rd .

Extension of the solution (case g ≡ 0)

In what has gone before we have not had the use of an entropy. In fact, the anal-ysis has not used the hyperbolicity of the inviscid system (when we suppress theviscosity). However, the entropy plays an essential role in the extension of thesolution to R

+ × Rd . We suppose that the inviscid system has a strongly convex

entropy, denoted by E , of flux F (hence this system is symmetrisable hyperbolic by

Theorem 3.4.2). Free to replace the entropy E by u �→ E(u)−E(u)−dE(u) · (u−u),we can suppose that E is positive, and zero at u. Hence, there exists a number δ > 0such that, for all a ∈ B(u; r ), we have

δ‖a − u‖2 ≤ E(a) ≤ δ−1‖a − u‖2.

The solution of the Cauchy problem, since it is smooth, satisfies

E(u)t + div F(u)+ c(u)‖B∇xu‖2 ≤∑

α

∂αeα.

Similarly, on B(u; r ), c(u) satisfies c(u)‖Bm‖2 ≥ γ ‖m‖2, ∀m ∈ Md×n where γ

is a positive constant. The expressions eα, of the form∑

β(dE · Bαβ)(u)∂βu, tend

to zero at infinity since u ∈ L∞ ∩ H p for p large enough (let us say p > 12d + 1).

Similarly for F(u) which satisfies F(u)= 0 (we are allowed this choice) anddF(u) = dE(u) d f (u) = 0. The integration over R

d thus gives

d

dt

∫R

dE(u) dx + γ

∫R

d‖∇xu‖2 dx ≤ 0.

In particular, t �→ ∫R

d E(u) dx decreases on (0, T0] and hence also on [0, T0]since u is continuous in L2(Rd ). Thus∫

Rd

E(u(t, x)) dx ≤∫

Rd

E(u0(x)) dx,

from which it follows that

‖u(t)− u‖2 ≤ 1

δ‖u0 − u‖2.

Also, if 0 < t∗< t1 ≤ T0, the upper bound (6.18) and Sobolev’s inequality whichcorresponds to the injection Hm ⊂ C

0 for m > 12n give

‖u(t1)− u‖∞ ≤ ‖u(t1)− u‖1−θ2

∥∥∇mx u(t1)

∥∥θ

2

≤ δθ−1‖u0 − u‖1−θ2 Pm

(‖u0 − u‖2,

1√t∗

)θ

(t1 − t∗)−θ/2

with θ = θ (m, n) ∈ (0, 1]. The numbers r0, r , t∗ and t1 being fixed, there exists anumber r1 > 0 such that if ‖u0 − u‖ ≤ r1, then the right-hand side is less than r0.But then the Cauchy problem, made up of the system (6.2) and the initial conditionu(t1) at t = t1, possesses a smooth solution in the interval (t1, t1+T0) which has allthe properties stated in Theorem 6.2.5. The solution sought is therefore defined atleast on the interval (0, t1 + T0). We can take t∗ = 1

4 T0 and t1 = 12 T0. The number

r1 depends only on the choice of r0 and of r , with the result that we can extendthe solution of the Cauchy problem to all the intervals of the form (0, 1

4qT0) byrepeating the same argument. The final result is therefore the following.

6.3 Smooth convergence as ε→ 0+ 203

Theorem 6.2.7 (g ≡ 0) Let u ∈U and the numbers r > r0 > 0 be such that theblock B(u; r ) is contained inU. There exists r1 > 0 such that if u0 ∈ (L2∩L∞(Rd ))n

and if ‖u0− u‖∞ ≤ r0, ‖u0− u‖2 ≤ r1, then the Cauchy problem for (6.11) (whereg ≡ 0) possesses a global solution u ∈ Cb(R+; (L2(Rd ))n)∩ L∞(R+ ∩R

d )n. Thissolution is unique in this class and satisfies the estimates of Theorem 6.2.5 with inaddition

‖u − u‖∞ 2 ≤ δ(r )‖u0 − u‖2.

Existence with a small diffusion

The above theorem provides global existence for small data and Theorem 6.2.5ensures the local existence for all smooth data when ε > 0 is fixed. In practice, aswe consider the sequence (uε)ε>0 when ε tends to zero, we wish to know that thesesolutions are defined in a common strip (0, S) × R

d . We ought therefore to showthat the time of existence Tε of uε does not tend to zero.

In fact, neither of the two above theorems leads to this conclusion. To applythem, we consider a fixed value of ε, let us say ε= 1, by the change of variables(t, x) �→ (εt, εx). Hence, we apply them to the given initial condition

u0ε(x) := u0(εx).

This always satisfies ‖u0ε − u‖ ≤ r0 and the local theorem yields a solution uε inthe strip (0, T0(r ))×R

d . But the solution uε(t, x) = u(t/ε, x/ε) is only defined for0 < t ≤ Tε = εT0. Besides, the global theorem 6.2.7 does not apply for small ε as‖u0ε − u‖2 = ε−d/2‖u0 − u‖2 > r1.

We shall see in the following section a sharper estimate which makes use of theregular solution of the hyperbolic problem (6.1) (which is clearly hyperbolic sinceit has a convex entropy) and which permits us to prove the existence of uε andthe convergence of the sequence (uε)ε>0 to that value in a strip (0, S)× R

d for anS > 0. However, we shall restrict ourselves to the case of a single space dimension.

6.3 Smooth convergence as ε→ 0+

In this section, we shall prove a convergence result when d = 1, if the diffusion hasconstant coefficients.

We suppose still that the system (6.1) possesses a strongly convex entropy Eof flux F . Hence, it is symmetrisable hyperbolic. Let 0 < r0 < r , u and u0 ∈L∞(R; B(u; r0)) as in the preceding section. There exists a unique local smoothsolution of the Cauchy problem for (6.2). We denote this solution by uε and its time


of existence by Tε > 0. We have seen that Tε ≥ εT0(r ). We define

Sε = inf{τ ≤ Tε; ‖uε(t)− u‖∞ > r0, ∀t ∈ [τ, τ + εT0] ∩ [τ, Tε)}.There exist times arbitrarily close to Sε for which ‖uε(t) − u‖∞ ≤ r0. From thelocal existence theorem, we thus have that Tε ≥ Sε + εT0 and all the estimates thatwe have stated are valid for uε when t ∈ (0, εT0 + Sε) as u(t) can be consideredas the solution of the Cauchy problem after a time less than εT0 and for a giveninitial condition with values in B(u; r0). In fact, we shall use only the estimate‖uε(t)− u‖∞ ≤ r .

Finally, suppose that u0 ∈ H2(R)n . Then Theorem 3.6.1 assures us that theCauchy problem for the system (6.1) possesses a unique regular solution u in astrip (0, T )× R which satisfies u ∈ C ([0, T ); H2(R)) ∩ C

1([0, T )× R). The aimof this section is to prove the convergence of uε to u when ε tends to zero.

For that, we begin by establishing and energy estimate.

The energy estimate

This estimate having an interest in its own right, we present it for very generaldiffusion tensors.

Theorem 6.3.1 Let v �→ B(v) be a diffusion tensor satisfying the inequality

(D2E(v)ξ | B(v)ξ ) ≥ c(v)‖B(v)ξ‖2, ∀ξ ∈ Rn,

where v �→ c(v) > 0 is a continuous function.We suppose that the Cauchy problem for the system (6.2) has a smooth solution

uε with values in B(u; r ) for all (t, x) ∈ [0, tε]× R. We suppose finally that u, thesolution of the Cauchy problem for (6.1), has values in a compact set inU (this istrue, even if it entails reducing the value of T ).

Then there exists a constant C > 0 such that the upper bounds below are validfor all t ∈ [0, min(T, tε)] :

‖uε(t)− u(t)‖2 ≤ C√

εt, (6.19)∫ t

0

∫R

∥∥B(uε)uεx

∥∥2dx ds ≤ Ct. (6.20)

It is clear that these estimates cannot remain valid if u is a weak solutionwith discontinuities. For example (6.20), when B is invertible, implies, that uε

x

remains in a bounded set of (L2((0, t)×R))n . By equation (6.2), we deduce thatuε

t remains in a compact set of L2(0, t ; (H−1loc (R))n). By a classical compactness

lemma, it follows that (uε(t))ε is a relatively compact sequence for the topology


of uniform convergence, for almost all t , and this prevents the convergence(even in L∞(0, t ; L2(R))) to a discontinuous function, in contradiction to thefirst upper bound (6.19).

We shall note also that this theorem does not require that B is invertible. It constitutesa uniform estimate with respect to the diffusion.

Proof For every function g defined onU, we write g = g(u) and gε = g(uε). Weintroduce the expressions

�(t, x) := Eε − E − dE · (uε − u),

δ(t, x) := Fε − F − dE · ( f ε − f ).

We have

�t + δx = εdEε · (Bεuεx

)x − dE · ((uε − u)t + ( f ε − f )x )

−(dE)t · (uε − u)− (dE)x · ( f ε − f )

= ε(dEε − dE) · (Bεuεx

)x − D2E(ut , uε − u)− D2E(ux , f ε − f )

= ε((dEε − dE) · Bεuε

x

)x − ε

(D2Eεuε

x − D2Eux | Bεuεx

)+D2E(d f · ux , uε − u)− D2E(ux , f ε − f ).

Let γ > 0 be a lower bound of c(v) on B(u; r ), then

�t + δx + γ ε∥∥Bεuε

x

∥∥2 ≤ ε((dEε − dE) · Bεuε

x

)x + ε‖D2E ux‖

∥∥Bεuεx

∥∥−D2E(ux , f ε− f − d f · (uε − u)),

where we have also used the symmetry of d f relatively to the quadratic form D2E(Theorem 3.4.2).

On the compact set B(u; r ), the expression � is greater than c1(r )‖uε − u‖2

where c1(r ) > 0. As f ε− f − d f · (uε − u) = O(‖uε − u‖2), we therefore have

�t + δx + γ ε∥∥Bεuε

x

∥∥2 ≤ ε((dEε − dE) · Bεuε

x

)x + c2(r )‖ux‖∞�

+ ε‖D2E ux‖∥∥Bεuε

x

∥∥.

Using the Cauchy–Schwarz inequality, we find that this gives

�t+δx+ γ ε

2

∥∥Bεuεx

∥∥2 ≤ ε((dEε−dE) · Bεuε

x

)x+c2(r )‖ux‖∞�+ ε

2γ‖D2E ux‖2.

Let us integrate this inequality over R. As uε − u ∈ L2(R) and ux ∈ L2(R), theintegrals of δx and of ε((dEε − dE) · Bεuε

x )x are zero. There thus remains

d

dt

∫R

� dx + γ ε

2

∫R

∥∥Bεuεx

∥∥2dx ≤ c3

(ε +

∫R

� dx

). (6.21)


Since �(0, · ) ≡ 0, the Gronwall inequality yields∫R

� dx ≤ ec3t c3εt ≤ c4εt,

from which ‖uε(t) − u(t)‖22 ≤ c4εt/c1. Finally, integrating (6.21) from 0 to t , we

obtain

γ ε

2

∫ t

0

∫R

∥∥Bεuεx

∥∥2dx ds ≤ c3ε

(t + 1

2c4t2

)≤ c5εt,

which completes the proof of the theorem.

The essential point of this theorem is that the constant C depends only on r andon the norms of u in H1(R)n and in W 1,∞(R)n . But it does not depend on ε > 0or on the time of existence Sε + εT0. In particular, we deduce immediately thatuε converges to u in L∞(0, S; L2(R)n) where S := min(T, liminfε→ 0 Sε). For thisresult to have a significance, it must be shown that S > 0. That will be shown lateron. But first we examine two particular cases.

Two most favourable cases

We return to the estimates of the preceding section to prove the following equality:

�t + δx = ε(dEε − dE) · (Bεuεx

)x +D2E(ux , f ε − f − d f · (uε − u)). (6.22)

Now we use differently the dissipation:

�t + δx + ε(D2Eε(uε − u)x |Bε(uε − u)x )

= ε((dEε − dE) · Bεuε

x

)x + ε((D2E − D2Eε)ux |Bε(uε − u)x )

+ ε(dE − dEε)x · Bεux + D2E(ux , f ε − f − d f · (uε − u)).

Thus

�t + δx + γ ε‖Bε(uε − u)x‖2 ≤ ε((dEε − dE)Bεuε

x

)x + ε(dE − dEε)x · Bεux

+ ε((D2E − D2Eε)ux |Bε(uε − u)x )+ c2(r )‖ux‖∞�.

Using Young’s inequality, we have

�t + δx + γ ε

2‖Bε(uε − u)x‖2 ≤ ε[(dEε − dE) · Bε(uε − u)x ]x

+ c2(r )‖ux‖∞�+ c3‖ux‖2∞ε�+ ε(dE − dEε) · (dBε · uε

x

)ux . (6.23)

Let us decompose the last term into two parts. The first,

ε(dE − dEε) · (dBεux )ux ′

is integrable since u ∈ L∞(0, T ; H2(R)) and its integral is bounded above by

6.3 Smooth convergence as ε→ 0+ 207∫R

� dx + c4(‖u‖H2(R))ε2. The second is

ε(dE − dEε) · (dBε · (uε − ux ))ux ′

which is a bad one, in general. There are, however, two favourable cases. On theone hand, when the tensor B has constant coefficients, since this term is then zero.On the other hand the parabolic case (B is thus invertible), since then

ε(dE − dEε) · (dBε · (uε − u)x )ux ≤ εc‖ux‖ ‖uε − u‖ ‖Bε(uε − u)x‖≤ γ ε

4‖Bε(uε − u)x‖2 + εc5(‖ux‖∞)�.

Finally, the inequality (6.23), integrated with respect to x , leads to

d

dt

∫R

� dx + γ ε

4

∫R

‖Bε(uε − u)x‖2 ≤ c6

∫R

� dx + c4ε2.

The estimate of Theorem 6.3.1 is therefore improved in

Theorem 6.3.2 We suppose that

(1) either the diffusion tensor has constant coefficients,(2) or the perturbation is parabolic (B is invertible and D2E ·B is positive definite).

We suppose that the Cauchy problem for the system (6.2) has a smooth solutionuε with values in B(u, r ) for all (t, x) ∈ [0, tε] × R. We suppose, finally, that thesolution of the Cauchy problem for (6.1) has values in a compact set ofU (whichis true even if it means a reduction in the value of T ).

Then, there exists a constant C > 0 such that the upper bounds below are validfor all t ∈ [0, min(T, tε)] :

‖uε(t)− u(t)‖2 ≤ Cε√

t, (6.24)∫ t

0

∫R

‖B(uε)(uε − u)x‖2 dx ds ≤ Cεt. (6.25)

Uniformity of the existence times

We return to the case where the diffusion B is invertible and with constant coeffi-cients. The inequality (6.20) is therefore an estimate or uε

x − ux in L2((0, t)× R).Even if we have to replace r0 by a number r1 ∈ (r0, r ), we can suppose that

‖u0 − u‖∞ < r0. Then, we denote by T1 > 0 the time during which the solutionu of the hyperbolic problem remains with its values in B(u; 1

2 (r0 + ‖u0 − u‖∞)).If liminfε→ 0 Sε < T1, then Sε + εT0 < T1 for arbitrarily small values of ε. On theinterval [Sε, Sε+εT0], we have ‖uε(t)− u‖∞ ≥ r0 and therefore ‖uε(t)−u(t)‖∞ ≥r0 − ‖u(t) − u‖∞ ≥ 1

2 (r0 − ‖u0 − u‖∞) which is a strictly positive constant. In

addition, the classical inequality ‖v‖2∞ ≤ 2‖v‖2‖vx‖2, valid for all v in H1(R),gives ‖uε(t)− u(t)‖4∞ ≤ 4c(r )εt‖(uε − u)x‖2

2. Finally,

εT0

(r0 − ‖u0 − u‖∞

2

)4

≤∫ Sε+εT0

Sε

‖uε − u‖4∞ dt

≤ c1(r )ε(Sε + εT0)∫ Sε+εT0

Sε

‖(uε − u)x‖22 dt

≤ 2c1(r )ε(Sε + εT0)∫ Sε+εT0

Sε

(∥∥uεx

∥∥22 + ‖ux‖2

2

)dt

≤ c2(r )ε(Sε + εT0)2,

where we have used Theorem 6.3.1 and the C (0, T ; H1) smoothness of u. From thisinequality, we deduce an explicit lower bound of the time during which uε existsand ‖uε − u‖∞ stays less than r :

Sε + εT0 ≥ min

((T0

c2(r )

)1/2 (r0 − ‖u0 − u‖∞

2

)2

, T1

).

The final result is therefore the following.

Theorem 6.3.3 Letvt+ f (v)x = 0 be a system of conservation laws equipped with astrongly convex entropy E (thus, it is symmetrisable hyperbolic). Let B ∈ Mn(R) bea matrix with constant coefficients satisfying (D2E(u)η | Bη) ≥ c(u)‖η‖2, η ∈ R

n,with u �→ c(u) > 0 continuous. Finally, let u0 ∈ H2(R)n, with values in a compactset K ofU, this compact set being invariant for the equation vt = Bvxx .

We denote by u the local smooth solution of the Cauchy problem{ut + f (u)x = 0,

u(0, · ) = u0.

For ε > 0, we denote by uε the local smooth solution of the Cauchy problem{uε

t + f (uε)x = εBuεxx ,

uε(0, · ) = u0.

Then there exist a time T > 0 and a constant c(K ) > 0 such that u and uε (for0 < ε < 1) are defined on [0, T )× R and satisfy

‖uε(t)− u(t)‖2 ≤ C(K )√

εt, ∀t ∈ [0, T ),∫ t

0

∥∥uεx

∥∥22 ds ≤ C(K )t, ∀t ∈ [0, T ).

In particular, u = limε→ 0+ uε for the norm of L∞(0, T ; (L2(R))n).


Comments It is not, in general, clear that Tc, the time during which we have theconvergence of uε to u, is equal to the time of the existence Te of u. But as thesole obstacle to the energy estimates is the growth of uε in L∞(R)n , we clearlyhave Tc = Te once a maximum principle yields a set K ofU in which uε remainsindefinitely. The most obvious case is that of a scalar equation. Let us take also asan example the Keyfitz and Kranzer system

ut + (ϕ(‖u‖)u)x = 0,

which we perturb in a diagonal manner:

uεt + (ϕ(‖uε‖)uε)x = εuε

xx .

The expression ρε := 12‖uε‖2 satisfies the inequatility

ρεt + (A(ρε))x ≤ ερε

xx

for a suitable function A: R+ �→R. Thus ρε(t, x) ≤ 1

2‖u0‖2∞ and we deduce thatuε tends to u in L∞([0, Te); (L2(R))n).

We are restricted to invertible diffusion tensors with constant coefficients becausethis allows the use of Duhamel’s formula and the kernel of a parabolic linearequation, with all the explicit estimates which result. It is, however, plausible fromTheorem 6.3.1 that an existence result, uniform with respect to ε ∈ (0, 1], must arisefor the diffusions v �→ B(v) satisfying the inequality

(D2E(v)η | B(v)η) ≥ c(v)‖ B(v)η‖2, ∀η ∈ Rn,

where c(v) > 0. The difficulty in the general case is that we must proceed with theestimates of derivatives of order α for 0 ≤ |α| ≤ m with m > 1+ 1

2d, exactly as inthe proof of Theorem 3.6.1, treating in addition the diffusion term.

Even in the case of an invertible diffusion tensor with constant coefficients,we are restricted to the one-dimensional case because of the inequality ‖v‖2∞ ≤2‖v‖2‖vx‖2 which is precisely what we need to obtain a lower bound for Sε. Herealso, we should need the estimates of higher order derivatives when d ≥ 2.

The convergence of uε to a weak entropy solution of the system (6.1) is a muchmore delicate question. On the one hand, the estimates, if they exist, must be valid forsufficiently weak norms, for example L p norms. On the other hand, we do not have atheorem giving a priori the existence of entropy solutions (that of Glimm, restrictedto small data, is not satisfactory). It is just this convergence which has been usedto construct such solutions. The main method used to establish this procedure hasbeen that of compensated compactness (see the fundamental articles by Tartar [101]and DiPerna [17, 18]). This method is restricted to 2× 2 systems (more generallyto the systems called rich) and gives no information concerning the smoothness ofthe entropy solution.


The arguments of this section have been used in a more complex body of prob-lems, that of a problem with boundary conditions of Dirichlet type, in [30], [31].See Chapter 15.

6.4 Scalar case. Accuracy of approximation

In the scalar case, we are given the very strong properties such as the maximumprinciple, the uniqueness of the Cauchy problem and the contraction property inL1(Rd ) (see Theorem 2.3.5): for two solutions u and v of the same equation ut +div f (u) = 0, we have

‖u(t)− v(t)‖1 ≤ ‖u(0)− v(0)‖1.

These properties and their generalisations to solutions of the perturbed equation

uεt + div f (uε) = ε�uε (6.26)

allow us (cf. [58]) to bound the error ‖uε(t)− u(t)‖1 due to the approximation.

Theorem6.4.1 (Kuznetsov) There exists a constantC > 0 such that, if u0 ∈BV(Rd)and if u(0) = uε(0) = u0, then

‖uε(t)− u(t)‖1 ≤ C√

εt TV(u0).

This statement can be improved in the genuinely non-linear case in.

Theorem 6.4.2 We suppose that d = 1 and that infR f ′′ > 0. Then, for u0 ∈ BV(R)and with compact support, we have

‖u(t)− uε(t)‖1 ≤ C(u0)ε1/2t1/4.

Remark (1) Of course, this last estimate is only better than that of Kuznetsov fort � 1. In addition, if u0 ∈ L1(Rd ), the two estimates are only useful when they arebetter than the trivial bound

‖u(t)− uε(t)‖1 ≤ ‖u(t)‖1 + ‖uε(t)‖1 ≤ 2‖u0‖.The interesting times are therefore

(a) t � ε−1, in the general case,(b) 1 � t � ε−2, in the one-dimensional genuinely non-linear case.

(2) Neither of these two results is uniform with respect to the time and indeedthey could not be. In the linear case, with f ≡ 0, we have u(t, x) = u0(x) while

6.4 Scalar case. Accuracy of approximation 211

‖uε(t)‖∞ tends to zero when t tends to infinity. We thus have that

limt→ inf+∞‖u(t)− uε(t)‖1 ≥ ‖u0‖1,

which is independent of ε.The genuinely non-linear case (in which d = 1 and u0 ∈ L1) is subtler and is

supported by the asymptotic description of u and of uε with ε > 0 and fixed. Wecan suppose that f ′(0) = 0 and f ′′(0) = 1. First of all, u(t) is asymptotic in L1 toan N-wave (see [19], Theorem 9.1):

N (x, t) ={

x/t, −√(2pt) ≤ x ≤ √(2qt),

0, otherwise,

where

p := − infx∈R

∫ x

−∞u0(ξ ) dξ, q := sup

x∈R

∫ +∞

xu0(ξ ) dξ.

In addition, uε(t) is asymptotic in L1 to a non-linear diffusion wave of the form

v(t, x) = 1√tV

(x√t

),

where V is positive (if∫

u0 dx > 0, negative otherwise). Thus

limt→ inf+∞‖u(t)− uε(t)‖1 ≥ lim

t→+∞

∫ 0

−√2pt

|x |t

dx = p,

which is independent of ε.(3) As ‖uε(t)− u(t)‖∞ ≤ 2‖u0‖∞ ≤ 2 TV(u0) if inf u0≤ 0≤ sup u0, we can de-

duce from Kuznetsov’s theorem and the Holder inequality the following estimate:

‖uε(t)− u(t)‖p ≤ C(εt)1/2p TV(u0)

for p ≥ 1.(4) If we measure the error in a norm other than L1(R), we can obtain a power of

ε different from 12 ; the above remark is an illustration of this. But the exponent can

approach the optimal value 1 in the favorable cases. Tadmor [100] has shown thatif inf f ′′> 0 and if the initial condition has a Lipschitz increasing part, that is if

∃M ; x < y =⇒ u0(y)− u0(x)

y − x≤ M,

then

‖(uε(t)− u(t)) ∗ ϕ‖∞ ≤ K (t, u0)ε‖ϕx‖∞for every test function ϕ ∈ D (R).


Before proving these theorems, we are going to state some properties of theparabolic equation (6.26). First of all, the Cauchy problem has a unique locallysmooth solution. This satisfies the maximum principle since the equation can alsobe written as a transport–diffusion equation vt + f ′(v) · ∇v = ε�v. Thus, uε re-mains with values in the interval I = [infx∈R u0(x), supx∈R u0(x)] which entailsthat the smooth solution is defined for all time t ≥ 0. If v is another solution of thesame equation (6.26), we have

(uε − v)t + div( f (uε)− f (v)) = ε�(uε − v). (6.27)

Now, we use the following formulae. If ϕ: R2→R is a Lipschitz function and if

a, b are two functions belonging to W 1,1loc (Rd+1) (in particular if a and b are smooth)

then ϕ(a, b) ∈ W 1,1loc (Rd+1) and we have

∂

∂x jϕ(a, b) = ∂ϕ

∂a

∂a

∂x j+ ∂ϕ

∂b

∂b

∂x j, 0 ≤ j ≤ d. (6.28)

In addition if ψ : R→R is convex and Lipschitz then we have

�(ψ ◦ a) ≥ ψ ′(a)�a. (6.29)

Multiplying equation (6.27) by sgn(uε − v) and applying the preceding formulaewith ϕ(a, b) := |a − b|, ϕ(a, b) := sgn(a − b)( f α(a)− f α(b)) and ψ(a) := |a|, wededuce

|uε − v|t + div {sgn(uε − v)( f (uε)− f (v))} ≤ ε�|uε − v|.

Integrating over [0, T ]× Rd , we obtain

‖uε(t)− v(t)‖1 ≤ ‖uε(0)− v(0)‖1, ∀t ≥ 0.

In particular, if uε(0)= u0 and v(0, x)= u0(x−h), then v is nothing but a translationof uε: v(t, x) = uε(t, x − h), with the result that

‖uε(t)− uε(t, · − h)‖1 ≤ ‖u0 − u0( · − h)‖1.

Dividing by h > 0 and letting h tend to zero, we arrive at the decay of the totalvariation of uε,

TV(uε(t)) ≤ TV(u0). (6.30)

We can now proceed with the proof of Theorem 6.4.1.

Proof From the entropy inequality for u we have for all (s, y) ∈ R+ × R

d

|u − uε(s, y)|t + divx{sgn(u − uε(s, y))( f (u)− f (uε(s, y)))} ≤ 0. (6.31)

In fact, it is necessary to see this inequality in its integral form, including the initialcondition which uses test functions. Similarly, again making use of the formulae(6.28) and (6.29), we have for all (s, y) ∈ R

+ × Rd

|uε − u(s, y)|t + divx (sgn(uε − u(s, y))( f (uε)− f (u(s, y))))≤ ε�|uε − u(s, y)|. (6.32)

Let α > 0 and β > 0 be two parameters which we shall adjust in a moment. Weuse a smoothing kernel ωα(z) = α−1ω(z/α) where ω∈D

+(R) is even and satisfies∫R

ω dz= 1. We also use the smoothing kernel �β(x) :=ωβ(x1) . . . ωβ(xd ) on Rd .We put

gαβ(s, τ, x, y) = ωα(s− τ )�β(x − y).

Let us define for every smooth function h ∈ D (Rd+1) and every a ∈ R

θ t (h, u, a) :=∫∫

(0,t)×Rd

{∂h

∂s(s, x)|u(s, x)− a|

+ ∇xh · sgn(u − a)( f (u)− f (a))

}dx ds

+∫

Rd

h(0, x)|u0(x)− a| dx −∫

Rd

h(t, x)|u(t, x)− a| dx .

Since u is an entropy solution of the Cauchy problem of the unperturbed equation,we have θ t (h, u, a) ≥ 0, provided that h ≥ 0. Let us substitute in this inequalityg(τ, y, ·, ·) for h and uε(τ, y) for a. Then let us integrate the resulting expressionwith respect to τ and y in the strip (0, t)×R

d . Using the formulae ∂g/∂τ = −∂g/∂sand ∇yg = −∇x g we find that δt (u, uε) ≤ 0 where

δt (u, uε) :=∫∫∫∫

0<s,τ<t

{∂g

∂τ|u(s, x)− uε(τ, y)| + sgn(u(s, x)

−uε(τ, y))∇yg · ( f (u(s, x))− f (uε(τ, y)))

}dx dy ds dτ

−∫∫∫

0<τ<tg(τ, 0, x, y)|u0(x)− uε(τ, y)| dx dy dτ

+∫∫∫

0<τ<tg(τ, t, x, y)|u(t, x)− uε(τ, y)| dx dy dτ.

since t �→ TV(u(t)) is decreasing. Thus

‖u(t)− uε(t)‖1≤ 2Cβ TV(u0)+ ε

∫∫∫0<s<t

∇y�β(x − y) · ∇y|uε(s, y)

− u(s, x)| dx dy ds.

Let us accept temporarily the following result.

Lemma 6.4.3 Let p: R2d →R be a measurable function and µ a positive measure

of finite mass on Rd . We suppose on the one hand that y �→ p(x, y) is bounded and

smooth, and on the other hand that x �→ p(x, y) is of bounded variation and that|∇x p(x, y)| ≤ dµ(x) in the sense of the measures.

Then ∣∣∣∣∫∫

R2d∇y�β(x − y) · ∇y p(x, y) dx dy

∣∣∣∣ ≤ C

β

∫R

ddµ(x).

Let us apply the lemma with p(x, y) := |u(s, x)− uε(s, y)| and dµx := |∇xu(x, s)|to obtain

‖u(t)− uε(t)‖1 ≤ 2Cβ TV(u0)+ Cε

β

∫ t

0TV(u(s)) ds

≤ 2Cβ TV(u0)+ Cεt

βTV(u0), (6.34)

since TV(u(s)) ≤ TV(u0). Choosing β = √(εt), we obtain the result sought:

‖uε(t)− u(t)‖1 ≤ C√

εt TV(u0).

Proof of the lemma

If p is smooth with respect to both of its variables, we can carry out two integrationsby parts where we have made use of ∇2

x �β = ∇2y�β to arrive at∫∫

R2d∇y�β(x − y) · ∇y p(x, y) dx dy =

∫∫R

2d∇x�β(x − y) · ∇x p(x, y) dx dy.

We have an upper bound on the absolute value of this integral by∫∫R

2d|∇x�β(x − y)| dy dµ(x) =

∫∫R

2d|∇z�β(z)| dz dµ(x) = Cε

β

∫R

ddµ(x).

In the general case, we approximate p by a sequence (pn)n∈N of smooth functionswhich satisfy the hypotheses uniformly with respect to n, then we pass to the limitin the above inequality when n tends to infinity. It remains to give the

We recall the inequality (6.34) and we make use of the asymptotic estimate ofDafermos [15] (see Exercise 2.19):

TV(u(s)) ≤ C(u0)s−1/2,

where C depends on infR f ′′. Then

‖u(t)− uε(t)‖1 ≤ Cβ TV(u0)+ Cε

β

∫ t

0

ds√s

≤ Cβ TV(u0)+ Cε

β

√t .

Then we choose β2 = ε√

t .

6.5 Exercises

6.1 For the following systems, show that if the sequences (uε)ε>0 and ( f (uε))ε>0

are bounded in L1(ω) (where ω is a bounded open set in R+×R

d ) and convergein the sense of distributions to u and f (u) respectively, then

limε→ 0

∫ω

‖uε − u‖ dx dt = 0.

The Burgers equation.Isentropic gas dynamics. Here, we suppose that in addition the sequences

(ρε)ε>0 and ((ρε)−1)ε>0 are bounded in L∞(ω) and also that the pressureis of the form p(ρ) = ργ with γ > 1.

Show that this property is false for the system ut + (ϕ(‖u‖)u)x = 0 even whenϕ′ �= 0 and ϕ′′ �= 0, as well as for the system of isothermal gas dynamics(p(ρ) = cρ, where c is a constant).

6.2 We consider the scalar equation

ut = 0,

with its perturbation

uεt = εuε

xx .

We choose the initial condition

a(x) = −sgn x ={

1, x < 0,

−1, x > 0.

6.5 Exercises 217

Show that the viscous solution is of the form uε(x, t) = v(x/√

εt) where weshall make v explicit. Deduce that

∫R|uε(t) − u(t)| dx = C

√εt where C is

a constant. Thus, from the qualitative point of view, Kuznetsov’s theorem isoptimal.

6.3 Some readers might find the preceding exercise unconvincing since the initialcondition is not itself integrable. We recall therefore the equation ut = 0 andits perturbation ut = εuxx . The initial condition is now

a(x) = K s(x) = 1√s

K

(x√s

),

where K is the kernel of the heat equation:

dK s

ds= d2K s

dx2.

Show that uε(t) = K εt+s . We choose s = εt ; show that ‖u(t)−uε(t)‖1 dependsneither on ε nor on t , but that TV(u0) = c(εt)−1/2. Deduce that the best constantCV in the upper bound ‖u(t)− uε(t)‖1 ≤ CV (ε, t) TV(u0) is at least equal toC√

εt . Kuznetsov’s theorem is therefore optimal.Similarly, calculate ‖a‖1 and conclude that the best constant C1(ε, t) in the

inequality ‖u(t)− uε(t)‖1 ≤ C1(ε, t)‖u0‖1 satisfies C− ≤ C1(ε, t) ≤ 2 whereC− > 0 is independent of ε and of t .

6.4 We consider the Burgers equation,

ut +(

1

2u2

)x= 0,

with its perturbation

uεt + uεuε

x = εuεxx .

We choose the same initial conditions as in Exercise 2.

(1) Calculate the entropy solution u.(2) Show that the viscous solution is of the form uε(t, x) = u1(x/ε, t/ε). We

write z = u1.(3) Show that the restriction of z to the quarter-plane (R+)2 is the solution of

the parabolic problem with Dirichlet condition,

zt + zzx = zxx ,

z(0, x) = −1,

z(t, 0) = 0.

(4) let x �→ Y be the solution of the differential equation Y ′ = 12 (Y 2 − 1)

satisfying Y (0) = 0. Also, let b(t) ∈ [0, 1) be the solution of the equationlog|1 − b| + b = −t . Show that Y and b are defined on R

+, that Y < 0,Y ′ < 0, b′ > 0, b(0) = 0, b(+∞) = 1, Y (+∞) = −1.

(5) We define y(t, x) := Y (x/b(t)). Show that yt + yyx ≥ yxx , y(t, 0) = 0and y(0, x) = −1. Deduce from the maximum principle that

−1 ≤ z(t, x) ≤ y(t, x).

(6) Show that ∫R

|uε(t)− u(t)| dx ≤ 2∫

R

∣∣∣∣1+ Y

(σ

b(

tε

))∣∣∣∣ dσ,

then that ∫R

|uε(t)− u(t)| dx ≤ cεb

(t

ε

)

where C is a constant.(7) Compare this result with Kuznetsov’s theorem, respectively for t = o(ε)

and for t−1 = O(ε−1) (it seems natural that the uniform (in time) estimate∫R|uε(t)− u(t)| dx = O(ε) is true for a genuinely non-linear conservation

law (that is with f ′′ > 0) when limx→+∞ u0(x) < limx→−∞ u0(x)).

6.5 We consider the smooth solutions (u, v) of the isentropic Navier–Stokes equa-tion in which the kinematic viscosity ε > 0 is chosen to be constant. The spatialdimension is d = 1 and the equations are written in lagrangian coordinates:

vt = ux ,

ut + p(v)x = ε(ux/v)x ,

}t > 0, x ∈ (0, M).

The total mass M is finite. For simplicity, we assume that the fluid spreads outfreely in R. The boundary conditions are thus

εux

v= p(v), x = 0, M.

The pressure is a given smooth function which satisfies p(0) = +∞, p(+∞) =0, p > 0. We write e(v) :=−∫ v

1 p(w) dw and we suppose that e(0) = +∞,e(+∞) > −∞.

(1) Establish the energy estimate∫ M

0

(1

2u2 + e(v)

)dx + ε

∫ t

0

∫ M

0

u2x

vdx ds =

∫ M

0

(1

2u2

0 + e(v0)

)dx .

6.5 Exercises 219

(2) Let h(t, x) := ∫ x0 u(t, ξ ) dξ − ε log v. Show that ht = −p(v).

(3) Deduce that h(t, x) ≤ h(0, x), then calculate an explicit lower bound K ofv(t, x)/v0(x).

(4) Deduce also that h(t, x) ≥ h(0, x) − tq(Kv0(x)) where q is a suitablechosen function. Then calculate an explicit upper bound of v(x, t).

(5) Are the above estimates uniform with respect to ε? Can we deduce a boundfor the sequence (uε, vε)ε>0?

6.6 We consider again the smooth solutions of the Navier–Stokes equations, but foran isothermal fluid: p(v) = v−1. We write τ = εu − x . Verify that τ satisfiesa diffusion equation τt = (ατx )x where α will be identified. Deduce from themaximum principle an explicit bound of u. Is this bound uniform with respectto ε?

7

Viscosity profiles for shock waves

This chapter treats of an admissibility condition for discontinuous solutions of agiven hyperbolic system. This approach is restricted to a single space dimension(d = 1).

We regard the hyperbolic system

ut + f (u)x = 0 (7.1)

as ‘the limit’ of the perturbed system

ut + f (u)x = ε(B(u)ux )x (7.2)

when ε tends to zero. By ‘the limit’, we understand that weak solutions of (7.1) areadmissible if and only if they are pointwise limits almost everywhere of sequencesof solutions (uε)ε>0 of (7.2) which are locally uniformly bounded. The aim of thischapter is the study of the progressive waves for a system of the form (7.2) fromthe point of view of existence and asymptotic stability. These waves are used ascriteria of admissibility for shock waves of the system (7.1).

7.1 Typical example of a limit of viscosity solutions

A simple case of a family of solutions (uε)ε>0 of (7.2) is that of a progressive waveof speed s independent of ε whose profile is the same for each value of ε, within achange of scale:

uε(t, x) = U

(x − st

ε

). (7.3)

The condition that (7.3) defines a solution of (7.2) is that U : R → U ⊂ Rn

is a solution of the differential-algebraic system (differential system if B(U ) is

220

7.1 Typical example of a limit of viscosity solutions 221

invertible) (B(U )U ′)′ = ( f (U ))′ − sU ′, or equivalently

B(U )U ′ = f (U )− sU + const. (7.4)

In another way, to say that uε(x, t) converges almost everywhere and is locallybounded when ε tends to zero means that U is bounded and has limits, which wedenote by uL and uR, at±∞. The limit of uε is then a step function with two values:

u(t, x) ={

uL, x < st,

uR, x > st,

}(7.5)

where uL and uR satisfy the Rankine–Hugoniot condition

[ f (u)] = s[u]. (7.6)

Finally the equation satisfied by U is

B(U )U ′ = f (U )− f (uL)− s(U − uL) (= f (U )− f (uR)− s(U − uR)). (7.7)

Definition 7.1.1 We say that a discontinuous solution of (7.1) of the form (7.5)admits a viscosity profile U if U is a bounded solution of the system (7.7) (calledthe profile equation) which tends to uL at −∞ and to uR at +∞.

We note that this definition depends a priori on the viscosity tensor that we haveadopted.

Profiles vs. Lax’s entropy condition

As we have said above, a discontinuous solution, represented by (uL, uR, s) andwhich admits a viscosity profile, will often be considered as admissible (in fact weshall shortly impose a supplementary condition of stability which is frequently sat-isfied). The existence of a viscosity profile is thus seen here as a sufficient conditionof admissibility. On the other hand, if E is a strongly convex entropy (of flux F),for which the diffusion B is dissipative, Lax’s entropy condition [F(u)] ≤ s[E(u)]is a necessary condition since it expresses the inequality E(u)t+F(u)x ≤ 0, itselfthe consequence of the inequality

E(uε)t + F(uε)x ≤ ε(dE(uε) · B(uε)uε

x

)x .

Moreover we verify directly that the existence of a profile implies the entropycondition since if uR �= uL and if (D2

u Eη | B(u)η) ≥ c(u)‖B(u)η‖2, then

(F(U )− sE(U ))′ = dE(U ) · ( f (U )− sU )′ = dE(U ) · (B(U )U ′)′

222 Viscosity profiles for shock waves

and therefore

(F(U )− sE(U )− dE(U ) · B(U )U ′)′

= −(D2E U ′ | BU ′)≤− c(U )‖BU ′‖2. (7.8)

In practice, the last term cannot be identically zero without U being stationary. Thusξ �→ F(U )− sE(U )− dE(U ) · B(U )U ′ is strictly decreasing and its evaluation at±∞gives [F(U )] < s[E(U )]. The existence of a viscosity profile is thus a sufficientcondition of admissibility which is not necessary since a contact discontinuitycannot have such a profile (the contact discontinuities satisfy [F(U )] = s[E(U )]for every entropy). We shall see in §7.2 a subtler version of this remark, but rightnow we can conclude that the truth concerning the admissibility of discontinuoussolutions of (7.1) lies somewhere between the existence of a viscosity profile andLax’s entropy condition.

Profile vs. Lax’s shock condition

Pushing ahead of the analysis we can ask if a discontinuous wave (uL, uR, s) admit-ting a profile for a viscosity tensor, but whose profile does not persist under a smallperturbation of the data (for example B �→ B , f �→ f , or (uL, uR; s) �→ (uL, uR; s)with the constraint [ f (u)] = s[u]), is admissible. Actually, the system itself, oragain the states and the speed of a wave, are never perfectly known and the impre-cision of these data contributes to the instability of the profile, and always hindersobserving this in practice. In addition, we therefore shall impose on the viscosityprofiles the condition that they persist when the profile equation is perturbed. Thisis the structural stability of the profile.

In general, the system (7.7) is a differential-algebraic system since B(U ) is notnecessarily invertible. It is correct in general to assume that p, the rank of B(u),is constant. Then, we isolate n − p algebraic equations, applying a projectionparallel to the image of B. In general, these define a sub-manifold V (uL; s) in U ofdimension p, on which the profile equation consists of the dynamical system

v′ = g(uL, s; v) (7.9)

provided that ker B(u) is supplementary to both Im B(u) and the tangent spaceat u to V (uL; s). The fact that R

n = ker B(u) ⊕ Im B(u) follows for examplefrom the existence of strongly convex entropy E for which B is dissipative, inthe sense in which there exists a number c(u) > 0 such that (D2

u Eξ, B(u)ξ ) ≥c(u)‖B(u)ξ‖2. Indeed, if z ∈ ker B(u)2 and y= Bz, then for all α ∈R, we have(D2

u E(y + αz), αy) ≥ α2‖y‖2, which, in the limit α→ 0, gives (D2u Ey, y)= 0,

hence y = 0: z ∈ ker B(u). Finally, ker B(u)2 = ker B(u), which is the expected

7.1 Typical example of a limit of viscosity solutions 223

property. Of course, uL and uR are in V (uL; s) and are stationary points of thereduced system (7.9). Hence, the profile U is a trajectory of this system, which linksthe critical points uL and uR. This is contained in the stable manifold of uR, denotedby W s(uR), and in W i(uL), the unstable manifold of uL. It is structurally stable ifand only if W s(uR) and W i(uL) are transverse to each other, that is to say when ata point U (ξ ) of the trajectory their tangent spaces satisfy (in an obvious notation)

T sR(U (ξ ))+ T i

L(U (ξ )) = TU (ξ )V (uL; s). (7.10)

This condition does not depend on the choice of the point U (ξ ) on the trajectory,but only on the trajectory itself. Since these two tangent spaces contain in commonthe tangent to the trajectory, the condition of tranversality implies in particular aninequality between dimensions,

dim W s(uR)+ dim W i(uL) ≥ p + 1. (7.11)

Let us look, in more detail, at the case of an invertible diffusion (p= n). We haveV (uL; s) = U. Let us suppose that, in addition, the system (7.1) possesses a stronglyconvex entropy E for which B is dissipative. This system is therefore hyperbolicand we denote by λ1(u), . . . , λn(u) the eigenvalues of d f (u), arranged in increasingorder. Furthermore, we suppose them to be of constant multiplicities.

Let us re-write the profile equation in the form

u′ = g(U ) =: B(U )−1( f (U )− f (uR)− s(U − uR)).

First of all, we prove a lemma.

Lemma 7.1.2 We suppose that the speed s satisfies λk(uR) < s < λk+1(uR) for acertain index 1≤ k ≤ n. Then the endomorphism dg(uR) does not have an eigen-value with real part zero. The sum of the multiplicities of eigenvalues with negativereal part is k.

Proof Let us write C = d f (uR)−s In which is invertible, and S = D2E(uR) whichis symmetric and positive definite. Since B is dissipative, we have

(Sη | B(uR)η) > 0, ∀η ∈ Rn, η �= 0.

In particular,

(Sη | B(uR)η) > 0, ∀η ∈ Cn, η �= 0.

Let η be an eigenvector of dg(uR) = B−1C , associated with an eigenvalue λ. Thisis not null and we have

λ(Sη | Bη) = (Sη |Cη).

The right-hand side of this formula is real because, E being an entropy of the system(7.1), the matrix SC is symmetric. If λ = 0 we therefore have 0 = (Sη | Bη)which is false. Therefore λ �= 0.

This conclusion remains true when we replace B by In +αB with α ∈ R+ since

this is again a dissipative tensor for E . By continuity, the number of eigenvalues n(α)of (In+αB)−1C in the half-plane z < 0, counted with their orders of multiplicity,does not depend on α. Hence, it is equal to n(0) = k. Dividing by α and letting α

tend to infinity we obtain the stated result.

When s is an eigenvalue of d f (uR), we obtain a similar result by applying thelemma with s1 �= s and letting s1 tend to s: if λ j (uR) < s= λ j+1(uR) = · · · =λk(uR) < λk+1(uR) the number of eigenvalues of dg(uR) in the half-plane (z) < 0(respectively in the half-plane (z) > 0) is equal to j (respectively n − k). Thedimension of W s(uR) is therefore between j and k. Similarly, that of W i(uL) isbounded above by n − m where λm(uL) < s ≤ λm+1(uL). Then, the inequality(7.11) shows that k − m ≥ 1. We deduce Lax’s shock condition.

Theorem 7.1.3 We consider a hyperbolic system whose characteristic speeds areof constant multiplicities, provided with a strongly convex entropy E. We considera diffusion tensor B, strictly dissipative for E. Let (uL, uR; s) be a discontinuity of(7.1) for which there exists a structurally stable viscosity profile. Then there existsan index 1 ≤ k ≤ n such that we have

λk(uR) ≤ s ≤ λk(uL). (7.12)

Naturally, Lax’s condition (7.12) is necessary but not sufficient for a viscosityprofile to be stable. However, the strict inequalities λk(uR) < s < λk(uL) constitutea sufficient condition when uR is close enough to uL. In fact, in this case, λk cannotbe linearly degenerate (as uR is on the kth Hugoniot curve which originates at uL,(see Theorem 4.2.1), and we have λk(uR) �= λk(uL)) so it must be simple. Therefore,we have λk−1(uL) < s < λk+1(uR). Thus, dim W s(uR)= k and dim W i(uL) = n −k+1, W s(uR) being tangent to the invariant sub-space Y−R of dg(uR) associated witheigenvalues of strictly negative real part and W i(uL) being tangent to the invariantsub-space Y+L of dg(uL) associated with eigenvalues of strictly positive real part.As uR is near to uL, s is also near to λk(uL). Let us denote by X−, X+ and X0 theinvariant sub-spaces of B(uL)−1(d f (uL) − λk(uL)In) associated with eigenvalueswhose real parts are positive, negative and zero respectively. Then X0 is a straightline and we have R

n = X− ⊕ X0 ⊕ X+. In addition, Y−R is near to X− ⊕ X0 andY+L is near to X+ ⊕ X0. Since X− ⊕ X0 and X+ ⊕ X0 are transverse to each other,the same is true of Y−R and Y+L . At every point of the trajectory, the tangent spaces

7.2 Existence of the viscosity profile for a weak shock 225

T sR(U (ξ )) and T i

L(U (ξ )) are close to Y−R and Y+L , therefore they too are transverseand the profile is structurally stable.

It now remains to show that such viscosity profiles exist. This we shall do in §7.2under the non-linearity condition (dλk · rk)(uL) �= 0.

7.2 Existence of the viscosity profile for a weak shock

Being given two distinct states uL and uR in U and a number s satisfying theRankine–Hugoniot condition (we thus have s = σ (uL, uR)), we want to know ifthere exists a trajectory of the system (7.7) going from uL to uR. We have seen thatif the tensor B is strictly dissipative for a strongly convex entropy E , of flux F , thenthe existence of a viscosity profile from uL to uR implies the entropy inequalityF(uR)− F(uL) < s(E(uR)− E(uL)) and hence excludes the possibility of a profilegoing from uR to uL. In what follows, we shall suppose that B is invertible,that there exists a strongly convex entropy E and that B is strictly dissipative:(D2Eξ, B(u)ξ ) ≥ c(u)‖ξ‖2 with c > 0. Finally, we abandon the use of capitalletters in the notation of a profile.

We begin with the study of the scalar case, which is the simplest and the mostcomplete.

The scalar case

If n = 1, B is a numerical function, denoted by u �→ b(u), which satisfies b > 0.The profile equation is here

u′ = g(u) =:f (u)− f (uL)− s(u − uL)

b(u).

We have g(uL) = g(uR) = 0. The trajectories of the equation are strictly monotonicand there exists one which links uL to uR if and only if (uR − uL)g(u) is strictlypositive for all u lying between uL and uR. By this means we recover Oleınik’scondition, where the inequalities in the broad sense have been replaced by the strictinequalities:

uL < uR: there is a viscosity profile if and only if the graph of the restriction off to (uL, uR) is strictly above its chord.

uL > uR: there is a viscosity profile if and only if the graph of the restriction off to (uR, uL) is strictly below its chord.

We note that the existence of a viscosity profile does not depend on the choice of theviscosity b. On the other hand, if f is not affine on any non-trivial interval, everyshock (uL, uR; s) satisfying Oleınik’s condition can be seen as the juxtaposition ofshocks for which there exist viscosity profiles.

The case of weak shocks with B = b(u)I n

Let us return to the general case of a system of the form (7.1), which we assumeto be strictly hyperbolic and to be provided with an eigenvalue λk genuinely non-linear at uL. The Hugoniot locus H (uL) is locally the union of sub-manifolds whichare tangent at uL to the eigenspaces of d f (uL). Since λk is genuinely non-linear atuL, it is a simple eigenvalue. We denote by �k the Hugoniot curve tangent to theeigenvector rk(uL) at uL, which is normalised by dλk · rk ≡ 1. We shall denote bylk the differential eigenform normalised by lk · rk ≡ 1.

Let us recall that uL locally separates �k(uL) into two connected components onwhich we have

if u ∈ �+k , λk(uL) < σ (uL, u) < λk(u),if u ∈ �−k , λk(u) < σ (uL, u) < λk(uL).

For a scalar diffusion, we have the following result, which is due to Foy [23].The proof which follows is that of Goodman [36].

Theorem 7.2.1 We suppose that λk is genuinely non-linear at uL and that B(u) =b(u)In with b(u) > 0. For every neighbourhood V of uL, there exists a neighbour-hood W of uL such that if uR ∈ W ∩�k(uL), the discontinuity (uL, uR, σ (uL, uR))admits a viscosity profile with values in V if and only if this is a shock (that is tosay if and only if uR ∈ �−k (uL)).

Proof Even if it entails a change of the parametrisation of the profile equation, wecan suppose that b ≡ 1. The existence of a profile for this discontinuity is equivalentto the existence of a heteroclinic orbit joining (uL, σ ) to (uR, σ ) for the augmentedsystem

d

dξ

(v

s

)=

(g(v, s)

0

)=: G(v, s),

g(v, s) = f (v)− f (uL)− s(v − uL). (7.13)

At the stationary point P = (uL, λk(uL)), 0 is an eigenvalue of multiplicity 2 for

dG(P) =(

d f (uL)− λk(uL)In 0

0 0

)

and ker dG(P) is of dimension 2, spanned by (rk(uL), 0) and (0, 1). In addition,there is no other eigenvalue with real part zero, since they are all real. The theoremof the centre manifold [108] ensures the existence of a sub-manifold M locally

7.2 Existence of the viscosity profile for a weak shock 227

invariant for (7.13), of dimension 2 and which contains all the orbits which remainin a sufficiently small neighbourhood V1 ⊂ V of uL. This manifold is tangent atuL to the kernel of dG(P). Since (v, s) �→ (s, x =: lk(uL) · (v− uL)) is a system ofaffine coordinates on dG(P), we can choose the same coordinates on M in V1. Forthese the flow of (7.13), restricted to M , is vertical (s = const.). See Fig. 7.1.

Let us pass in review certain trajectories that M is bound to contain. First ofall, we must have the zeros of G in V1. There are two kinds, which form twosmooth curves according to Theorem 4.2.1; firstly those of the form (uL, s), s ∈ R,then those of the form (u, σ (uL, u)) for u ∈ �k(uL). For the latter, we have theformula σ (uL, u) − λk(uL) ∼ 1

2 lk(uL) · (u − uL), that is, s − λk(uL) ∼ 12 x . This

curve is therefore transverse on the one hand to the preceding curve ({uL} × R)and on the other hand to the flow. It follows that, in a neighbourhood of P ,each vertical line s = σ �= λk(uL) contains exactly two critical points of theflow of (7.13), let us say (uL, σ ) and (uR, σ ) with σ = σ (uL, uR). The segmentwhose extremities are these two points, invariant by the flow, is therefore a het-eroclinic trajectory from one to the other, whose direction of motion remains tobe determined. We note that between these two points, the only heteroclinic tra-jectories which remain in V1 are obtained from the preceding by a shift of theparameter.

To know the direction of the motion of this trajectory, it is enough to know ifthe critical point (s= σ, x = 0) is attractive or repulsive for the flow restricted tothe vertical s = σ (uL, uR) of M . The trajectory goes from (uL, σ ) to (uR, σ ) (andhence corresponds to the profile sought) if and only if (s, 0) is repulsive. The flowon this line can be described by the differential equation dx/dξ = h(σ, x) whereh(s, x) =: lk(uL) · ( f (u)− f (uL)− s(u−uL)). We have u−uL = xrk(uL)+O(x2)since M contains the straight line {uL} × R and is tangent to (rk(uL), 0). Thush(s, x) = (λk(uL)− s)x +O(x2) and (dh/dx)(s, 0) = λk(uL)− s. Hence the point(s, 0) is repulsive if and only if λk(uL) − σ (uL, uR) > 0, that is to say if and onlyif uR ∈ �−k .

Extensions of Theorem 7.2.1 (B �= bIn)

Since uR is near to uL, the viscosity profile which we have constructed in thecase where B= b(u)In is structurally stable (see §7.1). We deduce that if the vis-cosity tensor at uL is nearly a strictly positive scalar matrix, then a weak shock(uL, uR; σ (uL, uR)) again admits a viscosity profile, still structurally stable.

When B is not near to a scalar matrix, we use the dissipative hypothesis of B(u)relatively to the strongly convex entropy E . Recall the proof of Theorem 7.2.1 with

Fig. 7.1: The flow of G on the centre manifold at (uL, λk(uL)).

now

G B(v, s) =:

(B(v)−1g(v, s)

0

).

The critical points of G B are the same as those of G. We have

dG B(P) =(

B−1C 0

0 0

),

with C = d f (uL) − λk(uL)In . The matrix B−1C has only one eigenvalue whosereal part is zero, λ = 0, which is simple (see Lemma 7.1.2 and the subsequentcomments). Hence there is again a centre manifold MB of dimension 2, which hasthe same tangent space as M at uL. Therefore, the flow is again transverse to thetwo curves from critical points and there is always a trajectory which links (uL, σ )and (uR, σ ), when uR is close to uL and σ = σ (uL, uR). As

[F(u(ξ ))− σ E(u(ξ ))]ξ=+∞ξ=−∞ < 0,

the trajectory goes from (uL, σ ) to (uR, σ ) if and only if (uL, uR, σ ) satisfies Lax’sentropy condition. Finally:

Theorem 7.2.2 We suppose that the system (7.1) is provided with a strongly convexentropy E for which the diffusion B is strictly dissipative. Let u �→ λk(u) be a simpleeigenvalue of d f , genuinely non-linear at uL: dλk(uL) · rk(uL) �= 0. Then for everyneighbourhood V of uL, there exists a neighbourhood W of uL such that if uR ∈�k(uL) ∩W , the discontinuity (uL, uR, σ (uL, uR)) admits a viscosity profile with

7.3 Profiles for gas dynamics 229

values in V if and only if it satisfies Lax’s entropy condition: F(uR)− σ E(uR) <

F(uL) − σ E(uL). In addition, this profile is unique to within a translation of theparametrisation.

7.3 Profiles for gas dynamics

The reader may verify by himself (Exercise 7.6) that the existence of a viscosityprofile does not depend on the choice of variables (eulerian or lagrangian), providedthat the diffusion is not carried by the equation of the conservation of mass. Hence,we treat only the case of the equations in lagrangian coordinates.

Isentropic fluid with viscosity

For an isentropic fluid, the natural perturbation is the newtonian viscosity:

vt = zx ,

zt + p(v)x = ε(b(v)zx )x .

}

For a discontinuity (uL, uR, s) of the system without viscosity, hence, which satisfiesthe Rankine–Hugoniot condition

[z]+ s[v] = 0, [p(v)] = s[z],

the profile equation is

b(v)z′ = p(v)− p(vL)− s(z − zL),

z − zL = −s(v − vL).

}

The existence of the viscosity profile is equivalent to that of a hetero-clinic orbitlinking vL to vR for the reduced equation (in which s �= 0)

−sb(v)v′ = p(v)− p(vL)+ s2(v − vL). (7.14)

The study of this equation is entirely analogous to that of the scalar case. We denoteby I the open interval whose extremities are vL and vR.

Theorem 7.3.1 We suppose that b > 0 and p are smooth functions of v. Let(uL, uR; σ (uL, uR)) be a discontinuity of isentropic gas dynamics.

Case σ (uL, uR)(uL − uR) > 0 : There exists a viscosity profile if and only if thegraph of v �→ p(v) restricted to I is situated strictly above its chord.

Case σ (uL, uR)(uL − uR) < 0 : There exists a viscosity profile if and only if thegraph of v �→ p(v) restricted to I is situated strictly below its chord.

Remark In the above statement we have not made the hypothesis p′ < 0 (hyperbol-icity in the inviscid model). The system without viscosity can be elliptic or simplybe able to change type according to the value of v. If p′ takes positive values, theinviscid Cauchy problem is ill-posed, in the sense of Hadamard, with the result thatthe system is not a reasonable model for gas dynamics. The states v > 0 for whichp′(v) > 0 must be excluded by a mathematical criterion which describes faithfullythe physics of the problem. As the preceding calculation allows the construction ofviscosity profiles between two states uL and uR for which p′(uL) and p′(uR) are ofopposite signs, it seems that viscosity alone is unable to provide such a criterion. Weshall see in Exercise 7.3 a more complete approach which gives plausible results,thanks to the introduction of a capillary force. When p has a local minimum at β

and a local maximum at γ > β, the model represents a fluid able to occupy a liquidphase and a gaseous phase; the typical equation of state is that of Van der Waals.

7.4 Asymptotic stability

Generalities on the stability of profiles

Viscosity profiles are of importance only inasmuch as we can observe them in ex-periments or in numerical simulation. This is due to their stability in many differentcontexts, stability which is general for scalar problems and frequent for systems.For example, we have shown the structural stability of weak shocks associated withgenuinely non-linear characteristic fields, it was a matter of stability relatively toa perturbation of the profile equation. In this section and the following one weshall consider another type of stability, where the viscosity is definitively fixed(hence we can take ε= 1) but where the viscosity profile is perturbed at the ini-tial instant and where we make the time t tend to infinity. Then we seek if thisprofile is asymptotically stable when t tends to infinity. This general problem hasprincipally been the object of two important studies, that of Sattinger [88] in thescalar case and that of Liu [70] for systems.1 Sattinger treats the scalar case bya spectral analysis of the linearised problem and shows the exponential decay ofthe error (that is of the difference between the solution u(t) and the profile). In thecase of a uniformly convex flux f , this result had been obtained previously by Il’in and Oleınik [47]. In general the decay is shown in spaces with weights (theseweights enable us, in the general procedure of Sattinger, to shift the spectrum ofthe linearised operator to a convenient half-plane). The weights chosen are of the

1 See also the valuable recent work by R. Gardner and K. Zumbrun (Comm. Pure Appl. Math. 51 (1998), 797–855).

7.4 Asymptotic stability 231

form eα|x |. With the spaces L2((1 + x2)αdx), Kawashima and Matsumura havealso shown the algebraic decay of the error. However, Osher and Ralston [82] haveshown the convergence in L1(dx) by making use of the contraction properties ofthe semi-group S(t). In that which concerns the systems, Liu [70] uses the energyestimates and a precise analysis of the waves associated with each characteristicfamily and with the diffusion. Pego [83] has shown that a spectral analysis of thelinearised operator is again possible for certain shocks. But, in all the cases, thestability of the viscosity profiles has been shown only for shocks satisfying strictlyLax’s shock condition

λk(uR) < σ < λk(uL).

At the present time, it seems that there is no known stability theorem for viscosityprofiles including the case s = λk(uL) or s = λk(uR). However, for a scalar equation,Ming Mei [76] obtains a decay rate for small initial perturbation, supposing thatf ′′′ does not vanish between uL and uR. We give below a result concerning thescalar case without a hypothesis concerning either the flux f or the perturbation.Obviously, this does not contain an estimate of the speed of convergence towardsa profile since this convergence might be very slow.

In the next section, we shall give, without proof, a description of the results dueto Liu for systems.

The stability problem is posed in the following way. We consider a hyperbolicsystem in space of one dimension,

ut + f (u)x = 0,

augmented by a diffusion term of the second order:

ut + f (u)x = (B(u)ux )x .

We suppose that a discontinuity (uL, uR, σ ) of the hyperbolic part admits a viscosityprofile, denoted by ξ �→ U (ξ ). We then consider a perturbationϕ ∈ L1(R)∩ L∞(R)of the profile, that is to say that we solve the Cauchy problem with diffusion for theinitial condition

u0(x) = U (x)+ ϕ(x).

We ask if the solution (supposed to exist globally in time and to be unique) convergesin a suitable space, let us say L p(R), to the profile when t →∞. Since the profileis not unique (every shift gives rise to another one) and since L p is Hansdorff, theanswer is obviously no (it is enough to choose ϕ =: U (· + x0)−U and to note thatwe still have ϕ ∈ L1 ∩ L∞) and the stability sought is rather an orbital stability.Therefore, we ask if there exists a phase shift x0 such that the solution of the Cauchy


problem satisfies

limt→+∞‖u(t)−U (· −σ t − x0)‖ = 0

for a suitable norm. We shall see, and it is essential in this study, that the phaseshift can be calculated explicitly as a function of

∫R

ϕ(x) dx , as a result of theconservation laws when (uL, uR, σ ) is a Lax shock. In particular, it is for thisreason that the perturbation must be integrable.

The scalar case

The scalar case is the simplest one and we lay out for it the most complete results.First of all, let us note that the theory of monotonic operators lets us construct asemi-group (S(t))t≥0 which solves the Cauchy problem for the equation

ut + f (u)x = uxx (7.15)

when the initial datum u0 is in L∞(R). As the equation (7.15) satisfies the maximumprinciple, this semi-group enjoys the following properties.

(SG1) (smoothness) For all α ∈ L∞(R), S(t) a is infinitely differentiable on R forall t > 0.

(SG2) (maximum principle) If a ≤ b almost everywhere, then S(t) a ≤ S(t) b, forall t > 0.

(SG3) (conservation of mass) If a− b ∈ L1(R), then S(t) a− S(t) b ∈ L1(R), andwe have for all t > 0∫

R

(S(t) a − S(t) b) dx =∫

R

(a − b) dx .

(SG4) (contraction) Under the same hypothesis as (SG3), the mapping t �→‖S(t)a − S(t)b‖ is decreasing.

It is not necessary for the understanding of this section to prove the above assertions.They are classical.

We consider now a viscosity profile U for the equation (7.15), joining two valuesuL and uR. Denoting by c the speed of the shock wave between uL and uR, we areprovided with a one-parameter family of progressive waves

(x, t) �→ uh(x, t) =: U (x + h − ct).

We consider the stability of u0. Let ϕ ∈ L1(R) ∩ L∞(R) be a perturbation of theinitial condition which thus becomes u0(x) = U (x)+ϕ(x). Throughout this sectionwe write u(t) = S(t)u0. We ask if there exists a number h such that ‖u(t)−uh(t)‖1

7.4 Asymptotic stability 233

tends to zero when t tends to infinity. If such is the case, then the property (SG3)shows that

∫R

(u0 − uh(0)) dx = 0, that is to say∫R

(U (x)+ ϕ(x)−U (x + h)) dx = 0.

However, Lebesgue’s theorem shows that h → ∫R

(U (x + h)−U (x)) dx is differ-entiable with derivative

∫R

U ′(x + h) dx = uR − uL. We therefore have∫

R(U (x +

h)−U (x)) dx = h(uR − uL) and the phase shift, if it exists, is determined by

h = 1

uR − uL

∫R

ϕ(x) dx . (7.16)

It remains to show that for this value of h, u and uh are asymptotically equivalent forthe distance defined by ‖·‖1 which we shall now state for a reasonable perturbation.

Theorem 7.4.1 Let U be a viscosity profile which joins uL to uR. Let a perturbationϕ be a measurable function on R, such that U + ϕ is contained between twotranslates of U (there exist a, β ∈ R such that U (x+α) ≤ U (x)+ϕ(x) ≤ U (x+β)almost everywhere). Then the solution u(t) = S(t)(U + ϕ) of the Cauchy problemfor (7.15) satisfies

limt→+∞‖u(t)−U (· + h − ct)‖1 = 0,

h being defined by the formula (7.16).

We notice that from the hypothesis of this theorem ϕ is integrable and bounded onR. It is probable that this result remains true if ϕ ∈ L1 ∩ L∞, but no such resultexists at the moment.2 In any case, we can report on the work of H. Weinberger[111] in the case where the Riemann problem between uL and uR also involvesrarefaction waves.

Proof First of all, even if it means making the change of variables (t, x) �→ (t, x −ct), we can suppose that the shock is stationary: c= 0. Hence, the functions (t, x) �→U (x + α) and (t, x) �→U (x + β) are stationary solutions of (7.15). Using themaximum principle and the hypothesis concerning the perturbation (which ensuresthat ϕ is bounded), we have U (x + α) ≤ u(t, x) ≤ U (x + β). Let us write v(t) =u(t) − U . Then v(t) is included between two integrable functions which do notdepend on the time so remains within a bounded set of L1(R). In addition, thecontraction property yields the inequality ‖v(t, · + r ) − v(t)‖1 = ‖u(t, · + r ) −2 H. Freistuhler and the author have succeeded in proving stability in L1 for every initial perturbation ϕ ∈ L1(R).

This result has appeared in Communications in Pure and Applied Mathematics 51 (1998), 291–301.


u(t)‖1 ≤ ‖u(0, · + r ) − u(0)‖1 = ‖ϕ(· + r ) − ϕ‖1 which tends to zero with r .By the compactness theorem of Frechet and Kolmogorov, the family (v(t))t≥0 istherefore relatively compact in L1(R). The ω-limit set A =: U+⋂

s≥0 Bs where Bs

is the closure in L1(R) of {v(t) : t > s} is non-empty since A−U is the decreasingintersection of non-empty compact sets. This set is that of all cluster points for thedistance d(z, w) = ‖z − w‖1 of sub-sequences (u(tn))n∈N where tn �→ ∞.

The ω-limit set is invariant under the semi-group S since if a ∈ A, with a =limn→∞ u(tn), then S(t) a = limn→∞ u(t + tn). For the same reason, S(t): A →A is onto as we also have a= S(t) b where b is a cluster point of the sequence(u(tn− t))n∈N. The smoothness property (SG1) therefore implies that A is includedin C

∞.Now, let k ∈R. The decreasing function t �→ ‖u(t) − U (· − k)‖1 admits a limit

denoted by c(k) when t →∞. If a ∈ A, we deduce that ‖a − U (· − k)‖1 = c(k).However, S(t) a again belongs to A, so that it follows that the function t �→ ‖S(t) a−U (·−k)‖1 is constant. Let us write provisionally w(t) = S(t) a and z(t) = S(t) a−U (· − k). We have

0 = d

dt‖z(t)‖1 =

∫R

zt sgn z dx .

Now zt + ( f (w) − f (U (· − k)))x = zxx ′ were we have used the profile equationfor U : Uxx = f (U )x . Multiplying this by sgn z, we deduce that

|z|t + (( f (w)− f (U (· − k))) sgn z)x = zxx sgn z,

which after an integration over R gives

d

dt

∫R

|z| dx =∫

R

zxx sgn z dx .

Finally,

0 =∫

R

zxx sgn z dx . (7.17)

However, the a priori estimates made at the time of the construction of the semi-group S show that wxx is integrable over R and hence so also is zxx . Therefore,using the theorem of dominated convergence, we have

0 = limε→0

∫R

zxx j ′ε(z) dx

where jε(τ ) =√

(ε2 + τ 2). Integrating by parts, we have

0 = limε→0

∫R

z2x j ′′ε (z) dx .

7.5 Stability of the profile for a Lax shock 235

Let x0 be a point where z vanishes and let γ > |zx (x0)|. For ε > 0, sufficientlysmall, we have‖z‖ < ε on (x0− γ ε, x0+ γ ε) since z is differentiable. Now j ′′ε (τ ) =ε−1 J (τ/ε) with J (τ ) = (1+ τ 2)−3/2. Thus∫

R

z2x j ′′ε (z) dx ≥ 1

ε

∫ x0+γ ε

x0−γ ε

J (1)z2x dx,

of which the right-hand side tends to 2γ J (1)zx (x0)2 when ε tends to zero. We deducethat zx (x0)= 0. Finally we have proved (taking t = 0 in the preceding calculation)that

∀a ∈ A, ∀k ∈ R, (a(x) = U (x − k)) ⇒ (a′(x) = U ′(x − k)). (7.18)

To conclude, we note first of all that a lies between U (·+α) and U (·+β) as limitof such functions, hence a takes its values strictly between uL and uR. Thus thefunction x �→ k(x) =: x −U−1 ◦ a(x) is well-defined and smooth (recall that U isstrictly monotonic). By construction, a(x) = U (x−k(x)), which on differentiationgives a′(x) = U ′(x − k(x))(1− k ′(x)). Using (7.18) we find that

U ′(x − k(x))k ′(x) = 0

and hence that k ′(x)= 0 since U ′ does not vanish. Finally, k is a constant anda = U (· − k). However, the elements of A satisfy∫

R

(a −U − φ) dx = 0,

which, as we have seen, fixes the value of k: we have k = −h.Hence, we have proved that the ω-limit set is reduced to a single-element

U (· + h). Since the family (v(t))t≥0 is relatively compact in L1(R) and as it hasonly a single limiting value when t →+∞, it is convergent, that is

limt→+∞‖u(t)−U (· + h)‖1 = 0.

7.5 Stability of the profile for a Lax shock

We consider now a system with a parabolic conservation law

ut + f (u)x = uxx , (7.19)

for which the first order part is strictly hyperbolic. To simplify the notation, wesuppose that all the eigenvalues of d f (u)= A(u) are simple (that is, including thoseof the linearly degenerate fields). We denote them by λ1(u) < · · · < λn(u). Theeigenvectors to the right and left are denoted respectively by r j (u) and l j (u) with

l j · rk = δkj and with the normalisation dλ j · r j ≡ 1 for all the genuinely non-linear

fields. It is solely for the simplicity of the exposition, related to concentrating ourattention on a single concept at a time, that we have chosen the matrix B(u) ≡ In .In the last sub-section of this section we shall consider other diffusion tensors.

Transport vs. diffusion

Let p be the index of a genuinely non-linear characteristic field. We consider aLax p-shock (uL, uR; c) which admits a viscosity profile (for example a weakshock, according to Theorem 7.2.1), denoted by U . For a perturbed initial con-dition u(0, ·) = U + ϕ where ϕ is given in (L1 ∩ L∞(R))n we seek again theasymptotic behaviour, when t →+∞, of the solution u(t) of the Cauchy problemfor (7.19). In particular, one stage of the argument consists of proving that thissolution is defined for all time t > 0. But as the asymptotic description makes useof the a priori estimates of u(t) − U in L1 ∩ L∞ and of the fact that the estimatein L∞ is sufficient for showing that the solution remains defined, the two stages(existence, asymptotic behaviour) are done together.

The description cannot be as simple as in the scalar case. In fact, if there existsa phase shift h ∈R such that limt→+∞ ‖u(t) − U (· + h − ct)‖1 = 0, then theconservation of mass, expressed as∫

R

(u(t, x)−U (x + h − ct)) dx = const.,

entails that ∫R

(U (x + h)−U (x)) dx =∫

R

ϕ(x) dx,

that is, that h(uR− uL) = ∫R

ϕ(x) dx . Now the mass m = ∫R

ϕ dx is a vector in Rn

which has no reason to be collinear with [u] = uR − uL.

Therefore, it is necessary to introduce waves of another type which carry aconstant and calculable mass. These waves, called diffusion waves, are of smallamplitude, of the order of t−1/2, with the result that we shall have, despite all,limt→+∞ ‖u(t)−U (· + h − ct)‖∞ = 0, for an appropriate phase shift. They willbe (asymptotically) localised in one of two sectors x − ct � 1 and x − ct � 1,in which U (x − ct) is approximately constant. Let us look at the case of the sectorx − ct � 1, where U and hence u has a value very close to uR. The hyperbolicpart of the system (7.19) allows waves of speeds λ j (uR) to propagate. If j < p,these waves merge with the viscosity profile, where u is no longer nearer to uR;these waves cannot therefore be present in an asymptotic description. It is the samefor j = p because of Lax’s shock inequality λp(uR) < c. Finally, for j > p, these

waves lengthen the zone occupied by the profile and do not interact with it; hence,we shall observe them for all times large enough. Of course, the diffusion dampensdown these waves (which explains their amplitude being O(t−1/2)) at the sametime as they spread out. But this spreading out is a very slow process, of the samenature as the brownian motion in which the location of a particle has expectation(λ j (uR)−c)t which is linear in time, whereas its standard deviation is only of ordert1/2. Similarly, these waves will become asymptotically uncoupled from each othersince they get farther apart with non-zero speeds λ j (uR)−λk(uR). Finally, we shallobserve asymptotically n uncoupled waves, of which none is small in L1(R) thoughn − 1 are in L∞(R):

the viscosity profile of the shock (uL, uR; c), which has a phase shift h to bedetermined,

the diffusion waves of speed λk(uR) for k > p, which lengthen the shock zone tothe right,

the diffusion waves of speed λk(uL) for k < p, which lengthen the shock zone tothe left.

The asymptotic behaviour will then be able to be predicted if each one of thesewaves depends on a real parameter and if the principle of the conservation of massleads to their calculation. We shall see in a later sub-section (Liu’s theorem) underwhat condition, of a geometrical nature, that is possible.

Non-linear diffusion waves

We now construct the diffusion waves. As we anticipate that these will be of smallamplitude and localised in a domain where the solution is nearly constant webegin by linearising (7.19), for example, around uR (we shall see however that thelinearisation is not precise enough):

vt + A(uR)vx = vxx . (7.20)

Using the eigenvectors of the matrix A(uR), we decouple this system of n transport–diffusion equations. We express v in terms of the basis of eigenvectors, v(t, x) =∑

i wi (t, x)ri (uR), and we obtain

∂twi + λi (uR)∂xwi = ∂2x wi , 1 ≤ i ≤ n. (7.21)

The change of variables (t, x) �→ (t, x − λi (uR)t) transforms this equation into theheat equation wt = wxx , for which we know the asymptotic behaviour in L1(R),governed by the total mass m(w). If m(w) = ∫

Rw dy then w(t) ∼ m(w)k(t) where


k(t) is the fundamental solution

k(t, y) = 1√tK

(y√t

), K (ξ ) = 1

2√

πexp

(−ξ 2

4

).

Let us now see why the linearisation is not satisfactory. The error due to thelinearisation of (7.19) is of the order of ∂x (D2 f (uR)v ⊗ v) which, in particular,includes ∂x (w2

i )Xi where Xi is the vector D2 f (uR)ri (uR) ⊗ ri (uR). Now ∂x (w2i )

is a function of the form t−3/2F(t−1/2(x − λi (uR)t)), of the same order as the twoterms (∂t + λi∂x )w and ∂2

x wi of (7.21). Hence, we cannot neglect the quadraticterm (on the other hand, the terms of higher order are of negligible amplitudeand mass owing to these). Therefore, we must consider a priori the quadraticsystem

vt + A(uR)vx + 1

2D2 f (uR)(v ⊗ v)x = vxx . (7.22)

We observe that there do not exist explicit solutions of this equation because ofthe coupling due to the quadratic term. Indeed, we have to solve equations of theform

(∂t + λi∂x )wi + 1

2

n∑j,k=1

ci jk∂x (w jwk) = ∂2x wi , 1 ≤ i ≤ n,

where ci jk =: li (uR) · D2 f (uR)r j (uR) ⊗ rk(uR). However, if w j behaves, as weexpect, as t−1/2Fj (t−1/2(x − λ j t)), with λ j �= λk , where Fj decreases rapidly (asdoes its first derivative), then the terms ∂x (w jwk) are exponentially small for j �= k,simultaneously in L∞ and in L1 when t →+∞. There remains for us to examinethe role of square terms ∂x (w2

j ).

The role of the terms ∂x (w2j ), j �= i

To understand the effect of such a term in the i th equation when w j is asymptoticallyequivalent to t−1/2Fj (t−1/2(x − λ j t)) with λ j �= λi , Fj being rapidly decreasing,we are led, by x �→ x − λi t and the elimination of the terms of the equation whichare not essential to our understanding, to the case of a heat equation with a sourceterm:

wt − wxx = g(t, x).

Here, g = t−1/2∂x g, g = G(t−1/2(x−λt)) with λ �= 0, G being rapidly descreasing.A rather tiresome calculation shows that the right-hand side of this equation isresponsible for a contribution to w of the form t−1h(t, x) where h is bounded,rapidly decreasing and ‖t−1h‖1 = O(t−1/2). Finally, the term ∂x (w2

j ) ought not to


be taken into account for the calculation of the dominant terms of the asymptoticexpansion which we seek. We are then led to a list of uncoupled equations ofBurgers–Hopf type:

(∂t + λi (uR)∂x )wi + bi

2∂x

(w2

i

) = ∂2x wi , 1 ≤ i ≤ n, (7.23)

where bi = ciii = (dλi · ri )(uR) takes the value 1 if the i th field is genuinely non-linear, and the value 0 if it is linearly degenerate.

Calculation of the diffusion waves

Equation (7.23), although non-linear, again possesses self-similar solutions of theform

wi (t, x) = 1√tW

(x − λi t√

t

).

The profiles W are the integrable solutions of the differential equation W ′′ =bi W W ′ − 1

2 (ξW ′ +W ) which has a first integral

2W ′ = bi W2 − ξW. (7.24)

The constant of integration is zero, as otherwise, W with a behaviour at infinity ofthe order of ξ−1 would not be integrable. The solutions of (7.24) are written

W (ξ ) = e−ξ2/4

C − bi2

∫ ξ

−∞ e−s2/4 ds= κe−ξ2/4

1− bi2 κ

∫ ξ

−∞ e−s2/4 ds(7.25)

where C = κ−1, the constant of integration, belongs to R\[0, bi√

π ]. We link Cwith the mass m of W by the following calculation (if bi �= 0):

m =∫

R

e−ξ2/4 dξ

C − bi2

∫ ξ

−∞ e−s2/4 ds=

∫ 2√

π

0

de

C − bi2 e= 2

bilog

(C

C − bi√

π

).

Finally, the mass being given, we find the value of the parameter

κ = 1− e−bi m/2

bi√

π(bi �= 0) or κ = m

2√

π(bi = 0). (7.26)

We denote by Wi (·, κ) the profile defined in (7.25).

Remarks (1) As for the viscosity profiles, we can envisage composing a diffusionwave for a translation of the space variable (or of the time). However, such anoperation does not change the asymptotic expansion since if wi = t−1/2Wi (·, κ),we have ‖wi (· + h)− wi‖1 = O(t−1/2) and ‖wi (· + h)− wi‖∞ = O(t−1/2).

(2) The asymptotic behaviour for the system (7.19) is extremely different fromthat of a hyperbolic system since the diffusion waves are of constant sign anddepend on only a single parameter, their mass. Tai-Ping Liu has shown ([71] and[68]) that the solution of a hyperbolic system whose initial function has compactsupport and is small enough behaves for large times as a superposition of N-waves.The N-waves, thus called because of their form in the scalar case,3 take oppositesigns on opposite sides of their mean position and depend on two scalar parameters.Also, the N-waves represent only the effect of genuinely non-linear fields and thesituation is even worse for linearly degenerate fields. For these, their oscillationsare not damped in the absence of dissipation and their profiles are arbitrary insteadof depending on a finite number of parameters.

Liu’s theorem

Now, we are able to calculate the terms of the asymptotic expansion by supposingthat its main terms are of the form

U (· + h − ct)+∑jp

wRj (t)r j (uR),

with

wLj (t, x) = 1√

tW j

(x − λ j (uL)t√

t; κ j

), wR

j (t, x) = 1√tW j

(x − λ j (uR)t√

t; κ j

).

The parameters to be determined are the phase shift h and the numbers κ j linkedto the masses of the terms wL

j and wRj . Let us suppose that this expansion is valid

in L1(R), that is to say that

limt→+∞‖u(t)−U (· + h − ct)−

∑jp

wRj (t)r j (uR)‖1 = 0.

(7.27)

Then the conservation of mass, expressed by∫R

(u(t, x)−U (x − ct)) dx = const.,

implies ∫R

φ(x) dx =∫

R

(u(t, x)−U (x − ct)) dx (∀t > 0)

= limt→+∞

∫R

(u(t, x)−U (x − ct)) dx

3 However, for the Burgers equation, this form is rather that of a cyrillic vowelN

than an N.

=∫

R

(U (y + h)−U (y)+

∑jp

W j (y; κ j )r j (uR)

)dy

= h(uR − uL)+∑jp

m jr j (uR),

where the relation between m j and κ j is given by the formula (7.26).The above equality enables us to calculate the n-tuple (m1, . . . , m p−1, h,

m p+1, . . . , mn) when the family of vectors Bp(uL, uR)= (r1(uL), . . . , rp−1(uL),uR − uL, rp+1(uR), . . . , rn(uR)) is a basis of R

n . Finally, knowing the m j , we candeduce the κ j by the formula (7.26).

The geometrical condition which we are about to state generalises in a certainway the hyperbolicity hypothesis. In fact, if the shock (uL, uR; c) is weak, then thedirection of uR − uL is close to that of rp(uL) while the vectors r j (uR) are close tor j (uL) respectively. Therefore, we have

det(Bp(uL, uR)) ∼ l p(uL) · (uR − uL) det(r1(uL), . . . , rn(uL)) �= 0

and Bp(uL, uR) is really a basis. On the other hand, for a shock of large amplitude,this condition is not necessarily satisfied and can be used to eliminate the shockswhich are not stable for generic perturbations: A. Majda has shown that this con-dition plays a role in the stability (local in time, without diffusion) of shock wavesfor hyperbolic systems [73, 74]. It expresses Lopatinski’s condition for a mixedproblem which is equivalent to the linearisation of the system which governs theevolution of the shock front (see Chapter 14).

Although the property (7.27) has not yet been proved (in fact Tai-Ping Liu assertsthat it is false in general as it lacks a term of zero mass), it provides a good method ofcalculating the parameters, at least that which concerns the phase shift. The result,obtained by Liu [70] and completed by Szepessy and Xin [99], is

Theorem 7.5.1 (Liu) We suppose that the eigenvalues of d f (u) are simple, andthat dλp(uL) · rp(uL) �= 0. There exist a neighbourhood V of uL in U and a numberc1 > 0 such that if (uL, uR; c) is a p-shock with uR ∈ V , then

(1) there exists a viscosity profile U of equation (7.19) linking uL to uR,(2) for all ϕ ∈ L1(R)n, there exists one and only one n-tuple (m1, . . . , m p−1,

h, m p+1, . . . , mn) such that∫R

ϕ(x) dx = h(uR − uL)+∑jp

m jr j (uR),

(3) if in additionϕ∈H1(R),∑

j �=p |m j | ≤ c1‖uR− uL‖and∫

R(1+ (x − h)2)‖ϕ(x)

+U (x)−U (x − h)‖2 dx ≤ c1, then the Cauchy problem for (7.19), providedwith the initial condition u(0) = U +ϕ, has one and only one smooth solution,global in time, which satisfies in addition

limt→+∞‖u(t)−U (· + h − ct)‖2 = 0,

limt→+∞‖u(t)−U (· + h − ct)‖∞ = 0.

We note that no estimate of the decay of the norms of the error in L2 or in L∞ isknown. However, Pego [83] has shown that the viscosity profile is linearly stablein that which concerns the 1-shocks or the n-shocks provided that the perturbationdecreases exponentially at infinity. The limitation of this study to the shocks ofthe fastest characteristic fields is not troublesome in applications, such as in gasdynamics or in the elasticity of an elastic string. However, a family of shocks inmagnetohydrodynamics is not covered by this study.

7.6 Influence of the diffusion tensor

We now consider the Cauchy problem for a general diffusion,

ut + f (u)x = (B(u)ux )x . (7.28)

We are given a Lax shock (uL, uR; c) which we suppose to have a viscosity profileU :

B(U )U ′ = f (U )− f (uL)− c(U − uL),

U (−∞) = uL,

U (+∞) = uR.

The perturbation ϕ of the initial condition (u(0) = U + ϕ) is given, smooth andsufficiently small, at least as in Theorem 7.5.1. However, we shall not give here aprecise statement, leaving the reader to consult [70]. We again look for a descriptionof u(t) in the form

U (· + h − ct)+∑jp

wRj (t)r j (uR)

with

wL,Rj = 1√

tW j

(x − λ j (uL,R)t√

t

),

W j being a rapidly decreasing function which has to be determined.

7.6 Influence of the diffusion tensor 243

Owing to the conservation of mass, the phase shift and the masses of the diffusionwaves are again determined by the formula∫

R

ϕ(x) dx = h(uR − uL)+∑jp

m jr j (uR).

Hence, they do not depend on the diffusion chosen. However, to justify the con-struction, it is necessary to be able to calculate the terms (!), here the diffusionwaves W j . For j > p, they are essentially supported by the zone x � ct , located tothe right of the profile, where the value of u is very close to uR. The same argumentas that of the §7.5 leads to the retention, on the left-hand side of (7.28), only of theterms (∂t + λ j (uR)∂x )w j and 1

2b j∂x (w2j ). As for the right-hand side, it develops in

the following way:

l j (uR) · ∂x (B(uR)∂xu) = l j (uR) · B(uR)∂2x u + O(t−2)

= (l j B)(uR)∂2x U +

∑k �=p

l j (uR)B(uR)rk(uR)∂2x wk + O(t−2).

The remaining terms are O(t−2) in uniform norm and in L1-norm. They havea negligible influence on the correction to the asymptotic expansion; for exam-ple, they are negligible owing to the term (l j Br j )∂2

x w j . Concerning the terms ofthe sum for which k �= j, p, they are of the form t−3/2Wk(t−1/2(x − λkt)) withλk �= λ j . Hence they have an influence of the same order as those of the terms∂x (w jwk) which we have already neglected in §7.2. It is the same for the term(li B)∂2

x U since it is transported with speed c �= λ j . In the two cases (k �= j, pand k = p), it is essential that the terms considered should have a zero mean withrespect to x ; this is the case since these are derivatives of functions vanishing atinfinity.

Hence, there only remains the term corresponding to k = j and we again havea system of uncoupled Burgers–Hopf equations:

(∂t + λ j (uL)∂x )w j + 1

2b j∂x

(w2

j

) = α j∂2x w j , α j = (l j Br j )(uL), j < p,

(∂t + λ j (uR)∂x )w j + 1

2b j∂x

(w2

j

) = α j∂2x w j , α j = (l j Br j )(uR), j > p.

This equation has integrable diffusion waves if and only α j > 0, that is if and onlyif these equations are well-posed for increasing time. Hence, we require that thediffusion satisfy the general condition

(l j Br j )(u) > 0, ∀u ∈ U, 1 ≤ j ≤ n. (7.29)

We notice that this condition is satisfied when the diffusion is strictly dissipativefor a strongly convex E , or simply when it is dissipative and, in addition, satisfies

Br j �= 0 (this allows us to apply the theory to the case of gas dynamics [50]).In fact, the eigenvector basis (r j )1≤ j≤n is orthogonal for the quadratic form D2E ,that is, there exist numbers e j (u) such that D2Er j = e j l j . These numbers arestrictly positive since D2E(r j , r j ) > 0. In addition, if D2E(Bη, η) ≥ c(u)‖Bη‖2

(dissipation hypothesis) we have

0 < c(u)‖Br j‖2 ≤ e j l j Br j .

Finally, l j Br j > 0.

Example: gas dynamics

Let us consider the system of gas dynamics in lagrangian coordinates, with vis-cosity and heat conduction (we shall also envisage the case where the viscosity isnegligible):

vt = zx ,

zt + p(v, e)x = (b(v)zx )x ,(e + 1

2z2

)t+ (pz)x = (bzzx + k(v, e)Tx )x .

Denoting by c = √(ppe − pv) the speed of sound, the eigenvectors of d f are

r1 =

1

c

zc − p

, r2 =

pe

0

−pv

, r3 =

1

−c

−zc − p

,

while the eigenforms are

l1 = 1

2c2(−pv, c + zpe,−pe),

l2 = 1

c2(p,−z, 1),

l3 = 1

2c2(−pv,−c + zpe,−pe).

Finally,

Br2 =

0

0

−kTe pv

, Br1,3 =

0

±bc

k(Tv − pTe)± czb

.

We have seen that the hyperbolicity of the system of gas dynamics is equivalent tothe convexity of the entropy (opposite to the physical entropy), which from now on

7.7 Case of over-compressive shocks 245

we shall suppose realised, and also that the tensor of viscous and thermal diffusionis dissipative for this case. We deduce that the condition l j Br j > 0 holds if and onlyif Br j �= 0. For j = 2, this is satisfied by all real gases as Te > 0 and pv < 0. Forj = 1 or j = 3, this is again true when the viscosity is present (b �= 0). If it isnot (b ≡ 0), it will be enough that pTe − Tv is not zero. In this case, we shall havepTe − Tv > 0 since α1 = α3 = 1

2c−2kpe(pTe − Tv) and pe > 0 for all real gases.This inequality is generally satisfied. For example, for a polytropic gas, Tv = 0,and Te > 0. Again, other ways of writing this inequality are

∂S

∂v

∣∣∣∣T=const.

> 0, or∂T

∂p

∣∣∣∣S=const.

> 0.

7.7 Case of over-compressive shocks

Definition 7.7.1 We call an over-compressive shock of a strictly hyperbolic systemut + f (u)x = 0 every triplet (uL, uR; c) which satisfies the Rankine–Hugoniot con-dition and is such that λ j−1(uL) < c < λ j (uL), λk(uR) < c < λk+1(uR) with j < k(we recall that if j = k, (uL, uR; c) is a Lax shock).

Let us consider an over-compressive shock (uL, uR; c) with the fixed perturbationuxx and profile equation U ′ = f (U ) − f (uL) − c(U − uL). The stable manifoldW s(uR) and the unstable manifold W i (uL) have dimensions k and n − j + 1respectively. Their intersection V is thus in general a sub-manifold of dimensionk− j+1 ≥ 2, for example in the structurally stable case where these manifolds aretransverse to each other. The set of viscosity profiles for this shock is identified withV by the mapping U �→ b(U ) := U (0). Concerning the diffusion waves whichlengthen the zone of the shock (and hence which are able to be superposed on aprofile in an asymptotic expansion) they have speeds λi (uL) for i ≤ j−1 or λi (uR)for i ≥ k + 1. Hence there are n + j − k − 1 families of diffusion waves, eachbeing parametrised by mi , miri (uL,R) being the mass of the wave. The asymptoticexpansions which we use to express the behaviour of a solution u(t) when t →+∞,when the initial condition is a perturbation U (·; b0) + ϕ of a viscosity profile, aretherefore of the form

U (·; b)+∑

i≤ j−1

1√tWi

(x − λi (uL)t√

t; κi

)ri (uL)

+∑

i≥k+1

1√tWi

(x − λi (uR)t√

t; κi

)ri (uR)+ o(1).


In this formula, the remainder is small in L1 ∩ L∞ and there are n parametersto be determined by making use of the conservation of mass. If we denote byδ(b; b0) = ∫

R(U (ξ ; b)−U (ξ, b0)) dξ the difference of mass between two profiles,

the equation we have to solve to determine b and the κi (or equivalently the massesmi ) is

δ(b; b0)+∑

i≤ j−1

miri (uL)+∑

i≥k+1

miri (uR) =∫

R

ϕ(x) dx . (7.30)

The mapping b �→ δ(b; b0) is differentiable at b0 and the range π (b0) of its dif-ferential contains the straight line generated by uR − uL since

∫R

(U (ξ + h; b0) −U (ξ ; b0))dξ = h(uR − uL). When the spaces π (b0), ⊕i≤ j−1Rri (uL), and ⊕i≥k+1

Rri (uR), the sum of whose dimensions equals n, are in direct sum, equation (7.30)has a solution

(m1, . . . , m j−1, b, mk+1, . . . , mn)

when the mass m = ∫R

ϕ(x) dx is small enough. More generally, this equationhas a solution for all m belonging to a cylinder C =ω + X , where X =R(uR −uL) + ⊕i≤ j−1Rri (uL) + ⊕i≥k+1Rri (uR). When π (b0) + X = R

n (which held fora Lax shock with weak amplitude because X = R

n), C is a neighbourhood of theorigin.

When the equation (7.30) has a solution, we can hope that limt→+∞ ‖u(t) −U (· − ct ; b)‖∞ = 0. Such a result is obviously as difficult to prove as for a Laxshock and must necessitate supplementary hypotheses concerning the perturba-tion ϕ.

The essential difference with the case of a Lax shock is that the cylinder C is not,in general, the whole of R

n . Hence, there are perturbations ϕ of the initial conditionfor which the equation (7.30) does not have a solution. For these, the asymptoticbehaviour of u(t) cannot be described by a viscosity profile. We shall illustrate thisby an example.

Example 7.7.2

The simplest system which possesses super-compressive shocks is that which hasbeen popularised by Keyfitz and Kranzer (which has n = 2):

ut + (r2u)x = 0, r = ‖u‖. (7.31)

The study which follows draws its inspiration from the work of T.-P. Liu and H.Freistuhler [25, 72].

Over-compressive shocks

The characteristic speeds of the system (7.31) are λ(u) = r2 and µ(u) = 3r2.The first corresponds to a linearly degenerate field. Being given a state uL �= 0,uL = rLeL, the shocks (here, we exclude contact discontinuities) (uL, uR; c) are oftwo kinds:

Regular shocks: uR = rReL and c = r2L + rLrR + r2

R,Irregular shocks: uR = −rReL and c = r2

L − rLrR + r2R.

Also, we require that the shocks have a viscosity profile for the parabolic perturba-tion uxx .

Now, let us fix rL > 0 and the unit vector eL and choose a shock speed c ∈( 3

4r2L, r2

L). For such a choice, there is no shock such that uR = rReL. On the otherhand, there are two states u1 and u2 of the form −rReL with 0 < r1 < r2. We haver2 < rL. The eigenvalues of the system, calculated with the states u1, u2, uL, havethe following properties:

λ(u1), λ(u2), µ(u1) < c < µ(u2), λ(uL), µ(uL).

Thus, (uL, u2; c) is a Lax shock and (uL, u1; c) is an over-compressive shock.

The profile equation

The profile equation is written

u′ = G(u) =: (r2 − c)u − (r2

L − c)rLeL. (7.32)

Its critical points are u1, u2 and uL, respectively attractive focus, saddle point andrepulsive focus. None among these is degenerate.

As the vector field G is emerging from the disk D(0; rL), every trajectory whichpasses into this disk at an instant ξ0 remains there for the times ξ < ξ0. As this diskis compact, the Poincare–Bendixson theorem ensures that trajectories tend to a sta-tionary point or are wound round a limit cycle (which can contain critical points)when t →−∞. We are going to show that this latter eventuality is impossible.

Let � be the axis ReL. Let us consider a limit cycle γ , which is a piece-wiseC

1 curve, invariant for the flow of G. Since the axis � is invariant for the flow, γ

is contained in one of the half-planes delimited by �. Since γ encircles a simplyconnected invariant compact set, this must contain a stationary point of the flow,that is u1, u2 or uL. As these points are on �, we deduce that in fact γ passes throughone of these points. Since γ is oriented by the flow, this stationary point cannotbe either attractive or repulsive, this is therefore the saddle point u2. But then γ

contains one of the two trajectories of W i(u2), which must be bounded. Now ofthese two trajectories, contained in �, one goes off to infinity while the other endsin u1. Finally, γ contains u1, which is absurd since u1 is attractive.


Fig. 7.2: The set, open but bounded, of the profiles between uL and u1, over-compressiveshock.

Every trajectory which passes into D(0; rL) has therefore issued from one of thestationary points of G. There are only four cases, of which the last is generic:

The trajectory u ≡ u1.The trajectory u ≡ u2.The two trajectories contained in � and issuing from u2.The other trajectories, which have issued from uL.

An important application of this result is that the two trajectories which end in u2

(they are symmetric with respect to � and their union forms W s(u2)) have issuedfrom uL. They encircle a compact set K (see Fig. 7.2) with non-empty interior,invariant by the flow. In the interior of K , all the trajectories leave uL and end upin u1, with the exception of the trajectories u ≡ ui and of that which goes fromu2 to u1 following the axis. Since K is a neighbourhood of u1, invariant by theflow, there is no other trajectory going from uL to u1. Finally, the image V of theviscosity profiles for the shock (uL, u1; c) is the interior of K , without the segment[u1, u2].

Here, the cylinder C has ReL for direction. Let us take, for example, b0 on thesegment [uL, u1]. The profile that we are going to disturb is thus that which followsthe axis �. Without loss of generality, we can put eL= (1, 0). The cylinder C isdefined by a relation y2 ∈ J , y2 being the coordinate along to the vector (0, 1). Theinterval J is the set of values taken by

δ2(b; b0) =∫

R

(U2(ξ ; b)−U2(ξ ; b0)) dξ =∫

R

U2(ξ ; b) dξ.

Let us write the profile equation in polar coordinates (U = r exp iθ):

r ′ = (r2 − c)r − (r2

L − c)rL cos θ,

rθ ′ = (r2

L − c)rL sin θ.

As each profile lies wholly in a half-plane U2 > 0 or U2 < 0, we have J = −Jand we can treat only the case U2 > 0. Hence, the range of the angle θ is [0, π ].Thus θ ′ > 0, and we can parametrise the trajectory by θ . It turns out that

U2 dξ = r sin θ dξ = r2θ ′dξ(r2

L − c)rL= r2(

r2L − c

)rL

dθ.

Thus,

δ2(b; b0) = 1(r2

L − c)rL

∫ π

0r2 dθ.

This expression does not depend on b but only on the trajectory on which b occurs.We can parametrise the trajectories by θ �→ r (θ, β) where (0, β) is a point onthe vertical axis. By the Cauchy–Lipschitz theorem, the mapping β �→ r (θ, β)is continuous and strictly increasing. It follows that J = [−Y, Y ] where Y is theintegral of U2 when U is the viscosity profile which links uL to u2 (and not to u1)situated in the upper half-plane (it is the upper boundary of K ). As K ⊂ D(0; rL),we have r < rL in the integral and so Y < πrL/(r2

L − c). Similarly the vector fieldG is entering into the disk D(0, r2) (verify that r2

2 < c). Hence, the trajectory iswholly outside of this disk and we again have Y > πr2/(c − r2

2 ). Finally, we haveshown the following result.

Theorem 7.7.3 Being given an over-compressive shock (uL, u1; c) of the system(7.31) (see the above construction) and an initial condition such that

∫−∞(a(x)−

uL) dx and∫∞(a(x)− u1) dx converge, the following conditions are equivalent:

(1) There exists a viscosity profile U between uL and u1 such that∫

R(U (x) −

a(x)) dx = 0;(2)

∣∣∫R

a2(x) dx∣∣ < Y.

The number Y satisfies

πr2

c − r22

< Y <πrL

r2L − c

.

The asymptotic behaviour of the solution of the parabolic Cauchy problem forthe initial condition a can therefore be described by a viscosity profile when theinequality (2) is satisfied (a necessary but not a sufficient condition).

Instability of the over-compressive shock

As T.-P. Liu notes, the useful notion of stability is that of uniform stability withrespect to the coefficient ε > 0 of the parabolic perturbation of a hyperbolic systemof a conservation law. In other words, being given an initial condition a ∈ C b(R),such that

∫−∞(a(x)− uL) dx

∫ +∞(a(x)− uR) dx converge, we ask if the solutionuε of the Cauchy problem for the parabolic system

ut + f (u)x = ε(B(u)ux )x ,

u(0, x) = a(x)

has the under-noted properties:

(1) uε is defined on R+ × R,

(2) uε(t) is asymptotic, in L1(R)∩ L∞(R), to a progressive wave U (ε−1[x − ct −x0(ε)]; ε) when t →+∞,

(3) the profile y �→ U (y − x0(ε); ε)) possesses a uniform limit when ε → 0.

In the example presented above, Theorem 7.7.3 shows that the condition (2) doesnot hold when

∫R

a2(x) dx �= 0. Indeed, the condition for the existence of a profileU between uL and uR having the desired mass defect is that |∫

Ra2(x) dx |< εY ,

this is obtained by making the change of variables (t, x) �→ (εt, εx) to lead to thecase where ε = 1. It is this instability of the shock, when the velocity is sufficientlysmall, which leads T.-P. Liu to reject this. He notes finally that by excluding theover-compressive shocks, we can solve the Riemann problem for the system (7.31)uniquely.

7.8 Exercises

7.1 We consider again the proof of Theorem 7.2.1 in the case where (dλk ·rk)(uL) =0. But as contact discontinuities are not able to admit profiles, we exclude thecase where dλk · rk is constant on an integral curve of rk . More precisely, wesuppose that γk := (d(dλk · rk) · rk) (uL) �= 0.

(1) Show that the curve of the critical points of G of the form (uR, σ (uL, uR))for uR ∈ �k(uL), uR �= uL, is locally situated on one side of the straightline s = λk(uL), this side depending on the sign of γk .

(2) Depending on this sign, deduce that there exist viscosity profiles for allthe triplets (uL, uR, σ (uL, uR)) or none, as we stay in a neighbourhood of(uL, uL, λk(uL)).

7.8 Exercises 251

7.2 We consider the p-system

vt = zx ,

zt + p(v)x = 0,

}

with p ∈ C3(R), supv∈R p′(v) < 0. We suppose that the points where p′′ is

zero satisfy p′′′ �= 0. We adopt the admissibility condition of shock waves suchas is stated in Theorem 7.3.1.

(1) A simple 2-wave is by definition a self-similar solution

(x, t) �→ (v(x/t), z(x/t)),

piecewise smooth, where we have connected two constant states u− and u+by 2-rarefaction-waves (x/t = √−p′(v) ) and 2-admissible-shock-waves(s > 0).

Let uL = (vL, zL) ∈ R2. Show that the states uR = (vR, zR) to which

uL can be linked by a simple 2-wave form a curve parametrised by v anddefined in the following way. Denoting by I the interval with extremitiesuL and uR, we denote by p∗I the envelope of the restriction of p to I , lowerconvex (if uR < uL) or upper concave (if uR > uL). Then

zR = zL −∫ vR

vL

√−dp∗I

dvdv =: ϕ(vR; vL).

Show that v �→ ϕ(v; vL) is continuous.(2) Similarly, study the curves of the simple 1-wave and show that the Riemann

problem admits one and only one solution made up of one 1-wave and one2-wave separated by a constant state.

7.3 We consider the isentropic gas dynamics with viscosity and capillarity. Thislatter is expressed by a perturbation of the law of conservation of momentum(a is a positive constant):

vt = zx ,

zt + p(v)x = ε(b(v)zx )x − aε2vxxx .

}

After the elimination of the speed, the profile equation becomes

av′′ + sb(v)v′ + p(v)− p(vL)+ s2(v − vL) = 0. (7.33)

So far as the question (5) inclusive, we assume that b > 0.

(1) We suppose that (uL, uR, σ (uL, uR)) admits a visco-capillary profile (wemust properly call it thus!). Show that the expression∫ vR

vL

(p(v)− p(vL)+ σ 2(v − vL)) dv

is of opposite sign to that of σ and that it is identically zero if and only ifσ = 0. Deduce that if σ �= 0, one at least of the discontinuities (uL, uR; σ )and (uR, uL; σ ) does not admit a visco-capillary profile.

(2) Show that this inequality is always satisfied be an entropy shock when p′′

is of constant sign (we can take Lax’s entropy condition or Lax’s shockcondition since, here, they are equivalent).

(3) We suppose that p′ is of the sign of (β − v)(v − α) where −∞ < α <

β < +∞. We consider the stationary discontinuities ((vL, 0), (vR, 0); 0),that is the couples (non-ordered) (vL, vR) for which p(vL) = p(vR). Showthat there exists one and only one which satisfies the condition∫ vR

vL

(p(v)− p(vL)) dv = 0.

(4) For this couple, show that (uL, uR; 0) and (uR, uL; 0) each admit a visco-capillary profile.

(5) We denote this couple by (v−, v+) with v− < v+. Show that v− < α <

β < v+.(6) We consider the case without viscosity (b ≡ 0). Show that for all vL

in the neighbourhood of v−, there exists a single couple (vR, σ ) in theneighbourhood of (v+, 0) satisfying

p(vR)− p(vL)+ σ 2(vR − vL) = 0,∫ vR

vL

(p(v)− p(vL)+ σ 2(v − vL)) dv = 0.

Show that for all zL ∈ R, there exists a capillary profile in each directionbetween (vL, zL) and (vR, zL + δz) where δz = σ (vL − vR).

7.4 We consider the dynamics of a perfect gas with state equation pv = (γ−1)e andfor sole perturbation a thermal diffusion obeying Fourier’s law (k = k(v, e) >

0). As here the temperature can be taken equal to e, the equations are equivalentto

vt = zx ,

zt + px = 0,(e + 1

2z2

)t+ (pz)x = ε(k(v, e)ex )x .

7.8 Exercises 253

(1) Show that the profile equation leads to a single differential equation of theform f (v)′ = g(v) where g(vl,r) = 0 and f ′(v) < 0 for v > v∗ and f ′(v) > 0for v < v∗, v∗ being defined by

v∗ = 1

2σ 2(pL + σ 2vL) = 1

2σ 2(pR + σ 2vR).

(2) Verify that g is quadratic, hence of constant sign between vL and vR. Deducethat if vL and vR are on opposite sides of v∗, then those singular points ofthe reduced equation f (v)′ = g(v) are of the same nature (attractive orrepulsive) and hence that there is not a thermal profile for the discontinuity(uL, uR; σ (uL, uR)) in this case.

(3) On the other hand, show that if vL and vR are situated on the same side withrespect to v∗, there is a thermal profile from one state towards the other, inthe direction which respects Lax’s entropy condition.

7.5 We consider once again gas dynamics in lagrangian variables with the perfectgas equation of state pv = (γ−1)e with γ > 1, but with viscosity (b = b(v, e))and without thermal conduction. Hence the equations are

vt = zx ,

zt + px = ε(b(v, e)zx )x ,(e + 1

2z2

)t+ (pz)x = ε(b(v, e)zzx )x .

(1) Show that for this perturbation, the contact discontinuities (zR = zL, pR =pL, s = 0) have viscosity profiles.

(2) Let (uL, uR; σ (uL, uR)) be a discontinuity for which σ �= 0. Show that theexistence of a viscosity profile is equivalent to the existence of a heteroclinictrajectory from zL to zR for a differential equation

vb(v, e)z′ = q(z)

where q is a quadratic polynomial which depends on σ , is zero at zL andat zR and where v, e are expressed as functions of z.

(3) Deduce that there exists a viscosity profile between uL and uR but only ina single sense, that which respects Lax’s entropy condition.

7.6 Let us consider non-isentropic gas dynamics with diffusion, whose generalform in eulerian variables is

ut + f (u)x = ε(B(u)ux )x .


The first of these equations is, on supposing that there is no diffusion for themass,

ρt + (ρz)x = 0, (7.34)

where we have denoted the density by ρ and the velocity by z. We recall that thelagrangian coordinates (t, y) are defined by the formula dy = −ρz dt + ρ dx ,justified by the equation (7.34).

(1) Show that outside of the vacuum, the equations of motion are written, inlagrangian coordinates, in the form(

u

ρ

)t+ ( f (u)− zu)y = ε(ρB(u)uy)y,

which contains a trivial equation 1t + 0y = 0, which we replace by theequation vt = zy which itself comes from the trivial equation 1t + 0x = 0(we denote by v = ρ−1 the specific volume).

(2) Let (uL, uR; s) be a discontinuity satisfying the Rankine–Hugoniot condi-tion for the eulerian system:

[ f ] = s[u].

We suppose that zR �= zL (we thus exclude contact discontinuities). Showthat there corresponds a discontinuity ((vL, zL), (vR, zR); σ ) of the euleriansystem, whose speed of propagation σ is given by the formula

σ = s − zL

vL= s − zR

vR.

(3) Write the profile equations for the lagrangian system and for the euleriansystem.Show that the existenceof an eulerianviscosity profile for (uL, uR; s)is equivalent to that of a lagrangian viscosity profile for (vL, uL, vR, uR; σ )and that we pass from one to the other by a change of parameter.

Bibliography

[1] Bergh, J. and Lofstrom, J. Interpolation spaces, Springer Verlag, Berlin, 1976.[2] Boillat, G. La Propagation des ondes, Gauthier-Villars, Paris, 1965.[3] Boillat, G. ‘Chocs caracteristiques’ C.R. Acad. Sci. Paris, Serie I 274 (1972),

1018–21.[4] Brenner, P. ‘The Cauchy problem for the symmetric hyperbolic systems in L p.’

Math. Scand 19 (1966), 27–37.[5] Carasso, C., Rascle, M., and Serre, D. ‘Etude d’un modele hyperbolique en

dynamique des cables’ Mod. Math. Anal. Numer. 19 (1985), 573–99.[6] Chen, Gui-Qiang. ‘Convergence of the Lax–Friedrichs scheme for isentropic gas

dynamics (i)’ Acta Math. Scientia 6 (1986), 75–120.[7] Chen, Gui-Qiang, Lu, Y.-G. ‘Convergence of the approximate solutions to

isentropic gas dynamics’ Acta Math. Scientia 10 (1990), 39–45.[8] Ciarlet, P. Mathematical elasticity: vol. I, three-dimensional elasticity, North

Holland, Amsterdam, 1988.[9] Coleman, B. D. and Dill, E. H. ‘Thermodynamic restrictions on the constitutive

equations of electromagnetic theory’ Z. Angew. Math. Phys. 22 (1971), 691–702.[10] Collela, P. ‘Glimm’s method for gas dynamics’ SIAM J. Sci. Statist. Comp., 3

(1982), 76–109.[11] Courant, R. and Friedrichs, K. O. Supersonic flow and shock waves, John Wiley &

Sons – Interscience, New York, 1948.[12] Courant, R. and Hilbert, D. Methods of mathematical physics, vol. II, John Wiley &

Sons – Interscience, New York, 1962.[13] Crandall, M. and Ligett, T. M. ‘Generation of semi-groups of nonlinear

transformations on general Banach spaces’ Amer. J. Math. 93 (1971), 265–98.[14] Crandall, M. ‘The semi-group approach to first order quasilinear equations in

several space variables’ Israel J. Math. 12 (1972), 108–32.[15] Dafermos, C. ‘Characteristics in hyperbolic conservation laws, a study of the

structure and asymptotic behaviour of solutions’ In R. J. Knops (editor), Nonlinearanalysis and mechanics, vol. 39, Research Notes in Mathematics, Pitman, London,1977, pp. 1–58.

[16] Dafermos, C. and Ling Hsiao ‘Development of singularities in solutions of theequations of nonlinear thermoelasticity’ Quart. Appl. Math. 44 (1986), 463–74.

[17] DiPerna, R. J. ‘Convergence of approximate solutions to conservation laws’ Arch.Rat. Mech. Anal. 82 (1983), 27–70.

255

256 Bibliography

[18] DiPerna, R. J. ‘Convergence of the viscosity method for isentropic gas dynamics’Comm. Math. Phys. 91 (1983), 1–30.

[19] Ekeland, I. and Temam, R. Analyse convexe et problemes variationnels, Dunod,Paris, 1974.

[20] Evans, L. C. Weak convergence methods for nonlinear partial differentialequations. vol. 74, CBMS Regional Conference Series in Math., Amer. Math. Soc.,Providence, R.I., 1990.

[21] Feynman, R. P., Leighton, R. B., and Sands, M. Feynman lectures on physics,Addison-Wesley, Reading, Mass., 1963.

[22] Fife, P. ‘Mathematical aspects of electrophoresis’ in R. J. Knops (editor),Reaction–diffusion equations, Oxford University Press, 1988.

[23] Foy, L. ‘Steady state solutions of hyperbolic systems of conservation laws withviscosity terms’ Comm. Pure Appl. Math. 17 (1964), 177–88.

[24] Freistuhler, H. ‘Rotational degeneracy of hyperbolic systems of conservation laws’Arch. Rat. Mech. Anal. 113 (1991), 39–64.

[25] Freistuhler, H. and Liu, Tai-Ping. ‘Nonlinear stability of overcompressive shockwaves in a rotationally invariant system of viscous conservation laws’ Comm.Math. Phys. 153 (1993), 147–58.

[26] Friedrichs, K. O. and Lax, P. D. ‘Systems of conservation laws with a convexextension’ Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1686–8.

[27] Garcke, H. ‘Travelling wave solutions as dynamic phase transitions in shapememory alloys’ J. Diff. Eqns. 121 (1995), 203–31.

[28] Garding, L. ‘Problemes de Cauchy pour les systemes quasi-lineaires d’ordre unstrictement hyperboliques’ Les EdPs. Colloques Internationaux du CNRS 117(1963), 33–40.

[29] Gilquin, H. ‘Une famille de schemas numeriques TVD pour les lois deconservation hyperboliques’ Math. Mod. Numer. Anal. 20 (1986), 429–60.

[30] Gisclon, M. ‘Etude des conditions aux limites pour un systeme strictementhyperbolique, via 1’approximation parabolique’ J. Math Pures Appl. 75 (1996),485–508.

[31] Gisclon, M. and Serre, D. ‘Etude des conditions aux limites pour un systemestrictement hyperbolique via 1’approximation parabolique’ C.R. Acad. Sci. Paris,Serie I 319 (1994), 377–82.

[32] Glimm, J. ‘Solutions in the large for nonlinear hyperbolic systems of equations’Comm. Pure Appl. Math. 18 (1965), 697–715.

[33] Glimm, J. and Lax, P. D. Decay of solutions of systems of nonlinear hyperbolicconservation laws, Memoir 101, A.M.S., Providence, R.I., 1970.

[34] Godlewski, E. and Raviart, P. -A. Hyperbolic systems of conservation laws,Ellipses, Paris, 1991.

[35] Godunov, S. K. ‘Lois de conservation et integrales d’energie des equationshyperboliques’ In C. Carasso, P.-A. Raviart, and D. Serre (editors), Nonlinearhyperbolic problems, St Etienne, vol. 1270, Springer Lecture Notes inMathematics, Berlin, 1987, pp. 135–49.

[36] Goodman, J. ‘Nonlinear asymptotic stability of viscous shock profiles forconservation laws’ Arch. Rat. Mech. Anal. 95 (1986), 325–44.

[37] Hagan, R. and Slemrod, M. ‘The viscosity–capillarity condition for shocks andphase transitions’ Arch. Rat. Mech. Anal. 83 (1983), 333–61.

[38] Hammersley, J. M. and Handscomb, D. C. Monte Carlo methods, Methuen,London, 1965.

[39] Heibig, A. ‘Regularite des solutions du probleme de Riemann’ Comm. Partial Diff.Eqns. 15 (1990), 693–709.

Bibliography 257

[40] Heibig, A. ‘Unicite des solutions du probleme de Riemann’ C.R. Acad. Sci. Paris,Serie I 312 (1991), 793–7.

[41] Heibig, A. and Serre, D. ‘Une approche algebrique du probleme de Riemann’ C.R.Acad. Sci. Paris, Serie I 309 (1989), 157–62.

[42] Hoff, D. ‘Invariant regions for systems of conservation laws’ Trans. Amer. Math.Soc. 289 (1985), 591–610.

[43] Hoff, D. and Smoller, J. ‘Solutions in the large for certain nonlinear parabolicsystems’ Ann. Inst. Henri Poincare 3 (1985), 213–35.

[44] Hoff, D. and Smoller, J. ‘Global existence for systems of parabolic conservationlaws in several space variables’ J. Diff. Eqns. 68 (1987), 210–20.

[45] Hormander, L. Lectures on nonlinear hyperbolic differential equations,Springer-Verlag, Berlin, 1997.

[46] Hsiao, Ling and Tong Zhang. The Riemann Problem and interaction of waves ingas dynamics, Longman, London, 1989.

[47] Il’in, A. M. and Oleınik, O. A. ‘Behaviour of the solutions of the Cauchy problemfor certain quasilinear equations for unbounded increase of time’ AMS Translations42 (1964), 19–23.

[48] Ivrii, V. Ya. Linear hyperbolic equations, vol. 33, Encyclopedia of mathematicalsciences, Springer-Verlag, Heidelberg, 1993.

[49] Joly, J.-L., Metivier, G., and Rauch, J. ‘A nonlinear instability for 3× 3 systems ofconservation laws’ Comm. Math. Phys. 162 (1994), 47–59.

[50] Kawashima, S. and Matsumura, A. ‘Asymptotic stability of travelling wavesolutions of systems for one-dimensional gasmotion’ Comm. Math. Phys. 101 (1985), 97–127.

[51] Kazhikhov, A. V. ‘Cauchy problem for viscous gas equations’ Sibirsk. Mat. Zh. 23(1982), 60–4.

[52] Keyfitz, B. L. and Kranzer, H. C. ‘A system of non-strictly hyperbolicconservation laws arising in elasticity theory’ Arch. Rat Mech. Anal. 72 (1980),219–41.

[53] Klainerman, S. ‘Global existence for nonlinear wave equations’ Comm. Pure Appl.Math. 33 (1980), 43–101.

[54] Klainerman, S. ‘Uniform decay estimates and the Lorentz invariance of theclassical wave equation.’ Comm. Pure Appl. Math. 38 (1995), 321–32.

[55] Kreiss, H.-O. and Lorenz, J. Initial–boundary value problems and theNavier–Stokes equations, Academic Press, San Diego, 1989.

[56] Kruzkov, S. N. ‘Generalized solutions of the Cauchy problem in the large fornonlinear equations of first order’ Dokl. Akad. Nauk SSSR 187 (1969), 29–32.

[57] Kunik, M. ‘A solution formula for a non-convex scalar hyperbolic conservation lawwith monotone initial data’ Math. Methods Appl. Sci. 16 (1993), 895–902.

[58] Kuznetsov, N. N. ‘Accuracy of some approximate methods for computing the weaksolution of a first order quasi-linear equation’ USSR Comp. Math. Math. Phys. 16(1976), 105–19.

[59] Lax, P. D. ‘Hyperbolic systems of conservation laws II’ Comm. Pure Appl. Math.10 (1957), 537–66.

[60] Lax, P. D. Hyperbolic systems of conservation laws and the mathematical theory ofshock waves vol. 11, CBMS-NSF Regional Conference Series in AppliedMathematics, SIAM, Philadelphia, 1973.

[61] Lax, P. D. and Levermore, D. ‘The small dispersion limit for the KdeV equation’Comm. Pure Appl. Math. 36 (1983), I 253–90, II 571–93, III 809–30.

[62] Leveque, R. Numerical methods for conservation laws, Lectures in Mathematics,Birkhauser, Basle, 1990.

258 Bibliography

[63] Levermore, C. D. ‘The hyperbolic nature of the zero dispersion KdeV limit’ Comm.Partial Diff. Eqns. 13 (1988), 495–514.

[64] Li Ta-Tsien. ‘Solutions regulieres globales des systemes hyperboliquesquasi-lineaires d’ordre 1’ J. Math. Pures Appl. 4 (1982), 401–9.

[65] Li Ta-Tsien and Yu Wen-Ci. Boundary value problems for quasilinear hyperbolicsystems, vol. V, Math. Series, Duke University, Durham, N.C., 1985.

[66] Liu, Tai-Ping. ‘The Riemann problem for general 2× 2 conservation laws’ Trans.Amer. Math. Soc. 199 (1974), 89–112.

[67] Liu, Tai-Ping. ‘The Riemann problem for general systems of conservation laws’ J.Diff. Eqns. 18 (1975), 218–34.

[68] Liu, Tai-Ping. ‘Decay to N-waves of solutions of general systems of nonlinearhyperbolic conservation laws’ Comm. Pure Appl. Math. 30 (1977), 585–610.

[69] Liu, Tai-Ping. ‘The deterministic version of Glimm’s scheme’ Comm. Math. Phys.57 (1977), 135–48.

[70] Liu, Tai-Ping. Nonlinear stability of shock waves for viscous conservation laws,Memoir 328, Amer. Math. Soc., Providence, R.I., 1985.

[71] Liu, Tai-Ping. ‘Pointwise convergence to N-waves for solutions of hyperbolicconservation laws’ Bull. Inst. Math. Acad. Sinica 15 (1987), 1–17.

[72] Liu, Tai-Ping. ‘On the viscosity criterion for hyperbolic conservation laws’ In M.Shearer (editor), Viscous profiles and numerical methods for shock waves, SIAM,Philadelphia, 1991, pp. 105–14.

[73] Majda, A. The existence of multi-dimensional shock fronts, Memoir 281, Amer.Math. Soc., Providence, R.I., 1983.

[74] Majda, A. The stability of multi-dimensional shock fronts, Memoir 275, Amer.Math. Soc., Providence, R.I., 1983.

[75] Majda, A. Compressible fluid flow and systems of conservation laws in severalspace variables, Springer-Verlag, Berlin, 1984.

[76] Mei, Ming. ‘Stability of shock profiles for nonconvex scalar viscous conservationlaws’ Math. Mod. Methods Appl. Sci. 5 (1995), 279–96.

[77] Mihailescu, M. and Suliciu, I. ‘Riemann and Goursat step data problems forextensible strings with non-convex stress–strain relation’ Rev. Roum. Math. Pureset Appl. 20 (1975), 551–9.

[78] Mihailescu, M. and Suliciu, I. ‘Riemann and Goursat step data problems forextensible strings’ J. Math. Anal. Appl. 52 (1975), 10–24.

[79] Morawetz, C. S. Notes on time decay and scattering for some hyperbolic problems,vol. 19, Regional Conference Series in Applied Mathematics, SIAM, Providence,R.I., 1975.

[80] Nishida, T. ‘Global solutions for an initial boundary value problem of a quasilinearhyperbolic system’ Proc. Jap. Acad. 44 (1968), 642–6.

[81] Oleınik, O. A. ‘Uniqueness and stability of the generalized solution of the Cauchyproblem for a quasi-linear equation’ Uspekhi Mat. Nauk. 14 (1959), 165–70.

[82] Osher, S. and Ralston, J. ‘L1 stability of travelling waves with application toconvective porous media flow’ Comm. Pure Appl. Math. 35 (1982), 737–51.

[83] Pego, R. L. ‘Linearized stability of extreme shock profiles in systems ofconservation laws with viscosity’ Trans. Amer. Math. Soc. 280 (1983), 431–61.

[84] Poupaud, F., Rascle, M. and Vila, J.-P. ‘Global solution to the isothermalEuler–Poisson system with arbitrarily large data’ J. Diff. Eqns. 123 (1995), 93–121.

[85] Rauch, J. ‘BV estimates fail for most quasilinear hyperbolic systems in dimensiongreater than one’ Comm. Math. Phys. 106 (1986), 481–4.

[86] Richtmyer, R. D. and Morton, K. Difference methods for initial value problems,Wiley–Interscience, New York, 1967.

Bibliography 259

[87] Sakamoto, R. Hyperbolic boundary value problems, Cambridge University Press,1982.

[88] Sattinger, D. H. ‘On the stability of waves of nonlinear parabolic systems’Advances in Math. 22 (1976), 312–55.

[89] Schatzman, M. ‘Fonctionnelle de Glimm continue’ C.R. Acad. Sci. Paris, Serie I291 (1980), 519–22.

[90] Schatzman, M. ‘Continuous Glimm functionals and uniqueness of the solution ofthe Riemann problem’ Indiana Univ. Math. J. 34 (1985), 533–89.

[91] Schecter, S. and Shearer, M. ‘Transversality for undercompressive shocks inRiemann problems’ In M. Shearer (editor), Viscous profiles and numerical methodsfor shock waves, SIAM, Philadelphia, 1991, pp. 142–54.

[92] Serre, D. ‘Temple’s fields and integrability of hyperbolic systems of conservationlaws’ In Guangchang Dong and Fanghua Lin (editors), International conferenceon nonlinear PDEs, International Academic Publishers, New York, 1993, pp.233–51.

[93] Sevennec, B. Geometrie des systemes de lois de conservation, vol. 56, Memoires,Soc. Math. de France, Marseille, 1994.

[94] Shearer, M. ‘The Riemann problem for the planar motion of an elastic string’ J.Diff. Eqns. 61 (1986), 149–63.

[95] Slemrod, M. ‘Global existence, uniqueness and asymptotic stability of classicalsmooth solutions in one-dimensional, non-linear thermoelasticity’ Arch. Rat. Mech.Anal. 76 (1981), 97–133.

[96] Slemrod, M. ‘Dynamic phase transitions in a Van der Waals fluid.’ J. Diff. Eqns. 52(1984), 1–23.

[97] Smoller, J. Shock waves and reaction diffusion equations, Springer-Verlag, Berlin,1983.

[98] Sod, G. A. Numerical methods in fluid dynamics, Cambridge University Press,1985.

[99] Szepessy, A. and Zhouping Xin. ‘Nonlinear stability of viscous shock waves’ Arch.Rat. Mech. Anal. 122 (1993), 53–103.

[100] Tadmor, E. ‘Local error estimates for discontinuous solutions of nonlinearhyperbolic equations’ SIAM J. Numer. Anal. 28 (1991), 891–906.

[101] Tartar, L. ‘Compensated compactness and applications to partial differentialequations’ In R. J. Knops (editor), Nonlinear analysis and mechanics, vol. 39,Research Notes in Mathematics, Pitman, London, 1979, pp. 136–92.

[102] Taub, A. H. ‘Relativistic Rankine–Hugoniot equations’ Phys. Rev. 74 (1948)328–34.

[103] Temple, B. ‘Systems of conservation laws with invariant manifolds’ Trans. Amer.Math. Soc. 280 (1983), 781–95.

[104] Temple, B. ‘No L1-contractive metrics for systems of conservation laws’ Trans.Amer. Math. Soc. 288 (1985), 471–80.

[105] Temple, B. and Young, R. ‘The large time stability of sound waves’ Comm. Math.Phys. 179 (1996), 417–66.

[106] Tsarev, S. P. ‘On Poisson brackets and one-dimensional systems of hydrodynamictype’ Dokl. Akad. Nauk SSSR 282 (1985), 534–37.

[107] Tsarev, S. P. ‘The geometry of hamiltonian systems of hydrodynamic type. Thegeneralized hodograph method.’ Izv. Akad. Nauk SSSR 54 (1990), 1048–68.

[108] Vanderbauwhede, A. ‘Centre manifolds, normal forms and elementarybifurcations.’ Dynamics Reported 2 (1989), 89–169.

[109] Vichnevetsky, R. and Bowles, J. B. Fourier analysis of numerical approximationsof hyperbolic equations, SIAM, Philadelphia, 1982.

260 Bibliography

[110] Wagner, D. H. ‘Equivalence of the Euler and Lagrangian equations of gas dynamicsfor weak solutions’ J. Diff. Eqns. 68 (1987), 118–36.

[111] Weinberger, H. F. ‘Long-time behaviour for a regularized scalar conservation law inthe absence of general nonlinearity’ Ann. Inst. Henri Poincare 7 (1990), 407–25.

[112] Whitham, J. Linear and nonlinear waves, John Wiley & Sons, New York, 1974.

Index

absolute temperature, 136accretive operator, 48adiabatic exponent, 3admissible discontinuity, 40admissible solution, 34Alfven waves, 16Ampere’s law, 11asymptotic stability, 230

barotropic model, 6Boillat’s theorem, 81bounded variation, 37, 146Burgers’ equation, 30, 61, 66, 216, 217Burgers–Hopf equations, 239, 243BV-space, 37

capillarity, 251Cauchy problem: scalar equations in d = 1,

25–43approximate solutions, 32blow-up in finite time, 27classical solutions, 25discontinuous solutions, 30entropy solutions, 34; piecewise smooth, 40existence and uniqueness of solution,

36irreversibility, 36linear, 25maximum principle, 37non-linear, 26non-uniqueness of solutions, 31weak solutions, 27

characteristic curve, 26characteristic field, 80

genuinely non-linear, 113linearly degenerate, 81

characteristic foliation, 81chromatography, 24compression waves, 139conservation law, xiiicontact discontinuity, xvii, 43, 116contraction semi-group, 38Courant–Friedrich–Levy (CFL) condition, 149

differential form, xivdifferential eigenform, 81

diffusion tensor, 191, 220, 242diffusion waves, 236

non-linear, 237discontinuities, 40dispersive equation, 19

singular limit, 19dissipation by viscosity, 187dissipative tensor, 190domain

of dependence, 38of influence, 38

duality method, 58

elastic string, 119electric current, 11electric induction, 11electric permittivity, 13electromagnetism, 11

Maxwell’s equations, 12plane waves, 13Poynting’s formula, 12

entropy, xx, 5, 82convex, 82dissipation of, 188physical, xx, 82specific, 5

entropy balance, 111entropy flux, xx, 36, 82entropy inequalities, 33, 160entropy production, 139, 182entropy solution, xx, 34

piecewise smooth, 40equidistributed sequence, 154Euler equations, 4Euler–Darboux–Poisson equation,

102

Faraday’s law, 11flow in a shallow basin, 8Fourier’s law, 4frame indifference, 18

261

262 Index

Garding–Leray theorem, 91gas dynamics

in eulerian variables, 1, 119isentropic, 7, 216isothermal, 7, 216in lagrangian variables, 9, 85, 119, 245

generic explosion by a cusp, 61Glimm functional

continuous, 182linear, 162quadratic, 164

Glimm scheme, 149–60compactness, 156consistency, 153convergence, 156description of scheme, 149stability, 174

Glimm’s theorem, 151Godunov’s scheme, 180

heat flux, 11Helly’s theorem, 247Hugoniot locus, 108

local description, 107Huygens’ principle, 101hyperbolic system, 71

linear, 69partially, 192quasi-linear, 74symmetrisable, 73

hyperelastic materials, 17

inadmissible discontinuities, 42incompressible fluid, 8interaction potential, 153internal energy, 2irreversibility, 36isentropic model, 7isothermal model, 7

j-discontinuities, 110Joule effect, 13jump, xix, 25

Keyfitz and Kranzer system, 119, 246kinetic energy, 2Korteweg–de Vries (KdV) equation, 19Kreiss matrix theorem, 72, 79Kruzkov’s theorem, 36

existence proof by semi-group method,47

uniqueness, 51Kunik’s formula, 47Kuznetsov’s theorem, 210, 216

Lax entropy condition, 87, 111, 116Lax formula, 45Lax shock condition, 41, 114, 116, 224Lax theorem, 124Lax–Friedrichs scheme, 180Legendre–Hadamard condition, 19, 189

Leroux system, 183Liu’s theorems, 155, 241

magnetic induction, 11magnetic permittivity, 133magnetohydrodynamics (M.H.D.), 14

plane waves, 15; simplified model of, 16maximum principle, 37, 50, 232Maxwell’s equations, xiv, 75membrane, 19method of characteristics, 26, 42mixed problem, xvmodel

barotropic, 6isentropic, 7isothermal, 7

N-wave, 63, 240Navier–Stokes equations, 4

isentropic, 218isothermal, 219

Neumann condition, 6Nishida’s example, 167Nishida’s theorem, 173numerical scheme

Glimm, 149Godunov, 180Lax–Friedrichs, 180

Oleınik’s condition, 41Oleınik’s inequality, 41, 45, 57

perfect gas, 2, 3physical system, 83, 89polytropic gas, 7profile

for isentropic fluid with viscosity, 229for Keyfitz–Kranzer system, 247for Lax shock, 235

profile equation, 221

Rankine–Hugoniot condition, xvii, 28, 88, 90,107, 109

rarefaction, 43rarefaction wave, 119, 134relativistic model of a gas, 8rich system, 181Riemann invariant, 117

weak, 118Riemann problem for d = 1, 106–45

gas dynamics 132–43; rarefaction waves, 133;shocks, 135; wave curves, 140

p-system, 127–31; rarefaction waves, 127; shocks,128; wave curves, 129

road traffic, xv, 10

St Venant–Kirchhoff law, 19Schrodinger equation, 19second order perturbations, 186–219semi-groups, 48

contraction, 48

Index 263

Sevennec’s theorem, 118shock

characteristic shock, 43Lax j-shock, 114over-compressive, 245semi-characteristic, 43, 62; weak, 62

state law, xivperfect gas, 2polytropic gas, 7

stress tensor, 4string, 19system

conservative, 80hyperbolic, xx; with constant coefficients,

71; linear, 69; quasi-linear, 79; strictly, 73;symmetrisable, 73

Keyfitz and Kranzer, 119, 246p-system, 107physical, 83

rich, 131Temple system, xvi

Tokamak, 14total variation, 37

viscosity profile for shock waves, 220–64vs. Lax entropy condition, 221vs. Lax shock condition, 222

waveplane wave, 74pressure wave, 76shear wave, 76simple, 107

wave curvedirect, 120reverse, 120

wave equation, 101

denis serre systems of conservation laws 1_ hyperbolicity, entropies, shock waves 1999.pdf

Documents