Waves in Compressible Fluids: Viscous Shock,Rarefaction, and Contact Waves

Akitaka Matsumura

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Generalized Burgers’ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Global Asymptotics Toward Rarefaction Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Asymptotic Stability of Viscous Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Viscous Contact Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Isentropic/Isothermal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Dissipative Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Riemann Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Global Asymptotics Toward Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Asymptotic Stability of Viscous Shock Wave, a Density-Dependent

Viscosity Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Ideal Polytropic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 Riemann Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Viscous Contact Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Asymptotic Stability of Viscous Contact Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Abstract

This short article focuses on several one-dimensional model systems which oftenappear in the field of compressible viscous fluids, and discusses their Cauchyproblems with prescribed far-field states. In particular, it describes the asymptotic

A. Matsumura (�)Department of Pure and Applied Mathematics, Graduate School of Information Science andTechnology, Osaka University, Toyonaka, Osaka, Japane-mail: akitaka@math.sci.osaka-u.ac.jp

© Springer International Publishing Switzerland 2016Y. Giga, A. Novotny (eds.), Handbook of Mathematical Analysis in Mechanicsof Viscous Fluids, DOI 10.1007/978-3-319-10151-4_60-1

1

2 A. Matsumura

behavior of the solutions in relation to the Riemann problems for the hyperbolicparts with the same far-field states. It has been expected that the solutions tendtoward various asymptotic wave patterns as time goes to infinity, that is, variouscombinations of viscous shock, rarefaction, and contact waves. Many cases havebeen mathematically justified, but many others still remain open. The intent ofthis article is to give the reader introductory insights into how various primitiveenergy methods have contributed to the mathematical justifications, throughsome specific topics.

1 Introduction

In the mathematical study of one-dimensional motions of compressible viscousfluids, there often appears a system of nonlinear partial differential equations, knownas “viscous conservation laws,” which takes the form:

zt C f .z/x D .B.z/zx/x .t > 0; x 2 R/; (1)

where z is a vector valued conserved quantity, f .z/ represents its convectiveflux, and B.z/zx its diffusive flux given by a viscous effect. In what follows, theconvective flux function f .z/ is simply referred to as “flux function.” Basically, theflux function f .z/ is supposed to satisfy that the inviscid system zt C f .z/x D 0 isstrictly hyperbolic on a domain � in the phase space of z under consideration, andthe coefficient matrix B.z/ of the viscosity term is also supposed to satisfy that thewhole system does have a dissipative structure for z 2 �. A general class of suchdissipative systems of viscous conservation laws is well known as “Kawashima-Shizuta systems,” whose characterization is given by [20,24]. The typical examplesoften seen in concrete problems are Burgers’ equation and its variants (generalizedBurgers’ equation), an isentropic/isothermal model system for viscous gas, and anideal polytropic model system for viscous and heat-conductive gas. One of thefundamental mathematical problems for the system (1) is the Cauchy problem withprescribed initial and far-field conditions

z.0; x/ D z0.x/ .x 2 R/; limx!˙1

z.t; x/ D z˙ .t � 0/: (2)

In particular, two important issues both from the mathematical and physical pointsof view are the existence of a solution for large time and its asymptotic behavior withrespect to the time, in connection with the flux function f and the far-field statesz˙. On this subject, it has been known that even if the system (1) is fully parabolic,the asymptotic aspect of wave propagation should be closely related to that of theweak solution of the following Cauchy problem for the inviscid part (hyperbolicpart) of (1):

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 3

(zt C f .z/x D 0 .t > 0; x 2 R/;

z.0; x/ D zR0 .x/ .x 2 R/;(3)

where the initial data zR0 is defined by

zR0 .x/ D

�z� .x < 0/;

zC .x > 0/:

The Cauchy problem (3) is the so-called “Riemann problem” and its solution the“Riemann solution” (cf. [26, 46]).

Over the last half century, much research has been performed on solutions ofthe viscous system that corresponds to the various wave patterns of the Riemannsolution to the hyperbolic system. Of particular note is the pioneering work byIl’in-Oleinik [14], where they studied the scalar case with a genuinely nonlinearflux function and showed that if the Riemann solution admits a rarefaction wave(resp. a shock wave), the global solution in time of the Cauchy problem (1), (2)tends toward the rarefaction wave (resp. a corresponding smooth traveling wavesolution of (1), “viscous shock wave”). Since the main tool for the proofs was themaximum principle, there was no progress to systems until Matsumura-Nishihara[39] and Goodman [5] independently found in the mid-1980s that an L2-energymethod is applicable even to some systems for the study of asymptotic stabilityof viscous shock wave. Since then, many progress have been made concerning theasymptotic behavior of the solution for the viscous problem, in connection with thewave patterns of the Riemann solution. In particular, various energy methods haveplayed essential roles to establish “a priori estimates.”

This article aims to give the reader introductory insights into how various L2-energy methods have played important roles to obtain a priori estimates, throughsome specific topics on the generalized Burgers’ equation in Sect. 2, the isen-tropic/isothermal model system of viscous gas in Sect. 3, and the ideal polytropicmodel system of viscous and heat-conductive gas in Sect. 4. No attempt is madehere to present the entire mathematical theory of viscous conservation laws.

The rest of the introduction devotes some further remarks to a better understand-ing of the following sections. First is a heuristic argument on how the asymptoticbehavior of the solution z of (1), (2) is related with the Riemann solution zR of (3).Here and in what follows, the Riemann solution of (3) will be written as zR wheneverneeded to distinguish it from the solution z of the viscous system. If the variables.t; x/ are rescaled to .�; �/ as .t; x/ D .�="; �="/ and the new unknown variable Qzis defined by Qz D Qz".�; �/ D z.�="; �="/ for any " > 0, the Cauchy problem for z isrewritten as (

Qz� C f .Qz/� D ".B.Qz/Qz� /� .� > 0; � 2 R/;

Qz.0; �/ D z0.�="/ .� 2 R/:

4 A. Matsumura

Hence, it can be expected that Qz D Qz" ! zR ."!C0/, which also suggests that foralmost every � 2 R, the asymptotic value of z.t; �t/ D Qz1=t .1; �/ as t !1 is givenby zR.1; �/. This shows an aspect of the relationship between the Riemann solutionand the asymptotic behavior of the solution of (1) and (2).

Next, some remarks are made on the strategies which have been employed inmany works to show the existence of the global solution in time and its asymptoticbehavior. For this important issue, it is hoped that the unique global existence wouldbe generally established for any smooth and bounded initial data, and then theasymptotic behavior would be investigated in each case. However, there has been nosuch general theory on the global existence, except for some special cases: the scalarcase (the maximum principle can be used) and the isothermal model of viscous gas(cf. [15]). Thus, the strategies employed in many practical works are as follows:

• Predict the asymptotic behavior from mathematical/physical considerations orfrom inspiration.

• Construct an approximate solution Z with the desired asymptotic behavior, as inthe form

Zt C f .Z/x � .B.Z/Zx/x D R .t > 0; x 2 R/;

where R represents a residual term with suitable decay properties.• Rewrite the problem around the approximate solutionZ in terms of the deviation� D z�Z, and look for the global solution in time � with the asymptotic behavior� ! 0 .t ! 1/ for a suitable class of the initial data, making use of whateverspecific properties of Z.

If this process fortunately works, not only the global existence in time but also theasymptotic behavior can be simultaneously shown in a neighborhood of Z. Thetrivial example is the case where z� D zC .DW Nz/, that is, the constant state Nz isexpected as the asymptotic state. In this case, there have been many works, includingthe global asymptotic stability for some concrete models and the asymptotic stabilityfor more general systems like Kawashima-Shizuta system (cf. [20, 24]), and evenmuch deeper results on the higher level of the approximation (arguments ondiffusion waves) also have been known (cf. [19, 20, 28, 45]). Since there aren’tenough pages to step further into this topic, only some remarks shall be given inthe following sections. Now, in the cases z� ¤ zC which are the main subjects inthis article, it should be remarked firstly on how the suitable approximate solutionZis constructed, in connection with the corresponding Riemann solution. As for theRiemann problem, it is known that under a proper assumption on the flux function,there are basic three kinds of simple wave solution, “shock wave” or “rarefactionwave’’ along the genuinely nonlinear characteristic field and “contact discontinuity”along the linearly degenerate characteristic field, and then the Riemann solutiongenerally forms a multi-wave pattern given by a various linear combination of thesethree simple waves (cf. [26, 46]). How the viscous effect influences these three

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 5

simple waves is inferred as follows. As for the rarefaction wave, the viscous effect isexpected to make the rarefaction smooth, but not to be strong enough to influence theasymptotic behavior because the rarefaction wave is spreading much faster than thediffusion process. In fact, a smoothed rarefaction wave with the same asymptoticbehavior can be approximately constructed. As for the shock wave, because thediscontinuity of the shock is formed by compression of waves (concentration ofparticle passes), the physically reasonable viscous system is expected to have acorresponding smooth traveling wave solution with a thin transition layer, which iscalled a “viscous shock wave.” On the other hand, for the contact discontinuity, sincethere are no nonlinear interactions across the discontinuity, the viscous system isexpected to have an approximate solution which relaxes the discontinuity by a lineardiffusion process, which is called a “viscous contact wave”. Then, the asymptoticbehavior of z is predicted, and also the approximate solution Z is constructedby replacing the every shock wave by the corresponding viscous shock wave,contact discontinuity by viscous contact wave, and rarefaction wave by smoothedrarefaction wave in the wave pattern of zR. Finally, here is a word on how to showthe global solution in time for the reformulated problem in terms of the deviation� D z � Z. In all the topics in this article, the global solution is constructed bycombining the unique existence of the local solution in time, together with thea priori estimate, which is obtained by various energy methods according to theindividual features of the problem.

Some Notations. Denote by C generic positive constants unless they need tobe distinguished. For function spaces, L2 D L2.R/ and Hk D Hk.R/ denote theusual Lebesgue space of square integrable functions and k-th order Sobolev spaceon the whole space R with norms jj � jj and jj � jjk , respectively.

2 Generalized Burgers’ Equation

This section picks up some topics on the asymptotic behavior of global solutionsin time of the generalized Burgers’ equation, which often appears as a basic scalarmodel in fluid dynamics. Through the topics, the basic strategy and techniques ofvarious energy methods to show the global existence in time and the asymptoticbehavior are presented. The Cauchy problem under consideration is described by

8̂̂̂<ˆ̂̂:

ut C f .u/x D �uxx .t > 0; x 2 R/;

u.0; x/ D u0.x/ .x 2 R/;

limx!˙1

u.t; x/ D u˙ .t � 0/;

(4)

where u D u.t; x/ is the unknown scalar function of t � 0 and x 2 R, the so-calledconserved quantity, f D f .u/ is the flux function, � is the viscosity coefficient,u0 is the given initial data, and u˙ 2 R are the prescribed far-field states. The

6 A. Matsumura

flux function f .u/ is assumed to be a given smooth function and � a positiveconstant. Also the far-field states u˙ are assumed to satisfy u� < uC without loss ofgenerality, and the initial data satisfy u0.˙1/ D u˙ as the compatibility condition.As stated in the introduction, it has been known that the asymptotic behavior shouldbe closely related to the corresponding Riemann solution, say uR, of the Riemannproblem

(ut C f .u/x D 0 .t > 0; x 2 R/;

u.0; x/ D uR0 .x/ .x 2 R/;(5)

where the initial data uR0 is given by

uR0 .x/ D uR0 .x I u�; uC/ D

(u� .x < 0/;

uC .x > 0/:

In fact, when the smooth flux function f is genuinely nonlinear on the whole R,i.e., f 00.u/ ¤ 0 .u 2 R/, Il’in-Oleinik [14] showed the following by using themaximum principle: if f 00.u/ > 0 .u 2 R/, that is, the Riemann solution consistsof a single rarefaction wave solution, the global solution in time of the Cauchyproblem (4) tends toward the rarefaction wave; if f 00.u/ < 0 .u 2 R/, that is,the Riemann solution consists of a single shock wave solution of the so-called Laxtype, the global solution of the Cauchy problem (4) tends toward the correspondingsmooth traveling wave solution (“viscous shock wave”) of (4). The Sect. 2.1 picksup the case f 00.u/ > 0 .u 2 R/ with some extra nonlinearity conditions and showshow to get the global asymptotics toward the rarefaction wave by using only anelementary energy method, without using the maximum principle. The Sect. 2.2picks up more general case where the Riemann solution consists of a single shockwave solution of the so-called Oleinik type, including the Lax type, and showsthe asymptotic stability of the corresponding viscous shock wave by introducinga technical weighted energy method. In the Sect. 2.3, a viscous contact wave inthe case the flux function is linearly degenerate is introduced, and finally, furtherremarks are given in the Sect. 2.4.

2.1 Global Asymptotics Toward Rarefaction Wave

In this subsection, the flux function f .u/ is further assumed to satisfy for somep > 1

f 00.u/ > 0 .u 2 R/; jf .u/j � jujp; jf 0.u/j � jujp�1 .u!˙1/: (6)

The typical example f .u/ D u2=2, Burgers’ equation, satisfies (6) with p D 2. It isshown that the global solution in time of the viscous problem (4) tends toward the

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 7

rarefaction wave of the inviscid part by using only an elementary energy method. Itis well known that under the assumptions f 00.u/ > 0 .u 2 R/ and u� < uC, theRiemann solution of (5), which is both mathematically and physically reasonable,is given by

uR D ur .x=t I u�; uC/ D

8̂<:̂

u� .x � �.u�/ t/;

��1.x=t/ .�.u�/ t � x � �.uC/ t/;

uC .x � �.uC/ t/;

(7)

where �.u/ D f 0.u/ (cf. [46]). This simple wave, which consists of two constantstates and one centered rarefaction wave, is called a “rarefaction wave” connectingfar-field states u˙. Then the main theorem in this subsection is as follows.

Theorem 1. Assume the flux f to satisfy (6), the far-field states u� < uC, and theinitial data u0�uR0 2 L

2 and u0;x 2 L2. Then, the Cauchy problem (4) has a uniqueglobal solution in time u, satisfying

(u � uR0 2 C.Œ 0;1/IL

2/;

ux 2 C.Π0;1/IL2/ \ L2loc.0;1IH1/;

and the asymptotic behavior

supx2R

ˇ̌u.t; x/ � ur .x=t I u�; uC/

ˇ̌! 0 .t !1/:

In order to show Theorem 1, the several preparations concerning the rarefactionwave are needed. In particular, it is needed to construct a smooth approximatesolution for the rarefaction wave (7), because the non-smoothness of ur at the edgesof the centered rarefaction causes a trouble to handle the second derivative of thesolution in the process of the a priori estimate. Start with the the rarefaction waveto the Riemann problem for the inviscid Burgers’ equation, with the given far-fieldstates w˙ 2 R .w� < wC/:8̂<

:̂wt C

�12w2�xD 0 .t > 0; x 2 R/;

w.0; x/ D wR0 .x Iw�;wC/ D

�w� .x < 0/;

wC .x > 0/:

(8)

In this simplest case, the rarefaction wave is exactly given by

wr .x=t I w�;wC/ D

8̂̂<ˆ̂:

w� .x � w�t /;

x=t .w�t � x � wCt /;

wC .x � wCt /:

(9)

8 A. Matsumura

Under the assumptions f 00.u/ > 0 .u 2 R/ and u� < uC, it is well known that therarefaction wave solution of the Riemann problem (5) is given by

ur .x=t I u�; uC/ D ��1�wr .x=t I ��; �C/

�(10)

which is nothing but the definition (7), where �˙ D �.u˙/ D f 0.u˙/. Here,following the arguments in [40], a smooth approximation W .t; xIw�;wC/ ofwr .x=t Iw�;wC/ is defined by the solution of the Cauchy problem

(Wt C

�12W 2

�xD 0 .t > 0; x 2 R/;

W .0; x/ D W0.xIw�;wC/ WDwCCw�

2C

wC�w�2

tanh x .x 2 R/:(11)

Then, corresponding to the formula (10), a smooth approximation U r.t; xI u�; uC/of ur .x=t I u�; uC/ is defined by

U r.t; x I u�; uC/ D ��1�W .t; x I ��; �C/

�: (12)

The next lemma shows the basic properties of U r .

Lemma 1. 1. U r is the smooth global solution in time of the Cauchy problem

8̂̂<ˆ̂:U rt C f .U

r/x D 0 .t � 0; x 2 R/;

U r.0; x/ D ��1��CC��

2C

�C���2

tanh x�

.x 2 R/;

limx!˙1

U r.t; x/ D u˙ .t � 0/:

2. u� < U r.t; x/ < uC and U rx .t; x/ > 0 .t � 0; x 2 R/.

3. For q 2 Œ1;1�, there exists a positive constant Cq such that

kU rx .t/kLq � Cq.1C t /

�1C 1q .t � 0/;

kU rxx.t/kLq � Cq.1C t /

�1 .t � 0/:

4. limt!1

supx2R

ˇ̌U r.t; x/ � ur .x=t/

ˇ̌D 0:

The proof of the lemma is given by elementary calculations. In fact, it is provedby the facts that the method of characteristic curve gives a representation formulaof the solution W of (11) as

(W .t; x/ D W0.x0.t; x// D

��C�C2C

�C���2

tanh.x0.t; x//;

x D x0.t; x/CW0.x0.t; x// t;(13)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 9

and the Jacobian of the variable transformation between x and x0 is given by

@x0

@xD

1

1CW 00 .x0/ t> 0: (14)

The details are omitted (refer to [40]). Then, from Lemma 1, U r is expected tobe a suitable approximate solution with the desired asymptotics for the viscousproblem (4) as

U rt C f .U

r/x � �Urxx D R; (15)

where the residual term R is given by R D ��U rxx . Now, by setting

� D u � U r

and using (15), the problem (4) is reformulated in terms of � as

(�t C .f .U

r C �/ � f .U r//x � ��xx D �R .t > 0; x 2 R/;

�.0; x/ D �0.x/ WD u0.x/ � U r.0; x/ .x 2 R/;(16)

and the global solution in time � 2 C.Œ 0;1/IH1/ is looked for. The correspondingtheorem for � to Theorem 1 is as follows.

Theorem 2. For any initial data �0 2 H1, the Cauchy problem (16) has a uniqueglobal solution in time �, satisfying

� 2 C.Œ 0;1/IH1/; �x 2 L2.0;1IH1/;

and the asymptotic behavior

supx2R

j�.t; x/ j ! 0 .t !1/:

The proof of Theorem 2 is given by combining the unique existence of the localsolution in time, together with the a priori estimate. To state the local existenceprecisely, the Cauchy problem at general initial time � � 0 with the given initialdata �� 2 H1 is formulated:

(�t C .f .U

r C �/ � f .U r//x � ��xx D �R .t > �; x 2 R/;

�.�; x/ D ��.x/ .x 2 R/:(17)

10 A. Matsumura

Proposition 1 (local existence). For any M > 0, there exists a positive constantt0 D t0.M/ not depending on � such that if �� 2 H1 and k��k1 � M , thenthe Cauchy problem (17) has a unique solution � on the time interval Œ� ; � C t0�satisfying

8<:� 2 C.Œ�; � C t0�IH

1/ \ L2.�; � C t0IH2/;

supt 2Œ� ;�Ct0�

k�.t/k1 � 2M:

The proof of Proposition 1 is very standard, so omitted.

Proposition 2 (a priori estimate). For any initial data �0 2 H1, there existsa positive constant C0.�0/ depending only on k�0k1 such that if the Cauchyproblem (16) has a solution � 2 C.Œ 0; T �IH1/ \ L2.0; T IH2/ for some T > 0,then it holds that

k�.t/k21 C

ˆ t

0

.kpU rx �k

2 C k�xk21/ d� � C0.�0/ .t 2 Œ 0; T �/: (18)

Once Propositions 2 is established, by combining the local existence Proposi-tion 1 with M D M0 WD

pC0.�0/, � D nt0.M0/, and �� D �.nt0.M0// .n D

0; 1; 2; : : : / together with the a priori estimate with T D .nC 1/t0.M0/ inductively,the unique solution of (17) � 2 C.Œ0; nt0.M0/�IH

1/ for any n 2 N is easilyconstructed, that is, the global solution in time � 2 C.Œ0;1/IH1/. Then, the apriori estimate again asserts that

supt�0

k�.t/k1 <1;

ˆ 10

.k�x.�/k21 C k

pU rx �.�/k

2/ d� <1: (19)

By using (19) and the equation, it is easy to see

ˆ 10

ˇ̌̌ dd�k�x.�/k

2ˇ̌̌d� <1: (20)

Hence, it follows from (19) and (20) that

k�x.t/k ! 0 .t !1/:

Due to the Sobolev’s inequality, the desired asymptotic behavior in Theorem 2 isthus obtained as

supx2R

j�.t; x/ j �p2 k�.t/ k

12 k�x.t/ k

12 ! 0 .t !1/:

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 11

Now Proposition 2 is shown for p � 2 for simplicity. The case p 2 .1; 2/ issimilarly proved with a slight modification. By multiplying (16) by � and integratingthe resultant formula with respect to x and t , it holds after integration by parts that

1

2k�.t/k2C

ˆ t

0

ˆ.f .U r C �/ � f .U r/ � f 0.U r/ �/U r

x dxd�

C�

ˆ t

0

ˆj�xj

2 dxd� D1

2k�0k

2 C

ˆ t

0

ˆ��U r

xx dxd�:

(21)

By using the Sobolev’s and Young’s inequalities, the second term in the right handside of (21) is estimated as follows:

ˇ̌̌ˇˆ t

0

ˆ��U r

xx dxd�

ˇ̌̌ˇ � p2�

ˆ t

0

k�k12 k�xk

12 kU r

xxkL1 d�

��

2

ˆ t

0

ˆj�xj

2 dxd� C C

ˆ t

0

.k�k2 C 1/.1C �/�43 d�;

(22)

where the decay estimate of U rxx in Lemma 1 is used. Then, plugging (22) to (21)

and using the Grownwall’s inequality and the assumptions on the nonlinearity (6)give

k�.t/k2 C

ˆ t

0

ˆ �.j�j2 C j�jp/U r

x C j�xj2�dxd� � C.1C k�0k

2/: (23)

By multiplying the equation of (16) by ��xx and using the Young’s inequality, theestimate of the derivative �x is obtained as

1

2k�x.t/k

2 C

ˆ t

0

ˆ�

2j�xxj

2 dxd�

�1

2k�0;xk

2 C C

ˆ t

0

ˆ.j.f .U r C �/ � f .U r//xj

2 C jU rxxj

2/ dxd�:

(24)

The second term in the right hand side of (24) is estimated as follows:

ˆ t

0

ˆj.f .U r C �/ � f .U r//xj

2 dxd�

� C

ˆ t

0

ˆ �jf 0.U r C �/j2j�xj

2 C jf 0.U r C �/ � f 0.U r/j2jU rx j2�dxd�

� C

ˆ t

0

ˆ �.1C j�j2.p�1//j�xj

2 C .j�j2 C j�j2.p�1//jU rx j2�dxd�;

(25)

12 A. Matsumura

and

ˆ t

0

ˆjU rxxj

2 dxd� �

ˆ t

0

kU rxxk

2L2d� � C

ˆ 10

.1C �/�2 d� <1: (26)

In order to dispose the right hand side of (25) without using the maximum principlefor u D U r C �, multiplying (16) by j�j2.p�1/� yields as in (21)

1

2p

ˆj�.t/j2p dx C .2p � 1/

ˆ t

0

ˆ � ˆ �

0

.f 0.U r C s/

� f 0.U r//jsj2.p�1/ ds�U rx dxd� C �.2p � 1/

ˆ t

0

ˆj�j2.p�1/j�xj

2 dxd�

D1

2p

ˆj�0j

2p dx C

ˆ t

0

ˆ�j�j2.p�1/�U r

xx dxd�: (27)

As in (22), due to the Sobolev’s and Young’s inequalities, the second term in theright hand side of (27) can be estimated as follows:

ˇ̌̌ˆ t

0

ˆ�j�j2.p�1/�U r

xx dxd�ˇ̌̌�

ˆ t

0

ˆ�.j�jp/

2p�1p jU r

xxj dxd�

�p2�

ˆ t

0

kj�jpk2p�12p k.j�jp/xk

2p�12p kU r

xxkL1 d�

��.2p � 1/

2

ˆ t

0

ˆj�j2.p�1/j�xj

2 dxd�

C C

ˆ t

0

�1C

ˆj�j2p dx

�.1C �/

�4p

2pC1 d�: (28)

Hence, by plugging (28) to (27) and using the Grownwall’s inequality together withthe assumption (6), it holds

ˆj�.t/j2p dxC

ˆ t

0

ˆ..j�j2p C j�j3p�2/U r

x C j�j2.p�1/j�xj

2/ dxd�

� C

�ˆj�0j

2p dx C 1

�:

(29)

Thus, combining all the estimates (23)�(26) and (29) completes the desired a prioriestimate (18).

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 13

2.2 Asymptotic Stability of Viscous Shock Wave

This subsection picks up a topic on the asymptotic stability of viscous shock wavefor the Cauchy problem (4). Through this topic, another type of a priori estimateand a technical weighted energy method are presented, which are useful to treatsmall initial perturbations. As stated in the introduction, when the Riemann solutionconsists of a single shock wave

uR D us.x � Ns t I u�; uC/ WD

(u� . x � Ns t /;

uC . x � Ns t /;

which is mathematically and physically reasonable, the equation with viscosity isexpected to have a corresponding smooth traveling wave solution with the sameshock speed Ns and the far-field states u˙. Here, the shock speed Ns is determinedby the so-called the Rankine-Hugoniot condition, and for the solution to be unique,some additional condition (e.g., the so-called “Lax’s shock (entropy) condition”,“Oleinik’s shock (entropy) condition”) depending on the nonlinearity of f isimposed. In this subsection, apart from the Riemann solution for a moment, itstarts with studying the necessary and sufficient conditions on the existence of thetraveling wave solution for (4). First, assume that a smooth traveling wave solutionof the form

u D U.�/; � D x � st (30)

satisfies the equation and the far-field conditions in (4), where s 2 R is a propagatingspeed. Plugging (30) to (4) gives

(.�sU C f .U / � �U 0/0 D 0 .� 2 R/;

U .˙1/ D u˙;(31)

where 0 D d=.d�/. In order for the solution of (31) to exist, it is easy to see that itnecessarily holds U 0.˙1/ D 0 and

� sU C f .U / � �U 0 D �su˙ C f .u˙/ .� 2 R/; (32)

which implies the “Rankine-Hugoniot condition”

� s.uC � u�/C .f .uC/ � f .u�// D 0: (33)

Then, under the “Rankine-Hugoniot condition” (33), it follows from (32) that

(�U 0 D f .U / � f .u˙/ � s.U � u˙/ DW h.U / .� 2 R/;

U .˙1/ D u˙;(34)

14 A. Matsumura

which is equivalent to (31). In order for the solution of (34) to exist, it is again easyto see that when u� < uC (resp. u� > uC) it necessarily holds

h.u/ D f .u/ � f .u˙/ � s.u � u˙/ > 0 .u 2 .u� < u < uC//�resp. h.u/ D f .u/ � f .u˙/ � s.u � u˙/ < 0 .u 2 .uC < u < u�//

�;

(35)which is nothing but the “Oleinik’s shock condition” (or “Oleinik’s entropycondition”). It is noted that in the case where f 00.u/ ¤ 0 .u 2 R/, the condition (35)is equivalent to f 0.u�/ > f 0.uC/ which is known as the “Lax’s shock condition”(or “Lax’s entropy condition”). Thus it should be emphasized that the Rankine-Hugoniot and Oleinik’s shock conditions which are usually imposed for theRiemann solution of the inviscid equation to uniquely exist are naturally obtainedas the necessary ones for the existence of a traveling wave solution of the viscousequation. Conversely, it can be proved that the Rankine-Hugoniot and Oleinik’sshock conditions are sufficient for the existence of a traveling wave solution of (4).In fact, under these two conditions, the existence of the solution of (34) which isunique up to shift of � can be proved by standard theory of ordinary differentialequations. This traveling wave solution is called a “viscous shock wave” connectingfar-field states u˙.

Now, for a fixed viscous shock wave U , the asymptotic stability is the nextproblem to be considered. It firstly should be noted that the usual definition ofasymptotic stability needs to be modified a bit because for any ˛ 2 R, the spatiallyshifted function U.x� st C˛/ also gives a viscous shock wave of (4). In fact, if theinitial data is taken as u0.x/ D U.x C ˛/, which is a perturbed one from U.x/ byU.xC˛/�U.x/, then the corresponding perturbed solution u D U.x�stC˛/ nevertends toward the unperturbed one U.x � st/, no matter how ˛ is small. It suggeststhat for general small perturbations, the solution may tend toward a spatially shiftedU . Here a heuristic argument is shown to see how the shift is determined. To dothat, it is assumed that the perturbation u � U˛ has every nice decay property withrespect to t and x, where U˛.�/ D U.� C ˛/. Since U˛ is an exact solution of (4),it holds

.u � U˛/t C .f .u/ � f .U˛//x � �.u � U˛/xx D 0: (36)

Integration of (36) with respect to x gives an expectation

ˆ.u � U˛/.t; x/ dx D

ˆ.u0.x/ � U.x C ˛// dx ! 0 .t !1/;

which implies

ˆ.u0.x/ � U.x C ˛// dx D 0: (37)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 15

Thus, by an elementary calculation for (37), the shift ˛ is expected to be given by

˛ D

´.u0.x/ � U.x// dx

uC � u�; (38)

and the solution is expected to eventually tends toward the spatially shifted U˛ . Inwhat follows, it shall be proved true for small initial perturbations. For the proof,assume the initial data to satisfy

u0 � U 2 H1 \ L1; (39)

and the ˛ is given by (38) so that (37) holds. By setting

�0.x/ D

ˆ x

�1

.u0 � U˛/.y/ dy; (40)

further assume

�0 2 L2: (41)

Note that (41) is equivalent to �0 2 H2 under the condition (39). Then the maintheorem in this subsection is stated as follows.

Theorem 3. Suppose the Rankine-Hugoniot condition (33) and the Oleinik’s shockcondition (35) with f 0.u˙/ ¤ s, and U.x � st/ to be a viscous shock waveof (4) connecting the far-field states u˙. Furthermore, suppose the initial data u0satisfy (39) and (41) with the shift ˛ given by (38). Then there exists a positiveconstant "0 such that if k�0k2 � "0, the Cauchy problem (4) has a unique globalsolution in time u, satisfying

u � U˛ 2 C.Œ0;1/I H1/ \ L2.Œ0;1/IH2/;

and the asymptotic behavior

supx2R

ju.t; x/ � U.x � st C ˛/j ! 0 .t !1/:

Before the proof of Theorem 3, a heuristic explanation is given on how to treatthis problem by an energy method. For a moment, u� < uC and f 00.u/ ¤ 0 .u 2 R/

are assumed for simplicity. Then, Lax’s shock condition and (34) yields

f 00.u/ < 0 .u 2 R/; u� < U.�/ < uC; U0.�/ > 0 .� 2 R/: (42)

Also assume that the deviation v D u � U˛ from U˛ is small and has every nicedecay properties as needed. Then, it is expected that the asymptotic behavior of v is

16 A. Matsumura

described by the linearized problem at U˛(vt C .f

0.U˛/v/x � �vxx D 0 .t > 0; x 2 R/;

v.0; x/ D v0.x/ WD u0.x/ � U˛.x/ .x 2 R/:(43)

As for the the L2-estimate for v, as in (21), multiplying the equation of (43) by vand integration by parts easily give

1

2kv.t/k2 C

ˆ t

0

ˆ1

2U 0˛f

00.U˛/jvj2 dxd� C �

ˆ t

0

ˆjvxj

2 dxd� D1

2kv0k

2:

(44)Since U˛ 0f 00.U˛/ < 0 from (42), the second term in (44) causes a difficulty to havea dissipative property of v by standard energy estimate. This difficulty can also beexplained by a fact that U 0˛ is an exact solution of the linearized equation in (43),which suggests that unless an extra condition on the v0 is used, v never tends towardzero. To overcome the difficulty, recall ˛ is chosen so that the integral of v0 is zerowhich induces ˆ

v.t; x/ dx D

ˆv0.x/ dx D 0 .t � 0/: (45)

This property suggests that v has a form

v D u � U˛ D �x .t � 0/ (46)

for a function � which also has nice properties as needed. In fact, for the Fourierimage of v, say Ov.�/, the property (45) asserts Ov.0/ D 0 which makes therepresentation Ov.�/ D i�. Ov.�/=.i�// D i� O�.i�/ have a sense. By plugging theform (46) into (43) and integrating it with respect to x once, it results in

(�t C f

0.U˛/�x � ��xx D 0 .t > 0; x 2 R/;

�.0; x/ D �0.x/ .x 2 R/;(47)

where �0 is as in (40). Again, multiplying the equation of (47) by � gives

1

2k�.t/k2 �

ˆ t

0

ˆ1

2U 0˛f

00.U˛/j�j2 dxd� C �

ˆ t

0

ˆj�xj

2 dxd� D1

2k�0k

2:

(48)Since the second term in (48) does have the right sign this time, it is expectedthat the L2-energy method works well for an integrated problem (47) and even forthe original nonlinear problem. This basic idea to apply the L2-energy method forthe antiderivative � of v under the integral zero condition on v was first proposedin Matsumura-Nishihara [39] and Goodman [5] independently to investigate theasymptotic stability of viscous shock waves for some systems of viscous conserva-tion laws. This method is often referred as to “antiderivative method.”

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 17

Now, turn to the proof of Theorem 3. As suggested in the above heuristicargument, the solution of (4) is also expected to be given in the form

u D U˛ C �x: (49)

Plugging (49) into (36) and integrating it with respect to x give the following Cauchyproblem in terms of �:

(�t C .f .U˛ C �x/ � f .U˛// � ��xx D 0 .t > 0; x 2 R/;

�.0; x/ D �0.x/ .x 2 R/:(50)

Once the problem (50) is formulated even by heuristic argument, the small globalsolution in time � 2 C.Œ 0;1/IH2/ of (50) with the asymptotic behaviork�x.t/kL1 ! 0 .t ! 1/ is looked for. Then u is newly defined by (49) forthe existence of the solution of (4) with the desired asymptotic behavior, and lastlythe uniqueness of u�U˛ in the class C.Œ0;1/I H1/\L2.Œ0;1/IH2/ is confirmed.Thus, the proof of Theorem 3 can be completed via the following theorem in termsof �.

Theorem 4. Under the same assumptions in Theorem 3, there exists a positiveconstant "0 such that if k�0k2 � "0, the Cauchy problem (50) has a global solutionin time �, satisfying

� 2 C0.Œ0;1/I H2/ \ L2loc.Œ0;1/IH3/;

and the asymptotic behavior

supx2R

j�x.t; x/j ! 0 .t !1/:

The proof of Theorem 4 is also given by combining the unique existence of thelocal solution for the given initial data at general initial time � � 0, together with thea priori estimate, as in the Sect. 2.1. The description of the local existence theoremcorresponding to Proposition 1 is omitted, because it is the same except replacingthe initial data class byH2 and the solution class by C.Œ �; �C t0 �IH2/\L2.�; �C

t0IH3/.

The a priori estimate to obtain the small global solution in time is as follows.

Proposition 3 (a priori estimate). There exist positive constants ı0 and C0depending only on the shape of the flux function f , the viscous coefficient �and the far-field states u˙ such that if the Cauchy problem (50) has a solution� 2 C.Œ 0; T �IH2/ \ L2.0; T IH3/ for some T > 0 and

supt2Œ0;T �

k�.t/k2 � ı0;

18 A. Matsumura

then it holds that

k�.t/k22 C

ˆ t

0

�k

qjU 0˛j�k

2 C k�xk22

�d� � C0k�0k

22 .t 2 Œ0; T �/: (51)

This type of a priori estimate to obtain small global solutions in time wasfirst well formulated and applied to multidimensional quasilinear dissipative waveequations by Matsumura [35] and later applied to the system for multidimensionalcompressible and heat-conductive gas by Matsumura-Nishida [38]. Once the a prioriestimate Proposition 3 is established, choosing "0 D ı0=.2

pC0/ and combining the

local existence together with the a priori estimate Proposition 3 with M D M0 WD

ı0=2 and T D nt0.M0/ .n 2 N/ step by step give the small global solution in time� 2 C.Œ0;1/IH2/, and the desired asymptotic behavior is shown in the same wayas in the last subsection.

Here the proof of Proposition 3 is given by following the arguments in [42].Assume u� < uC without loss of generality, and then assume the Oleinik’s shockcondition (35). Then it follows from (34) that

u� < U.�/ < uC; U0.�/ > 0 .� 2 R/: (52)

Next, rewrite the equation of (50) so that all the linearized terms at U˛ are collectedon the left hand side and nonlinear term on the right as in the form

�t C f0.U˛/�x � ��xx D N.�x/; (53)

where the nonlinear term N is given by

N.�x/ D �.f .U˛ C �x/ � f .U˛/ � f0.U˛/�x/:

In the case f satisfies the Lax’s shock condition, simple multiplication of (53) by �gives a nice estimate as in the previous discussion. However, since a more generalcondition, the Oleinik’s condition, is assumed here, a weighted energy method isnewly proposed, where the weight function w is chosen as a positive function of thetarget asymptotic state U˛ itself, that is, w D w.U˛/. This way of choosing weightfunction was first proposed in Matsumura-Nishihara [39] to study the asymptoticstability of the viscous shock wave for an isentropic/isothermal model of viscousgas. By multiplying the equation (53) by w� and integrating with respect to x andt , it holds after integration by parts

ˆ1

2wj�.t/j2 dx C

ˆ t

0

ˆ1

2..s � f 0/w/0j�j2U 0˛ dxd�

C

ˆ t

0

ˆ.�w0U 0˛��x C �wj�xj

2/ dxd� D

ˆ1

2wj�0j

2 dx C

ˆ t

0

ˆw�N dxd�:

(54)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 19

From �U 0 D h.U /, f 0.U /�s D h0.U / and taking integration by parts with respectto x again, it follows that

ˆ1

2wj�.t/j2 dx �

ˆ t

0

ˆ1

2.hw/00j�j2U 0˛ dxd�

C

ˆ t

0

ˆ�wj�xj

2 dxd� D

ˆ1

2wj�0j

2 dx C

ˆ t

0

ˆw�N dxd�:

(55)

Now define w.u/ by

w.u/ D

8̂̂<ˆ̂:�.u � u�/.u � uC/

h.u/.u� < u < uC/;

�u˙ � u�f 0.u˙/ � s

.u D u˙/:(56)

Then it follows from the assumptions in Theorem 3 that w is smooth in .u�; uC/,continuous on Œu�; uC�, and in particular uniformly bounded both from below andabove, that is,

C�1 � w.u/ � C .u 2 Œu�; uC�/

for a positive constant C . Due to the identity .wf /00.U / D �2 in (55), it thus holdsthat

k�.t/k2 C

ˆ t

0

�kpU 0˛�k

2 C k�xk2�d� � C

�k�0k

2 C

ˆ t

0

ˆj�N j dxd�

�:

(57)To have the L2-estimate of the derivative �x , multiply the equation by ��xx in thesame way as in the last subsection. Also, for the second derivatives �xx , differentiatethe equation once and multiply it by ��xxx . The details are omitted. Then it resultsin

k�.t/k22C

ˆ t

0

.kpU 0˛�k

2 C k�xk22/ d�

�Ck�0k22 C C

ˆ t

0

ˆ �j�N j C jN j2 C jNxj

2�dxd�:

(58)

Finally, due to the Sobolev’s inequality, the nonlinear terms on the right hand sideof (58) are easily estimated as

ˆ t

0

ˆ �j�N j C jN j2 C jNxj

2�dxd� � C"0

ˆ t

0

k�xk21 d�; (59)

20 A. Matsumura

provided k�k2 � "0 � 1. Thus, plugging (59) to (58) and choosing "0 suitably smallcomplete the desired a priori estimate, Proposition 3.

2.3 Viscous Contact Wave

This subsection introduces a “viscous contact wave” of (4) and gives relatedseveral remarks on it. First, consider the case where the flux function f is linearlydegenerate on whole R, that is, f 00.u/ D 0 .u 2 R/. Then, since f have the formf .u/ D �uC c for some constants � and c, it is known that the Riemann solutionof (5) is given by

uR D uc.x � � t I u�; uC/ D

(u� . x � � t /;

uC . x � � t /:(60)

This simple wave with a discontinuity whose propagation speed and characteristicspeeds of the both sides are all given by the same � is called a “contact discontinu-ity” connecting far-field states u˙. Then, since the corresponding viscous equation(linear convective heat equation)

ut C � ux D � uxx

with the Riemann initial data uR0 has the exact solution

u D U vc

�x � � tptI u�; uC

�; (61)

where U vc.� I u�; uC/ is defined by

U vc.� I u�; uC/ D u� CuC � u�p

ˆ �

�1

e�2

d; (62)

the solution of (4) is expected to tend toward the exact solution U vc as time goesto infinity. In fact, because the equation is linear, it is easily proved that for anyinitial data satisfying u0 � uR0 2 L

2, it does hold. This kind of solution U vc (moregenerally, could be approximate solution) which has a diffusive structure along amoving frame .x��t; t/ as in (61) is called a “viscous contact wave” connecting far-field states u˙. It is emphasized that the viscous shock wave U in the last subsectionand the contact wave U vc have quite different feathers even when the correspondingRiemann solutions are same each other.

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 21

2.4 Further Remarks

A remark to the Sect. 2.1. In the trivial case u� D uC.DW Nu/ of the Riemannproblem, the constant state Nu is expected to be asymptotically stable for any initialperturbation u0 � Nu 2 H1. In this case, if the wave ur is regarded as the identicalconstant Nu, the entirely same energy method in the proof of Theorem 1 is applicable.Thus, the global asymptotic stability of the constant state is also proved.

A remark to the Sect. 2.2. In the heuristic argument to determine the shift ˛,the expectation

´.u � U˛/ dx ! 0 .t !1/ is used. However, Theorem 3 doesn’t

assure whether the solution constructed by Theorem 3 satisfies this property or not.Furthermore, in the usage of the antiderivative method, the initial perturbation isassumed to be suitably small. On this issue, Freistühler-Serre [4] should be referredto, where they showed the global asymptotic L1-stability even for the Oleinik’s typeviscous shock wave using a kind of the maximum principle.

Remarks to the Sect. 2.3. Consider a case where the flux function f is smooth(2 C1) and genuinely nonlinear on the whole R except a finite interval I D .a; b/ �R and linearly degenerate on I , that is,

(f 00.u/ > 0

�u 2 .�1; a � [ Œ b;C1/

�;

f 00.u/ D 0�u 2 .a; b/

�:

(63)

Under the condition (63), it is known that unless the interval .a; b/ is disjoint fromthe interval .u�; uC/, the Riemann solution consists of the contact discontinuity withthe jump from u D a to u D b and the rarefaction waves, depending on the choiceof a, b, u�, and uC. For example, if

u� < a < b < uC; (64)

the Riemann solution is given by a multi-wave pattern given by the linear combina-tion of the three simple waves as in the form

uR D ur�xtI u�; a

�C uc.x � N� t I a; b/C ur

�xtI b; uC

�� a � b; (65)

where

N� Df .b/ � f .a/

b � aD f 0.a/ D f 0.b/:

22 A. Matsumura

This is a very interesting example where the Riemann solution forms a multi-wave pattern even along the single characteristic field. Then, corresponding to theform (65), the asymptotic state of the solution of (4) is expected to be given by

U D ur�xtI u�; a

�C U vc

x � N� tptI a; b

!C ur

�xtI b; uC

�� a � b: (66)

In fact, Matsumura-Yoshida [44] recently succeeded in showing this conjectureby combining a technical energy estimate and also the maximum principle. Theyproved the solution u tends toward U uniformly with respect to x as t !1.

Another more generic case where the Riemann solution forms a multi-wavepattern is one where the flux function f is smooth and genuinely nonlinear on thewhole R except a finite interval I D .a; b/ � R and satisfies the Oleinik’s shockcondition (35) (for u� D a; uC D b; s D Ns WD .f .b/ � f .a//=.b � a/) on I , thatis,

(f 00.u/ > 0 .u 2 .�1; a � [ Œ b;C1// ;

f .u/ � f .a/ � Ns.u � a/ > 0 .u 2 .a; b// :(67)

Like in (65), if u� < a < b < uC, the Riemann solution has a multi-wave pattern

uR D ur�xtI u�; a

�C us.x � Ns t I a; b/C ur

�xtI b; uC

�� a � b: (68)

Accordingly, the asymptotic state of the solution of (4) is expected to be given by

U D ur�xtI u�; a

�C U vs.x � Ns t C ˛ I a; b/C ur

�xtI b; uC

�� a � b (69)

for some shift parameter ˛ 2 R. On this problem, Weinberger [48] should bereferred to, where the author gave a mathematical justification to the relationbetween the Riemann solution uR and the asymptotic behavior of the solution of (4)on which the heuristic argument explained in the introduction, that is, for almostall � 2 R, u.t; �t/ ! uR.1; �/ .t ! 1/. Note that U in (69) also satisfiesU.t; �t/ ! uR.1; �/ .t ! 1/. However, it should be emphasized that along theline x D �t , any behavior of the viscous shock wave inside the transition layeris asymptotically ignored. Thus, the uniform asymptotics with respect to x is stillinteresting open problem.

3 Isentropic/Isothermal Model

This section proceeds to the topics on a model system which describes one-dimensional isentropic/isothermal motion of viscous gas in the Lagrangian masscoordinates as in the form

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 23

8̂<:̂vt � ux D 0;

ut C p.v/x D .�uxv/x .t > 0; x 2 R/;

p.v/ D av�� ;

(70)

where the unknown functions v > 0 and u are the specific volume, the fluid velocity,while the constants � > 0 and � .� 1/ denote the viscosity coefficient and adiabaticconstant, and a is a positive constant. The Cauchy problem to the system (70) isstudied with initial and far-field conditions8<

:.v; u/.0; x/ D .v0; u0/.x/ .x 2 R/;

limx!˙1

.v; u/.t; x/ D .v˙; u˙/ .t � 0/;(71)

where v˙ .> 0/ and u˙ 2 R are the given far-field states and it is assumed.v0; u0/.˙1/ D .v˙; u˙/ as the compatibility conditions. As in Sect. 2, this sectioninvestigates the asymptotic behavior of the global solution in time of the Cauchyproblem (70), (71) in relation to the corresponding Riemann solution, say .vR; uR/,of the Riemann problem

8̂̂ˆ̂<ˆ̂̂̂:

vt � ux D 0;

ut C p.v/x D 0 .t > 0; x 2 R/;

.v; u/.0; x/ D .vR0 ; uR0 /.x/ D

�.v�; u�/ .x < 0/;.vC; uC/ .x > 0/:

(72)

Section 3 is organized as follows. The Sect. 3.1 presents an introductory dis-cussion in terms of energy inequalities to realize a basic dissipative structureunderlaid in the system (70) through the linearized system at the constant state.v; u/ D .1; 0/. After recalling the basic results in the theory of conservationlaws on the Riemann problem (72) in the Sect. 3.2, a survey of the known resultson the asymptotic behaviors in time is given in the Sect. 3.3. Then, two specifictopics are presented. One from Matsumura-Nishihara [41] on the global asymptoticstability of the rarefaction waves is presented in the Sect. 3.4 and the other fromMatsumura-Wang [43] on the asymptotic stability of large viscous shock wave witha density-dependent viscosity in the Sect. 3.5. Through these two specific topics,how the various weighted energy methods work for systems is presented.

3.1 Dissipative Structure

Linearizing the system (70) at the trivial constant state .v; u/ D .1; 0/ gives thelinear system �

vt � ux D 0;ut � a�vx � �uxx D 0 .t > 0; x 2 R/:

(73)

24 A. Matsumura

Although the system (73) is not uniform parabolic system, it can be shown thatthe whole system is dissipative. For the basic L2-energy estimate, multiplying thesecond equation of (73) by u and integrating with respect to x and t as usual, withthe aid of the first equation, easily yield the basic energy equality

ˆ1

2.juj2 C a� jv � 1j2/ dx

ˇ̌̌t0C �

ˆ t

0

ˆjuxj

2 dxd� D 0: (74)

Note that the first term of (74) corresponds to the conservation of total energy andthe second one the dissipation effect for u due to the viscosity. Now, by using thefirst equation ux D vt , the second one of (73) can be rewritten in the form

.�vx � u/t C a�vx D 0: (75)

Then, multiply (75) by vx to have the another energy equality

ˆ ��2jvxj

2 � uvx�dxˇ̌̌t0C a�

ˆ t

0

ˆjvxj

2 dxd� �

ˆ t

0

ˆjuxj

2 dxd� D 0: (76)

With the aid of (74), the second term of (76) assures the dissipation effect of v,and also the first term gives the uniform point-wise boundedness of v from above.Finally, multiply the second equation again by �uxx to have

ˆ1

2juxj

2 dxˇ̌̌t0C a�

ˆ t

0

ˆvxuxx dxd� C �

ˆ t

0

ˆjuxxj

2 dxd� D 0: (77)

Thus, the energy equalities (75), (76) and (77), with the help of the Young’sinequality, eventually induce

k.v � 1; u/.t/k21 Cˆ t

0

. kvxk2 C kuxk

21/ d� � Ck.v0 � 1; u0/k

21 .t � 0/ (78)

for any initial data .v0 � 1; u0/ 2 H1. A nonlinear counterpart to the abovearguments was first introduced by Kanel [18] to show the global asymptotic stabilityof the constant state .v; u/ D .1; 0/ for (70), (71), provided initial data satisfies.v0 � 1; u0/ 2 H1; inf v0 > 0. This Kanel’s argument will be used in the Sect. 3.4.It is also noted that the similar arguments were developed in Matsumura-Nishida[38] to show the asymptotic stability of constant states in the multidimensionalcases. Furthermore, these kinds of arguments have been much sophisticated evenfor general hyperbolic-parabolic systems by Kawashima [20], Kawashima-Shizuta[24], etc.

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 25

3.2 Riemann Problem

This subsection quickly recalls the classification of the Riemann solutions of (72)(cf. [46]). Now, by setting

z D

�v

u

�; f .z/ D

��up.v/

�;

the equation in (72) is written in the form

zt C f .z/x D 0 .t > 0; x 2 R/: (79)

Then it is easy to see that for any z 2 O WD ft .v; u/j v > 0; u 2 R g, the Jacobimatrix of f is given by

Dzf D

�0 �1

p0.v/ 0

�; (80)

its two eigenvalues by

�1.v/ D �.�p0.v//1=2; �2.v/ D .�p

0.v//1=2 (81)

and the corresponding right eigenvectors by

r1.v/ D

�1

��1.v/

�; r2.v/ D

�1

��2.v/

�: (82)

Since p0.v/ D �a�v���1 < 0; p00.v/ D a�.� C 1/v���2 > 0 .v > 0/, it followsfrom (81) and (82) that �1.v/ < 0 < �2.v/ .v > 0/ and

rz�i � ri D �.�1/i 1

2.�p0.v//�1=2p00.v/ ¤ 0; .v > 0; i D 1; 2/;

which means the system (79) is strictly hyperbolic (regularly hyperbolic) and itsboth characteristic fields are uniformly genuinely nonlinear on any bounded domainin the phase space O . In what follows, for any fixed z� 2 O , the Riemann solutionszR of (72) are classified in terms of the location of zC 2 O , where zR D t .vR; uR/and z˙ D t .v˙; u˙/. It is firstly noted that for each i D 1; 2, the integral curve ofri , say Ii .z�/, which passes through z� is given by

Ii .z�/ D

�.v; u/ 2 O j u D u� �

ˆ v

v�

�i .s/ ds

;

and along the integral curve in the phase space, the smooth simple wave solutionof (79) is reduced to the inviscid Burgers’ equation �i;tC.�2i =2/x D 0 in terms of �i ,

26 A. Matsumura

provided the characteristic field is genuinely nonlinear. Then, from the arguments inthe Sect. 2.1, the so-called rarefaction curves starting from z� are naturally definedby

Ri.z�/ D fz 2 Ii .z�/ j �i .v/ � �.v�/g .i D 1; 2/;

and for zC 2 Ri.z�/, the Riemann solution zR D zri .x=t I z�; zC/ Dt .vri ; u

ri /.x=t I z�; zC/ of (72), the so-called a i -rarefaction wave connecting far-

field states z˙, is given by

.vri ; uri /.x=t I z�; zC/ D

8̂̂ˆ̂<ˆ̂̂̂:

.v�; u�/ .x � �i .v�/t/;(vri D �

�1i .x=t/;

uri D u� �´ vriv��i .s/ds

.�i .v�/t � x � �i .vC/t/;

.vC; uC/ .x � �i .vC/t/:(83)

Next, when a quarter region R1R2.z�/ of O is defined by

R1R2.z�/ D

�z 2 O j u > u� �

ˆ v

v�

�1.s/ ds; u > u� �ˆ v

v�

�2.s/ ds

;

(84)it can be shown that if zC 2 R1R2.z�/, then there exists a unique intermediate statezm 2 R1.z�/ satisfying zC 2 R2.zm/, and the Riemann solution is given by

zR D zr .x=t I z�; zC/ D zr1.x=t I z�; zm/C zr2.x=t I zm; zC/ � zm; (85)

which has a multi-wave pattern consisting of two rarefaction waves.In the cases where the Riemann solution consists of a simple wave of shock type

with a shock speed s 2 R as

zR D zs.x � st I z�; zC/ D

�z� x < st;

zC x > st;(86)

it is known that for zs of (86) to be the Riemann solution of (72), it necessarily holdsthe Rankine-Hugoniot condition

��s.vC � v�/ � .uC � u�/ D 0;�s.uC � u�/C .p.vC/ � p.v�// D 0:

(87)

Then (87) implies the speed s is determined in two ways s D si .v�; vC/ .i D 1; 2/by

s1.v�; vC/ D �

s�p.vC/ � p.v�/

vC � v�< 0; s2.v�; vC/ D �s1.v�; vC/ > 0;

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 27

and for the fixed z� 2 O and i D 1; 2, uC is represented by

uC D u� � .vC � v�/si .v�; vC/:

Furthermore, under the Rankine-Hugoniot condition, for the solution zs of (86) withs D si .v�; vC/ to be mathematically and physically reasonable, the Lax’s shockcondition

�i .v�/ > �i .vC/

is usually imposed for genuinely nonlinear characteristic field. Thus, the so-calledshock curves starting from z� are naturally defined by

Si .z�/ D˚z 2 O j u D u� � .v � v�/si .v�; v/; �i .v/ � �i .v�/

;

and for zC 2 Si .z�/, the Riemann solution zR D zsi .x�si t I z�; zC/ D t .vsi ; usi /.x�

si t I z�; zC/ of (72), the so-called a i -shock wave connecting far-field states z˙, isgiven by the formula (86) with s D si .v�; vC/. Like R1R2.z�/, another quarterregion S1S2.z�/ of O is defined by

S1S2.z�/ D˚z 2 O j u < u� � .v � v�/s1.v�; v/; u < u� � .v � v�/s2.v�; v/

;

and it can be shown that for zC 2 S1S2.z�/, there exists a unique intermediate statezm 2 S1.z�/ satisfying zC 2 S2.zm/, and then the Riemann solution is given by

zR D zs.t; xI z�; zC/ WD zs1.x � s1t I z�; zm/C zs2.x � s2t I zm; zC/ � zm; (88)

which has a multi-wave pattern consisting of two shock waves.For the remaining cases, another two quarter regions are defined by

R1S2.z�/ D

�z 2 O j u < u� �

ˆ v

v�

�1.s/ ds; u > u� � .v � v�/s2.v�; v/

;

and

S1R2.z�/ D

�z 2 O j u > u� � .v � v�/s1.v�; v/; u < u� �

ˆ v

v�

�2.s/ ds

:

Then as in the previous cases, for zC 2 R1S2.z�/ .resp. zC 2 S1R2.z�//, thereexists a unique intermediate state zm 2 R1.z�/ .resp. zm 2 S1.z�// satisfying zC 2S2.zm/ .resp. zC 2 R2.zm//, and then the Riemann solution is given by

zR D zr1.x=t I z�; zm/C zs2.x � s2t I zm; zC/ � zm (89)

�resp. zR D zs1.x � s1t I z�; zm/C zr2.x=t I zm; zC/ � zm

�; (90)

28 A. Matsumura

which has a multi-wave pattern consisting of both shock and rarefaction waves.Thus, for any fixed z� 2 O , the Riemann solutions are classified in eight

cases except the trivial case zC D z�, depending on the sets where zC is located,R1.z�/; R2.z�/; S1.z�/, S2.z�/; R1R2.z�/; S1S2.z�/; R1S2.z�/; and S1R2.z�/.

3.3 Historical Remarks

This subsection gives a brief survey on the known results and related remarks on theasymptotic behaviors of the solution of (70), (71) in connection with the Riemannproblem (72). In what follows in this subsection, a far-field state z� 2 O is fixed.

First, in the trivial case zC D z� .DW Nz/, as already mentioned before, Kanel [18]showed the global asymptotic stability of the constant state Nz for any smooth initialdata satisfying z0 � Nz 2 H1 and inf v0 > 0, by using a technical idea to obtain theuniform boundedness of v.

Second, in the cases zC 2 R1.z�/ [ R2.z�/ [ R1R2.z�/, the solution of (70),(71) is expected to tend toward the corresponding Riemann solution given by (85)

zR D zr1.x=t I z�; zm/C zr2.x=t I zm; zC/ � zm:

Here, in the case zC 2 R1.z�/ (resp. zC 2 R2.z�/), zr2 (resp. zr1) is regarded asthe trivial constant state zm D zC (resp. zm D z�). Matumura and Nishihara [40]showed this asymptotics, provided the wave amplitude jzC � z�j and initialperturbations are suitably small, by using the dissipative structure as explained inthe Sect. 3.2 and the spatial monotonicity of the rarefaction waves. Later, they [41]obtained the global result without any smallness conditions, combining with theKanel’s idea. A sketch of the proof is given in the next Sect. 3.4.

Third, in the case zC 2 Si .z�/ .i D 1; 2/, corresponding to the single shockwave zsi .x�si t/, the unique existence of the viscous shock waveZi.x�si t/ of (70)up to spatial shift is easily shown, and the asymptotic state of the solution of (70),(71) is expected to be Zi.x � si t C ˛/ with a suitable shift ˛. Contrary to the scalarcase, note that the shift ˛ satisfying

´.z0.x/ �Zi.x C ˛// dx D 0 can’t be always

chosen because z is vector valued. Matsumura-Nishihara [39] showed the followingby antiderivative method : there exists a positive constant C.z�; �/ which satisfiesC.z�; �/!1 .� ! 1/ such that if jzC � z�j � C.z�; �/,

´.z0.x/�Zi.x// dx D

0 and the antiderivative of z0 � Zi is suitably small in H2, the viscous shock waveZi is asymptotically stable. Here note that for the isothermal model � D 1, there isno restriction on the wave amplitude jzC � z�j. For more general initial data whichsatisfies

´.z0.x/ � Zi.x// dx ¤ 0, Liu [28] proposed a general criterion how the

shift is determined and showed how the diffusion waves of the other characteristicfields play important roles in order to apply antiderivative method. For example, forzC 2 S2.z�/, Liu’s criterion is written as

ˆ.z0.x/ �Zi.x// dx D ˛.zC � z�/C ˇr1.v�/; (91)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 29

where ˛ and ˇ are uniquely determined because the vectors zC;�z�, and r1.v�/are linearly independent. Since then, much efforts to prove whether the criterionis correct or not have been made, and eventually Mascia-Zumbrun [34] and Liu-Zeng [33] showed it does hold for the suitably small amplitude jzC � z�j andinitial perturbations. On the other hand, the asymptotic stability for large amplitudeviscous shock wave had been a long standing open problem, except the case� D 1 as noted above. On this problem, Barker-Humpherys-Laffite-Rudd-Zumbrun[1] and Humpherys-Laffite-Zumbrun [13] established the asymptotic stability ofsufficiently large amplitude viscous shock wave for a properly rescaled systemof (70) and also carried out numerical studies which indicate the asymptoticstability for intermediate amplitude also holds. However, except for the small andlarge amplitude limits, it still remains open whether viscous shock wave can beanalytically shown to be asymptotically stable. The Sect. 3.5 picks up a case wherethe viscosity coefficient is depending on the density in a physically reasonable wayand shows any large amplitude viscous shock is asymptotically stable for suitablysmall initial perturbations with integral zero. This topic presents how antiderivativemethod is applied to the system, and how another type of technical energy methodworks well.

Fourth, in the case zC 2 S1.z�/S2.z�/, corresponding to the Riemann solution

zR D zs1.x � s1t I z�; zm/C zs2.x � s2t I zm; zC/ � zm;

the asymptotic state is expected to be a multi-wave pattern of two viscous shockwaves

Z˛1;˛2 D Z1.x � s1t C ˛1I z�; zm/CZ2.x � s2t C ˛2I zm; zC/ � zm

with suitable shifts ˛1 and ˛2. In this case, due to the facts

ˆ.z0 �Z˛1;˛2/.x/ dx D

ˆ.z0 �Z0;0/.x/ dx � ˛1.zm � z�/ � ˛2.zC � zm/;

and the vectors .zm � z�/ and .zC � zm/ are linearly independent, the shifts ˛1 and˛2 are uniquely determined so that

´.z0 � Z˛1;˛2/.x/ dx D 0. Thus, antiderivative

method is applicable to z � Z˛1;˛2 , and the parallel arguments as in [9, 39] canshow that Z˛1;˛2 is asymptotically stable, provided the wave strengths jzm � z�jand jzC � zmj is suitably small with the same order and also the initial perturbationz0�Z0;0 is suitably small. It should be emphasized that the assumption of thesame-order smallness of the two waves is technically needed to control a residual termwhich comes from the factZ˛1;˛2 is not an exact solution of (70). Since this difficultyis closely related to how the shift of the single viscous shock is determined by theformula (91), the zero limit problem with respect to jzC � zmj under the fixedjzm � z�j seems a very interesting problem.

Last, in the cases zC 2 R1S2.z�/ [ S1R2.z�/, the asymptotic state is expectedto be a multi-wave pattern which includes both rarefaction wave and viscous shockwave. However, the energy methods which have been used can’t be simply applied,

30 A. Matsumura

because antiderivative method works well for viscous shock wave, but not forrarefaction wave. Another difficulty comes from the facts that the rarefaction waveis not an exact solution of the viscous system and any spatial shift of the rarefactionwave has the same asymptotic state. These facts suggest that the influence of therarefaction wave to other characteristic fields might be stronger than other waves(refer to Liu-Yu [31] on this subject), in particular, the interaction of the tails of therarefaction wave and the viscous shock wave very subtly influences the shift of theviscous shock wave. Thus, these cases are very challenging open problems.

3.4 Global Asymptotics Toward Rarefaction Waves

This subsection treats the cases zC 2 R1.z�/[R2.z�/[R1R2.z�/ and shows thatthe global solution in time of (70), (71) globally tends toward the Riemann solutionzr defined by (83) and (85), that is,

zr .x=t I z�; zC/ D zr1.x=t I z�; zm/C zr2.x=t I zm; zC/ � zm (92)

as time goes to infinity, where in the case zC 2 R1.z�/ .resp. zC 2 R2.z�//, regardzm D zC and zr2.x=t I zm; zC/ � zC (resp. zm D z� and zr1.x=t I z�; zm/ � z�). Themain theorem in this subsection is as follows.

Theorem 5. Suppose zC 2 R1.z�/[R2.z�/[R1R2.z�/, and the initial data satisfyz0 � zR0 2 L

2, z0;x 2 L2, and inf v0 > 0: Then, the Cauchy problem (70), (71) has aunique global solution in time z, satisfying

8̂̂<ˆ̂:

z � zR0 2 C.Œ 0;1/IL2/;

vx 2 C.Π0;1/IL2/ \ L2loc.0;1IL

2/;

ux 2 C.Π0;1/IL2/ \ L2loc.0;1IH1/;

and the asymptotic behavior

supx2R

j z.t; x/ � zr .x=t I z�; zC/ j ! 0 .t !1/:

To prove Theorem 5, as in the Sect. 2.1, a smooth approximation ofzri .x=t I z�; zC/ is firstly constructed for zC 2 Ri.z�/ .i D 1; 2/. This time,following the arguments in [41], a smooth approximation W .t; xIw�;wC/ ofthe rarefaction wave wr .x=t Iw�;wC/ in (9) for the inviscid Burgers’ equation isconstructed by the solution of the Cauchy problem8̂<:̂Wt C

�12W 2

�xD 0;

W .0; x/ D W0.xIw�;wC/ WDwC C w�

2C

wC � w�2

�q

ˆ "x

0

.1C y2/�q dy;

(93)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 31

where q > 1=2; " > 0, and �q are the constants chosen by �q´10.1Cy2/�q dy D 1.

This choice of the initial data is introduced so that it can give a better decay rate ofkWxx.t/k than that in Lemma 1 and also can control the derivative Wx small bytaking " small. Then, corresponding to the formula (83), a smooth approximationZ1 D

t .V1; U1/ (resp. Z2 D t .V2; U2/) of the rarefaction wave zr1 Dt .vr1; u

r1/ (resp.

zr2 Dt .vr2; u

r2/) is constructed as follows:

8̂<:̂V1.t; xI z�; zC/ D �1

�1�W .t; xI�1.v�/; �1.vC//

�;

U1.t; xI z�; zC/ D u� �ˆ V1.t;x/

v�

�1.s/ ds

�resp.

8̂<:̂V2.t; xI z�; zC/ D �2

�1 .W .t; xI�2.v�/; �2.vC/// ;

U2.t; xI z�; zC/ D u� �ˆ V2.t;x/

v�

�2.s/ ds

�:

(94)

Next, for zC 2 R1R2.z�/, define the smooth approximation Z D t .V; U / of theRiemann solution zr .x=t/ in (92) by

.V; U / D .V1; U1/.t; xI z�; zm/C .V2; U2/.t; xI zm; zC/ � .vm; um/: (95)

Then, it turns out that .V; U / approximately satisfies the viscous system (70) as

8<:Vt � Ux D 0;

Ut C p.V /x ���Ux

V

�xD R;

(96)

where the residual term R is given by

R D�p.V / � p.V1/ � p.V2/C p.vm/

�x���Ux

V

�x:

Like Lemma 1, the basic properties of .V; U / can be obtained as follows, bychoosing q suitably large if needed.

Lemma 2. (1) Vt D Ux > 0; jVxj � C jVt j � C" .t � 0; x 2 R/.(2) For p 2 Œ1;1� and " > 0 , there exists a positive constant Cp;" such that

k.V; U /x.t/kLp � Cp;".1C t /�1C 1

p .t � 0/;

k.V; U /xx.t/kLp � Cp;".1C t /�1�

p�12pq .t � 0/:

(3)ˆ 10

kR.t/k dt <1.

32 A. Matsumura

(4) limt!1

supx2R

j.V; U /.t; x/ � .vr ; ur /.x=t/j D 0:

Now, by setting

.v; u/ D .V C �;U C / (97)

and using (96), the problem (70), (71) is rewritten as in the form

8̂̂̂<ˆ̂̂:�t � x D 0;

t C .p.V C �/ � p.V //x � ��UxC xVC�

� UxV

�xD �R .t > 0; x 2 R/;

.�; /.0; x/ D .�0; 0/.x/ WD .v0.x/ � V .0; x/; u0.x/ � U.0; x// .x 2 R/;(98)

where note that .�0; 0/ 2 H1 and inf .�0 C V .0; �// D inf v0 > 0 under theassumptions of Theorem 5. Thus, for the proof of Theorem 5, it suffices to lookfor the global solution in time .�; / 2 C.Œ0;1/IH1/ with the desired asymptoticbehavior k.�; /.t/kL1 ! 0 .t !1/. The solution is obtained by combining theunique existence of the local solution in time together with the a prior estimate inthe same way as in the previous sections. To do that, the Cauchy problem (98) isreformulated as usual to one at the general initial time t D � � 0 with the giveninitial data .�; /.�/ D .�� ; � / 2 H

1 satisfying inf .�� C V .�; �// > 0. Denotethe generalized problem by (98)� . With keeping in mind that the system becomessingular as v D � C V ! C0, the solution space for any positive constants M;m,and interval I � R is introduced by

XM;m.I / D˚.�; / 2 C.I I H1/ j �x 2 L

2.I I L2/; x 2 L2.I I H1/;

supt2I

k.�; /.t/k1 �M; inft2I;x2R

.V C �/.t; x/ � m;

and in particular, for M D1 and m D 0,

X1;0.I / D˚.�; / 2 C.I I H1/ j �x 2 L

2.I I L2/; x 2 L2.I I H1/;

inft2I;x2R

.V C �/.t; x/ > 0:

Proposition 4 (local existence). For anyM;m > 0, there exists a positive constantt0 D t0.M;m/ not depending on � such that if .�� ; � / 2 H1, k.�� ; � /k1 � Mand infx2R.V .�; x/ C ��.x// � m, then the Cauchy problem (98)� has a uniquesolution .�; / 2 X2M;m=2.Œ� ; � C t0�/.

The proof of Proposition 4 is given by the standard iteration method by construct-ing the approximate Cauchy sequence .�.n/; .n// 2 X2M;m=2.Œ� ; � C t0�/ .n 2 N/by

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 33

8̂̂ˆ̂<ˆ̂̂̂:

�.n/t D

.n/x ;

.n/t � �

� .n/x

V .VC�.n�1//

�xD �.p.V C �.n�1//� p.V //x � �

�Ux�

.n�1/

V .VC�.n�1//

�x�R .t > �/;

.�.n/; .n//.�/ D .�� ; � /;

with .�.0/; .0// D .�� ; � /. The details are omitted. Now, the a priori estimate toshow the desired global solution in time is as follows.

Proposition 5 (a priori estimate). For any initial data .�0; 0/ 2 H1 satisfyinginfx2R.V .0; x/ C �0.x// > 0, there exists a positive constant C0 D C0.�0; 0/

such that if the Cauchy problem (98) has a solution .�; / 2 X1;0.Œ0; T �/ for someT > 0, then it holds that

C�10 � V .t; x/C �.t; x/ � C0 .t 2 Œ0; T �; x 2 R/; (99)

and

k.�; /.t/k21 C

ˆ t

0

�kpUx�k

2 C k�xk2 C k xk

21

�d� � C0 .t 2 Œ 0; T �/:

(100)

In what follows, a sketch of the proof of Proposition 5 is given. First, multiplyingthe second equation in (98) by and using also the first equation give

ˆ �1

2 2 Cˆ.v; V /

�dxˇ̌̌t0C

ˆ t

0

ˆQdxd� D �

ˆ t

0

ˆR dxd�; (101)

where

ˆ.v; V / D p.V /.v � V / �

ˆ v

V

p.s/ds; v D V C �; (102)

and

Q D � 2x

v� �

Ux x�

vVC .p.v/ � p.V / � p0.V /�/Ux: (103)

Here it should be noted that if E.z/ denotes the physical total energy juj2=2 C´ vp.s/ ds of gas, then j j2=2Cˆ.v; V / is represented by the formula

1

2j j2 Cˆ.v; V / D E.z/ �E.Z/ � rzE.Z/.z �Z/; (104)

which may allow the form j j2=2 C ˆ.v; V / be called a “relative total energy” tothe background state Z. This kind of relative total energy is very often used for

34 A. Matsumura

many other systems to obtain the basic energy estimate (e.g., see the Sect. 4.4). Onthe other hand, the term Q somehow represents the dissipation of the total energydue to the effect of viscosity and the rarefaction property Ux > 0. To see the non-negativity of Q, set

f D f .v; V / Dp.v/ � p.V / � p0.V /.v � V /

.v � V /2;

and regard Q as the quadratic form ofp�. x=

pv/ and

pf Ux� as

Q D

�p� xpv

�2�p�

pUx

Vpvf

�p� xpv

��pf Ux�

�C�p

f Ux��2:

Since the discriminant of the quadratic form is given by

D D �Ux

V 2vf� 4;

and 1=.V 2vf .v; V // is uniformly bounded for v > 0, the quadratic form Q turnsout to be strictly positive definite if " is suitably chosen small (use (1) of Lemma 2).Next, the right hand side of (101) is estimated as

ˇ̌̌ˇˆ t

0

ˆ Rdxd�

ˇ̌̌ˇ �

ˆ t

0

k kkRk d� �1

2

ˆ t

0

.k k2 C 1/kRk d�:

Thus, due to Lemma 2 and the Gronwall’s inequality for (101), it holds the basicenergy estimate

k .t/k2C

ˆˆ.v; V / dx

C

ˆ t

0

ˆ � 2x

vCˇ̌Ux x�vV

ˇ̌C .p.V C �/ � p.V / � p0.V /�/Ux

�dxd�

� C.�0; 0/

(105)

for some positive constant C.�0; 0/ depending only on the initial data. Note thatthis estimate corresponds to (74) in the Sect. 3.1. To proceed to the stage of (75),(76) and employ the Kanel’s method [18], rewrite the basic estimate (105) by settingQv D v=V and plugging the concrete formula p.v/ D av�� as

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 35

k .t/k2C

ˆV 1�� Q̂ . Qv/ dx

C

ˆ t

0

ˆ � 2x

vC j

Ux x�

vVj C

Ux

V �. Qv�� � 1C �. Qv � 1//

�dxd�

� C.�0; 0/;

(106)

where

Q̂ . Qv/ D

(Qv � 1 � ln Qv .� D 1/;

Qv � 1C 1��1

. Qv1�� � 1/ .� > 1/:(107)

Since it holds

��

�Ux C x

V C ��Ux

V

�x

D ��

�vtV � vVt

vV

�x

D ��

�Qvt

Qv

�x

D ��

�Qvx

Qv

�t

;

the second equation in (98) can be rewritten as

��Qvx

Qv�

�t

C� Qvx

V � Qv�C1C

�Vx

V �C1

1 � Qv�

Qv�D R: (108)

Then, multiply (108) by Qvx= Qv to have

ˆ ��2

� QvxQv

�2�

� QvxQv

��dxˇ̌̌t0C

ˆ t

0

ˆ�

V �

j Qvxj2

Qv�C2dxd�

D �

ˆ t

0

ˆ�Vx

V �C1

1 � Qv�

Qv�Qvx

Qvdxd�

C

ˆ t

0

ˆ � j xj2v�Ux x�

vV

�dxd� C

ˆ t

0

ˆRQvx

Qvdxd�;

(109)

which corresponds to (76). Here the first term in the right hand side of (109) can beestimated as

ˇ̌̌ˆ t

0

ˆ�Vx

V �C1

1 � Qv�

Qv�Qvx

Qvdxd�

ˇ̌̌�

ˆ t

0

ˆj Qvxj

2

Qv�C2dxd�

C C

ˆ t

0

ˆUx. Qv

�� � 1C �. Qv � 1// dxd� C C

ˆ t

0

ˆjVxj

���1� QvxQv

�2dxd�

(110)

36 A. Matsumura

for any > 0, where the last term in (110) is not needed for � D 1, and the estimate

kVx.t/k�

��1

L1 � C.1Ct /�

���1 holds for � > 1 from Lemma 2. Hence, plugging (110)

to (109), choosing suitably small, and combining (109) and (105) with the help ofthe Gronwall’s inequality imply

�� QvxQv.t/��2 C ˆ t

0

ˆj Qvxj

2

Qv�C2dxd� � C.�0; 0/: (111)

Now, it can be shown that (105) and (111) give the point-wise estimate of v bothfrom below and above by using the idea of Kanel [18]. To do that, define ‰. Qv/ by

‰. Qv/ D

ˆ Qv1

Q̂ ./1=2d

: (112)

By the definition of Q̂ , (107), it is easy to see that ‰. Qv/ is monotonically increasingfor Qv > 0 and ‰. Qv/ ! �1 . Qv ! C0/ and ‰. Qv/ ! C1 . Qv ! C1/. Then, byusing (105) and (111), ‰. Qv/ is estimated as follows

j‰. Qv.t; x//j Dˇ̌̌ˆ x

�1

@

@y

�‰. Qv.t; y//

�dyˇ̌̌Dˇ̌̌ˆ x

�1

Q̂ . Qv.t; y//1=2Qvx

Qv.t; y/ dy

ˇ̌̌

��ˆQ̂ . Qv/ dx

�1=2� ˆ ˇ̌ QvxQv

ˇ̌2dx�1=2� C.�0; 0/:

(113)

Thus, by the property of ‰. Qv/, (113) implies the uniform boundedness of Qv (that is,v) from both below and above as

C.�0; 0/�1 � v.t; x/ D .V C �/.t; x/ � C.�0; 0/ (114)

for some positive constant C.�0; 0/. By (114), it immediately follows from (105)and (111) that

k .t/k2 C k�.t/k21 C

ˆ t

0

.kpUx�k

2 C k.�x; x/k2/ d� � C.�0; 0/: (115)

Finally, once the estimates (114) and (115) are established, by multiplying thesecond equation in (98) by � xx as in (77), it is easily shown that

k x.t/k2 C

ˆ t

0

k xxk2 d� � C.�0; 0/: (116)

Thus, by the estimates (114), (115), and (116), the proof of Proposition 5 iscompleted.

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 37

3.5 Asymptotic Stability of Viscous Shock Wave, aDensity-Dependent Viscosity Case

This subsection picks up a topic studied in Matsumura-Wang [43] where theasymptotic stability of large amplitude viscous shock wave for the isentropic modelwith a density-dependent viscosity is investigated and introduces another techniqueof weighted energy method.

First, recall the isentropic model with density-dependent viscosity has the form

8̂<:̂vt � ux D 0;

ut C p.v/x D .�.v/uxv/x .t > 0; x 2 R/;

p.v/ D av�� :

(117)

Here, an assumption on the dependency of viscous coefficient to the density � D1=v should be made so that it has a physically reasonable background. Accordingto the Chapman-Enskog expansion theory in rarefied gas dynamics (cf. [2,22]), theviscosity coefficient is given by a function of the absolute temperature � , dependingon the assumptions on molecular interaction. The typical two examples are given asfollows: (

� D N��12 ; Hard sphere Model;

� D N��12C

2s�1 ; Cut � off inverse power force Model;

where s .� 5/ and N� .> 0/ are some constants. Then, the above two models areunified as

� D N��ˇ .ˇ �1

2/: (118)

On the other hand, since the model is isentropic, it holds

p D R�� D R�

vD a v�� .R W gas constant/;

which implies

� Da

Rv�.��1/: (119)

Thus, by (118) and (119), it is assumed that the viscosity coefficient �.v/ has theform

� D �.v/ D �0 v�˛ .˛ �

1

2.� � 1/; �0 > 0 W constants/: (120)

38 A. Matsumura

Now, under the assumption (120), the Cauchy problem to (117) is considered withinitial and far-field conditions8<

:.v; u/.0; x/ D .v0; u0/.x/ .x 2 R/;

limx!˙1

.v; u/.t; x/ D .v˙; u˙/ .t � 0/:(121)

Before the precise statement of the theorem, the argument on the existence of theviscous shock wave .V; U /.x � st/ is recalled. Set � D x � st . Plugging .v; u/ D.V; U /.�/ into (117) gives the system of ordinary differential equations8<

:�sV� � U� D 0;

�sU� C p.V /� D �0.U�

V ˛C1/� .� 2 R/;

(122)

with the far-field condition .V; U /.˙1/ D .v˙; u˙/: If the existence of a solutionof (122) is assumed, then integrating the system (122) with respect to � induces8̂̂<

ˆ̂:sV C U D sv˙ C u˙;

�sU C p.V / � �0U�

V ˛C1D �su˙ C p.v˙/ .� 2 R/;

(123)

which in particular implies the well-known “Rankine-Hugoniot condition”

(�s.vC � v�/ � .uC � u�/ D 0;

�s.uC � u�/C p.vC/ � p.v�/ D 0:(124)

For any fixed .v�; u�/, regarding vC as a parameter, there exist two families of thesolution .s; uC/ of (124) defined by

s D s˙ WD ˙

s�p.vC/ � p.v�/

vC � v�; uC D u� � s˙.vC � v�/: (125)

In what follows, only the case s D sC > 0 (second shock) is considered forsimplicity. Now plugging the first equation of (123) into the second one gives theequation for V

8<:V� D

V ˛C1

�0sh.V / .� 2 R/;

V .˙1/ D v˙;

(126)

where

h.V / D s2.v� � V /C p.v�/ � p.V /: (127)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 39

Since h.V / > 0 for V taking the value between v� and vC, in order for theproblem (126) to have the solution, it must hold for the case s > 0 that

v� < vC; .i:e:; u� > uC/: (128)

The assumption (128) is also well known as the “entropy condition.” Conversely,if .v˙; u˙/ and s > 0 satisfy the Rankine-Hugoniot and entropy conditions (124),(128), it is easily shown that the solution .V; U / of (126) uniquely exists up to theshift of � , satisfying

V�.�/ > 0; v� < V .�/ < vC .� 2 R/: (129)

Now, for a fixed viscous shock wave .V; U /, the Cauchy problem (117), (121) isconsidered around a neighborhood of .V; U /. As for the initial data, it is assumedthat

.v0 � V; u0 � U/ 2 H1 \ L1; inf

x2Rv0.x/ > 0;

ˆ.v0 � V /.x/ dx D

ˆ.u0 � U/.x/ dx D 0:

(130)

And, by setting

�0.x/ D

ˆ x

�1

.v0 � V /.y/ dy; 0.x/ D

ˆ x

�1

.u0 � U/.y/ dy; (131)

it is further assumed that

.�0; 0/ 2 L2: (132)

Note that (132) is equivalent to .�0; 0/ 2 H2 under the condition (130). Then themain theorem in this subsection is stated as follows.

Theorem 6. Suppose the initial data satisfy (130), (132), and ˛ � 12.� � 1/.

Then there exists a positive constant "0 such that if k.�0; 0/k2 � "0, the Cauchyproblem (117), (121) has a unique global solution in time .v; u/, satisfying

.v�V; u�U/ 2 C.Œ0;1/IH1/; v�V 2 L2.Œ0;1/IH1/; u�U 2 L2.Œ0;1/IH2/;

and the asymptotic behavior

supx2R

j.v; u/.t; x/ � .V; U /.x � st/j ! 0 .t !1/:

40 A. Matsumura

For the proof of the theorem, antiderivative method is employed as in thearguments in the Sect. 2.2, that is, set

v D V C �x; u D U C x: (133)

Plugging the relation (133) into (117) and integrating it with respect to x with theaid of the equation (122) for .V; U / give the following system in terms of .�; /:

8<:�t � x D 0;

t C p.V C �x/ � p.V / D �0

� .U C x/x

.V C �x/˛C1�

Ux

V ˛C1

�.t > 0; x 2 R/:

(134)Then, the Cauchy problem to (134) is considered with initial data

.�; /.0/ D .�0; 0/ 2 H2; (135)

where .�0; 0/ is defined by (131), and the small global solution in time .�; / islooked for. To do that, for any interval I � R, the solution space X.I / is defined by

X.I / D f .�; /2C.I IH2/ j�x2L2.I IH1/; x2L

2.I IH2/; supt2I

k.�; /.t/k2

�1

2v� g:

As in the previous arguments, the small global solution in X.Œ0;1// is constructedby the combination of the local existence and the a priori estimate. Since the localsolution is well understood in the previous works, only the a priori estimate is statedas follows.

Proposition 6 (a priori estimate). Suppose ˛ � 12.� � 1/, and .�0; 0/ 2 H2.

Then there exist positive constants ı0 and C0 such that if .�; / 2 X.Œ0; T �/ is asolution of the Cauchy problem (134), (135) for some T > 0 and

supt2Œ0;T �

k.�; /.t/k2 � ı0;

it holds that

k.�; /.t/k22 C

ˆ t

0

.k�xk21 C k xk

22/ d� � C0k.�0; 0/k

22 .t 2 Œ0; T �/:

Once Proposition 6 is obtained, the following global existence theorem can beshown, and it implies Theorem 6 by defining v D V C �x; u D U C x .

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 41

Theorem 7. Suppose ˛ � 12.� � 1/, and .�0; 0/ 2 H2. Then there exists a

positive constant "0 such that if k.�0; 0/k2 � "0, the Cauchy problem (134), (135)has a unique global solution in time .�; / 2 X.Œ0;1// satisfying the asymptoticbehavior

supx2R

j.�; /x.t; x/j ! 0 .t !1/:

For the proof of the a priori estimate, it is assumed that .�; / 2 X.Œ0; T �/ is asolution of the Cauchy problem (134), (135) for some T > 0 and ˛� 1

2.��1/, and

the system (134) is rewritten so that all the linearized terms at .�; / D .0; 0/

are collected on the left hand side and nonlinear terms on the right as in theform (

�t � x D 0;

t �K.V /�x ��0

V ˛C1 xx D G;

(136)

where

K.V / D �p0.V /C .˛ C 1/h.V /

V;

and G stands for the nonlinear terms. Here, recall that

s > 0; v� < V < vC; Vx DV ˛C1

�0sh.V / > 0; p0.V / D ��

p.V /

V; (137)

and note that (137) easily implies

K.V / � �p.vC/

vC:

Now it is ready to show the following basic energy estimate.

Lemma 3. There exists a positive constant C such that it holds

k.�; /.t/k2 C

ˆ t

0

.k xk2 C k

pVx k

2/ d�

� C�k.�0; 0/k

2 C

ˆ t

0

ˆj jjGj dxd�

�.t 2 Œ0; T �/:

(138)

42 A. Matsumura

Since Lemma 3 is the most essential, only the proof of Lemma 3 is given, and theestimates for the higher derivatives and the nonlinear terms are omitted. For details,refer to [43].

Proof of Lemma 3. Following the idea in [6,37], the positive weight functions �1 D�1.V / and �2 D �2.V / are introduced, which are properly determined later, so thatthe variable .�; / is renormalized to . Q�; Q / by

� D �1.V / Q�; D �2.V / Q : (139)

Plugging (139) into (136) yields

8<:.�1

Q�/t � .�2 Q /x D 0;

.�2 Q /t �K.�1 Q�/x ��0

V ˛C1.�2 Q /xx D G:

(140)

Introduce the another set of positive weight functions W1 D W1.V / and W2 D

W2.V / of V which are also determined later. Then multiplying the first equationof (140) by W1

Q�, the second equation by W2Q and summing the resultant formulas

together result in, after integration by parts with respect to x,

d

dt

ˆ1

2.W1�1j Q�j

2 CW2�2j Q j2/ dx

C

ˆs

2

�.W 01�1 �W1�

01/jQ�j2 C .W 02�2 �W2�

02/jQ j2�Vx dx

C

ˆ �.W 01�2 �KW2�

01/VxQ� Q C .W1�2 �KW2�1/ Q�x Q

�dx

C

ˆ�0.

W2

V ˛C1Q /x.�2 Q /x dx D

ˆW2Q G dx: (141)

In order for the coefficients of the cross terms Q� Q and Q�x Q in (141) to vanish, it isimposed that

W 01�2 �KW2�01 D 0; W1�2 �KW2�1 D 0;

and then the functions �1; �2;W1 and W2 are chosen as

�1.V / D W1.V / D 1; �2.V / D K.V /W .V /; W2.V / D W .V /; (142)

for any positive function W .V / of V . By plugging (142) into (141) and using thefact Vx D V ˛C1h.V /=.�0s/, it holds after further integration by parts,

d

dt

ˆ1

2.j Q�j2 CKW 2j Q j2/ dx C

ˆ.A.V /Vxj Q j

2 C �0KW 2

V ˛C1j Q xj

2/ dx

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 43

D

ˆW Q G dx; (143)

where

A.V / D �s

2K 0W 2 C

1

s.W

V ˛C1/0.KW /0V ˛C1h �

1

2s..KW 2

V ˛C1/0V ˛C1h/0: (144)

Due to

p0.V / D ��p.V /

V; h0.V / D �s2 C �

p.V /

V;

A.V / is reformulated as a quadratic form of h as in the form

A.V / DW

2s.A0.V /C A1.V /h.V /C A2.V /h

2.V //; (145)

where

A0.V / D�2.� C 1/p2

V 3W � 2.

�2p2

V 2� �s2

p

V/W 0;

A1.V / D .�.� C 2/.2˛ C 1 � �/p

V 3� 2s2

.˛ C 1/

V 2/W

� 2.�.2˛ C 1 � �/p

V 2� 2s2

.˛ C 1/

V/W 0 � 2�

p

VW 00;

A2.V / D 2.˛ C 1/.�W

V 3CW 0

V 2�W 00

V/:

(146)

In order for A2.V / to be non-negative, W is chosen as

W .V / D V; (147)

which implies A2.V / D 0. Plugging (147) into (146) and (145) induces

A.V / D1

2s

n2s2�p.V /C�2.� � 1/

p2.V /

V

C2�2.˛ �1

2.� � 1//

p.V /

Vh.V /C 2.˛ C 1/s2h.V /

o:

(148)

Hence, due to the assumption ˛ � 12.� � 1/, it follows from (148) that

A.V / � s�p.vC/; V 2 Œv�; vC�: (149)

44 A. Matsumura

Thus, integrating (143) with respect to t and recalling the renormalization � DQ�; D VK.V / Q finally give the desired inequality (138). Thus, the proof ofLemma 3 is completed.

4 Ideal Polytropic Model

This section lastly proceeds to some topics on a model system which describesone-dimensional motion of viscous and heat-conductive ideal gas in the Lagrangianmass coordinates:8̂<

:̂vt � ux D 0;ut C px D �.

uxv/x;

.e C u2

2/t C .pu/x D .�

�xvC � uux

v/x .t > 0; x 2 R/

(150)

where the unknown functions v > 0, u, � > 0, e > 0 and p are the specific volume,fluid velocity, internal energy, absolute temperature, and pressure, respectively,while the constants � > 0 and � > 0 denote the viscosity and heat conductionCoefficients, respectively. Here the ideal and polytropic gas is studied, that is, p ande are given by the state equations

p DR�

v; e D

R

� � 1�

where � > 1 is the adiabatic exponent and R > 0 is the gas constant. The Cauchyproblem to the system (150) is considered with initial and far-field conditions

8<:.v; u; �/.0; x/ D .v0; u0; �0/.x/ .x 2 R/;

limx!˙1

.v; u; �/.t; x/ D .v˙; u˙; �˙/ .t � 0/;(151)

where v˙ .> 0/; u˙ 2 R; �˙ .> 0/ are the given far-field states and .v0; u0; �0/.˙1/ D .v˙; u˙; �˙/ is assumed as compatibility conditions. As in Sects. 2 and 3,the asymptotic behavior of global solutions in time of the Cauchy problem (150),(151) is investigated in relation to the corresponding Riemann problem for thehyperbolic part of (150) :

8̂̂ˆ̂̂̂<ˆ̂̂̂̂̂:

vt � ux D 0;ut C px D 0;

.e C u2

2/t C .pu/x D 0 .t > 0; x 2 R/;

.v; u; �/.0; x/ D .vR0 ; uR0 ; �

R0 /.x/ WD

�.v�; u�; ��/ .x < 0/;.vC; uC; �C/ .x > 0/:

(152)

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 45

Section 4 is organized as follows. After the Riemann problem (152) is brieflyrecalled in the Sect. 4.1, a survey of the known results on the asymptotic behaviorsin time is given in the Sect. 4.2. Then, a topic on the asymptotic stability of viscouscontact wave is picked up, which is the major characteristic of the full systemcompared with the isentropic/isothermal case. The Sect. 4.3 shows how to constructa suitable approximate solution which has the desired properties of viscous contactwavelike (61) with � D 0, and finally the Sect. 4.4 presents rough ideas how toobtain the asymptotic stability.

4.1 Riemann Problem

The system of conservation laws in (152) has three distinct real eigenvalues forpositive v and �

�1 D �p�p=v < 0; �2 D 0; �3 D ��1 > 0;

which implies the first and third characteristic fields are genuinely nonlinear andthe second field is linearly degenerate. Then it is known that for any fixed z� thereexists a neighborhood O of z� such that if zC 2 O , the Riemann solutions of (152)consist of the various combinations of three elementary nonlinear waves, that is,either shock or rarefaction wave along each genuinely nonlinear characteristic fieldand contact discontinuity along the linearly degenerate one (in total, 17 cases,except the trivial case z� D zC) (cf. [46]). In what follows, the abbreviationsz D .v; u; �/; Qz D .v; u; E/; z˙ D .v˙; u˙; �˙/, etc., are used where E is the totalenergy E D e C juj2=2. Also in all the cases, the strength of the Riemann solutionis assumed suitably weak, that is, jzC � z�j is assumed suitably small.

4.2 Historical Remarks

First, in the trivial case z� D zC .DW Nz/, Kazhikhov [25] first showed the existenceof the unique global solution in time of (150), (151) for any initial data satisfyingz0 � Nz 2 H1 with inf v0 > 0; inf �0 > 0. However, the estimates of the solutionare local in time, so any information of the asymptotic behavior of the solution wasnot obtained. Later, Jiang [16, 17] showed the uniform boundedness of the densityfrom below and above with respect to both the space and time variables and thenshowed the solution tends toward some constant as time goes to infinity locally inspace. Recently, Li-Liang [27] obtained a conclusive result that the temperature isalso uniformly bounded from below and above, and the solution uniformly tendstoward the constant state Nz as time goes to infinity, by using only energy method.It would suggest possibilities of obtaining the global asymptotic stability of evennontrivial nonlinear waves.

46 A. Matsumura

Second, in the case where the Riemann solution consists of a single rarefactionwave zri .x=t/.i D 1; 3/ corresponding to the i -characteristic field, Kawashima-Matsumura-Nishihara [23] and also Liu-Xin [30] showed that in a suitably smallneighborhood of the smoothed rarefaction wave, the global solution in time of (150),(151) exists and asymptotically tends toward the rarefaction wave zri .x=t/ of thehyperbolic part. For the proof, they fully made use of a relative total energy and thefact that the velocity component of the smoothed rarefaction wave is monotonicallyincreasing as in the Sect. 3.4. The case where the Riemann solution has a multi-wave pattern consisting of two rarefaction waves zr1.x=t/ and zr3.x=t/ can be treatedsimilarly, and the global solution in time is proved to tend toward the linearcombination zr1.x=t/ C zr3.x=t/ � zm as in the Sect. 3.4. Here zm is the uniquelydetermined intermediate constant state so that zr1.x=t/ connects z� to zm and zr3.x=t/connects zm to zC.

Third, in the case where the Riemann solution consists of a single shock wavezsi .x � si t/.i D 1; 3/ connecting z˙ with the shock speed si .s1 < 0 < s3/,corresponding to the i -characteristic field, it is known that the system (150) hasthe viscous shock wave Zi.x � si t/, and it is expected that in a small neighborhoodof Zi.x/ the global solution in time of (150), (151) exists and tends toward theZi.x � si t C ˛i / with a suitable shift ˛i . Kawashima-Matsumura [21] first showedthe asymptotic stability provided the integral of the initial perturbation in terms ofQz is zero, where Qz D .v; u; e C juj2=2/. By taking into account that the velocitycomponent of the viscous shock wave is monotone decreasing, the proof is given bythe antiderivative method as in the Sect. 2.4. For more general initial perturbationwhose integral in terms of Qz is not necessarily zero, Szepessy-Xin [47] replacedthe viscous term by some artificial one and showed the asymptotic stability. Liuand his collaborators [29, 32, 33] and also Zumbrun and his collaborators [34, 49–51] extended the result to the original physical system (150). They developedso much deep analysis on the linearized system at viscous shock wave (e.g.,point-wise estimates by constructing the approximate Green function, or spectralanalysis for the eigenvalue problem by Evans function arguments) together withenergy estimates to close the a priori estimates (see their latest papers [26, 43] andreferences therein). However, it still seems very meaningful to give a simpler proof,in order to attack many other open problems.

Fourth, in the case where the Riemann solution has a multi-wave patternconsisting of two shock waves zs1.x�s1t/ and zs3.x�s3t/, the global solution in timeof (150), (151) is expected to tend toward a linear combination of the correspondingcombination of viscous shock waves

Z˛1;˛3 D Z1.x � s1t C ˛1/CZ3.x � s3t C ˛3/ � zm

with suitable shifts ˛1 and ˛3. Here zm is the uniquely determined intermediateconstant state so that zs1.x � s1t/ connects z� to zm and zs3.x � s3t/ connectszm to zC. Huang-Matsumura [9] showed that this asymptotic stability does holdin a small neighborhood of Z0;0, provided the strengths of the two shock wavesare small with the same order. The proof is technically given by constructing a

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 47

good approximation of the linear diffusion wave around the constant state zm andcombining the arguments by Liu [28] on how the shifts ˛1; ˛3, and the strength ofthe diffusion wave are determined, together with the antiderivative method used inKawashima-Matsumura [21].

Fifth, the case where the Riemann solution has a multi-wave pattern includingboth shock and rarefaction waves is entirely open as in the isentropic/isothermalcase.

Last, in the case where the Riemann solution consists of a single contactdiscontinuity corresponding to the two-characteristic field, Huang-Matsumura-Shi[10] first introduced a corresponding viscous contact wave Zvc.x=

p1C t / which

approximately satisfies the system (150), and Huang-Matsumura-Xin [11] showedthat in a suitably small neighborhood of Zvc , the global solution in time of (150),(151) exists and tends toward Zvc provided the integral of the initial perturbationin terms of Qz is zero, by using the antiderivative method. Then, Huang-Xin-Yang[12] extended the result to more general initial perturbation whose integral in termsof Qz is not necessarily zero. Note that since they use the antiderivative method,their arguments can’t be applied to the case the Riemann solution has a multi-wavepattern consisting of contact discontinuity and rarefaction waves. To attack this casewithout using the antiderivative method, the main difficulty lies on the fact that thevelocity component of the viscous contact wave is not monotone, contrary to therarefaction wave. Huang-Li-Matsumura [8] overcame this difficulty by introducinga new weight function technique and succeeded in showing the asymptotic stabilityof the multi-wave pattern consisting of contact discontinuity and rarefaction waves.Since the viscous contact wave is a main feature of the full system, compared withthe isentropic/isothermal model, the following Sect. 4.3 presents how the viscouscontact wave is constructed, and the Sect. 4.4 shows the essential ideas of the proof.The cases where the Riemann solution has a multi-wave pattern consisting of thecontact discontinuity and shock waves are interesting open problems.

4.3 Viscous Contact Wave

This subsection presents how a viscous contact wave is constructed. It is known thatthe contact discontinuity of the Riemann problem (152) is given in the form

z D zc.x/ D .vc; uc; �c/.x/ WD

�.v�; u�; ��/ .t > 0; x < 0/;.vC; uC; �C/ .t > 0; x > 0/;

(153)

provided that

u� D uC .DW Nu/; p� D pC .DW Np/; (154)

where p˙ D R�˙=v˙ (cf. [46]). A corresponding viscous contact wave Zvc D

.V; U;‚/ which has the similar diffusive property as in the Sect. 2.3 is constructed

48 A. Matsumura

as follows. Since the pressure is expected to be almost constant, set

R‚

VD Np; (155)

and ignore the second equation in (152) because Ut and .Ux=V /x are expectedto have better decay estimates. Then, the first and third equations in (152) arereduced to 8̂̂<

ˆ̂:R

Np‚t � Ux D 0;

R

� � 1‚t C NpUx D �

�‚x

V

�xC �jUxj

2

V:

(156)

By ignoring the last term jUxj2 again and inserting the first equation to the secondone in (156), a quasilinear diffusion equation for ‚ is obtained with far-fieldcondition: 8̂<

:̂‚t D a

�‚x

‚

�x

.t > 0; x 2 R/;

‚.t;˙1/ D �˙ .t > 0/;

(157)

where a D � Np.� � 1/=.�R2/ > 0. It is known that (157) has a self-similarity-typesolution ‚ D ‚.�/; .� D x=

p1C t / which is unique among the solutions of

the form ‚ D ‚.�/ (cf. [3, 7]). Furthermore, it is known that there exist positiveconstants Nı; ; C such that for ı WD j�C � ��j � Nı, it holds that

.1C t /j‚xxj C .1C t /1=2j‚xj � Cıe

� x2

1Ct .t > 0; x 2 R/; (158)

and

j‚ � �˙j � Cıe� x2

1Ct .t � 0; x ? 0/: (159)

Once ‚ is determined, .V; U / is determined by (155) and (156) as follows:

V DR

Np‚; U D NuC

�.� � 1/

�R

‚x

‚: (160)

Thus, a “viscous contact wave” of (150), Zvc , corresponding to the contactdiscontinuity zc is defined by Zvc D .V; U;‚/, which was first introduced in [10].Here it should be emphasized that U is not monotone with respect to x, contraryto the cases of smoothed rarefaction wave or viscous shock wave, which causesa difficulty later. Then, the viscous contact wave Zvc turns out to approximatelysatisfy (150) as

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 49

8̂̂̂ˆ̂̂̂̂<ˆ̂̂̂̂̂ˆ̂:

Vt � Ux D 0;

Ut C Px � ��UxV

�xD R1;

� R

� � 1‚C

1

2U 2�tC .PU /x �

��‚x

VC �

UUx

V

�xD R2;

(161)

where P D R‚=V D Np, and the residual terms R1 and R2 satisfy the followingpoint-wise estimates from (158) and (159)

jR1.t; x/j; jR2.t; x/j � Cı.1C t /�3=2e�

x2

1Ct : (162)

Note that due to (162), the residual terms have good estimates

ˆ 10

.kR1.�/k C kR2.�/k/ d� � Cı: (163)

Finally, a remark is given on the quasilinear equation (157). If �� D �C.WD N�/,the solution of (157) is expected to be further approximated by the self-similarsolution, say ‚D , of the linear heat equation ‚t D .a= N�/‚xx with the far-field condition ‚.˙1/ D N� . Then, it should be noted that the “diffusion wave”.V D; UD;‚D/ WD .R

Np‚D; NuC �.��1/

�R N�‚Dx ;‚

D/ played an essential role in the case

where the Riemann solution consists of two shock waves (refer to the last remark inthe Sect. 4.2).

4.4 Asymptotic Stability of Viscous Contact Wave

This last subsection presents some essential ideas to prove the asymptotic stability ofthe viscous contact wave Zvc constructed in the last subsection. As in the previoussections, set

.�; ; �/ D .v � V; u � U; � �‚/; (164)

and rewrite the Cauchy problem (150), (151) as in the form

50 A. Matsumura

8̂̂̂ˆ̂̂̂<ˆ̂̂̂̂ˆ̂:

�t � x D 0;

t C .p � Np/x � ��uxv�Ux

V

�xD �R1;

R��1

�t C pux � NpUx � ���xv�‚x

V

�xD �R2 C UR1;

.�; ; �/.0/ D .�0; 0; �0/ WD .v0 � V .�/; u0 � U.�/; �0 �‚.�// 2 H1:(165)

In order to obtain the small global solution in time .�; ; �/ 2 C.Œ0;1/I H1/ withthe desired asymptotic behavior k.�; ; �/.t/kL1 ! 0 .t ! 1/, it suffices toestablish the following a priori estimate which is of similar type as Propositions 3and 6.

Proposition 7 (a priori estimate). For any fixed .v�; u�; ��/, there exist posi-tive constants "0; ı0 and C0 such that if .�; ; �/ 2 C.Œ0; T �I H1/; . ; �/ 2

L2.0; T I H1/; is a solution of (165) for some T > 0, and

sup0�t�T

k.�; ; �/.t/k21 � "0; ı D j�C � ��j � ı0;

then it holds that, for t 2 Œ0; T �,

k.�; ; �/.t/k21 C

ˆ t

0

.k�xk2 C k. x; �x/k

21/ d� � C0.k.�0; 0; �0/k

21 C ı

1=2/:

(166)

The most essential estimate to show Proposition 7 is the basicL2-energy estimateas in the proofs of Propositions 4 and 5. In what follows, only some essential ideasto have the basic L2-energy estimate are presented, and the arguments on the higherderivatives are omitted. By multiplying the first equation by P .1�V =v/, the secondone by , and the third one by .1 � ‚=�/, and then adding the resultant formulastogether, it holds after integration by parts that

ˆ �12j j2CR‚ˆ

� vV

�C

R

� � 1‚ˆ

� �‚

��dxˇ̌̌t0C

ˆ t

0

ˆ ��Vj xj

2 C�

V ‚j�xj

2�dxd�

C

ˆ t

0

ˆPUx

��ˆ� vV/Cˆ.

p

P/�dxd� D

ˆ t

0

ˆG dxd�;

(167)

where ˆ.y/ D y � 1 � logy � 0 .y > 0/, and G in the right hand of (167)represents all the residual terms which can be eventually controlled by choosing "0and ı0 suitably small and using the estimate (162). Here it should be noted that ifthe entropy s is introduced by the first thermodynamical law de D � ds � p dv

(s D R��1

log � CR log v C const: for the ideal polytropic gas) and the total energy

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 51

E D e C juj2=2 D R��1

� C juj2=2 is regarded as a function of Oz D .v; u; s/, thendue to the relation rOze D .�p; 0; �/, the relative total energy to a background stateOZ D .V; U; S/ is given by

E. Oz/ �E. OZ/ � rOzE. OZ/.Oz � OZ/

D1

2j j2 C

R

� � 1.� �‚/C P .v � V / �‚.s � S/

D1

2j j2 CR‚ˆ

� vV

�C

R

� � 1‚ˆ

� �‚

�(168)

which is nothing but the integrand in the first term of (167). It also should be notedthat if the background state is a smoothed rarefaction wave, the third term in theleft side of (167) has the right sign because of Ux > 0, which shows how the basicenergy estimate for the rarefaction wave case is well obtained; if the backgroundstate is a viscous shock wave, the sign is opposite because ofUx < 0, which suggeststhe antiderivative method is suitable instead; if the background state is the viscouscontact wave under consideration, because the sign of Ux changes, is needed a newtechnique to overcome it . Since it follows from the estimates (158), (159) for‚ andthe fact ˆ.y/ � jy � 1j2 .y ! 1/ that

ˆ t

0

ˆP jUxj

��ˆ� vV/Cˆ.

p

P/�dxd� � Cı

ˆ t

0

ˆe�

x2

1C�

.1C �/.j�j2 C j�j2/ dxd�

(169)for a suitable small "0, it suffices to estimate the right hand side of (169). To do that,the next key lemma plays an essential role, which was first announced in [8].

Lemma 4. Suppose 2 C.Œ0; T �IH1/, t 2 L2.0; T IL2/, and let a weightfunction ! be defined by

! D

ˆ p x

p2.1Ct /

�1

e�y2

dy

for a positive constant . Then, it holds that, for t 2 Œ0; T �,

ˆ1

2!22 dx

ˇ̌̌t0C1

8

ˆ t

0

ˆe�

x2

1C�

.1C �/2 dxd�

�

ˆ t

0

ˆjxj

2 dxd� C

ˆ t

0

ˆ!2t dxd�: (170)

The proof is very elementally obtained by integration by parts and the Young’sinequality as follows:

52 A. Matsumura

d

dt

�ˆ 1

2!22 dx

�D

ˆ!!t

2 dx C

ˆ!2t dx

D1

2

ˆ!!xx

2 dx C

ˆ!2t dx

D�1

2

ˆj!xj

22 dx C1

ˆ!!xx dx C

ˆ!2t dx

� �1

4

ˆj!xj

22 dx C

ˆjxj

2 dx C

ˆ!2t dx;

(171)

where the facts !t D 12 !xx and 0 � ! �

p are used. Thus, noting

j!xj2 D

2.1C t /e� x2

.1Ct / ;

and integrating (171) with respect to t complete the proof.Since there already has the nice estimate of dissipation for k. x; �x/k2 in the

left hand side of (167), Lemma 4 suggests that under the estimate of dissipationfor k�xk2 which is restored later by the dissipative structure as explained in theSect. 3.2, the right hand side of (169) could be controlled by taking a suitable in Lemma 4 and choosing ı0 suitably small. In fact, this process is realized, andthe following basic estimate eventually holds for suitably small "0 and ı0 (for thedetails, refer to [8]):

k.�; ; �/.t/k2 C

ˆ t

0

k. x; �x/k2 d�

� C�k.�0; 0; �0/k

2 C ı1=2 C ı1=2ˆ t

0

k�xk2 d�

�:

(172)

Thus, by starting with the estimate (172) and combining the estimates for the higherderivatives, the proof of Proposition 7 can be completed.

5 Conclusion

Only basic topics on the asymptotic wave patterns of the global solutions in timeof the Cauchy problems have been discussed through some specific examples, inparticular, in connection with the Riemann problems. Many other related interestingproblems (e.g., on decay rate toward asymptotic state, higher approximation viadiffusion waves, extension to initial boundary value problems, extension to otherphysical models, etc.) are omitted because of short pages. In particular, there is noroom to discuss the initial boundary value problems on the half space, where notonly the wave patterns discussed in the Cauchy problem but also their interactions

Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves 53

with the boundary have to be taken into account (refer to an introductory survey[36]).

Cross-References

� I-1 Derivation of Equations for Continuum Mechanics and Thermodynamics ofFluids

� III-30 Equations and Various Concepts of Solutions in the Thermodynamics ofCompressible Fluids

� III-31 Well-Posedness of the IBPV for the Viscous Barotiropic and Heat-Con-ducting Gas 1D Equations in the Lagrangian Coordinates

� III-32 Existence, Uniqueness and Asymptotic Behavior of Strong Solutions to the1D Non-stationary Equations of the Viscous Polytropic Gas and Heat ConductingGas

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