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RAREFACTIVE SOLUTIONS TO A NONLINEARVARIATIONAL WAVE EQUATION OF LIQUID CRYSTALSPing Zhang a & Yuxi Zheng ba Institute of Mathematics , Academia Sinica, Beijing, 100080, P. R. Chinab Department of Mathematics , Indiana University , Bloomington, IN, 47405, U.S.A.Published online: 07 Feb 2007.

To cite this article: Ping Zhang & Yuxi Zheng (2001) RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVEEQUATION OF LIQUID CRYSTALS, Communications in Partial Differential Equations, 26:3-4, 381-419, DOI: 10.1081/PDE-100002240

To link to this article: http://dx.doi.org/10.1081/PDE-100002240

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COMMUN. IN PARTIAL DIFFERENTIAL EQUATIONS, 26(3&4), 381–419 (2001)

RAREFACTIVE SOLUTIONS TO ANONLINEAR VARIATIONAL WAVEEQUATION OF LIQUID CRYSTALS

Ping Zhang1 and Yuxi Zheng2

1Institute of Mathematics, Academia Sinica,Beijing 100080, P. R. China

2Department of Mathematics, Indiana University,Bloomington, IN 47405

ABSTRACT

We study a nonlinear wave equation derived from a simplifiedliquid crystal model, in which the wave speed is a given functionof the wave amplitude. We formulate a viscous approximation ofthe equation and establish the global existence of smooth solutionsfor the viscously perturbed equation. For a monotone wave speedfunction in the equation, we find an invariant region in the phasespace in which we discover: (a) smooth data evolve smoothly for-ever; (b) both the viscous regularization and the smooth solutionsobtained through data smoothing for rough initial data yield weaksolutions to the Cauchy problem of the nonlinear variational waveequation. The main tool is the Young measure theory and relatedtechniques.

381

Copyright C© 2001 by Marcel Dekker, Inc. www.dekker.com

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I. INTRODUCTION

In this paper, we study the existence and regularity properties of weak solu-tions to the Cauchy problem{

∂2t u − c(u)∂x [c(u)∂x u] = 0, t > 0, x ∈ R

u|t=0 = u0(x), ∂t u|t=0 = u1(x),(1.1)

where c(·) is a given smooth and positive function and (u0(x), ∂x u0(x), u1(x))∈ L2(R). The equation in (1.1) is the Euler-Lagrange equation of the least actionprinciple

δ

δu

∫ ∫ {(∂t u)2 − c2(u)(∂x u)2

}dx dt = 0. (1.2)

The equation in (1.1) may be regarded as a generalization of equations for har-monic wave maps and arises in a number of different physical contexts, includingnematic liquid crystals ([37]), long waves on a dipole chain in the continuum limit([16], Zorski and Infeld [53], and Grundland and Infeld [17]), and in classical fieldtheories and general relativity ([16]). Let us take nematic liquid crystals for exam-ple. We know that the mean orientation of the long molecules in a nematic liquidcrystal is described by a director field of unit vectors, n ∈ S

2, the unit sphere. As-sociated with the director field n, there is the well-known Oseen-Franck potentialenergy density W given by

W (n, ∇n) = α|n × (∇ × n)|2 + β(∇ · n)2 + γ(n · ∇ × n)2. (1.3)

The positive constants α, β, and γ are elastic constants of the liquid crystal. Forthe special case α = β = γ, the potential energy density reduces to

W (n, ∇n) = α|∇n|2,which is the potential energy density used in harmonic maps into the sphere S

2.There are many studies on the constrained elliptic system of equations for n de-rived through variational principles from the potential (1.3), and on the parabolicflow associated with it, see [3][7][10][19][26][47] and references therein. In theregime in which inertia effects dominate viscosity, however, the propagation ofthe orientation waves in the director field may then be modeled by the least actionprinciple (Saxton [37])

δ

δu

∫{∂t n · ∂t n − W (n, ∇n)} dx dt = 0, n · n = 1. (1.4)

In the special case α = β = γ, this variational principle (1.4) yields the equationfor harmonic wave maps from (1 + 3)-dimensional Minkowski space into the twosphere, see [6][40][41] for example. For planar deformations depending on a single

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space variable x , the director field has the special form

n = cos u(x, t)ex + sin u(x, t)ey,

where the dependent variable u ∈ R1 measures the angle of the director field to

the x-direction, and ex and ey are the coordinate vectors in the x and y directions,respectively. In this case, the variational principle (1.4) reduces to (1.2) with thewave speed c given specifically by

c2(u) = α cos2 u + β sin2 u. (1.5)

The general problem of global existence and uniqueness of solutions to theCauchy problem of the nonlinear variational wave equation (1.1) is open. It hasbeen demonstrated in [15] that (1.1) is rich in structural phenomena associatedwith weak solutions. Writing the highest derivatives of (1.1) in conservative form

∂2t u − ∂x (c2∂x u) = −cc′(∂x u)2,

we see that the strong precompactness in L2 of the derivatives {∂x u} of a sequenceof approximate solutions is essential in establishing the existence of a global weaksolution. However, the equation has the phenomenon of persistence of oscillation([8]) and annihilation in which a sequence of exact solutions with bounded energycan oscillate forever so that the sequence {∂x u} is not precompact in L2, but theweak limit of the sequence is still a weak solution. Secondly the equation hasshort-time smooth solutions that blow up in finite time. Thirdly, from the study ofits asymptotic equation (see below), it is clear that a positive amount of energy ofa solution can concentrate in a set of measure zero, and there are multiple choicesfor the continuation of the solution beyond blow-up time. To put the equation andits particulars into context of nonlinear wave equations under current research, wecompare this equation with the equation

∂2t u − ∂x [p(∂x u)] = 0, (1.6)

where p(·) is a given function, considered by Lax [29]. The derivative ∂x u remainsbounded for (1.6), but we find that it is merely in L2 for (1.1). We note interest-ingly that solutions of (1.6)—with a “stronger” ∂x u-dependent nonlinearity—aremore regular than solutions of (1.1)—with an apparently “weaker” u-dependentnonlinearity. (This kind of behavior is well-known for nonlinear parabolic partialdifferential equations.) In each case it appears that singularities develop to themaximum extent permitted by the existence of global weak solutions. For furtherresearch into (1.6) and its generalizations, we refer the reader to Klainerman andMajda [28] and Liu [34]. Another related equation is

∂2t u − c2(u)u = 0 (1.7)

considered by Lindblad [31], who established the global existence of smoothsolutions of (1.7) with smooth, small, and spherically symmetric initial data in

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R3, where the large-time decay of solutions in high space dimensions is crucial.

The multi-dimensional generalization of equation (1.1),

∂2t u − c(u)∇ · (c(u)∇u) = 0, (1.8)

contains a lower order term proportional to cc′|∇u|2, which (1.7) lacks. This lowerorder term is responsible for the blow-up in the derivatives of u. Finally, we notethat equation (1.1) also looks related to the perturbed wave equation

∂2t u − u + f (u, ∇u, ∇∇u) = 0, (1.9)

where f (u, ∇u, ∇∇u) satisfies an appropriate convexity condition (for example,f = u p or f = a(∂t u)2 + b|∇u|2) or some nullity condition. Blow-up for (1.9)with a convexity condition has been studied extensively, see [2][14][18][23][25][30][38][43][44] and Strauss [45] for more reference. Global existence and unique-ness of solutions to (1.9) with a nullity condition depend on the nullity structureand large time decay of solutions of the linear wave equation in higher dimensions(see Klainerman and Machedon [27] and references therein). Therefore (1.1) withthe dependence of c(u) on u and the possibility of sign changes in c′(u) is familiaryet truly different.

We point out that, early in the study of (1.1), Hunter and Saxton ([21])derived an asymptotic equation

∂tv + u∂xv = −1

2v2, (v = ∂x u) (1.10)

for (1.1) via weakly nonlinear geometric optics. We mention with great interestthat the x-derivative of equation (1.10) appears in the high-frequency limit of thevariational principle for the Camassa-Holm equation ([1][4][5]), which arises inthe theory of shallow water waves. In [22], Hunter and Zheng established the globalexistence of weak solutions to (1.10) with initial data of bounded variations. In[52–54], the authors study the global existence, uniqueness, and regularity of theweak solutions to (1.10) with L2 initial data. The study of the asymptotic equationhas been very beneficial for both the blow-up result [15] and the current globalexistence result for the wave equation (1.1).

In this paper, we use the generalized compensated compactness (Gerard[13] or Tartar [47]), compensated compactness in L p (Shearer [42] and referencestherein and Schonbek [39]), the latest developments in the Young measure methodof Lions ([33]) and J. L. Joly, G. Metivier and J.Rauch ([24]), and the techniquesused in our earlier paper [52] to establish the global existence of weak solutionsfor (1.1). We first establish the global existence of weak solutions to a viscouslyperturbed equation for general c(u) with general data (Lemma 2). We then presentour discovery of an invariant region in the phase space for a monotone c(u). In theinvariant region, we show that smooth data evolve smoothly forever; and for weakdata we show that both the viscous approximation and the data regularization yield

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global weak solutions for (1.1) by using the aforementioned Young measure andrelated theories.

Before we present precisely our results, let us introduce some notations:R

+ = (0, ∞), H k are Sobolev spaces, and Lip stands for Lipschitz. We use

R := ∂t u + c(u)∂x u, S := ∂t u − c(u)∂x u, c(·) := 1

4ln c(·), (1.11)

so that c′(u) = c′(u)/[4c(u)]. We use := for definition.

Definition 1. We call u(t, x) an admissible weak solution of (1.1) if

1) u(t, x) ∈ L∞(R+, H 1loc(R)) ∩ Lip([0, ∞), L2

loc(R)), and∫R

(|∂t u|2 + |c(u)∂x u|2)dx ≤∫

R

(|u1|2 + |c(u0)∂x u0|2)dx ; (1.12)

2) for all test functions ϕ(t, x) ∈ C∞c (R+ × R), there holds∫ ∫

R+×R

(∂tϕ∂t u − ∂xϕc2(u)∂x u − ϕc′(u)c(u)(∂x u)2)dx dt = 0;

(1.13)

3) for any convex function η(·) with η′′(·) ∈ C∞c (R), there hold

∂tη(R) − c(u)∂xη(R) − c′(u)η′(R)(R2 − S2) ≤ 0;(1.14)

∂tη(S) + c(u)∂xη(S) − c′(u)η′(S)(S2 − R2) ≤ 0;

4) u(t, x) → u0(x) in Lip([0, ∞), L2(R)) and ∂t u(t, x) → u1(x) in the dis-tributional sense as t → 0+.

We explain the motivation of (1.14). As can be seen from [15], if u(t, x) is asmooth solution of (1.1), we can write (1.1) as the system of first order equations

∂t R − c∂x R = c′(u)(R2 − S2),

∂t S + c∂x S = c′(u)(S2 − R2),

∂x u = R−S2c(u) .

(1.15)

Similar to the viscous approximation of [22], we consider

∂t Rε − c(uε)∂x Rε = c′(uε)(R2ε − S2

ε ) + ε∂2x Rε,

∂t Sε + c(uε)∂x Sε = c′(uε)(S2ε − R2

ε ) + ε∂2x Sε,

∂x uε = Rε−Sε

2c(uε ) ,

(1.16)

where ε > 0. We will prove in the paper that (1.16) has global smooth solutionswith general Cauchy data. So for any convex function η(·), we multiply η(Rε) to

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both sides of the first equation of (1.16) to obtain

∂tη(Rε) − c∂xη(Rε) − c′(uε)η′(Rε)(R2

ε − S2ε

)= ε

(∂2

x η(Rε) − η′′(Rε)(∂x Rε)2) ≤ ε∂2

x η(Rε). (1.17)

Passing ε → 0 in (1.17), we obtain the first inequality of (1.14). It is similar toobtain the second of (1.14).

In the sequel, we always assume that

0 < C1 ≤ c(·) ≤ C2, and |c(l)(·)| ≤ Ml , for all l ≥ 1 (1.18)

for some positive constants C1, C2, and Ml . We also assume in most cases that

c′(·) ≥ 0. (1.19)

In some cases we even assume

c′(u) ≥ CM > 0, u ∈ [−M, M] (1.20)

for some positive constant CM and any M > 1.Now, we state precisely our main results of the paper.

Theorem 1 (Local classical solutions). Let u0 ∈ H k+1(R) and u1 ∈ H k(R) becompactly supported for some k ≥ 1. Then there exists a T ∗ ∈ (0, ∞] such that(1.1) has a unique solution u(t, x) ∈ L∞([0, T ],H k+1(R)) ∩ Lip([0, T ],H k(R))for any positive T < T ∗, and

limt→T ∗ (‖∂t u(t, ·)‖L∞ + ‖∂x u(t, ·)‖L∞ ) = ∞

if T ∗ < ∞.

Let R0 := u1 + c(u0)∂x u0 and S0 := u1 − c(u0)∂x u0.

Theorem 2 (Global rarefactive classical solutions). Assume u0 ∈ H k+1(R) andu1 ∈ H k(R) be compactly supported for some k ≥ 1. Assume further that c′ ≥ 0,R0 ≤ 0, and S0 ≤ 0. Then (1.1) has a global solution u(t, x) ∈ L∞(R+, H k+1(R))∩ Lip([0, ∞), H k(R)).

Remark 1. From the proof of Theorem 1 of [15], we find that there exists anexample where blow-up occurs in a solution of (1.1) with a smooth initial datumsatisfying c′ ≥ 0, R0 ≤ 0, S0 > 0. �

Theorem 3 (Global rarefactive L p solutions). Assume c′ ≥ 0, R0 ≤ 0, S0 ≤ 0,

and (R0, S0) ∈ L p(R) with compact support for some p > 3. Then (1.1) has aglobal admissible weak solution in the sense of Definition 1. The solution can

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be obtained either through initial data mollification or vanishing viscosity. More-over, there holds (R, S)(t, x) ∈ L∞(R+, L p(R)). Furthermore, if (1.20) holds,then ∂x u(t, x) ∈ L p+1([0, T ] × R) for any T > 0.

Theorem 4 (Global rarefactive L2 solutions). Assume (1.20) holds and R0 ≤0, S0 ≤ 0, (R0, S0) ∈ L2(R) with compact support. Then (1.1) has a global ad-missible weak solution in the sense of Definition 1. Moreover, there holds ∂x u ∈L2+α([0, T ] × R) for any α < 1 and T < ∞. Furthermore there exists a constantC > 0 such that

−C

t≤ R(t, x) ≤ 0, −C

t≤ S(t, x) ≤ 0, t ∈ (0, 1], x ∈ R, (1.21)

and (R, S) remain bounded from below by −C for all time t > 1.

We comment on Theorems 3-4. The L p regularity (p > 3) of the solutions inTheorem 3 is such that the quadratic nonlinearity of the equation is under control. InTheorem 4, we rely on the positivity of c′ to derive the nonlinear de-singularization(1.21) to control the quadratic nonlinearity.

We note that the wave speed c(u) in (1.5) does not have the monotonicityproperty required in Theorems 2–4. It is our wish to remove the monotonicitycondition in the future.

Remark 2. Theorems 2–4 hold similarly when the signs of c′, R0, and S0 arereversed. �

II. THE VISCOUS APPROXIMATION

In this section, we establish the global existence of smooth solutions to (1.16)with general initial data for general c(u). For convenience, we will just write (R, S)as (Rε, Sε). So, we consider the problem

∂t R − ∂x (c(u)R) = −c′(u)(R − S)2 + ε∂2x R,

∂t S + ∂x (c(u)S) = −c′(u)(R − S)2 + ε∂2x S,

∂x u = R−S2c(u) ,

limx→−∞ u(t, x) = 0,

(R, S)|t=0 = (R0, S0)(x).

(2.1)

where (R0, S0)(x) ∈ L2(R) ∩ L1(R) are given initial data, and ε > 0.

Lemma 1 (Solutions of the viscous system with smooth data). For given (R0, S0)(x) ∈ C∞

c (R), problem (2.1) has a global smooth solution (R, S, u)(t, x) ∈

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C∞(R+ × R) satisfying the identity∫(R2 + S2)(t, x)dx + 2ε

∫ t

0

∫(∂x R)2 + (∂x S)2 dx dt

=∫

(R0)2 + (S0)2dx . (2.2)

We give a detailed proof of Lemma 1 in the Appendix.

Remark 3. It is interesting to compare this existence result with the blow-upphenomena established by Fujita [12] for

∂t V = 1

2V 2 + ε∂2

x V,

with any initial function V0(x) such that V0(x0) > 0 for some point x0. �

Lemma 2 (Solutions of the viscous system with general data). For given (R0, S0)(x) ∈ L2(R) ∩ L1(R), problem (2.1) has a global weak solution (R, S, u)(t, x) ∈C∞(R+ × R) satisfying the inequalities∫ (

R2ε + S2

ε

)(t, x) dx + 2ε

∫ t

0

∫(∂x Rε)2 + (∂x Sε)2dx dt

≤∫

(R0)2 + (S0)2dx ; (2.3)

‖Rε(t, ·)‖L1(R) + ‖Sε(t, ·)‖L1(R) ≤ C(T, 1/ε), t ∈ [0, T ] (2.4)

for any T > 0, where C(T, 1/ε) depends also on the L1 ∩ L2(R) norm of (R0, S0).

Proof (sketch). Let

Rσ0 = (R0χσ ) ∗ jσ , Sσ

0 = (S0χσ ) ∗ jσ , (2.5)

where σ → 0+, jσ (x) is the standard Friedrichs’ mollifier, and χσ (x) = χ (xσ )is the scaled cut-off function where χ (x) = 1 on [−1, 1] and zero otherwise.Thus, by Lemma 1, we find that (2.1) has a global smooth solution (Rσ , Sσ , uσ )with this initial data (Rσ

0 , Sσ0 ). The solutions satisfy the energy estimate. We can

use Lions-Aubin Lemma (see Temam [48] for example) to establish the strongL2([0, T ] × R) precompactness of (Rσ , Sσ ) to show that we have a weak solutionin the limit σ → 0. We can use the heat kernel representation of the solution toestablish its C∞((0, ∞) × R) regularity.

Using the heat kernel E(t, x) = (4πεt)−12 e− |x |2

4εt and (2.1), we find

R(t, x) =∫

E(t, x − y)R0(y)dy

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+∫ t

0

∫c(u)R(s, y)∂x E(t − s, x − y)dy ds

−∫ t

0

∫c′(u)(R − S)2(s, y)E(t − s, x − y) dy ds.

Taking the L1 norm, we have

‖R(t, ·)‖L1(R) ≤ ‖R0‖L1(R) + Ct(‖R0‖2

L2(R) + ‖S0‖2L2(R)

)+ C2ε

− 12

∫ t

0‖R(s, ·)‖L1(R)(t − s)−

12 ds, (2.6)

from which the L1 bound follows, first by choosing a short time interval so thatthe last term in (2.6) is less than half of the left hand side of (2.6), and then thesame time interval can be repeated forward indefinitely. This completes the proofof Lemma 2. �

Lemma 3 (Precompactness of uε). Let (R0, S0) ∈ L2(R) ∩ L1(R). Then thereexists a subsequence {ε j } such that {uε j } of the solutions of (2.1) is uniformlyconvergent in every compact subset of [0, ∞) × R.

Proof: Firstly, we let F(uε) := 2∫ uε

0 c(ξ ) dξ . By the third equation of (2.1), wefind

∂x F(uε) = Rε − Sε are uniformly bounded in L∞(R+, L2(R)). (2.7)

While, by the fourth assumption of (2.1), the estimate (2.4), and (2.7), we have

F(uε) =∫ x

−∞(Rε − Sε)(t, y) dy. (2.8)

Then, by a simple calculation, we find

∂t F(uε) =∫ x

−∞∂t (Rε − Sε)(t, y) dy

= c(Rε + Sε) + ε(∂x Rε − ∂x Sε). (2.9)

Thus for any T > 0, t1, t2 ∈ R+, with t1 < t2 ≤ T, we find

‖F(uε(t1, ·)) − F(uε(t2, ·))‖L2

=∥∥∥∥

∫ t2

t1

∂s F(uε(s, ·))ds

∥∥∥∥L2

≤∫ t2

t1

‖(c(Rε + Sε) + ε∂x (Rε − Sε))(s, ·)‖L2 ds

≤ C(|t1 − t2| + ε

12 |t1 − t2| 1

2)(‖R0‖L2 + ‖S0‖L2 ). (2.10)

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On the other hand, by Rellich’s Theorem W 1,2loc (R) ↪→↪→ Cloc(R) ↪→ L2

loc(R) andLions-Aubin Lemma ([48]), we find that, for any η > 0, there exists a constantCη > 0 such that the inequality holds

‖F(uε(t1, ·)) − F(uε(t2, ·))‖L∞loc

≤ 2η supt

‖F(uε(t, ·))‖W 1,2loc

+ Cη,T |t1 − t2| 12 .

(2.11)

Now, by a diagonal process, we can extract a subsequence {F(uε j (tν, ·))} of{F(uε)}, which converges in L∞

loc for t in the rational number subset {tν |ν ∈ N}of R

+. For any t0 ∈ [0, T ] and δ > 0, there is some tν ∈ {tν |ν ∈ N} such that|t0 − tν | < δ, and∥∥(F

(uε j

) − F(uεk

))(t0, ·)

∥∥L∞

loc≤ ∥∥F

(uε j (t0, ·)

) − F(uε j (tν, ·)

)∥∥L∞

loc

+ ∥∥F(uε j (tν, ·)

) − F(uεk (tν, ·)

)∥∥L∞

loc

+ ∥∥F(uεk (t0, ·)

) − F(uεk (tν, ·)

)‖L∞loc

.

(2.12)

This shows that {F(uε j )} is uniformly convergent on any compact subset of[0, ∞) × R. While by (1.18), we find

∣∣F(uε j

) − F(uεk

)∣∣ =∣∣∣∣∫ 1

0F ′(τuε j + (1 − τ )uεk

)(uε j − uεk

)dτ

∣∣∣∣≥ c

∣∣uε j − uεk

∣∣.This completes the proof of Lemma 3. �

III. ESTIMATES

We specialize in the case c′ ≥ 0, R0(x) ≤ 0, and S0(x) ≤ 0 in this sectionand establish more estimates for Rε(t, x) and Sε(t, x) of the solution of (1.16) withdata (R0, S0)(x) ∈ L2(R) ∩ L1(R).

Lemma 4 (Invariant region). Assume (1.18) and c′ ≥ 0. Let R0(x) ≤ 0, S0(x)≤ 0, (R0(x), S0(x)) ∈ L2(R) ∩ L1(R). Then Rε(t, x) ≤ 0 and Sε(t, x) ≤ 0.

Proof: In fact, if c′(u) ≥ 0, by the first equation of (1.16), we immediately have{∂t R − c(u)∂x R ≤ c′(u)R2 + ε∂xx R,

R|t=0 = R0 ≤ 0.(3.1)

Next, we claim that

R(t, x) ≤ 0. (3.2)

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To prove (3.2), we denote −e−Ct R(t, x) =: f (t, x), where C is an upper bound of|c′(u)R| in [0, T ] × R for any T > 0, mollifying the data as in (2.5) if necessaryfor a finite C to exist. By (3.1), f (t, x) satisfies the inequality

∂t f − c(u)∂x f ≥ (−C + c′(u)R) f + ε∂xx f. (3.3)

On the other hand, we know that

lim|x |→∞

| f (t, x)| = 0. (3.4)

If (3.2) does not hold, then f (t, x) must attain its negative minimum in [0, T ] × R

at some point (t0, x0), (t0 > 0), where

∂t f ≤ 0, ∂x f = 0, ∂xx f ≥ 0, (−C + c′(u)R) f > 0. (3.5)

Trivially, (3.5) contradicts (3.3). Thus (3.2) holds. Exactly as above, we can proveS(t, x) ≤ 0 if S0(x) ≤ 0. This completes the proof of Lemma 4. �

Lemma 5 (Estimates). Assume (1.18) and c′ ≥ 0. Let R0(x) ≤ 0, S0(x) ≤ 0,

(R0(x), S0(x)) ∈ L p(R) ∩ L1(R) with p > 2. Then(a) (L p estimate) There holds

‖Rε‖pL p(R) + ‖Sε‖p

L p(R) ≤ ‖R0‖pL p(R) + ‖S0‖p

L p(R). (3.6)

(b) (L p+1 estimate) There holds∫ ∞

0

∫R

c′(uε)|∂x uε |p+1dx dt ≤ K p

∫R

(|R0|p + |S0|p)dx (3.7)

for some constant K p.

Proof: By Lemma 4, and for convenience, we let R′ε(t, x) := −Rε(t, x) ≥ 0,

S′ε(t, x) := −Sε(t, x) ≥ 0. Then, by the first equation of (1.16), we have

∂t R′ε − c∂x R′

ε = −c′((R′ε)2 − (S′

ε)2) + ε∂xx R′ε . (3.8)

Next, we multiply (R′ε)p−1 to both sides of the above equation to yield

1

p{∂t (R′

ε)p − c∂x (R′ε)p}

= − c′((R′ε)p+1 − (R′

ε)p−1(S′ε)2) + ε(R′

ε)p−1∂xx R′ε . (3.9)

Then, by the third equation of (1.16), we find

1

p{∂t (R′

ε)p − ∂x (c(R′ε)p}) − ε(R′

ε)p−1∂xx R′ε

= −(

1

4− 1

2p

)c′

c(R′

ε)p+1 + c′

4c(R′

ε)p−1(S′ε)2 − c′

2pcS′

ε(R′ε)p. (3.10)

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Exactly as the above procedure for S′ε(t, x), we have

1

p{∂t (S′

ε)p + ∂x (c(S′ε)p}) − ε(S′

ε)p−1∂xx S′ε

= −(

1

4− 1

2p

)c′

c(S′

ε)p+1 + c′

4c(S′

ε)p−1(R′ε)2 − c′

2pcR′

ε(S′ε)p. (3.11)

Thus, by summing up (3.10) and (3.11), and integrating over R, we have

1

p

d

dt

∫R

(R′ε)p + (S′

ε)p dx + ε(p − 1)∫

R

(R′ε)p−2(∂x R′

ε)2

+ (S′ε)p−2(∂x S′

ε)2 dx = −(

1

4− 1

2p

) ∫R

c′

c((R′

ε)2)((R′ε)p−1 − (S′

ε)p−1)

dx

− 1

2p

∫R

c′

cR′

ε S′ε(R′

ε − S′ε)((R′

ε)p−2 − (S′ε)p−2) dx

≤ 0. (3.12)

Hence, by integrating the above inequality over [0, t], we find (3.6). Moreover, by(3.6) and (3.12) and the identity,

((R′ε)2 − (S′

ε)2)((R′ε)p−1 − (S′

ε)p−1) = (R′ε − S′

ε)2((R′ε)p−1 + (S′

ε)p−1)

+ 2R′ε S′

ε(R′ε − S′

ε)((R′ε)p−2 − (S′

ε)p−2),

(3.13)

we find∫ ∞

0

∫R

c′

c{(Rε − Sε)2(|Rε |p−1 + |Sε |p−1) + C p Rε Sε ·

(|Rε | − |Sε |)(|Rε |p−2 − |Sε |p−2)} dx dt ≤ K ′p

(‖R0‖pL p(R) + ‖S0‖p

L p(R)

)(3.14)

where C p = 2 p−1p−2 and K ′

p = 4p−2 for p > 2. Since both terms in the integrand in

(3.14) are nonnegative, (3.14) implies

∫ ∞

0

∫R

c′

c(Rε − Sε)2(|Rε |p−1 + |Sε |p−1) dx dt

≤ K ′p

∫R

|R0|p + |S0|p dx . (3.15)

Using the third equation of (2.1) we obtain (3.7) from (3.15). �

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IV. CLASSICAL SOLUTIONS

Consider the problem

∂t R − ∂x (c(u)R) = −c′(u)(R − S)2,

∂t S + ∂x (c(u)S) = −c′(u)(R − S)2,

∂x u = R − S

2c(u),

limx→−∞ u(t, x) = 0,

(R, S)|t=0 = (R0, S0)(x)

(4.1)

where (R0, S0)(x) ∈ H k(R) ∩ L1(R), k ≥ 1, are given initial data.

Lemma 6. (Local classical solution). Assume (R0, S0)(x) ∈ H k(R) ∩ L1(R),k ≥ 1. Then there exists a T ∗ ∈ (0, ∞] such that problem (4.1) has a uniquesolution (R(t, x), S(t, x)) ∈ L∞([0, T ], H k(R)) for any positive T < T ∗, and

limt→T ∗ (‖R(t, ·)‖L∞ + ‖S(t, ·)‖L∞ ) = ∞ (4.2)

if T ∗ < ∞.

Remark 4. We remark that the catastrophe in (4.2) is associated with the type ofblow-up familiar from ordinary differential equation theory rather than associatedwith the formation of shock waves in the context of systems of conservation laws,see Chapter 2 of Majda [35]. �

Proof of Lemma 6. From standard theory on hyperbolic systems of equations(see Majda [35] for example), we know we can find a positive constant T suchthat problem (4.1) has a unique solution (R(t, x), S(t, x)) ∈ L∞([0, T ], H k(R)).The solution satisfies the estimate

‖R‖L∞([0,T ],H k (R)) + ‖S‖L∞([0,T ],H k (R)) ≤ C(k, ‖R0‖H k (R), ‖S0‖H k (R)). (4.3)

We note that although the equations in (4.1) do not consist of a bona fide symmetrichyperbolic system, the equivalent alternative form of using 2∂t u = R + S does andthe standard theory on hyperbolic systems applies.

Next, we claim that

‖(R, S)‖L∞([0,T ]×R) < ∞ implies ‖(R, S)‖L∞([0,T ],H k (R)) < ∞. (4.4)

In fact, by taking ∂ lx to both sides of the first equation of (4.1) for l ≤ k, we find

∂t (∂lx R) − c(u)∂x (∂ l

x R) = ∂ lx (c(u)∂x R) − c(u)∂ l+1

x R

+ ∂ lx (c′(u)(R2 − S2)). (4.5)

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By multiplying ∂ lx R to the above equation and integrating over R, we find

d

dt

∫R

1

2

(∂ l

x R)2

dx +∫

R

1

2c′(u)∂x u

(∂ l

x R)2

dx

=∫

R

[∂ l

x (c(u)∂x R) − c(u)∂ l+1x R

]∂ l

x R dx

+∫

R

∂ lx

[c′(u)(R2 − S2)

]∂ l

x R dx . (4.6)

Consider the case k = 1 first. If ‖(R, S)‖L∞([0,T ]×R) < ∞, then by (4.6), wefind

d

dt

∫R

(∂x R)2(t, x) dx ≤ CT

∫R

((∂x R)2 + (∂x S)2)(t, x) dx + CT . (4.7)

Exactly as the proof of (4.7), we can prove a similar inequality for the integralof (∂x S)2. Summing up the two inequalities and using Gronwall’s inequality, weprove that (4.4) holds for k = 1.

Consider the case k ≥ 2. By Moser-type calculus inequality (see p. 43 of[35]), we find∥∥[

∂kx (c(u)∂x R) − c(u)∂k+1

x R](t, ·)∥∥L2 ≤ Ck

[‖∂x c(u)(t, ·)‖L∞∥∥∂k

x R(t, ·)∥∥L2

+ ‖∂x R(t, ·)‖L∞∥∥∂k

x c(u)(t, ·)∥∥L2

].

(4.8)

Since ∥∥∂kx c(u)(t, ·)∥∥

L2 = ∥∥∂k−1x (c′(u)∂x u)

∥∥L2 ≤ C

(‖∂x u‖H k−1 + ∥∥∂k−1x c′(u)

∥∥L2

)≤ · · · ≤ Ck‖∂x u‖H k−1 ,

we further obtain that∥∥[∂k

x (c(u)∂x R) − c(u)∂k+1x R

](t, ·)∥∥L2

≤ Ck[∥∥∂k

x R(t, ·)∥∥L2 + ‖∂x R(t, ·)‖L∞‖∂x u(t, ·)‖H k−1

]. (4.9)

Similarly, we obtain that∥∥(∂k

x (c′(u)(R2 − S2)))(t, ·)∥∥

L2

≤ Ck{‖c′(u)(t, ·)‖L∞

∥∥∂kx (R2 − S2)

∥∥L2

+∥∥∂kx (c′(u))(t, ·)∥∥

L2‖(R2 − S2)(t, ·)‖L∞}

≤ Ck{∥∥∂k

x R(t, ·)∥∥L2 + ∥∥∂kx S(t, ·)∥∥L2 + ‖∂x u(t, ·)‖H k−1

}. (4.10)

By summing up (4.6), (4.9), and (4.10), we find

d

dt‖R(t, ·)‖2

H k ≤ Ck{‖R(t, ·)‖H k + ‖S(t, ·)‖H k

+ (1 + ‖∂x R‖L∞ )‖∂x u(t, ·)‖H k−1}‖R(t, ·)‖H k . (4.11)

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Exactly as the above procedure, we can prove a similar inequality for ‖S(t, ·)‖2H k .

On the other hand, by differentiating the first equation of (4.1) once, wefind

∂t∂x R − c(u)∂x (∂x R) − c′(u)∂x u∂x R = ∂x (c′(u)(R2 − S2)). (4.12)

Thus, by the characteristic method, we obtain

‖∂x R(t, ·)‖L∞ ≤ ‖∂x R0‖L∞ + C∫ t

0(‖∂x R‖L∞ + ‖∂x S‖L∞ + 1) ds. (4.13)

Exactly as the above we can prove a similar inequality for the L∞ norm of ∂x S.Summing up the two inequalities, we find by applying Gronwall’s inequality that

‖∂x R(t, ·)‖L∞ + ‖∂x S(t, ·)‖L∞ ≤ CT , t ∈ [0, T ] (4.14)

where CT = C(T, ‖R‖L∞ , ‖S‖L∞ , ‖∂x R0‖L∞ , ‖∂x S0‖L∞ ) is a constant dependingonly on the variables. The quantities (‖∂x R0‖L∞ , ‖∂x S0‖L∞ ) are finite since (R0, S0)∈ H 2(R).

Thus by summing up (4.11), (4.14), and Gronwall’s inequality, we prove(4.3). The proof of Lemma 6 is complete. �

Lemma 7 (Global classical solution). Assume c′ ≥ 0, R0 ≤ 0, S0 ≤ 0. Thenproblem (4.1) has a global solution (R, S )(t, x) ∈ L∞(R+, H k(R)) provided that(R0, S0) ∈ H k(R) ∩ L1(R), k ≥ 1.

Proof: By Lemma 5 (with ε = 0), if c′ ≥ 0, R0 ≤ 0, S0 ≤ 0, we have (3.6) holdsfor all p > 2. By taking p to infinity, we find that ‖R‖L∞ + ‖S‖L∞ is boundedby ‖R0‖L∞ + ‖S0‖L∞ which is finite since (R0, S0) ∈ H 1(R). By Lemma 6, wecomplete the proof of Lemma 7. �

Proof of Theorems 1–2. We show that the solutions of Lemmas 6-7 yield solu-tions for (1.1). From the conditions of Theorems 1–2, we have (R0, S0) ∈ H k(R),k ≥ 1, and both have compact supports. Then by Lemmas 6–7, (4.1) has a solution(R, S) ∈ L∞([0, T ], H k(R)), where T is finite or infinite accordingly. We alreadyhave R − S = 2c(u)∂x u, so we only need

R + S = 2∂t u. (4.15)

It can be shown easily that the solution (R, S) has compact support for eachtime t ≥ 0. From this and the third and fourth equations of (4.1), we see that uis supported in [Ct , ∞) in x for each time t ≥ 0 where Ct is some function of t .Now multiplying the third equation of (4.1) by 2c(u), taking t derivative, and using∂t (2c(u)∂x u) = ∂x (2c(u)∂t u), we find

∂x {2c(u)∂t u − c(u)(R + S)} = 0. (4.16)

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The support property and (4.16) implies (4.15). Hence by summing up the first twoequations of (4.1) and (4.15), we find that u(t, x) is indeed a solutionof (1.1). �

V. PRECOMPACTNESS

With the above preparation, we can now prove the precompactness of thesolution sequence {Rε(t, x), Sε(t, x)}, obtained either through the viscous approx-imation or the mollification of the initial data, in L2([0, T ] × R) for any T < ∞for initial data (R0, S0) ∈ L p(R) ∩ L1(R) with p > 3, by applying Young measuretheory ([46][50]), and the ideas used by the authors in the proof of global existenceof weak solutions to (1.10) in [52], and the ideas used by P. L. Lions ([33]) in theproof of the global existence of weak solutions to multidimensional compressibleNavier-Stokes equation, and by J. L. Joly, G. Metivier and J. Rauch ([24]) in therigorous justification of weakly nonlinear geometric optics for a semilinear waveequation. See also [51]. For the convenience of the reader, we quote the followinglemma from [24] (see also [8][11]) that we use in this paper.

Lemma 8 (General Young measure). Let U be an open subset of Rn , whose

boundary has zero Lebesgue measure. Given a bounded family {uε(y)} ⊂ Ls(U ),s > 1, of R

N −valued functions, then there exist a subsequence {ε j } and a mea-surable family of probability measures on R

N , {µy(·), y ∈ U }, such that for allcontinuous functions F(y, λ) with F(y, λ) = O(|λ|q ) as |λ| → ∞, where q < s,there holds

limε j →0

∫U

ϕ(y)F(y, uε j (y)) dy =∫

U

∫RN

ϕ(y)F(y, λ)dµy(λ) dy (5.1)

for all ϕ(y) ∈ Lr (U ) with compact support in the closure of U , where 1r + q

s = 1.Moreover,∫ ∫

|λ|s dµy(λ) dy ≤ limε j →0

‖uε j (y)‖sLs . (5.2)

Let (R0, S0) ∈ L2 ∩ L1. We now focus on the viscous regularization.The treatment for the initial data mollification is similar. By Lemmas 2 and 8,there exist three families of Young measures ν1

t,x (ξ ), ν2t,x (η), and µt,x (ξ, η) which

are associated with {Rε}, {Sε} and {Rε, Sε} respectively. Moreover, by modifyingthe proof of Lemma 3 in [52], we can prove the following lemma:

Lemma 9 (Time-distinguished Young measures). There exist a subsequence ofthe solution sequence {Rε(t, x), Sε(t, x)}, for convenience, we still denote it by{Rε(t, x), Sε(t, x)}, and three families of Young measures ν1

t,x (ξ ) and ν2t,x (η) on

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R and µt,x (ξ, η) on R2 such that for all continuous functions f (λ) ∈ C∞

c (R),ψ(x) ∈ C∞

c (R), g(ξ, η) ∈ C∞c (R2), and ϕ(t, x) ∈ C∞

c (R+ × R), there hold

limε→0

∫R

f (Rε(t, x))ψ(x) dx =∫ ∫

R×R

f (ξ )ψ(x) dν1t,x (ξ ) dx,

(5.3)limε→0

∫R

f (Sε(t, x))ψ(x) dx =∫ ∫

R×R

f (η)ψ(x) dν2t,x (η) dx,

uniformly in every compact subset of [0, ∞), and

limε→0

∫ ∞

0

∫R

g(Rε(t, x), Sε(t, x))ϕ(t, x) dx dt

=∫ ∞

0

∫R

∫ ∫R×R

g(ξ, η)ϕ(t, x) dµt,x (ξ, η) dx dt. (5.4)

Moreover,

t ∈ [0, ∞) �−→∫ ∫

R×R

f (ξ )ψ(x) dν1t,x (ξ ) dx is continuous;

(5.5)t ∈ [0, ∞) �−→

∫ ∫R×R

f (η)ψ(x) dν2t,x (η) dx is continuous.

In the sequel, we use the notation

g(R, S) =∫

R

g(ξ, η) dµt,x (ξ, η).

With Lemmas 2, 8–9, we can prove the decoupling of the Young measureµt,x (ξ, η) into the tensor product of the Young measures ν1

t,x (ξ ) and ν2t,x (η).

Lemma 10 (Decoupling of the Young measure). Let {Rε, Sε} be the solutions to(2.1) with data (R0, S0) ∈ L2 ∩ L1(R). Then the Young measures ν1

t,x (ξ ), ν2t,x (η),

and µt,x (ξ, η) have the property

µt,x (ξ, η) = ν1t,x (ξ ) ⊗ ν2

t,x (η).

Proof: Take any f ∈ C∞c (R) and g ∈ C∞

c (R). By (2.1) and a trivial calculation,we find that

∂t f (Rε) − ∂x (c(u) f (Rε)) = T ε1 + T ε

2 (5.6)

where

T ε1 : = ∂x ((c(uε) − c(u)) f (Rε)) + ε∂x ( f ′(Rε)∂x Rε),

T ε2 : = 2c′(uε)(Sε − Rε) f (Rε) + c′(uε)(R2

ε − S2ε ) f ′(Rε) (5.7)

− ε f ′′(Rε)(∂x Rε)2

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By Lemma 3, (c(uε) − c(u)) f (Rε) → 0 in L ploc(R+ × R) for any p < ∞. By

Lemma 2, ε f ′(Rε)∂x Rε → 0 in L2, thus {T ε1 } is precompact in H−1

loc (R+ × R). ByLemma 2 again, we find that {T ε

2 } is uniformly bounded in L1loc(R+ × R). Since

f (Rε) and c(uε) are uniformly bounded in L∞(R+ × R), thus by Murat Lemma[36] (or Corollary 1 on p. 8 of [11]), we find that {T ε

2 } is also a precompact subsetof H−1

loc (R+ × R). Summing up the above, we have proved the precompactness

{∂t f (Rε) − ∂x (c(u) f (Rε))} ⊂⊂ H−1loc (R+ × R). (5.8)

Exactly as the proof of (5.8), we can also prove that

{∂t g(Sε) + ∂x (c(u)g(Sε))} ⊂⊂ H−1loc (R+ × R). (5.9)

Hence, by (5.8), (5.9), and the generalized compensated-compactness Theorem in[13], we find that

f (Rε)g(Sε) ⇀ f (R) · g(S), as ε → 0, (5.10)

where ( f (R), g(S)) is the weak limit of ( f (Rε), g(Sε)). Thus, by Lemma 9, wehave proved that for any ϕ(t, x) ∈ C∞

c (R+ × R),

∫ ∫ ∫ ∫ϕ(t, x) f (ξ )g(η) dµt,x (ξ, η) dx dt

= limε→0

∫ ∫ϕ(t, x) f (Rε)g(Sε) dx dt

=∫ ∫

ϕ(t, x) f (R) · g(S) dx dt

=∫ ∫

φ(t, x)∫

f (ξ )g(η) dν1t,x dν2

t,x dx dt

=∫ ∫ ∫ ∫

ϕ(t, x) f (ξ )g(η) dν1t,x (ξ ) ⊗ ν2

t,x (η) dx dt. (5.11)

Since the above equality holds for any ϕ(t, x) ∈ C∞c (R+ × R), f (ξ ), g(η) ∈ C∞

c(R), the proof of Lemma 10 is complete. �

We prove in the next lemma that the two single-variable Young measuresare Dirac measures provided that p > 3.

Lemma 11 (Strong precompactness). Assume R0 ≤ 0, S0 ≤ 0, (R0, S0) ∈L p(R) with p > 3, and c′ ≥ 0. Let (R, S) be the weak-star limit of {(Rε, Sε)}in L∞(R+, L p(R)). Then, ν1

t,x (ξ ) = δR(t,x)(ξ ) and ν2t,x (η) = δS(t,x)(η).

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Proof 1: Firstly, as in [32], we introduce the notations

Tλ(ξ ) =

ξ, −λ ≤ ξ ≤ 0,

−λ, ξ ≤ −λ,

0, otherwise;Sλ(ξ ) =

12ξ 2, −λ ≤ ξ ≤ 0,

−λξ − 12λ2, ξ ≤ −λ,

0, otherwise.

(5.12)

That is, Tλ(ξ ) is a cut-off function of

ξ− ={

ξ, ξ ≤ 0,

0, ξ ≥ 0,

and Sλ(ξ ) is an anti-derivative of Tλ(ξ ). Next, we multiply Tλ(Rε) to both sides ofthe first equation of (1.16) to obtain

∂t Sλ(Rε) − ∂x (c(uε)Sλ(Rε)) = −c′(uε)[2(Rε − Sε)Sλ(Rε)

− Tλ(Rε)(R2ε − S2

ε )] + εTλ(Rε)∂xx Rε .

(5.13)

Now, we claim that

limε→0

ε

∫ ∫R+×R

ϕTλ(Rε)∂xx Rε dx dt ≤ 0, ∀ 0 ≤ ϕ ∈ C∞c (R+ × R). (5.14)

In fact, we have

−∫ ∫

R+×R

ϕTλ(Rε)∂xx Rε dx dt =∫ ∫

R+×R

ϕT ′λ(Rε)(∂x Rε)2 dx dt

+∫ ∫

R+×R

∂xϕTλ(Rε)∂x Rε dx dt.

(5.15)

Then claim (5.14) follows from this (5.15), the monotonicity T ′λ(ξ ) ≥ 0, the energy

bound (2.3), and the fact that |Tλ(ξ )| ≤ λ. Hence by taking ε → 0 in (5.13) andapplying Lemma 9 and (5.14), we find

∂t Sλ(R) − ∂x (c(u)Sλ(R)) ≤ c′(u)∫ ∫

R22(η − ξ )Sλ(ξ )

+ Tλ(ξ )(ξ 2 − η2) dµt,x (ξ, η). (5.16)

2. On the other hand, by applying Lemma 9 again and taking the limit forthe first equation of (1.16) as ε → 0, we find

∂t R − ∂x (c(u)R) = −c′(u)(R − S)2. (5.17)

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Convolving with the standard Friedrichs’ mollifier jε , we find

∂t Rε − ∂x (c(u)Rε) = −(

c′(u)(R − S)2)

∗ jε + rε, (5.18)

where Rε = ∫R

R(t, y) jε(x − y) dy and rε = jε ∗ ∂x (c(u)R) − ∂x (c(u)Rε). ByDiPerna-Lions folklore Lemma 2.3 of Lions [32] and Lebesgue dominated con-vergence theorem in the time direction, we have rε → 0 in L1

loc(R+ × R) (or seeLemma II.1 of [9]). Again, we multiply Tλ(Rε) to both sides of (5.18), we find

∂t Sλ(Rε) − ∂x (c(u)Sλ(Rε)) = [−(c′(u)(R − S)2) ∗ jε + rε + 2c′(u)

× (R − S)Rε]Tλ(Rε) − 2c′(u)(R − S)Sλ(Rε).

(5.19)

Thus, by taking ε → 0 in (5.19), we find

∂t Sλ(R) − ∂x (c(u)Sλ(R)) = −c′(u)[(R − S)2 − 2(R − S)R]Tλ(R)

− 2c′(u)(R − S)Sλ(R). (5.20)

While by Lemma 10

(R − S)2 =∫ ∫

R2(ξ − η)2 dµt,x (ξ, η)

=∫

R

ξ 2 dν1t,x (ξ ) +

∫R

η2 dν2t,x (η) − 2

∫R

ξ dν1t,x (ξ )

∫R

η dν2t,x (η)

= R2 + S2 − 2R S. (5.21)

By summing up (5.20) and (5.21), we find

∂t Sλ(R) − ∂x (c(u)Sλ(R))

= c′(u){−2(R − S)Sλ(R) + 2Tλ(R)R2 − Tλ(R)(R2 + S2)}. (5.22)

3. Now, we subtract (5.22) from (5.16) to obtain

∂t (Sλ(R) − Sλ(R)) − ∂x (c(u)(Sλ(R) − Sλ(R)))

≤ c′(u)

{ ∫R

[−2ξ Sλ(ξ ) + Tλ(ξ )ξ 2] dν1t,x (ξ ) + 2RSλ(R) − Tλ(R)R2

+∫ ∫

R22ηSλ(ξ ) dµt,x (ξ, η) − 2SSλ(R)

+∫ ∫

R2−Tλ(ξ )η2 dµt,x (ξ, η) + Tλ(R)S2 + Tλ(R)(R2 − R2)

}.

(5.23)

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Trivially, one has

Tλ(R)(R2 − R2) ≤ 0 (5.24)

since Tλ(R) ≤ 0 and R2 − R2 ≥ 0. Next, by Lemma 10, we have∫ ∫R2

ηSλ(ξ ) dµt,x (ξ, η) − SSλ(R) = S(Sλ(R) − Sλ(R)) ≤ 0,

(5.25)∫ ∫R2

−Tλ(ξ )η2 dµt,x (ξ, η) + Tλ(R)S2 = −(Tλ(R) − Tλ(R))S2 ≤ 0,

due to the fact that Sλ(ξ ) is a convex function of ξ , S ≤ 0, Tλ(ξ ) is convex in(−∞, 0), and Rε ≤ 0 (by Lemma 4). Further, we observe from the explicit struc-tures of Sλ and Tλ that

−2ξ Sλ(ξ ) + Tλ(ξ )ξ 2 = λξ (ξ + λ)|ξ≤−λ,(5.26)

2RSλ(R) − Tλ(R)R2 = −λR(R + λ)|R≤−λ ≤ 0.

We thus have from (5.23) to (5.24) that

∂t (Sλ(R) − Sλ(R)) − ∂x (c(u)(Sλ(R) − Sλ(R)))

≤ c′(u)∫

R

λξ (ξ + λ)|ξ≤−λ dν1t,x (ξ )

≤ c′(u)∫

R

|ξ |3|ξ≤−λ dν1t,x (ξ ). (5.27)

4. Next we prove that

limλ→∞

supt≥0

∥∥∥∥∫

R

|ξ |3|ξ≤−λ dν1t,x (ξ )

∥∥∥∥L1(R)

= 0, provided p > 3. (5.28)

In fact,∥∥∥∥∫

R

|ξ |3|ξ≤−λ dν1t,x (ξ )

∥∥∥∥L1(R)

≤ 1

λp−3supε>0

∫R

|Rε |p1Rε≤− λ2(t, x) dx . (5.29)

Thus, Lemma 5 and (5.29) implies (5.28).

5. So, by Lemma 9 and integrating (5.27) over R, we find

d

dt

∫R

(Sλ(R) − Sλ(R))(t, x) dx ≤∫ ∫

R2c′(u)|ξ |31ξ≤−λ dν1

t,x dx . (5.30)

Hence by (5.28), Lebesgue dominated convergence theorem, and taking λ → ∞in (5.30), we find∫

R

(R2 − R2)(t, x) dx = 0. (5.31)

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This implies that ν1t,x (ξ ) = δR(ξ ). Exactly as the above procedure, we can prove

that ν2t,x (η) = δS(η). This completes the proof of Lemma 11. �

Remark 5. We remark that the cut-off step of Tλ in the proof of Lemma 11 isnot needed for p > 3, although we choose to include it both to show that thecondition p > 3 is used only in one essential step and to use it for the proof ofTheorem 4.

Proof of Theorem 3. Now, let R0, S0 ≤ 0, (R0, S0) ∈ L p(R) for p > 3, andsupp(R0, S0) ⊂ [−a, a]. We have two cases.

Part (a) (Data mollification approximation). We mollify (R0, S0) to (Rε0, Sε

0 )with the standard Friedrichs’ mollifier. By Lemmas 6–7 and the proof of Theorems1-2, we find that (4.1) has a global smooth solution (Rε, Sε) which is uniformlybounded in L∞(R+, L p(R)) with supp(Rε, Sε) ⊂ (−a − C2t, a + C2t). Thus, byfollowing the proof of Lemma 4 and that of Lemmas 8-11, we can find a subse-quence of {Rε, Sε}, (for simplicity, we still denote them by {Rε, Sε}), and two func-tions (R, S) ∈ L∞(R+, L p(R)), such that (Rε, Sε) → (R, S) in Lq (R+, Lr (R)) forany q < ∞, r < p. Thus

supp(R(t, ·), S(t, ·)) ⊂ [−a − C2t − 1, a + C2t + 1]. (5.31)

Then repeating the support argument in the proof of Theorems 1–2, we can provethat (4.15) holds.

Part (b) (Equation regularization approximation). We need only to showthat the vanishing viscosity limit has compact support. Let ϕ(y) be a monotonedecreasing smooth function so that it is equal to 1 for y < −1 and equal to 0 fory > 0. We multiply the energy density equation

∂t (R2 + S2) + ∂x [c(S2 − R2)] = 2ε(R∂2x R + S∂2

x S)

with the test function ϕ(x + C2t + a), where the superscript ε is omitted for sim-plicity, to yield

∂t [(R2 + S2)ϕ] + ∂x [(S2 − R2)cϕ]

= 2ε(R∂2x R + S∂2

x S)ϕ + [C2(R2 + S2) + (S2 − R2)c]ϕ′.

Integrating it over [0, t] × R and using the support property of the initial data andϕ′ ≤ 0, we obtain∫

R

(R2 + S2)ϕ dx + 2ε

∫ t

0

∫R

ϕ((∂x R)2 + (∂x S)2) dx dt

≤ ε

∫ t

0

∫R

ϕ′′(R2 + S2)dx dt.

Sending ε → 0, we find that (R, S) is supported on [−C2t − a − 1, ∞) at each

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t ≥ 0. Similarly we can prove that (R, S) is supported on (−∞, C2t + a + 1].Thus (R, S) is supported on [−C2t − a − 1, C2t + a + 1].

Thus by summing up the first two equations of (4.1) and using (4.15), wefind that u(t, x) satisfies (1.13). Moreover, our solution u is finite in any finite timeinterval [0, T ] from the compact support property, the third and fourth equations of(2.1), and the energy bound. Then from condition (1.20), Lemma 5, and convexityof L p+1, we have ∂x u ∈ L p+1([0, T ] × R) for any T > 0 provided that (1.20)holds. This completes the proof Theorem 3. �

VI. SQUARE INTEGRABLE DATA

We prove Theorem 4 in this section. First we establish a lemma.

Lemma 12 (Local space-time higher integrability estimate). Let (R0, S0) ∈ L2 ∩L1(R). Let c satisfies (1.18). Let α = d2

d1∈ (0, 1) with d2 an even positive integer

and d1 an odd positive integer. Let a < b and χ (·) ∈ C∞c (R) with supp χ ⊂ (a, b).

Then for the viscous solution (Rε, Sε, uε) to (2.1), we have∣∣∣∣∣∫ T

0

∫ b

aχ (x)c′(uε)

[(1 − α

)(R1+α

ε − S1+αε

)

+ (1 + α)Rαε Sα

ε

(R1−α

ε − S1−αε

)](Rε − Sε) dx dt

∣∣∣∣∣ ≤ CT,a,b, (6.1)

where CT,a,b depends only on (T, a, b, χ, ‖R0‖L2 , ‖S0‖L2 ), (but not ε or α).

Proof: Take an odd function f (ξ ) ∈ C∞(R), f (ξ ) = 0 in [−1/2, 1/2], with

f ′(ξ ) ={

ξα, |ξ | ≥ 1,

0, |ξ | ≤ 12 .

Then

f (ξ ) =∫ ξ

0f ′(ζ ) dζ =

{1

1+αξ 1+α + C1, if |ξ | ≥ 1,

0, if |ξ | ≤ 1/2.(6.2)

Now, we multiply f ′(Rε)χ (x) to the first equation of (2.1) to obtain

∂t ( f (Rε)χ ) − cχ∂x f (Rε) = c′(R2ε − S2

ε

)χ f ′(Rε) + εχ f ′(Rε)∂xx Rε,

and integrate the above equation over [0, T ] × [a, b] to find∫ b

a( f (Rε)χ )(T, ·) − ( f (R0)χ ) dx

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+∫ T

0

∫ b

a[2c′(Rε − Sε)χ f (Rε) + cχ ′ f (Rε)] dx dt

=∫ T

0

∫ b

ac′(R2

ε − S2ε

)χ f ′(Rε) − ε∂x (χ f ′(Rε))∂x Rε dx dt. (6.3)

Then, by (6.2) and a trivial rearrangement of (6.3), we obtain∫ T

0

∫ b

a

(2c′(Rε − Sε)χ

(1

1 + αR1 + α

ε + C1

)− c′(R2

ε − S2ε

)χ Rα

ε

)1|Rε |≥1 dx dt

=∫ T

0

∫ b

a

(c′(R2

ε − S2ε

)χ f ′(Rε) − 2c′(Rε − Sε)χ f (Rε)

)1|Rε |<1 dx dt

− ε

∫ T

0

∫ b

a(χ f ′′(Rε)(∂x Rε)2 + χ ′ f ′(Rε)∂x Rε) dx dt

+∫ b

a( f (R0)χ ) dx −

∫ b

a( f (Rε)χ)(T, ·) dx −

∫ T

0

∫ b

acχ ′ f (Rε) dx dt.

(6.4)

Using the energy (2.3), we see that the absolute value of the left-hand side of (6.4)is bounded. Again by the energy estimate, we see that the restriction to |Rε | ≥ 1on the right-hand side can be removed. So

∣∣∣∣∣∫ T

0

∫ b

aχ c′

((Rε − Sε)

1

1 + αR1+α

ε − 1

2

(R2

ε − S2ε

)Rα

ε

)dx dt

∣∣∣∣∣≤ CT,a,b. (6.5)

Exactly as the proof of (6.5), we find∣∣∣∣∣∫ T

0

∫ b

aχ c′

(−(Rε − Sε)

1

1 + αS1+α

ε − 1

2

(S2

ε − R2ε

)Sα

ε

)dx dt

∣∣∣∣∣≤ CT,a,b. (6.6)

By summing (6.5) and (6.6), we obtain∣∣∣∣∣∫ T

0

∫ b

ac′χ

(1

1 + α

(R1+α

ε − S1+αε

)(Rε − Sε)

− 1

2

(R2

ε − S2ε

)(Rα

ε − Sαε

))dx dt ≤ CT,a,b. (6.7)

While a very simple calculation shows that

1

1 + α

(R1+α

ε − S1+αε

)(Rε − Sε) − 1

2

(R2

ε − S2ε

)(Rα

ε − Sαε

)

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=(

1

1 + α− 1

2

)(Rε − Sε)

(R1+α

ε − S1+αε

)+ 1

2Rα

ε Sαε (Rε − Sε)

(R1−α

ε − S1−αε

). (6.8)

Thus, we obtain∣∣∣∣∣∫ T

0

∫ b

aχ c′(Rε − Sε)

[(1

1 + α− 1

2

)(R1+α

ε − S1+αε

)(6.9)

+ 1

2Rα

ε Sαε

(R1−α

ε − S1−αε

)]dx dt

∣∣∣∣∣ ≤ CT,a,b.

Multiplying (6.9) by 2(1 + α), we obtain (6.1). This completes the proof of(6.1). �

From Lemma 12 and its proof, we have the corollary:

Corollary 13 (L2+α estimate). Let c′(·) ≥ 0. Let (R0, S0) ∈ L2(R) with compactsupport. Then for 0 < α < 1, there holds the estimate∫ T

0

∫ b

ac′(uε)|∂x uε |2+α dx dt ≤ Cα,T,a,b, (6.10)

where the constant Cα,T,a,b → ∞ as α → 1.

Proof: First take α = d2d1

with d2 an even natural number, and d1 an odd naturalnumber. We use (6.1) on the interval [a − 1, b + 1]. Since each term in the integral(6.1) is nonnegative, we immediately obtain∫ T

0

∫ b+1

a−1χ (1 − α)c′(uε)(Rε − Sε)2

(Rα

ε + Sαε

)dx dt ≤ CT,a,b. (6.11)

By the third equation of (2.1), (6.11), and choosing χ = 1 on [a, b], we imme-diately obtain (6.10). For any other α ∈ (0, 1), we can use interpolation. Thiscompletes the proof of the corollary. �

Now, let R0 ≤ 0, S0 ≤ 0, (R0, S0) ∈ L2(R) with supp (R0, S0) ⊂ (−a, a).Let c′ ≥ 0. We mollify (R0, S0) to (Rε

0, Sε0 ) with the standard Friedrichs’ mollifier.

Then by Theorem 2 and its proof on the compactness of the support of solutions, wefind that (1.15) has a global smooth solution (Rε, Sε) which is uniformly bounded inL∞(R+, L2(R)) with supp (Rε, Sε) ⊂ (−a − C2t, a + C2t), Rε ≤ 0, Sε ≤ 0, andsatisfies the estimates in Lemma 12 and Corollary 13. By Lemmas 8–10, we find,similar as before, that

µt,x (ξ, η) = ν1t,x (ξ ) ⊗ ν2

t,x (η), (6.12)

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where ν1t,x (ξ ), ν2

t,x (η), and µt,x (ξ, η) are the Young measures associated with{Rε}, {Sε}, and {Rε, Sε}, respectively. We establish short-time decay on R andS to control the quadratic terms.

Lemma 14 (Local decay). Assume that c has strict monotonicity (1.20). LetR0 ≤ 0, S0 ≤ 0, (R0, S0) ∈ L2(R) with compact support. For the smooth solutions(Rε, Sε) with mollified (R0, S0) and any T > 0, there exist a C and M0 (bothindependent of ε ∈ (0, 1]) such that for all M ≥ M0 there hold

−2M ≤ Rε(t, x) ≤ 0, −2M ≤ Sε(t, x) ≤ 0, t ∈[

C

M, T

]. (6.13)

Proof: For simplicity, we omit the subscript ε in the proof of (6.13).

1. Taking any b ∈ R, we introduce the plus and minus characteristics�±

t (b) as{dφ±

tdt = ±c(u(t, φ±

t )),

�±t |t=0 = b.

(6.14)

For the characteristic curve x = �−t (b), t > 0, we find by (1.18) that there exists

the inverse function t = t−(x) defined for all x < b. Similarly, we have the inversefunction t = t+(y) defined for all y > d (d ∈ R) for the plus characteristic x =�+

t (d), t > 0. By the proof on p. 56 of [15], we have the energy conservation in acharacteristic cone∫ �+

t (d)

dR2(t+(y), y) dy +

∫ b

�+t (d)

S2(t−(y), y) dy

= 1

2

∫ b

d

(R2

0(x) + S20 (x)

)dx, (6.15)

where d < b and t > 0 are such that �+t (d) = �−

t (b). In particular, (6.14) and(6.15) imply that∫ t

0c(u(s, �−

s (b)))S2(s, �−s (b)) ds ≤ ‖R0‖2

L2 + ‖S0‖2L2 , ∀b ∈ R. (6.16)

2. On the other hand, by (1.15), (6.14), and the fact that Rε ≤ 0, we find

d|R(t, �−t )|

dt= −c′(u)R2(t, �−

t ) + c′(u)S2(t, �−t ) (6.17)

where �−t is the abbreviation of �−

t (b). Our idea is to use the bound (6.16) of S2

and utilize the negative quadratic term −c′(u)R2 to show decay of |R|. We alreadyknow for fixed T > 0 that uε has a uniform bound for ε ∈ (0, 1] and t ∈ [0, T ]. Wechoose C0 to be the minimum of c′(uε(x, t)) in ε ∈ (0, 1], t ∈ [0, T ], and x ∈ R.

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Then

d|R(t, �−t )|

dt≤ −C0 R2(t, �−

t ) + c′(u)S2(t, �−t ). (6.18)

Thus, by (6.16) and (6.18), for any t2 > t1, we have

|R(t2, �−t2 )| ≤ |R(t1, �

−t1 )| +

∫ t2

t1

(c′(u)S2)(s, �−s ) ds

≤ |R(t1, �−t1 )| + C ′ (6.19)

where C ′ depends on the maximum of c′(·), the minimum of c(·) and the total initialenergy, but independent of (ε, b, t). We also choose C ′ ≥ 1 for later convenience.Now, we take M0 = C ′ + 4 and any M ≥ M0. The first case is |R0(b)| ≤ M , thenby directly applying (6.19), we find

|R(t, �−t )| ≤ M + C ′ ≤ 2M, ∀t ∈ R

+. (6.20)

3. The second case is when |R0(b)| ≥ M . We have two subcases here. Firstsubcase is when

|R(t, �−t )| ≥ M

2, for all t ∈

[0,

C

M

]. (6.21)

where C := 4C ′C0

. Then by (6.18) again, we find

1

|R(t, �−t )| ≥ 1

|R0(b)| + C0t −∫ t

0

(c′(u)

S2

R2

)(s, �−

s ) ds

≥ C0t − 4C ′

M2, t ∈

[0,

C

M

]. (6.22)

Let t0 := CM , we find by (6.22) that

∣∣R(t0, �

−t0

)∣∣ ≤ M

3C ′ ≤ M (6.23)

since M > 4 and C ′ ≥ 1. Hence, by summing up (6.19) and (6.23), we find that

|R(t, �−t )| ≤ M + C ′ ≤ 2M, if t ≥ t0. (6.24)

since M > C ′. The second subcase is when there exists some t ′ ∈ [0, CM ] such that

|R(t ′, �−t ′ )| < M

2 . Then, by (6.19), we have

|R(t, �−t )| ≤ M

2+ C ′ ≤ 2M, if t ≥ t ′. (6.25)

By summing up (6.24), (6.25), and (6.20), we have

|R(t, y)| ≤ 2M, for all t ∈[

C

M, T

]. (6.26)

We can estimate |S| similarly. The proof of Lemma 14 is complete. �

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We now prove that the Young measures are Dirac measures.

Lemma 15. Suppose (1.20) hold and R0 ≤ 0, S0 ≤ 0, (R0, S0) ∈ L2(R) withcompact support. Then the approximate solutions through initial data mollificationhave

ν1t,x (ξ ) = δR(ξ ), ν2

t,x (η) = δS(η).

Proof: 1. By the proof of Lemma 11 up to the first inequality of (5.27), we findthat

d

dt

∫(Sλ(R) − Sλ(R))(t, x) dx ≤ Cλ

∫ ∫R2

ξ 2 |ξ≤−λ dν1t,x dx, (6.27)

provided that we can prove that

(Rε − Sε)Sλ(Rε) ⇀

∫ ∫(ξ − η)Sλ(ξ ) dµt,x (ξ, η),

Tλ(Rε)(R2

ε − S2ε

)⇀

∫ ∫Tλ(ξ )(ξ 2 − η2) dµt,x (ξ, η), (6.28)

(Rε − Sε)2 ⇀

∫ ∫(ξ − η)2 dµt,x (ξ, η)

in the sense of distributions. But, by Corollary 13, we find that {Rε − Sε} is uni-formly bounded in L2+α([0, T ] × R) for any T < ∞ and α < 1 due to the fact thatsupp (Rε, Sε) ⊂ (−a − C2t, a + C2t). Thus, by Lemma 8 and a diagonal processfor the time T , we can always find a subsequence of {Rε, Sε} such that (6.28) holds.

2. Now, fixing a T > 0 for Lemma 14, noticing that supp ν1t,x (·) ⊂ [−2M, 0]

if t ≥ CM by (6.13) for any large M , we have∫ t

CM

∫ ∫R2

ξ 2|ξ≤−λ dν1t,x (ξ ) dx dt = 0, λ ≥ 2M. (6.29)

Thus, for λ ≥ 2M and by (6.27), we have∫(Sλ(R) − Sλ(R))(t, x) dx ≤

∫(Sλ(R) − Sλ(R))

(C

M, x

)dx, t ≥ C

M.

(6.30)

Thus again by (5.2), (6.30), and Lebesgue dominated convergence theorem, wefind by passing λ → ∞ in (6.30) that∫

R

( ∫R

ξ 2 dν1t,x (ξ ) − R2

)(t, x) dx

≤∫

R

( ∫R

ξ 2 dν1t,x (ξ ) − R2

)(C

M, x

)dx, t ≥ C

M. (6.31)

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Exactly as the proof of (6.31), we can prove that∫R

( ∫R

η2 dν2t,x (η) − R2

)(t, x) dx

≤∫

R

( ∫R

η2 dν2t,x (η) − R2

)(C

M, x

)dx, t ≥ C

M. (6.32)

3. Next, we claim that

limt→0

∫R

( ∫R

ξ 2 dν1t,x +

∫R

η2 dν2t,x

)dx =

∫R

(R2

0 + S20

)dx . (6.33)

In fact, by the convexity properties of Sλ(·), Lemma 9, and the compact supportand energy bound properties of (Rε, Sε), we find that∫

(Sλ(R) + Sλ(R))(t, x) dx ≤∫ [ ∫

Sλ(ξ ) dν1t,x (ξ ) +

∫Sλ(η) dν2

t,x (η)

]dx

= limε→0

∫(Sλ(Rε) + Sλ(Sε))(t, x) dx ≤ 1

2

∫R

(R2

0 + S20

)dx . (6.34)

Thus, by (5.2) and Lebesgue dominated convergence theorem, we find by tendingλ → ∞ in (6.34) that∫

(R2 + R2)(t, x) dx ≤∫

R

( ∫R

ξ 2 dν1t,x +

∫R

η2 dν2t,x

)dx

≤∫

R

(R2

0 + S20

)dx . (6.35)

While by the equation (5.17) satisfied by R and a similar equation for S and LemmaC.1 of Lions [32] (p. 177), we find that

(R(t, x), R(t, x)) ⇀ (R0(x), S0(x)) weakly in L2(R) as t → 0. (6.36)

Hence by (6.35) and (6.36), we have

limt→0

∫(R2 + R2)(t, x) dx =

∫R

(R2

0 + S20

)dx . (6.37)

Summing up (6.35) and (6.37), we prove (6.33). Incidentally, by (6.36), (6.37),and Theorem 1 on p. 4 of [11], we have

(R(t, x), S(t, x)) → (R0(x), S0(x)) strongly in L2(R) as t → 0. (6.38)

4. Let us take M → ∞. Summing up (6.31), (6.32), and (6.33), we find∫R

{(∫R

ξ 2 dν1t,x (ξ ) − R2

)+

(∫R

η2 dν1t,x (η) − R2

)}(t, x) dx ≤ 0, t > 0.

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410 ZHANG AND ZHENG

That is,∫R

{ ∫R

(ξ − R)2 dν1t,x (ξ ) +

∫R

(η − R)2 dν2t,x (η)

}dx = 0. (6.39)

Thus, ν1t,x (ξ ) = δR(t,x)(ξ ) and ν2

t,x (η) = δR(t,x)(η). The proof of Lemma 15 is com-plete. �

Proof of Theorem 4: By Lemma 15, we have

Rε − Sε → R − R in L1loc(R+ × R). (6.40)

But since by Corollary 13, {∂x uε} is uniformly bounded in L2+αloc (R+ × R), we in

fact prove that

Rε − Sε → R − R in L2+αloc (R+ × R), ∀α < 1. (6.41)

Thus, (R, R) is a weak solution of (1.15) and ∂x u(t, x) ∈ L2+α([0, T ] × R) forany T < ∞ and α < 1, due to the fact that both R(t, ·) and R(t, ·) have com-pact supports for any t ≤ T . To establish the local decay (1.21), we let T = 2 inLemma 14 and use (6.13) at t = C/M for t ∈ (0, C/M0]. The boundedness fort ≥ 1 in (1.21) follows from (6.19). The remaining of the proof of Theorem 4 isexactly as that of Theorem 3, which we omit here. This completes the proof ofTheorem 4. �

APPENDIX: PROOF OF LEMMA 1

We occasionally use subscripts to denote derivatives here.

Proof of Lemma 1. Step 1 (successive approximation). We note that the system{ut − εuxx = a1(t, x)(u − v) + f1(t, x),

vt − εvxx = a2(t, x)(u − v) + f2(t, x),

where ai (t, x), fi (t, x), i = 1, 2, are sufficiently smooth, has global smooth solu-tions for smooth initial data. Thus, we can solve (2.1) by successive approximation.Choose R0 = R0 and S0 = S0. Define (Rn, Sn, un) for n = 1, 2, . . . , by

Rnt − εRn

xx = (c(un−1)Rn−1)x − c′(un−1)(Rn−1 − Sn−1)(Rn − Sn),

Snt − εSn

xx = −(c(un−1)Sn−1)x − c′(un−1)(Rn−1 − Sn−1)(Rn − Sn),

unx = Rn−Sn

2c(un ) ,

limx→−∞ un(t, x) = 0,

Rn|t=0 = R0(x), Sn|t=0 = S0(x).

(7.1)

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Define u0 as follows. Given Rn−1 and Sn−1, we have by the third and fourthequations of (7.1),

F(un−1) :=∫ un−1

0c(v) dv = 1

2

∫ x

−∞(Rn−1 − Sn−1)(t, y) dy, (7.2)

Thus, by (1.18), we can uniquely determine un−1 by inverting F .2 (Energy estimate). Multiplying the first equation of (7.1) with Rn and

integrating over R, we find

1

2

d

dt

∫(Rn)2 dx + ε

∫ (Rn

x

)2dx

= −∫

c(un−1)Rn−1 Rnx − c′(un−1)(Rn−1 − Sn−1)(Rn − Sn)Rn dx

= −∫

c(un−1)Rn−1 Rnx + 1

2c(un−1)

[(Rn − Sn)x Rn + (Rn − Sn)Rn

x

]dx

≤ ε

4

∫ (Rn

x

)2dx + C

ε

∫(Rn−1)2 dx + 1

2

[ε

8

∫ (Rn

x − Snx

)2dx

+ C

ε

∫(Rn)2 dx + ε

4

∫ (Rn

x

)2dx + C

ε

∫(Rn − Sn)2 dx

].

Thus,

d

dt

∫(Rn)2 dx + ε

∫ (Rn

x

)2dx ≤ Cε

(yn +

∫(Rn−1)2 dx

)+ ε

4

∫ (Sn

x

)2dx

(7.3)

where yn(t) := ∫(Rn)2 + (Sn)2 dx . Here and in the sequel, we denote by Cτ a

constant depending only on τ. We have a similar estimate for Sn . By summing,we find

d

dtyn ≤ Cε(yn + yn−1).

By comparison theorem in ordinary differential equations with dy/dt = 2Cε y, wefind that for any T > 0, there exists a constant Cε(T ) > 0 such that∫

(Rn)2 + (Sn)2 dx ≤ Cε(T ), t ∈ [0, T ] (7.4)

3 (L p estimates). We adapt the method used by L. Hormander in [20] to prove

the convergence of the sequence {(Rn, Sn, un)}. Let E(t, x) = (4πεt)−12 e− |x |2

4εt ,

then, by (7.1), we find

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Rn(t, x) =∫E(t, x − y)R0(y) dy +

∫ t

0

∫(c(un−1)Rn−1)y E(t − s, x − y) dy ds

−∫ t

0

∫c′(un−1)(Rn−1 − Sn−1)(Rn − Sn)(s, y)E(t − s, x − y) dy ds

=∫

E(t, x − y)R0(y) dy +∫ t

0

∫c(un−1)Rn−1 Ex (t − s, x − y) dy ds

−∫ t

0

∫c′(un−1)(Rn−1 − Sn−1)(Rn − Sn)(s, y)E(t − s, x − y) dy ds.

(7.5)

Similarly, we have

Sn(t, x) =∫E(t, x − y)S0(y) dy −

∫ t

0

∫c(un−1)Sn−1 Ex (t − s, x − y) dy ds

−∫ t

0

∫c′(un−1)(Rn−1 − Sn−1)(Rn − Sn)(s, y)E(t − s, x − y) dy ds.

(7.6)

Thus, by (7.4) and Young’s inequality, we find

‖Rn(t, ·)‖L∞ ≤ ‖R0‖L∞ + Cε

[ ∫ t

0‖Rn−1(s, ·)‖L∞ (t − s)−

12

+ ‖(Rn−1 − Sn−1)(s, ·)‖L2‖(Rn − Sn)(s, ·)‖L2 (t − s)−12 ds

]

≤ Cε

∫ t

0‖Rn−1(s, ·)‖L∞ (t − s)−

12 ds + Cε(T ), for t ≤ T .

(7.7)

Choose B so that Cε(T ) ≤ B2 , then, it follows inductively that

‖Rn(t, ·)‖L∞ ≤ BeK t , for t ≤ T, (7.8)

provided that K is so large that Cε

∫ T0 s− 1

2 e−K s ds < 12 (see p. 42 of [20]). Similarly,

we can prove that Sn satisfies (7.8). Similar to the proof of (7.8), we can also provethe estimate

‖Rn(t, ·)‖L1 + ‖Sn(t, ·)‖L1 ≤ Cε(T ), for t ≤ T . (7.9)

Then, we find by (7.1) that

‖un(t, ·)‖L∞ ≤ 1

2C1(‖Rn(t, ·)‖L1 + ‖Sn(t, ·)‖L1 ) ≤ Cε(T ), for t ≤ T .

(7.10)

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4 (Cauchy property). Let vn = Rn − Rn−1, wn = Sn − Sn−1, we find by(7.5), (7.6), (7.8), and (7.10) that

|vn| ≤∫ t

0

∫(|c(un−1) − c(un−2)‖Rn−1| + c(un−2)|vn−1|)

· Ex (t − s, x − y) dy ds

+∫ t

0

∫[|(c′(un−1) − c′(un−2))‖Rn−1 − Sn−1‖Rn − Sn|

+ c′(un−2)|vn−1 + wn−1‖Rn − Sn|+ c′(un−2)|Rn−2 − Sn−2|(|vn| + |wn|)](s, y)E(t − s, x − y) dy ds

≤ C(T )

{ ∫ t

0

∫(|un−1 − un−2‖Rn−1| + |vn−1|)(s, y)

· Ex (t − s, x − y) dy ds

+∫ t

0

∫(|un−1 − un−2|(|Rn−1| + |Sn−1|) + |vn−1| + |wn−1|

+ |vn| + |wn|)(s, y)E(t − s, x − y) dy ds

}.

Hence, by Young’s inequality, we find

‖vn(t, ·)‖L1 ≤ C(T )

{ ∫ t

0[[‖Rn−1(s, ·)‖L1‖(un−1 − un−2)(s, ·)‖L∞

+ ‖vn−1‖L1 ](t − s)−12 + ‖(Rn−1 + Sn−1)(s, ·)‖L1‖(un−1 − un−2)(s, ·)‖L∞

+ ‖vn−1(s, ·)‖L1 + ‖wn−1(s, ·)‖L1 + ‖vn(s, ·)‖L1 + ‖wn(s, ·)‖L1 ]ds

}

≤ C(T )

{ ∫ t

0(‖vn−1(s, ·)‖L1 + ‖wn−1(s, ·)‖L1 )

[(t − s)−

12 + 1

]ds

+∫ t

0(‖vn(s, ·)‖L1 + ‖wn(s, ·)‖L1 )ds

}, (7.11)

since by (1.18) and (7.2), we have

‖(un−1 − un−2)(s, ·)‖L∞ ≤ 1

2C1‖(vn−1 + wn−1)(s, ·)‖L1 . (7.12)

And exactly as the proof of (7.11), we find

‖wn(t, ·)‖L1 ≤ C(T )

(∫ t

0(‖vn−1(s, ·)‖L1 + ‖wn−1(s, ·)‖L1 )

× ((t − s)−12 + 1) ds +

∫ t

0‖vn(s, ·)‖L1 + ‖wn(s, ·)‖L1 ds

).

(7.13)

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Summing up (7.11), (7.13), and letting yn(t) := ‖vn(t, ·)‖L1 + ‖wn(t, ·)‖L1 , wefind

yn(t) ≤ C(T )

(∫ t

0yn−1(s)((t − s)−

12 + 1) ds +

∫ t

0yn(s) ds

). (7.14)

We claim that

yn(t) ≤ Mn−11 Mn

2 tn−1

2

�( n+12 )

, n ≥ 1 (7.15)

where M1 = C(T )(1 + T C(T )eC(T )T ), M2 = max(2 + T12 , 4Cε(T )) with Cε(T )

given in (7.9). In fact, by (7.9) we have

y1(t) ≤ 4Cε(T ) ≤ M2.

This shows that (7.15) is true for n = 1. Now, suppose (7.15) is true for n = k − 1with k ≥ 2, then∫ t

0yk−1(s)

((t − s)−

12 + 1

)ds

≤∫ t

0

Mk−21 Mk−1

2 sk−2

2((t − s)−

12 + 1

)�

(k2

) ds

= Mk−21 Mk−1

2 tk−1

2

(B

(k2 , 1

2

)�

(k2

) + t12(

k2

)�

(k2

))

≤ Mk−21 Mk

2 tk−1

2

�(

k+12

) =: f (t). (7.16)

Using (7.16) in (7.14) and Gronwall′ inequality we find

yk(t) ≤ C(T )

[f (t) +

∫ t

0C(T ) f (s)eC(T )(t−s) ds

]

≤ C(T )(1 + C(T )T eC(T )T ) f (t)

≤ Mk−11 Mk

2 tk−1

2

�(

k+12

) . (7.17)

This completes the proof of (7.15).By (7.5),

∑∞n=1 yn(t) is convergent in C([0, T ]). This proves that {Rn} and

{Sn} are Cauchy sequences in C([0, T ], L1(R)) for any T > 0. Moreover, by (7.8),we in fact prove that {Rn} and {Sn} are Cauchy sequences in C([0, T ], L p(R))for any p < ∞. By (7.12), we trivially find that {un} is strongly convergent inC([0, T ] × R). Thus, the existence of a global weak solution to (2.1) is proved.

5 (Strong solution). To show that (2.1) has a smooth solution, we only needto obtain more estimates than (7.8) and (7.10). In fact, by applying (7.5) again, we

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find ∥∥Rnx (t, ·)∥∥

L∞ ≤ ‖∂x R0‖L∞ + C(T ) ·{ ∫ t

0

[∥∥un−1x (s, ·)∥∥

L∞‖Rn−1(s, ·)‖L∞

+ ∥∥Rn−1x (s, ·)∥∥L∞

](t − s)−

12

+ ∥∥un−1x (s, ·)∥∥

L∞‖(Rn−1 − Sn−1)(Rn − Sn)(s, ·)‖L∞

+ ∥∥(Rn−1

x − Sn−1x

)(s, ·)∥∥L∞‖(Rn − Sn)(s, ·)‖L∞

+ ‖(Rn−1 − Sn−1)(s, ·)‖L∞∥∥(

Rnx − Sn

x

)(s, ·)∥∥

L∞ ds

}

≤ C(T )

{1 +

∫ t

0

([∥∥Rn−1x (s, ·)∥∥L∞ + ∥∥Sn−1

x (s, ·)∥∥L∞]

× (1 + (t − s)−

12)+ ∥∥(

Rnx − Sn

x

)(s, ·)∥∥

L∞)

ds

}. (7.18)

Exactly as the proof of (7.18), we find

∥∥Snx (t, ·)∥∥

L∞ ≤ C(T ) + C(T )

{ ∫ t

0

([∥∥Rn−1x (s, ·)∥∥

L∞+ ∥∥Sn−1x (s, ·)∥∥

L∞]

× (1 + (t − s)−

12) + ∥∥(

Rnx − Sn

x

)(s, ·)∥∥L∞

)ds

}. (7.19)

Thus, if we let Y n(t) := ‖Rnx (t, ·)‖L∞ + ‖Sn

x (t, ·)‖L∞ , then by summing up (7.18)and (7.19), we find that

Y n(t) ≤ C(T ) + C(T )∫ t

0

{Y n−1(s)

[1 + (t − s)−

12] + Y n(s)

}ds (7.20)

where C(T ) is twice the C(T ) in (7.18) (7.19). Similar to (7.8), we can prove that

Y n(t) ≤ 4C(T )eAt n ≥ 0 (7.21)

for A ≥ 2C(T ) and A is so large that

4C(T )∫ T

0e−As

(1 + s− 1

2)

ds ≤ 1.

In fact, we have Y 0 ≤ C(T ). Using induction, we assume (7.20) holds for n − 1.We have∫ t

0Y n−1(s)

[1 + (t − s)−

12]ds ≤ 4C(T )eAt

∫ t

0e−As

[1 + s− 1

2]ds ≤ eAt .

Using Gronwall’s inequality in (7.20), we find

Y n(t) ≤ C(T )

[1 + eAt + C(T )

∫ t

0(1 + eAs)eC(T )(t−s)ds

].

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A further calculation yields (7.20). So (7.20) holds. Exactly as above, we canobtain the uniform estimates to the other derivatives of (Rn(t, x), Sn(t, x)).

6 (Energy estimate). Multiplying the first two equations of (1.16) with R andS respectively and using the third equation of (1.16) with a simple calculation, weobtain

∂t (R2 + S2) + ∂x (c(S2 − R2)) = 2ε(R Rxx + SSxx ) (7.22)

Thus, by integrating the above equation on [0, t] × R, we find (2.2). This completesthe proof of Lemma 1. �

ACKNOWLEDGMENTS

The work of Ping Zhang is supported by the Chinese Post-Doctor’s Foun-dation and the Morningside Center of Mathematics, and that of Yuxi Zheng issupported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fel-lows award. We would also like to thank Professors Tong Zhang and Ling Hsiaofor their constant help and encouragements, Professors Bob Glassey, Fanghua Lin,John Hunter, and Andy Majda for their interest in the work. This work was doneduring the first author’s visit to Indiana University, he would like to express hisappreciation of the hospitality of the Department of Mathematics.

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Received July 1999Revised September 1999

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