Hyperbolic Conservation Laws and the Compensated Compactness Method

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

128

HYPERBOLIC CONSERVATION LAWS AND THE COMPENSATED COMPACTNESS METHOD

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgässner, Universität Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

128

HYPERBOLIC CONSERVATION LAWS AND THE COMPENSATED COMPACTNESS METHOD YUNGUANG LU

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

2387 disclaimer Page 1 Thursday, August 22, 2002 10:58 AM

Library of Congress Cataloging-in-Publication Data Lu, Yunguang. Hyperbolic conservation laws and the compensated compactness method / Yunguang Lu. p. cm. — (Monographs and surveys in pure and applied mathematics) Includes bibliographical references and index. ISBN 1-58488-238-7 1. Conservation laws (Mathematics) 2. Differential equations, Hyperbolic. I. Title. II. Chapman & Hall/CRC monographs and surveys in pure and applied mathematics. QA377 .H965 2003 515′.353—dc21

2002073731

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2003 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-238-7 Library of Congress Card Number 2002073731 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

In memory of my father, Cai-qin Lu

Contents Preface

xi

1 Preliminary

1

2 Theory of Compensated Compactness 2.1 Weak Continuity of a 2 × 2 Determinant . . . . . . . . . 2.2 Measure Representation Theorems . . . . . . . . . . . . 2.3 Embedding Theorems . . . . . . . . . . . . . . . . . . .

9 9 16 20

3 Cauchy Problem for Scalar Equation 3.1 L∞ Solution . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lp Solution, 1 < p < ∞ . . . . . . . . . . . . . . . . . . 3.3 Related Results . . . . . . . . . . . . . . . . . . . . . . .

31 31 37 41

4 Preliminaries in 2 × 2 Hyperbolic System 4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . 4.2 L∞ Estimate of Viscosity Solutions . . . . . . . . . . .

43 43 45

5 A Symmetry System 5.1 Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . 5.2 Related Results . . . . . . . . . . . . . . . . . . . . . . .

47 49 52

6 A System of Quadratic Flux 6.1 Existence of Viscosity Solutions . . 6.2 Entropy-Entropy Flux Pairs of Lax −1 . . 6.3 Compactness of ηt + qx in Hloc 6.4 Reduction of ν . . . . . . . . . . . 6.5 Related Results . . . . . . . . . . .

53 56 57 64 67 70

vii

. . . . Type . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

CONTENTS

viii 7 Le Roux System

71

7.1

Existence of Viscosity Solutions . . . . . . . . . . . . . .

73

7.2

Entropy-Entropy Flux Pairs of Lax Type . . . . . . . .

75

−1 Hloc

7.3

Compactness of ηt + qx in

. . . . . . . . . . . . . .

79

7.4

Existence of Weak Solutions . . . . . . . . . . . . . . . .

81

7.5

Related Results . . . . . . . . . . . . . . . . . . . . . . .

82

8 System of Polytropic Gas Dynamics 8.1

85

Existence of Viscosity Solutions . . . . . . . . . . . . . . −1 Hloc

Compactness

89

8.2

Weak Entropies and

. . . . . . . . .

90

8.3

The Case of γ > 3 . . . . . . . . . . . . . . . . . . . . .

98

8.4

The Case of 1 < γ ≤ 3 . . . . . . . . . . . . . . . . . . . 106

8.5

Application on Extended River Flow System . . . . . . 114

8.6

Related Results . . . . . . . . . . . . . . . . . . . . . . . 119

9 Two Special Systems of Euler Equations 9.1

121

9.3

Existence of Viscosity Solutions . . . . . . . . . . . . . . 125
ρ Lax Entropy for P (ρ) = 0 s2 es ds . . . . . . . . . . . . . 126
ρ Lax Entropy for P (ρ) = 0 s2 (s + d)γ−3 ds . . . . . . . . 130

9.4

Related Results . . . . . . . . . . . . . . . . . . . . . . . 134

9.2

10 General Euler Equations of Compressible Fluid Flow 137 10.1 Existence of Viscosity Solutions . . . . . . . . . . . . . . 140 10.2 Lax Entropy and Related Estimates . . . . . . . . . . . 141 10.3 Existence of Weak Solutions . . . . . . . . . . . . . . . . 144 10.4 Related Results . . . . . . . . . . . . . . . . . . . . . . . 145 11 Extended Systems of Elasticity

147

11.1 Existence of Viscosity Solutions . . . . . . . . . . . . . . 150 11.2 Entropy-Entropy Flux Pairs of Lax Type . . . . . . . . 152 11.3 Existence of Weak Solutions . . . . . . . . . . . . . . . . 153 11.4 Related Results . . . . . . . . . . . . . . . . . . . . . . . 157

CONTENTS

ix

12 Lp Case to Systems of Elasticity

159

12.1 Lin’s Proof for Artificial Viscosity . . . . . . . . . . . . . 160 12.2 Shearer’s Proof for Physical Viscosity 12.3 System of Adiabatic Gas Flow

. . . . . . . . . . 163

. . . . . . . . . . . . . . 165

12.4 Related Results . . . . . . . . . . . . . . . . . . . . . . . 173 13 Preliminaries in Relaxation Singularity

175

14 Stiff Relaxation and Dominant Diffusion

179

14.1 Compactness Results . . . . . . . . . . . . . . . . . . . . 180 14.2 Proof of Theorem 14.1.1 . . . . . . . . . . . . . . . . . . 183 14.3 Applications of Theorem 14.1.1 . . . . . . . . . . . . . . 186 14.4 Proof of Theorem 14.1.2 . . . . . . . . . . . . . . . . . . 192 14.5 Applications of Theorem 14.1.2 . . . . . . . . . . . . . . 196 14.6 Related Results . . . . . . . . . . . . . . . . . . . . . . . 202 15 Hyperbolic Systems with Stiff Relaxation

203

15.1 Relaxation Limits for 2 × 2 Systems . . . . . . . . . . . 206 15.2 System of Extended Traffic Flows . . . . . . . . . . . . . 213 15.3 Related Results . . . . . . . . . . . . . . . . . . . . . . . 215 16 Relaxation for 3 × 3 Systems

217

16.1 Dominant Diffusion and Stiff Relaxation . . . . . . . . . 219 16.2 A Model System for Reacting Flow . . . . . . . . . . . . 222 16.3 Related Results . . . . . . . . . . . . . . . . . . . . . . . 228 Bibliography

229

Index

239

Preface I planned to write this book when I visited Professor Alan Jeffrey in Newcastle, England, in 1992. Since then, I have given seminars, courses and lectures on the applications of the compensated compactness method to hyperbolic conservation laws in many different universities. Among these are Stanford University, USA (1994), Heidelberg University, Germany (1995), International School for Advanced Studies (SISSA), Italy (1996), Federal University of Rio de Janeiro, Brazil (1998), National University of Colombia, Colombia (2000) and University of Science and Technology of China, China (2001). This book is a result of a one-year course for graduate students in applied mathematics, but it can also be a textbook for undergraduate students in their last year. The students should be familiar with the basic contents of introductory courses such as functional analysis, measure theory, Sobolev space, shock waves theory and so on. I want to thank Professor Alan Jeffrey for his continuing support and encouragement. Without his kind help, this book would never have been written. I must also thank Mr. Ding-hao Li, my mathematics teacher in middle school, whose excellent character has always been a great influence in my life. Thanks also go to my mother and all the members of my family, for their interest and sense of pride in every achievement in my career and who are my power resources to do mathematics. Finally, I would like to thank Dr. Ben-jin Xuan, my former student, who helped me graph the figures and resolve all the technical problems in the Latex file during the last three months of my typing the manuscript. After I finished the process, he carefully read all the pages and proposed many valuable suggestions. xi

Chapter 1

Preliminary Systems of hyperbolic conservation laws are very important mathematical models for a variety of physical phenomena that appear in traffic flow, theory of elasticity, gas dynamics, fluid dynamics and so on. In general, the classical solution of the Cauchy problem for nonlinear hyperbolic conservation laws exists only locally in time even if the initial data are small and smooth. This means that shock waves always appear in the solution for a suitable large time. Since the solution is discontinuous and does not satisfy the given partial differential equations in the classical sense, we have to study the generalized solutions, or functions which satisfy the equations in the sense of distributions. We consider the quasi-linear systems of the form (x, t) ∈ R × R+ ,

ut + f (u)x = 0,

(1.0.1)

where u = (u1 , u2 , ..., un )T ∈ Rn , n ≥ 1 is the unknown vector function standing for the density of physical quantities, f (u) = (f1 (u), ..., fn (u))T is a given vector function denoting the conservative term. These equations are commonly called conservation laws. Let us suppose for the moment, that u is a classical solution of (1.0.1) with the initial data u(x, 0) = u0 (x).

(1.0.2)

Let C01 be the class of C 1 function φ which vanishes outside of a compact subset. We multiply (1.0.1) by φ and integrate by parts over t > 0, to get


(uφt + f (u)φx )dxdt + u0 φdx = 0. (1.0.3) t=0

t>0

1

CHAPTER 1. PRELIMINARY

2

Definition 1.0.1 An Lp , 1 < p ≤ ∞, bounded function u(x, t) is called a weak solution of the initial-value problem (1.0.1) with Lp bounded initial data u0 , provided that (1.0.3) holds for all φ ∈ C01 (R × R+ ). An important aspect of the theory of nonlinear system of conservation laws is the question of existence of solutions to these equations. It helps to answer the question if the modelling of the natural phenomena at hand has been done correctly, and if the problem is well posed. To get a global weak solution or a generalized solution for given hyperbolic conservation laws, a standard method is to add a small parabolic perturbation term to the right-hand side of (1.0.1): ut + f (u)x = εuxx ,

(1.0.4)

where ε > 0 is a constant. We may first get a sequence of solutions {uε } of the Cauchy problem (1.0.4),(1.0.2) for any fixed ε by the following general theorem for parabolic equations: Theorem 1.0.2 (1) For any fixed ε > 0, the Cauchy problem (1.0.4) with the bounded measurable initial data (1.0.2) always has a local smooth solution uε (x, t) ∈ C ∞ (R × (0, τ )) for a small time τ , which depends only on the L∞ norm of the initial data u0 (x). (2) If the solution uε has an a priori L∞ estimate |uε (·, t)|L∞ ≤ M (ε, T ) for any t ∈ [0, T ], then the solution exists on R × [0, T ]. (3) The solution uε satisfies: lim uε = 0,

|x|→∞

if

lim u0 (x) = 0.

|x|→∞

(4) Particularly, if one of the equations in system (1.0.4) is in the form wt + (wg(u))x = εwxx ,

(1.0.5)

where g(u) is a continuous function of u ∈ Rn , then wε ≥ c(t, c0 , ε) > 0,

if

w0 (x) ≥ c0 > 0,

(1.0.6)

where c0 is a positive constant and c(t, c0 , ε) could tend to zero as the time t tends to infinity or ε tends to zero.

3 Proof. The local existence result in (1) can be easily obtained by applying the contraction mapping principle to an integral representation for a solution, following the standard theory of semilinear parabolic systems. Whenever we have an a priori L∞ estimate of the local solution, it is clear that the local time τ can be extended to T step by step since the step time depends only on the L∞ norm. The process to get the local solution clearly shows the behavior of the solution in (3). The details about the proofs of (1)-(3) in Theorem 1.0.2 can be seen in [LSU, Sm]. The following is the unpublished proof of (1.0.6) by Bereux and Sainsaulieu (cf. [Lu9, Pe]). We rewrite Equation (1.0.5) as follows: vt + g(u)vx + g(u)x = ε(vxx + vx2 ),

(1.0.7)

where v = log w. Then vt = εvxx + ε(vx −

g(u) 2 g2 (u) ) − g(u)x − . 2ε 4ε

(1.0.8)

The solution v of (1.0.8) with initial data v0 (x) = log(w0 (x)) can be 2 1 exp(− (x−y) represented by a Green function G(x − y, t) = √πεt 4εt ):
v=

∞

G(x − y, t)v0 (y)dy  ∞  g(u) 2 g2 (u) ) − − g(u)x G(x − y, t − s)dyds. ε(vx − 2ε 4ε −∞ (1.0.9)


−∞ t


+ 0

Since


∞ −∞


G(x − y, t)dy = 1,

∞

M |Gy (x − y, t)|dy ≤ √ , εt −∞

CHAPTER 1. PRELIMINARY

4

it follows from (1.0.9) that
∞ G(x − y, t)v0 (y)dy v ≥ −∞
t
∞ g2 (u) − g(u)x )G(x − y, t − s)dyds (− + 4ε
0 −∞ ∞

G(x − y, t)v0 (y)dy   g2 (u) G(x − y, t − s) dyds g(u)Gy (x − y, t − s) − + 4ε 0 −∞ 1 M t M1 t 2 − ≥ −C(t, c0 , ε) > −∞. ≥ log c0 − 1 ε ε2 (1.0.10) =

−∞
t
∞

Thus wε has a positive lower bound c(t, c0 , ε) for any fixed ε and t < ∞.

The solution obtained in Theorem 1.0.2 is called viscosity solution. After we have the sequence of viscosity solutions {uε }, ε > 0, if we furthermore suppose that {uε } are uniformly bounded in Lp (1 < p ≤ ∞) space with respect to the parameter ε, then there exists a subsequence (still labelled) {uε } such that uε (x, t)  u(x, t),

weakly in Lp ,

(1.0.11)

and also a subsequence {f (uε )} such that f (uε (x, t))  l(x, t),

weakly

(1.0.12)

under suitable growth conditions on f (u). If l(x, t) = f (u(x, t)),

a.e.,

(1.0.13)

then clearly u(x, t) is a weak solution of system (1.0.1) with the initial data (1.0.2) by letting ε tend to zero in (1.0.4). How could we obtain the weak continuity (1.0.13) of the nonlinear flux function f (u) with respect to the sequence of viscosity solutions {uε }? The theory of compensated compactness is just to answer this question. Why is this theory called Compensated Compactness? Roughly speaking, this term comes from the following fact:

5 If a sequence of functions satisfies wε (x, t)  w(x, t)

(1.0.14)

with either (wε )2 + (wε )3  w2 + w3 or (wε )2 − (wε )3  w2 − w3

(1.0.15)

weakly as ε tends to zero, in general, wε (x, t) is not compact. However, it is clear that any one weak compactness in (1.0.15) can compensate for another to make the compactness of wε . In fact, if we add them together, we get (wε )2  w2

(1.0.16)

weakly as ε tends to zero, which combining with (1.0.14) implies the compactness of wε . In this book, our goal is to introduce some applications of the method of compensated compactness to the scalar conservation law as well as some special systems of two or three equations. Moreover, applications to some physical systems with a relaxation perturbation parameter are also considered. The arrangement of this book is as follows: In Chapter 2, we introduce some elemental theorems in the theory of compensated compactness. Section 2.1 is about the weak continuity theorems of 2× 2 determinants, and the proofs come from [Ta]. Section 2.2 is about the Young measure representation theorems of weak limits and we use the proofs in [Lin]. Section 2.3 is about the Murat compact embedding theorems. In this part, we introduce two theorems. The proof of Theorem 2.3.2 is the same as that given in [DCL1] and the proof of Theorem 2.3.4 is copied from the French paper by Murat [Mu]. It is necessary to point out that Theorem 2.3.4 is independent of this book and the readers could pass over it without considering the details. We collect it here because it was used in some research papers (cf. [CLL, JPP]). In Chapter 3, we consider the Cauchy problem of the scalar equation with L∞ and Lp (1 < p < ∞) initial data, respectively. In this part, a simplified proof (cf. [CL1, Lu1]) without using the Young measure is given.

6

CHAPTER 1. PRELIMINARY

In the first part of Chapter 4, we introduce some basic definitions of systems of two equations, such as the strict hyperbolicity, genuine nonlinearity, linear degeneration, Riemann invariants, entropy-entropy flux pair and so on (cf. [La2, La3, Sm]). In the second part, a framework to obtain L∞ estimates of viscosity solutions for systems of two equations, called the Invariant Region Theory from [CCS], is introduced. In Chapter 5, we consider a special symmetric system of two equations ([Ch3]). This system is very similar to the scalar equation because one characteristic field is always linearly degenerate, although the other field is genuinely nonlinear. This system is of interest because along the genuinely nonlinear characteristic field, the compactness of viscosity solutions is obtained in L∞ space without any more regular condition. However, along the linearly degenerate characteristic field, some more regular conditions, such as BV estimates, must be added to ensure the strong compactness of the sequence of viscosity solutions. In Chapter 6, we consider a system of two equations with quadratic flux. This system is nonstrictly hyperbolic at the original point, one characteristic field is linearly degenerate on the positive half axis of u, while the other field is linearly degenerate on the negative half axis of u. Its entropy equation is the same as that of the system of polytropic gas dynamics with the adiabatic exponent γ = 2. The main difficulty in studying this system by the compensated compactness is that the entropy-entropy flux pairs are singular at the original point. Through a careful construction of exact solutions of the classical Fuchsian equation, we obtain the explicit entropy-entropy flux pair of Lax type for this system. Then the necessary estimates for the major terms of these entropies follow from the analysis of solutions of the Fuchsian equation (cf. [Lu4]). In Chapter 7, we extend the method given in Chapter 6 to the Le Roux system, which is also nonstrictly hyperbolic at the original point, but the entropy equation is the same as that of γ = 5/3 for the polytropic gas. This system is of interest because it is a typical system of Temple type, whose characteristic fields are both straight lines. The proof in this chapter is from [LMR]. In Chapter 8, we consider the most typical hyperbolic conservation system of two equations, the so-called system of the polytropic gas dynamics (or γ-law). For the case of γ > 3, our proof is copied from

7 the paper [LPT]. For the case of 1 < γ ≤ 3, using only four pairs of weak entropy-entropy flux, we give a short proof by assuming that the solution is away from vacuum and small (cf. [CL2]). In Chapter 9, the methods in Chapters 6 and 7 are again extended to study two special systems of one-dimensional Euler equations, which are nonstrictly hyperbolic on the vacuum line ρ = 0. For smooth solutions, they are equivalent to the systems of polytropic gas dynamics with the adiabatic exponents 3 < γ < ∞ and γ = ∞, respectively. Our proofs in this chapter come from [Lu2] and [Lu8]. In Chapter 10, we consider the general Euler equations of onedimensional, compressible fluid flow. This more general system is again nonstrictly hyperbolic on the vacuum line ρ = 0. To study this system by using the compensated compactness, one basic difficulty is how to construct entropy-entropy flux pairs and obtain the necessary estimates on these entropies. Since the method to construct entropy-entropy flux pairs of Lax type (cf. [La1]) to strictly hyperbolic systems does not work here, in this chapter we extend DiPerna’s method to nonstrictly hyperbolic systems. We introduce a special form of Lax entropy, in which the progression terms are functions of a single variable. The necessary estimates for the major terms are obtained by the singular perturbation theory of the ordinary differential equations of second order. The proof in this chapter comes from [Lu6]. In Chapter 11, we extend the method given in Chapter 10 to study some extended systems of elasticity in L∞ space. The proof is also from [Lu6]. In Chapter 12, some important results about Lp , 1 < p < ∞, weak solutions for the system of elasticity are introduced, which include a compactness framework of artificial viscosity solutions to this system by Lin [Lin] and a compactness framework of physical viscosity by Shearer [Sh]. An application of the latter compactness framework by Shearer on the system of adiabatic gas flow through porous media is also considered (cf. [LK1]). However, to avoid knotty mathematical formulas, we choose not to provide the proofs of these two compactness frameworks in this book, although they are very important and form a basis on relaxation problems of hyperbolic systems of three equations in Chapter 16. From Chapter 13 to Chapter 16, we introduce some applications of the compensated compactness on the relaxation problems.

8

CHAPTER 1. PRELIMINARY

In Chapter 13, a general description of the relaxation singular problem is introduced. In Chapter 14, singular limits of stiff relaxation and dominant diffusion for general 2 × 2 nonlinear systems of conservation laws are considered. These include the L∞ solutions of the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates and the extended models of traffic flows; the Lp , 1 < p < ∞, solutions for some physical models, without L∞ bounded estimates, such as the system of isentropic fluid dynamics in Lagrangian coordinates, and the models of traffic flows in different states. All proofs in this chapter can be found in [Lu9]. In Chapter 15, a framework for the singular limits of stiff relaxation for general 2×2 hyperbolic conservation systems (not necessary strictly hyperbolic) is introduced (cf. [CLL], [Lu10]). An application of this framework on a nonstrictly hyperbolic system, the so-called system of extended traffic flow, is also obtained. The proof is from [Lu10]. In Chapter 16, singular limits of stiff relaxation and dominant diffusion for the general 3×3 system of chemical reaction are considered (cf. [Lu12]). The pure relaxation limit (without viscosity) for a special case of this chemical reaction system is also introduced (cf. [LK1, LK2, Tz]).

Chapter 2

Theory of Compensated Compactness As a theory, the compensated compactness is a large subject. However, until now, all the applications on hyperbolic conservation laws were related to the theorems given in this chapter.

2.1

Weak Continuity of a 2 × 2 Determinant

Theorem 2.1.1 Let H −1 (Ω) be the dual of H01 (Ω) and Ω ⊂ RN be an open, bounded set. Suppose (H1 ) uε = (uε1 , uε2 , · · · , uεp )
u = (u1 , u2 , · · · , up ) weakly in L2 (Ω), p
N ∂uεj
−1 aijk are compact in the strong topology of Hloc , (H2 ) ∂xk j=1 k=1 where i = 1, 2, ...q. Then if Q = Q(λ), λ ∈ Rp is quadratic and satisfies Q(λ) ≥ 0 for all λ ∈ ∧, where p

p  N 

N

∧ = {λ ∈ R : ∃ξ ∈ R − {0}, s.t.

aijk λj ξk = 0

i = 1, 2, ...q};

j=1 k=1

(2.1.1) if Q(uε )
l in the sense of distributions (l may be a measure), then l ≥ Q(u)

in the sense of distributions. 9

(2.1.2)

10 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS Proof. Step 1: We make a translation viε = uεi − ui . Then (H1 ) and (H2 ) imply that vjε
0 weakly in L2 (Ω) for j = 1, 2, ...p, ∂vjε
−1 aijk are compact in the strong topology of Hloc for (2) ∂x k j,k i = 1, 2, ...q. (1)

Since Q is quadratic, there exists a bilinear form q(a, b) such that Q(a) = q(a, a). Therefore Q(v ε ) = Q(uε − u) = Q(uε ) − 2q(uε , u) + Q(u). But Q(uε )
l,

q(uε , u)
q(u, u)

weakly,

the second statement holding because q(a, b) is linear in a for fixed b. Therefore since q(u, u) = Q(u), Q(v ε )
l − Q(u),

weakly.

Step 2: Next we perform a localization as follows. We let wε = φv ε

with φ ∈ C0∞ (Ω).

Then supp wε ⊂ compact set of RN

(2.1.3)

wε
0 weakly in L2 (Ω).

(2.1.4)

and

Moreover  j,k

aijk

 ∂wjε ∂vjε  ∂φ =φ aijk + aijk vjε , ∂xk ∂xk ∂xk j,k

j,k

2.1. WEAK CONTINUITY OF A 2 × 2 DETERMINANT

11

and the first term in the right-hand side belongs to a compact set of H −1 (Ω). Therefore 

aijk

j,k

∂wjε ∈ compact set of H −1 (Ω). ∂xk

Hence extracting a possible subsequence we have lim



ε→0

aijk

j,k

∂wjε ∂xk

= 0 strongly in H −1 (Ω)

(2.1.5)

and Q(wε )
φ2 (l − Q(u))

weakly.

Step 3: To prove l − Q(u) ≥ 0, it is enough to show that  Q(wε )dx ≥ 0, lim ε→0

(2.1.6)

since this will prove that   φ2 (l − Q(u))dx ≥ 0 lim Q(φv ε )dx ≥ 0 ⇐⇒ ε→0

for all φ ∈ C0∞ (Ω), implying that l − Q(u) ≥ 0.

Step 4: Let us define the Fourier transformation of wjε as w jε

=

F (wjε )

 RN

wjε (x)e−2πi(ξ·x) dx.

The Plancherel formula gives   v(x)w(x)dx ¯ = RN

RN

¯ v(ξ)w(ξ)dξ,

(2.1.7)

(2.1.8)

where v and w ∈ L2 (RN ) are complex valued functions. We extend Q from Rp to Cp into an Hermitian form. Recall that Q is quadratic and takes the form  qjk λj λk Q(λ) = j,k

12 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS with real coefficients qjk = qkj . We define   qjk λj λk . Q(λ) = j,k

Hence we have  Re (Q(λ)) ≥0

if λ ∈ ∧ + i ∧ .

(2.1.9)

In fact, if λ = λ1 + iλ2 with λ1 , λ2 ∈ ∧, then  Q(λ) = (Q(λ1 ) + Q(λ2 )) + i(q(λ1 , λ2 ) + q(λ2 , λ1 )), where q was defined by Q(a) = q(a, a), and therefore  Re (Q(λ)) = Q(λ1 ) + Q(λ2 ) ≥ 0. By the Plancherel formula again,    ε ε  Q(w )dx = Q(w  )dξ = RN

RN

RN

 w Re Q( ε )dξ.

So (2.1.6) is equivalent to  lim

ε→0

RN

 w Re Q( ε )dξ ≥ 0.

(2.1.10)

But, since supp wjε ⊂ fixed compact set C of RN , we have  ε wjε (x)e−2πi(ξ·x) dx. w j (ξ) = C

But e−2πi(ξ·x) ∈ L2 (C) and since wjε
0 weakly in L2 (RN ), we deduce that jε (ξ) = 0 lim w

ε→0

strongly for all ξ,

and w jε (ξ) ≤ M.

(2.1.11)

Therefore jε = 0 locally in L2 (RN ). lim w

ε→0

Hence

 lim

ε→0 |ξ|≤r

 w Q( ε )dξ = 0

(2.1.12)

(2.1.13)

2.1. WEAK CONTINUITY OF A 2 × 2 DETERMINANT

13

for any fixed constant r > 0. Using the Fourier transformation of (2.1.5) leads to 1  aijk w jε (ξ)ξk = 0 strongly in L2 (RN ) for i = 1, ..., q. ε→0 1 + |ξ| lim

j,k

(2.1.14) Using the following lemma, we can finish the proof of Theorem 2.1.1. Lemma 2.1.2 Suppose that Q(λ) ≥ 0 for all λ ∈ ∧. Then for all α > 0, there exists a constant Cα such that  Re Q(λ) ≥ −α|λ|2 − Cα

q   2  aijk λj ηk  i=1

(2.1.15)

j,k

for all λ ∈ Cp , and for all η ∈ RN with |η| = 1. In fact, it follows from (2.1.14) that 1  aijk w jε (ξ)ξk = 0 strongly in L2 ({|ξ| ≥ 1}). ε→0 |ξ| lim

(2.1.16)

j,k

Using Lemma 2.1.2 and taking λ = w ε (ξ) and η = ξ/|ξ|, we have  w ε (ξ)|2 − Cα Re Q( ε (ξ)) ≥ −α|w

q  2    aijk w jε (ξ)ξk /|ξ| . (2.1.17)  i=1

Integrating this on |ξ| ≥ 1, we get   ε  Re Q(w  (ξ))dξ ≥ −α |ξ|≥1

 −Cα

j,k

|ξ|≥1

|w ε (ξ)|2 dξ

q  2    aijk w jε (ξ)ξk /|ξ| dξ. 

|ξ|≥1 i=1

j,k

But by (2.1.11) and (2.1.16) we deduce that   w Re Q( ε (ξ))dξ ≥ −αM lim ε→0 |ξ|≥1

(2.1.18)

(2.1.19)

14 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS for all α as small as desired. Hence   w Re Q( ε (ξ))dξ ≥ 0. lim ε→0

(2.1.20)

|ξ|≥1

Combining (2.1.13) with (2.1.20) gives the proof of ( 2.1.10), and hence (2.1.6), which implies the proof of Theorem 2.1.1. Proof of Lemma 2.1.2. To prove Lemma 2.1.2, we proceed by contradiction. Suppose there exists an α > 0 such that for all Cα = n there exist λn ∈ Cp with |λn | = 1 and η n ∈ RN with |η n | = 1 such that  n ) < −α|λn |2 − n Re Q(λ

q   2  aijk λn η n  . i=1

j k

(2.1.21)

j,k

Extract convergent subsequences such that λn → λ∞ and η n → η ∞ . Then q   2 C  aijk λnj ηkn  ≤ , n i=1

j,k

where C is a constant. Hence, passing to the limit, we deduce that  ∞ aijk λ∞ j ηk = 0, i = 1, · · · , q. j,k

Therefore λ∞ ∈ ∧ + i∧, and hence  ∞ ) ≥ 0. Re Q(λ But from (2.1.21) we have also  n ) ≤ −α < 0  ∞ ) = lim Re Q(λ Re Q(λ n→∞

and this is a contradiction. Corollary 2.1.3 If Q is quadratic and satisfies Q(λ) = 0 for all λ ∈ ∧, and if {uε } satisfies (H1 ) and (H2 ), then Q(uε )
Q(u) in the sense of distributions.

2.1. WEAK CONTINUITY OF A 2 × 2 DETERMINANT

15

Proof. Extract a subsequence such that Q(uε )
l, l may be a measure. Applying Theorem 2.1.1 to Q and then to −Q, we obtain l = Q(u) and hence the proof of Corollary 2.1.3 since this is true for all subsequences.

Theorem 2.1.4 (weak continuity theorem of a 2 × 2 determinant) Let Ω ⊂ R × R+ be a bounded open set and uε : Ω → R4 be measurable functions satisfying uε
u, in (L2 (Ω))4

(2.1.22)

and ∂uε1 ∂uε2 + , ∂t ∂x

∂uε3 ∂uε4 −1 + are compact in Hloc (Ω). ∂t ∂x

(2.1.23)

Then there exists a subsequence (still labelled) {uε } such that     ε   u1 uε2  
u1 u2  in the sense of distributions.  u3 u4  uε uε  3 4

(2.1.24)

Proof. From the conditions in (2.1.23), the set ∧ in Theorem 2.1.1 consists of all points λ ∈ L4 satisfying λ1 ξ1 + λ2 ξ2 = 0,

λ3 ξ1 + λ4 ξ2 = 0

for any ξ ∈ R × R+ − {0}, or equivalently ∧ = {λ ∈ L4 : λ1 λ4 − λ2 λ3 = 0}.

(2.1.25)

Let Q(λ) in Theorem 2.1.1 be λ1 λ4 − λ2 λ3 . Then clearly Q(λ) = 0 for all λ ∈ ∧. So (2.1.24) follows from Corollary 2.1.3. This completes the proof of Theorem 2.1.4.

16 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS

2.2

Young Measure Representation Theorems

Theorem 2.2.1 Let Ω ⊂ RN be measurable. Suppose that {un (x)} is a sequence of measurable functions from Ω to RS . Then there exists a subsequence {unk } of {un (x)} and a family of positive measures νx ∈ M(RS ), depending measurably on x ∈ Ω, such that for any f ∈ C0 (RS ),   nk f (λ)dνx (λ), (2.2.1) w − lim f (u ) =< f (λ), νx (λ) > RS

where M(RS ) is the dual space of C0 (RS ) and C0 (RS ) is the space of continuous functions which tend to zero at infinity, and w − lim denotes the weak-star limit in the L∞ space. Furthermore, if the range of un (x) is contained in G ⊂ RS , then so is the support of νx ; if the L∞ norm of un (x) is uniformly bounded or G is compact in RS , then νx are probability measures, i.e., the mass of νx is one. Proof. Let E = {f m } be a dense set in C0 (RS ). Then {f 1 (un )} is 1 bounded on Ω, and hence there exist a subsequence {unk } of {un } and (α(f 1 )(x) ∈ L∞ (Ω) such that 1

w − lim f 1 (unk ) = α(f 1 )(x).

(2.2.2)

1

Furthermore, {f 2 (unk )} is also bounded on Ω, and hence there exist a 2 1 subsequence {unk } of {unk } and α(f 2 )(x) ∈ L∞ (Ω) such that 2

w − lim f 2 (unk ) = α(f 2 )(x).

(2.2.3) nm k

Proceeding in this way we obtain a subsequence {u }, α(f m )(x) such that 1 2 3 (i) {unk } ⊃ {unk } ⊃ {unk } ⊃ · · · , and m (ii) for each fixed m, w − lim f m (unk ) = α(f m )(x). k We let {unk } = {unk }, the diagonal sequence. Then from (ii) we get that for each fixed m, w − lim f m(unk ) = α(f m )(x).

(2.2.4)

For each f m ∈ E, we define a bounded functional I(f m ) on L1 (Ω) by   ψα(f m )dx = lim ψf m (unk )dx, ∀ψ ∈ L1 (Ω). < I(f m ), ψ >= Ω

k→∞ Ω

(2.2.5)

2.2. MEASURE REPRESENTATION THEOREMS

17

Then for any given f ∈ C0 (RS ), suppose that f = lim f l in C0 (RS ), l→∞

where {f l } ⊂ E. We now prove that the following limit exists, and hence denote it by I(f ), namely,  ψf (unk )dx, ∀ψ ∈ L1 (Ω). (2.2.6) < I(f ), ψ >= lim k→∞ Ω

In fact, for any unk1 , unk2 , we notice that      ψ[f (unk1 ) − f (unk2 )]dx Ω

        n k1 l n k1 ψ[f (unk2 ) − f l (unk2 )]dx ≤  ψ[f (u ) − f (u )]dx +  Ω

Ω

    + ψ[f l (unk1 ) − f l (unk2 )]dx Ω

≤ 2f −

f l C0 ψ1

    +  ψ[f l (unk1 ) − f l (unk2 )]dx. Ω

(2.2.7) We first choose l large enough such that the first term on the righthand side of (2.2.7) is small. Then by (2.2.4) the second term on the right-hand side of (2.2.7) can be  small whenever nk1 and nk2 are large enough. Hence we prove that {

Ω

ψf (unk )dx} is a Cauchy sequence for

any fixed ψ ∈ L∞ (Ω), and so we have proved (2.2.6). Consequently, we obtain | < I(f ), ψ > | ≤ f C0 ψ1 ,

∀ψ ∈ L1 (Ω).

(2.2.8)

We notice, by (2.2.8), that I(f ) is a bounded functional on L1 (Ω), and hence by the Riesz representation theorem there exists α(f )(x) ∈ L∞ (Ω) such that  α(f )ψdx, ∀ψ ∈ L1 (Ω). (2.2.9) < I(f ), ψ >= Ω

We also have α(f1 + f2 ) = α(f1 ) + α(f2 )

∀fi ∈ C0 (RS ),

i = 1, 2,

18 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS α(kf ) = kα(f )

∀f ∈ C0 (RS ),

k ∈ R.

At this moment we suppose, without loss of generality, that every point x ∈ Ω is a Lebesgue point of each function α(f ). Then for any fixed x0 ∈ Ω, we set ψ(x) = (meas Br (x0 ))−1 χBr (x0 ) , where Br (x0 ) is the ball centered at x0 with radius r, and χBr (x0 ) is the characteristic function of Br (x0 ). By (2.2.8) and (2.2.9) we get  −1 α(f )dx| ≤ f C0 . |(meas Br (x0 )) Br (x0 )

We now pass to the limit as r → 0 to obtain |α(f )(x0 )| ≤ f C0 . Combining this with the fact that α(f ) is linear with respect to f we have that α(f )(x0 ) is a bounded functional on C0 (RS ). Therefore, applying the Riesz representation theorem we have a νx0 ∈ M (RS ) such that  f (λ)dνx0 . α(f )(x0 ) =< f (λ), νx0 >= RS

Since x0 is arbitrary we get  ψ < f (λ), νx > dx < I(f ), ψ >= Ω

∀ψ ∈ L1 (Ω),

M (RS )

for almost all x ∈ Ω. So we have proved (2.2.1). where νx ∈ Furthermore, we notice that for any positive f ∈ C0 (RS ), α(f )(x) =< f (λ), νx > ≥ 0 a.e. x ∈ Ω, which implies that νx is positive for almost all x ∈ Ω. It is obvious from (2.2.1) that the support of ν is the same as the range of un (x). In fact, we can choose any function f ∈ C0 (RS ) satisfying supp f ⊂ RS − {G}. Then clearly   nk f (λ)dνx (λ), 0 = w − lim f (u ) = RS

which implies that the support of ν is contained in G. Finally, if the L∞ norm of un (x) is uniformly  bounded or G is S compact in R , then we choose f ≡ 1 and get G dνx (λ) = 1 from (2.2.1). Thus the mass of νx is one, which shows that νx are probability measures. This completes the proof of Theorem 2.2.1.

2.2. MEASURE REPRESENTATION THEOREMS

19

Corollary 2.2.2 Suppose that {un (x)} is bounded in Lploc (RN ; RS ), where 1 ≤ p < ∞. Then there exist a subsequence {unk } of {un } and a family of positive measures νx ∈ M (RS ), x ∈ RN , such that for any bounded set A ⊂ RN , w − lim f (unk ) =< f (λ), νx > in L1 (A),

(2.2.10)

whenever f ∈ C(RS ) satisfies f (λ) = 0. |λ|→∞ |λ|p

(2.2.11)

lim

Proof. Without loss of generality we assume that f ≥ 0. Then we define f m ∈ C0 (RS ) by f m = θ m f , where θ m ∈ C0 (RS ) is defined by   for |λ| ≤ m, 1 θ m (λ) = 1 + m − |λ| for m ≤ |λ| ≤ m + 1,   0 for |λ| ≥ m + 1. We claim that for each φ ∈ L∞ (A),   m n φf (u )dx = φf (un )dx lim m→∞ A

(2.2.12)

A

uniformly in n. Indeed,      φ[f m (un ) − f (un )]dx ≤ φL∞ (A) 

{x∈A:|un |≥m}

A

f (un )dx

≤ φL∞ (A) uLp (A) max { |λ|≥m

f (λ) }, |λ|p (2.2.13)

which tends to 0 uniformly in n as m → ∞. On the other hand, by Theorem 2.2.1 there exist a subsequence of {un } and a family of positive measures νx ∈ M (RS ) such that for each m,   m nk φf (u )dx = φ < f m , νx > dx ∀φ ∈ L∞ (A). (2.2.14) lim {unk }

n→∞ A

A

20 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS Furthermore, from the monotone convergence theorem we get   φ < f m , νx > dx = φ < f, νx > dx. (2.2.15) lim n→∞ A

A

Combining (2.2.12), (2.2.14) and (2.2.15), we get (2.2.10) and complete the proof of Corollary 2.2.2. Theorem 2.2.3 Let 1 < p < ∞ and suppose that the support of νx in Corollary 2.2.2 is a point for a.e. x ∈ RN . Then there exists a subsequence {unk } of {un } which converges strongly to u in Lqloc (RN ) for 1 ≤ q < p. Furthermore, νx = δu(x) for a.e. x ∈ RN . Proof. From Corollary 2.2.2 and the assumption that the support of νx is a point, we have that νx = δv(x) for a.e. x ∈ RN and for some function v(x). Since 1 < p < ∞, we can let f (λ) = λi for i = 1, 2, · · · , S, respectively in Corollary 2.2.2, to obtain uni = g(un )
< f (λ), νx >=< λi , νx >= vi ,

(2.2.16)

i.e., un converges weakly to v and hence v = u because of the uniqueness of weak limit. Now define fi (λ) = |λi |q for any 1 ≤ q < p. Again using Corollary 2.2.2, we get for any φ ∈ C0 (RN ) supported on compact set K ∈ RN ,    n q φ(x)|ui | dx → φ(x) fi (λ)dνx (λ)dx K

K

 = K

RS

(2.2.17)

q

φ(x)|ui | dx

as n tends to ∞. Thus un → u strongly in Lqloc (RN ) for any 1 ≤ q < p, and this completes the proof of Theorem 2.2.3.

2.3

Embedding Theorems

In this section we shall establish a compactness embedding theorem which is a generalization of Murat Lemma [Mu] or just an interpolation inequality (cf. [Tr]). We first introduce the following lemma:

2.3. EMBEDDING THEOREMS

21

Lemma 2.3.1 Let Ω ⊂ RN be bounded and open, and f ∈ W −1,p (Ω) for 1 < p < ∞, supp f ⊂⊂ Ω. Let u be the solution of equation −∆u = f, in Ω, (2.3.1) u = 0, on ∂Ω. Then uW 1,p (Ω) ≤ Cf W −1,p (Ω) ,

(2.3.2)

where C is a positive constant independent of f . This lemma can be proved by the Lp -theory of elliptic equations (cf. [Si]). Theorem 2.3.2 Let Ω ⊂ RN be bounded and open, C be a compact −1,q −1,r (Ω), B be a bounded set in Wloc (Ω) for some constants set in Wloc q, p, r satisfying 1 < q ≤ p < r < ∞. Furthermore, let D ⊂ D(Ω) such −1,p (Ω) such that D ⊂ C ∩ B. Then there exists E, a compact set in Wloc that D ⊂ E. Proof. For any set D ⊂ C ∩ B, there exists a convergent subsequence {fk } ⊂ D such that fi W −1,r (Ω) ≤ M, loc

fi − fj W −1,q (Ω) → 0, loc

(i, j → ∞),

(2.3.3)

where M is independent of i. Furthermore, for any Ω1 ⊂⊂ Ω, we take a function φ ∈ C0∞ (Ω) satisfying φ|Ω1 ≡ 1 and define f¯i ≡ φfi . Then f¯i satisfies supp f¯i ⊂⊂ Ω and f¯i |Ω1 = fi |Ω1 . Now let ui be solutions of equations −∆ui = f¯i , in Ω, (2.3.4) ui = 0, on ∂Ω. Then ui W 1,q (Ω) ≤ Cf¯i W −1,q (Ω) ,

ui W 1,r (Ω) ≤ Cf¯i W −1,r (Ω) . (2.3.5)

Hence from (2.3.3) and (2.3.5), we have ui − uj W 1,q (Ω) ≤ Cf¯i − f¯j W −1,q (Ω) ≤ Cfi − fj W −1,q (Ω) → 0, loc

(i, j → ∞).

(2.3.6)

22 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS Substituting ui − uj into the interpolation inequality · vαW 1,q (Ω) , vW 1,p (Ω) ≤ v1−α W 1,r (Ω)

(2.3.7)

for a constant α ∈ (0, 1), we have · ui − uj α ui − uj W 1,p (Ω) ≤ ui − uj 1−α W 1,r (Ω) ≤ C1 ui − uj α

1,q (Ω) Wloc

→ 0,

1,q Wloc (Ω)

(i, j → ∞).

(2.3.8)

Therefore, fi − fj W −1,p (Ω1 ) = sup | < fi − fj , φ > | φ 1,p ≤1, supp φ⊂Ω1 w0 (Ω) | < f¯i − f¯j , φ > |f¯i − f¯j W −1,p (Ω) ≤ sup φ

≤1 1,p w (Ω) 0

= ui − uj W −1,p (Ω) ≤ ui − uj W 1,p (Ω1 ) → 0,

(2.3.9)

(i, j → ∞),

where p is the constant satisfying 1 1 + = 1. p p −1,p (Ω). So we get the proof This means that D is a compact set in Wloc of Theorem 2.3.2.

Lemma 2.3.3 Suppose that Ω ⊂ RN is a bounded domain with Lipschitz boundary ∂Ω. Then for any η > 0 small enough, 1 < p < q  < +∞, there exists a function ψ η ∈ D(Ω), such that for any ϕ ∈  W01,q (Ω), there holds (1 − ψ η )ϕW 1,p (Ω) ≤ ηϕW 1,q (Ω) . 0

0

(2.3.10)

Proof. Step 1: Partition of unity. Since Ω is bounded and ∂Ω is Lipschitz, there exists an open cov¯ such that ering β0 , β1 , · · · , βm of Ω, β0 ∩ ∂Ω = ∅, βj ∩ ∂Ω = ∅, (1 ≤ j ≤ m),

2.3. EMBEDDING THEOREMS

23

and for 1 ≤ j ≤ m, there exist an open ball Bj ⊂ RN and a bijective N = T (β ∩ Ω). Lipschitz mapping Tj : βj → Bj , such that Bj ∩ R+ j j N , (1 ≤ j ≤ m). Suppose Φ , · · · , Φ Let αj = βj ∩ Ω, Aj = Bj ∩ R+ 0 m is partition of unity associated with covering β0 , β1 , · · · , βm , i.e., Φi ∈ D(βi ), Φi ≥ 0, (0 ≤ i ≤ m), and

m 

¯ Φi = 1 on Ω.

i=0

Step 2: Construction of a function ψ¯η such that supp ψ¯η ⊂⊂ Ω. For any j(1 ≤ j ≤ m) and η > 0 small enough, define a function θjη on Aj (which depends only on xN ) as  0, if 0 ≤ xN ≤ η/2,     θjη = 1 (xN − η ), if η/2 ≤ xN ≤ η,  η 2    1, if η ≤ xN . Let ψ¯η = Φ0 +

m  (θjη ◦ Tj )Φj , j=1

where ◦ denotes the composition of two mappings. Then function ψ¯η is Lipschitz and supp ψ¯η ⊂⊂ Ω. Step 3: For any η > 0, there exist a function ψ η ∈ D(Ω) and a positive constant CN which only depends on N , such that for any  ϕ ∈ W01,q (Ω), (1 < p < q  < +∞), there holds (ψ η − ψ¯η )ϕW 1,p (Ω) ≤ CN ηϕW 1,q (Ω) . 0

0

(2.3.11)

In fact, let 1 < p < q  < +∞, define s as 1 1 1 + = , q s p hence p < s < +∞. For any η > 0, there exists a function ψ η ∈ D(Ω) such that ψ η − ψ¯η W 1,s (Ω) ≤ η. 0

(2.3.12)

24 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS 

Then for any ϕ ∈ W01,q (Ω), there holds (ψ η − ψ¯η )ϕW 1,p (Ω) ≤ CN

0

+

(ψ η N


k=1

N
∂ϕ − ψ¯η )ϕLp (Ω) + (ψ η − ψ¯η ) ∂x  p k L (Ω)

k=1  ∂(ψη −ψ¯η )  ∂xk ϕLp (Ω) .

From the H¨ older inequality, there hold (ψ η − ψ¯η )ϕLp (Ω) ≤ ψ η − ψ¯η Ls (Ω) ϕLq (Ω) ,

∂ϕ  p ≤ ψ η − ψ¯η Ls (Ω) ϕW 1,q (Ω) , k = 1, · · · , N (ψ η − ψ¯η ) ∂xk L (Ω) 0 and 

∂(ψ η − ψ¯η ) ϕLp (Ω) ≤ ψ η − ψ¯η W 1,s (Ω) ϕLq (Ω) , k = 1, · · · , N. 0 ∂xk

Thus, there holds (ψ η − ψ¯η )ϕW 1,p (Ω) ≤ CN ψ η − ψ¯η W 1,s (Ω) ϕW 1,q (Ω) . 0

0

0

(2.3.13)

(2.3.12) and (2.3.13) imply (2.3.11). Step 4: For any η > 0, there exists a positive constant CΩ , such  that for any ϕ ∈ W01,q (Ω), there holds (1 − ψ¯η )ϕW 1,p (Ω) ≤ CΩ η 1/s ϕW 1,q (Ω) . 0

0

(2.3.14)

m
(1 − θjη ◦ Tj )Φj , it suffices to show that for Since 1 − ψ¯η = j=1

any 1 ≤ j ≤ m, there exists a constant C > 0 such that for any  ϕ ∈ W01,q (Ω), there holds (1 − θjη ◦ Tj )Φj ϕW 1,p ≤ Cη 1/s ϕW 1,q (Ω) . 0

0

(2.3.15)

2.3. EMBEDDING THEOREMS

25 

For simplicity, we will drop the index j. For any ϕ ∈ W01,q (Ω), there holds (1 − θ η ◦ T )ΦϕW 1,p 0

≤ CN (1 − θ η ◦ T )ΦϕLp + +

N
k=1

N
k=1

 (1 − θ η ◦ T ) ∂(Φϕ) ∂xk Lp

 η ◦T )  ∂(1−θ ΦϕLp . ∂xk (2.3.16)

N is bi-Lipschitz, changing Since mapping T : α = β ∩ Ω → A = B ∩ R+ the variables of integration, there holds:  1/p   |1 − θ η ◦ T |p |Φϕ|p dx (1 − θ η ◦ T )ΦϕLp (α) = α  1/p  η p = |1 − θ | |(Φϕ) ◦ T −1 |p |Jac(T −1 )| dx A

1/p

≤ 1 − θ η Ls (A) (Φϕ) ◦ T −1 Lq (A) Jac(T −1 )L∞ (A) , where Jac(T −1 ) denotes the Jacobian of mapping T −1 . Let Aη = {x ∈ A | 0 < xN < η}. Then there holds 1 − θ η Ls (A) ≤ 1Ls (Aη ) = (meas Aη )1/s ≤ CA η 1/s . Thus, there holds (1 − θ η ◦ T )ΦϕLp (α) (2.3.17) ≤

CA η 1/s CT ΦϕLq (α)

≤

Cη 1/s ϕLq (Ω) ,

where C depends only on T and A. Similarly, for 1 ≤ k ≤ N , there holds  (1 − θ η ◦ T ) ∂(Φϕ) ∂xk Lp (α)

1/s ϕ   ≤ CA η 1/s CT  ∂(Φϕ) , ∂xk Lq (α) ≤ Cη W 1,q (Ω) 0

where C depends only on T, A and Φ. Finally, since θ η only depends on xN , there holds ∂θ η ∂TN ∂(θ η ◦ T ) = ◦T , ∂xN ∂xN ∂xN

(2.3.18)

26 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS  and ∂θ η ∂xN = 0 outside of Aη . Then from the H¨older inequality and changing the variables of integration, there holds 

∂(θ η ◦ T ) ΦϕLp (α) ∂xN p    ∂θ η  ∂T  1/p   N = ◦ T (Φϕ) ◦ T  |Jac(T −1 )| dx  Aη ∂xN ∂xN ≤

 ∂TN  ∂θ η Ls (A)  ◦ T L∞ (A) ∂xN ∂xN 1/p

·(Φϕ) ◦ T −1 Lq (Aη ) Jac(T −1 )L∞ (A) . From the definition of θ η , there holds 

2 2 ∂θ η  s ≤ (meas Aη )1/s ≤ CA η 1/s . ∂xN L (A) η η 

Note that for any g ∈ W01,q (A), there holds gLq (Aη ) ≤ η

∂g  q η . ∂xN L (A )

Let g = (Φϕ) ◦ T −1 . Then there holds (Φϕ) ◦ T −1 Lq (Aη ) N N

∂Tl−1 ∂(Φϕ) ∂(Φϕ) −1  ≤η  ◦ T Lq (A)  L∞ (A)   q ∂x ∂x ∂xl L (A) N l l=1 l=1

≤ ηCT

N
∂(Φϕ)   q . ∂xl L (α) l=1

Thus for 1 ≤ k ≤ N , there holds 

N
2 ∂(Φϕ) ∂(θ η ◦ T ) ΦϕLp (α) ≤ CA η 1/s CT η   q ∂xN η ∂xl L (α) l=1

(2.3.19)

≤ Cη 1/s ϕLq (Ω) , where C depends only on T, A and Φ. Then (2.3.16), (2.3.17), (2.3.18) and (2.3.19) imply (2.3.15), which finishes the proof of Lemma 2.3.3.

2.3. EMBEDDING THEOREMS

27

Theorem 2.3.4 (Murat theorem) Suppose that Ω ⊂ RN is an open domain, 1 < p < +∞, 1p + p1 = 1. If {f ε }ε>0 satisfies: f ε
f 0 weakly in W −1,p (Ω), as ε → 0,

(2.3.20)

f ε ≥ 0,

(2.3.21)

i.e., for any ϕ ∈ D(Ω), ϕ ≥ 0, there holds < f ε , ϕ >≤ 0,

(2.3.22) 

where < ·, · > denotes the pairing between W −1,p and W01,p . Then −1,q (Ω), as ε → 0, ∀ q < p. f ε → f 0 strongly in Wloc

(2.3.23)

If furthermore, Ω is bounded with Lipschitz boundary ∂Ω, then f ε → f 0 strongly in W −1,q (Ω), as ε → 0, ∀ q < p.

(2.3.24)

Proof. Step 1: For any K ⊂⊂ Ω, there exists CK such that for any ϕ ∈ D(Ω), supp ϕ ⊂ K, ε > 0, there holds | < f ε , ϕ > | ≤ CK ϕL∞ (Ω) .

(2.3.25)

In fact, suppose for any K ⊂⊂ Ω, there exists a function ΦK such that: ΦK ∈ D(Ω), ΦK ≡ 1 on K and ΦK ≥ 0 in Ω. Then for any ϕ ∈ D(Ω) with supp ϕ ⊂ K, there holds −ϕL∞ (Ω) ΦK (x) ≤ ϕ(x) ≤ ϕL∞ (Ω) ΦK (x), in Ω, i.e., ϕ(x) + ϕL∞ (Ω) ΦK (x), ϕL∞ (Ω) ΦK (x) − ϕ(x) ≥ 0, in Ω. From assumption (2.3.21) or (2.3.22), there hold < f ε , ϕ + ϕL∞ (Ω) ΦK > ≤ 0

(2.3.26)

< f ε , ϕL∞ (Ω) ΦK − ϕ > ≤ 0.

(2.3.27)

and

28 CHAPTER 2. THEORY OF COMPENSATED COMPACTNESS Since {f ε }ε>0 is weakly convergent in W −1,p (Ω), hence {f ε }ε>0 is bounded in W −1,p (Ω), (2.3.26) and (2.3.27) imply | < f ε , ϕ > | ≤ < f ε , ΦK > ϕL∞ (Ω) ≤ CK ϕL∞ (Ω) . Step 2: Suppose Φ ∈ D(Ω) and Ω ⊂ Ω is a bounded domain, such that supp Φ ⊂ Ω . If Ω is bounded, choose Ω = Ω. From assumptions (2.3.20) and (2.3.21), (2.3.25) implies Φf ε
Φf 0 weakly in W −1,p (Ω), as ε → 0;

(2.3.28)

and for any ϕ ∈ D(Ω), ϕ ≥ 0, there holds | < Φf ε , ϕ > | ≤ M ϕL∞ (Ω) ,

(2.3.29)

where M is a positive constant independent of ϕ. ¯  ) denote the space of functions which are continuous on Let C0 (Ω  ¯  ) is a Banach ¯ Ω and vanish on ∂Ω , with norm  ·  ∞ ¯  . Then C0 (Ω L (Ω )

space, and (2.3.29) implies that ¯  )) , Φf ε is bounded in (C0 (Ω

(2.3.30)

¯  )) denotes the dual space of C0 (Ω ¯  ). where (C0 (Ω Step 3: From the Sobolev imbedding theorem, for any r > N , ¯  ) is compact. Then by the duality princiimbedding W01,r (Ω ) → C0 (Ω   ¯ )) → W −1,r (Ω ) is compact for 1 ≤ r  = r ≤ ple, imbedding (C0 (Ω (r−1) N N −1 .

Thus (2.3.30) implies that: for any 1 ≤ r  ≤

N N −1 ,

there holds



Φf ε → Φf 0 strongly in W −1,r (Ω ), as ε → 0.

(2.3.31)

Step 4: From the classical interpolation theorem, for 0 < θ < 1, 1 < θ p0 , p1 < +∞, 1q = 1−θ p0 + p1 , there holds   W −1,p0 (RN ), W −1,p1 (RN )

θ,q

= W −1,q (RN ).

Particularly, if ϕ ∈ W −1,p0 (RN ) ∩ W −1,p1 (RN ), then ϕ ∈ W −1,q (RN ) and ϕθW −1,p1 (RN ) . ϕW −1,q (RN ) ≤ Cϕ1−θ W −1,p0 (RN )

(2.3.32)

2.3. EMBEDDING THEOREMS

29

 Let p0 = p, p1 = r  for some 1 < r < N (N − 1) and extend Φf ε = 0 outside Ω . Then for all 1 < q < p, (2.3.32) implies Φf ε − Φf 0 W −1,q (RN ) ≤ CΦf ε − Φf 0 1−θ Φf ε − Φf 0 θW −1,p1 (RN ) . W −1,p0 (RN ) Then (2.3.31) and (2.3.28) imply (2.3.23). Step 5: For any η > 0 small enough, ψ η ∈ D(Ω) is the function obtained in Lemma 2.3.3, i.e., ψ η satisfies (2.3.10) and (1 − ψ η )f ε ∈ W −1,p (Ω). Then for any ϕ ∈ D(Ω), there holds | < (1 − ψ η )f ε , ϕ > | = | < f ε , (1 − ψ η )ϕ > | ≤ f ε W −1,p (Ω) (1 − ψ η )ϕW 1,p (Ω) 0

≤ ηf ε W −1,p (Ω) ϕW 1,q (Ω) , q  > p , 0

i.e.,

(1 − ψ η )f ε W −1,q (Ω) ≤ ηf ε W −1,p (Ω) .

For η > 0 fixed, from (2.3.23), there holds ψ η f ε → ψ η f 0 strongly in W −1,q (Ω ), as ε → 0, ∀ q < p. On the other hand, there exists decomposition f ε = (1 − ψ η )f ε + ψ η f ε , f 0 = (1 − ψ η )f 0 + ψ η f 0 . Let η → 0. Then there holds f ε → f 0 strongly in W −1,q (Ω), as ε → 0, ∀ q < p. This completes the proof of Theorem 2.3.4.

Chapter 3

Cauchy Problem for Scalar Equation In this chapter, we shall consider the Cauchy problem for the scalar conservation law (n = 1). The study of the single equation has a long history and the existence and uniqueness of the generalized solution for this equation has been well studied (cf. [Ho, Kr, La1, Ol] or the references cited in [Sm]). As the simplest model of hyperbolic conservation laws, Tartar first introduced the theory of compensated compactness to the scalar equation and succeeded in obtaining a new method, called the compensated compactness method, to study the global existence of generalized solutions for hyperbolic conservation laws. The original proof of Tartar is given in [Ta]. In this chapter, we use a simplified proof from [CL1, Lu1] to study both L∞ (Sec. 3.1) and Lp , 1 < p < ∞ (Sec. 3.2) solutions, in which two pairs of entropy-entropy flux ((3.1.4), (3.1.5)) and the weak continuity theorem of a 2× 2 determinant (Theorem 2.1.4) play a more important role, but the idea of Young measures (Theorem 2.2.1) has been avoided.

3.1

L∞ Solution

In this section, we shall give a simple proof of existence of global generalized solutions to the Cauchy problem of scalar conservation law:
ut + f (u)x = 0, (3.1.1) u(x, 0) = u0 (x), 31

32 CHAPTER 3. CAUCHY PROBLEM FOR SCALAR EQUATION where the real-valued function f is of class C 2 and the initial data u0 (x) is bounded measurable. Theorem 3.1.1 Let Ω ⊂ R×R+ be a bounded open set and uε : Ω → R be a sequence of functions such that w − lim uε = u,

w − lim f (uε ) = v,

(3.1.2)

where f ∈ C 2 (− ||u0 (x)||L∞ , ||u0 (x)||L∞ ). Suppose that ηi (uε )t + qi (uε )x

−1 lies in a compact set of Hloc (Ω), (i = 1, 2) (3.1.3)

where (η1 (θ), q1 (θ)) = (θ − k, f (θ) − f (k)) and

 (η2 (θ), q2 (θ)) = (f (θ) − f (k),

θ k

(f  (s))2 ds),

(3.1.4)

(3.1.5)

where k is an arbitrary constant. Then, (1) v = f (u), a.e., for any f ∈ C 2 and (2) uε → u, a.e., if meas {u : f  (u) = 0} = 0.

(3.1.6)

Proof. Using (3.1.3) and the weak continuity theorem of a 2 × 2 determinant (Theorem 2.1.4), we have     η1 (uε ) q1 (uε ) η1 (uε ) q1 (uε )      = (3.1.7) w − lim     ε ε η2 (u ) q2 (u ) η (uε ) q (uε ) 2 2 here, and hereafter the weak-star limit is denoted by w − lim η(uε ) = η(uε ) and w − lim q(uε ) = q(uε ). But since   η1 (uε ) q1 (uε )  uε    = uε − k  (f  (s))2 ds − (f (uε ) − f (k))2 ,   k η (uε ) q (uε ) 2 2

(3.1.8)

3.1. L∞ SOLUTION

33

and

  η1 (uε ) q1 (uε )       η2 (uε ) q2 (uε )  uε ε (f  (s))2 ds − (f (uε ) − f (u))2 = (u − u) u  uε  u +(u − k) (f  (s))2 ds + (uε − k) (f  (s))2 ds

(3.1.9)

k

u

−(f (u) − f (k))2 − 2(f (uε ) − f (u))(f (u) − f (k)),

then there exists a zero measure set Ω1 , for any (x, t) ∈ Ω − Ω1 , from (3.1.7)-(3.1.9), we have

 (uε

− u)

uε u

(f  (s))2 ds − (f (uε ) − f (u))2

 +(u − k)

uε u



2

(f (s)) ds +

 (uε

− k)

u k

(f  (s))2 ds

−(f (u) − f (k))2 − 2(f (uε ) − f (u))(f (u) − f (k)) (3.1.10)

 = uε − k

uε k

 =

uε

−k

uε k

(f  (s))2 ds − (f (uε ) − f (k))2 (f  (s))2 ds − (f (uε ) − f (u))2

−(f (u) − f (k))2 − 2(f (uε ) − f (u))(f (u) − f (k)).

34 CHAPTER 3. CAUCHY PROBLEM FOR SCALAR EQUATION Since  uε

−k

k

(uε

=

uε

(f  (s))2 ds

 − u)

uε u

 +(u − k)  = (u − k)

(f  (s))2 ds

uε u

uε

(f  (s))2 ds + uε − k 

2

(f (s)) ds +

u

−k

u

k

 uε

(3.1.11)



u k

(f  (s))2 ds

(f  (s))2 ds,

we have from (3.1.10) and (3.1.11) that  (uε

− u)

uε u

(f  (s))2 ds − (f (uε ) − f (u))2 + (f (uε ) − f (u))2 = 0. (3.1.12)

Since both terms in the left-hand side of (3.1.12) are nonnegative, we have  (uε

− u)

uε u

(f  (s))2 ds − (f (uε ) − f (u))2 = 0

(3.1.13)

and (f (uε ) − f (u))2 = 0.

(3.1.14)

So v = f (u) is obtained by (3.1.14). It follows from (3.1.13) that  lim

ε→0 Ω

ε



(u − u)

uε u

(f  (s))2 ds − (f (uε ) − f (u))2 dxdt = 0

(3.1.15)

and hence,  lim

ε→0 Ω(|uε −u|>α)

ε

(u − u)



uε u

(f  (s))2 ds − (f (uε ) − f (u))2 dxdt = 0. (3.1.16)

3.1. L∞ SOLUTION

35

Since d (θ − u) dθ



θ u



2

2

(f (s)) ds − (f (θ) − f (u))



 = u

θ

(f  (θ) − f  (s))2 ds, (3.1.17)

and if f  (u) = 0, a.e., then 

ε

Ω((uε −u)>α)



(u − u)

uε u

(f  (s))2 ds − (f (uε ) − f (u))2 dxdt

≥ Cα meas (Ω((uε − u) > α)) (3.1.18) and  Ω((uε −u)<−α)

(uε − u)



uε u

(f  (s))2 ds − (f (uε ) − f (u))2 dxdt

≥ Cα meas (Ω((uε − u) < −α)) (3.1.19) for a suitable positive constant Cα , which is independent of ε. Therefore for any given constant α, we have lim meas (Ω(|uε − u| > α)) = 0,

ε→0

(3.1.20)

which implies the pointwise convergence of a subsequence of uε on Ω.

Theorem 3.1.2 If u0 (x) ∈ L∞ , f ∈ C 2 (− ||u0 (x)||L∞ , ||u0 (x)||L∞ ), then the sequence of viscosity solutions {uε } uniquely defined by the Cauchy problem (uε )t + f (uε )x = ε(uε )xx

(3.1.21)

with bounded measurable initial data u(x, 0) = u0 (x) satisfies the compactness (3.1.3).

(3.1.22)

36 CHAPTER 3. CAUCHY PROBLEM FOR SCALAR EQUATION Proof. Since the initial data (3.1.22) is bounded in L∞ space, by the standard maximum principle of parabolic equations, the viscosity solutions uε have an a priori L∞ estimate ||uε (x, t)||L∞ ≤ ||u0 (x)||L∞ ,

(3.1.23)

which implies the existence of uε for t > 0 (see Theorem 1.0.2). Let K ⊂ R × R+ be an arbitrary compact set and choose φ ∈ C0∞ (R × R+ ) such that φK = 1, 0 ≤ φ ≤ 1.

Multiplying Equation (3.1.21) by uε φ and integrating over R × R+ , we obtain  ∞ ∞ (uεx )2 φdxdt ε 0

−∞



∞ ∞

1

−∞

2

= 0

ε 2

ε

ε

(u ) φt + (u f (u ) −



uε 0

f (s)ds)φx

(3.1.24)

 1 + ε(uε )2 φxx dxdt ≤ M (φ), 2 and hence that ε(uεx )2 is bounded in L1loc (R × R+ ).

(3.1.25)

For any entropy η ∈ C 2 , we have from Equation (3.1.21) that η(uε )t + q(uε )x = εη(uε )xx − εη  (uε )(uεx )2 = I1 + I2 ,

(3.1.26)

where q is the entropy flux corresponding to the entropy η. by (3.1.25), it is easy to see that I2 is bounded in L1loc (R × R+ ), and hence compact −1,α −1 , for some exponent α ∈ (1, 2); and I1 is compact in Hloc . in Wloc −1,∞ because of the The left-hand side of (3.1.26) is bounded in Wloc boundedness of uε in L∞ . So Theorem 3.1.2 is proved by the embedding Theorem 2.3.2. Combining Theorem 3.1.1 and Theorem 3.1.2, we get the following main theorem in this section: Theorem 3.1.3 If u0 (x) ∈ L∞ , f ∈ C 2 (− ||u0 (x)||L∞ , ||u0 (x)||L∞ ), then the Cauchy problem (3.1.1) has a weak solution u ∈ L∞ in the

3.2. LP SOLUTION, 1 < P < ∞

37

sense of Definition 1.0.1. Moreover, if f satisfies the condition (3.1.6), then the weak solution u is also the entropy solution in the sense of Lax, namely  ∞ ∞ η(u)φt + q(u)φx dxdt ≥ 0, (3.1.27) 0

−∞

where the C 2 function pair (η, q) satisfies q  = η  f  , η  ≥ 0 and φ ∈ C0∞ (R × R+ − {t = 0}) is a positive function.

3.2

Lp Solution, 1 < p < ∞

In this section, we consider the Cauchy problem (3.1.1) again, but the nonlinear flux function and the initial data satisfy the following conditions: (C1 ) f (u) ∈ C 2 , and satisfies |f (u)| ≤ C|u|s+1 ,

|f  (u)| ≤ C|u|s ,

(s ≥ 0).

(3.2.1)

(C2 ) u0 (x)2(s+1) ≤ M . Similar to Section 3.1, we introduce the following parabolic equation: uεt + f (uε )x = εuεxx

(3.2.2)

uε (x, 0) = u0 (x) ∗ Gε = uε0 (x),

(3.2.3)

with the initial data

where Gε is a mollifier such that uε0 (x) are smooth, lim uε0 (x) = 0,

|x|→∞

uε0 (x) → u0 (x) a.e., as ε → 0,

(3.2.4)

and uε0 (x)L∞ ≤ M (ε),

uε0 (x)2(s+1) ≤ u0 (x)2(s+1) ≤ M,

for a positive constant M (ε) depending on ε.

(3.2.5)

38 CHAPTER 3. CAUCHY PROBLEM FOR SCALAR EQUATION Theorem 3.2.1 Let C1 , C2 hold. Then for any fixed ε and the time T , there is a smooth solution uε of the Cauchy problem (3.2.2), (3.2.3) on R × (0, T ] such that uε ∈ C ∞ (R × (0, T ]), lim uε (x, t) = 0

||u(·, t)||2(s+1) ≤ M,

|x|→∞

(3.2.6)

uniformly for t ∈ [0, T ]. Proof. Since the initial data |uε0 (x)| ≤ M (ε), then the existence is obtained similarly to Theorem 3.1.2. The solutions uε (x, t) → 0 as |x| → ∞ can be also obtained by Theorem 1.0.2. To prove the uniform L2(s+1) bound in the first part of (3.2.6), multiplying Equation (3.2.2) by 2(s + 1)(uε )2s+1 and then integrating in R × R+ , we get  ∞ ∞  ∞ ε 2(s+1) (u ) dx + ε2(s + 1)(2s + 1)(uε )2s (uεx )2 dxdt −∞  0 −∞ ∞ (uε0 (x))2(s+1) dx ≤ M, ≤ −∞

(3.2.7) which completes the proof of Theorem 3.2.1. Now we study the L2(s+1) weak solution or generalized solution for the Cauchy problem (3.1.1). Lemma 3.2.2 Let C1 , C2 hold. Then the solutions of the Cauchy problem (3.2.2), (3.2.3) satisfy: 1

ε 2 ∂x uε are uniformly bounded in L2loc (R × (0, ∞)).

(3.2.8)

Proof. Similar to the proof of (3.1.25) in Theorem 3.1.2, let K ⊂ S ⊂ R × (0, ∞) and choose φ ∈ C0∞ (R × R+ ) such that φK = 1, 0 ≤ φ ≤ 1 and S = supp φ. Multiplying Equation (3.2.2) by uε φ and integrating over R × R+ , we obtain  ∞ ∞ (uεx )2 φdxdt ε 0

−∞



∞ ∞

1

−∞

2

= 0

ε 2

ε

ε

(u ) φt + (u f (u ) −

 0

uε

f (s)ds)φx

 1 ε 2 + ε(u ) φxx dxdt ≤ M (φ, ||uε ||2(s+1) ), 2

(3.2.9)

3.2. LP SOLUTION, 1 < P < ∞

39

which implies the proof of Lemma 3.2.2. Lemma 3.2.3 Let C1 , C2 hold. Then for any fixed n, ∂ ∂ In (uε ) + fn (uε ), ∂t ∂x

∂ ∂ fn (uε ) + Fn (uε ) ∂t ∂x

(3.2.10)

−1 lie in a compact set of Hloc (Ω), where Ω ⊂ R × R+ is any open and bounded set, and functions In , fn , Fn are defined as follows:

In (u) = u, if |u| ≤ n, In ∈ C 2 ,

In = 0, if |u| ≥ 2n,

|In (u)| ≤ |u|, 

fn (u) = and

u 0

 Fn (u) =

0

u

|In (u)| ≤ 2,

In (s)f  (s)ds

fn (s)f  (s)ds.

Proof. Using Lemma 3.2.2 and noticing the boundedness of these entropy pairs as n fixed, we can get the proof of Lemma 3.2.3 in a fashion similar to Theorem 3.1.2.

Lemma 3.2.4 Let C1 , C2 hold. If f satisfies the condition (3.1.6), then there exists a subsequence (still labelled) uε such that uε → u strongly in Lh for all h, 0 < h < 2(s + 1).

(3.2.11)

Proof. From the boundedness of uε in L2(s+1) , there exists a subsequence (still labelled) uε such that uε  u, weakly in L2(s+1) .

(3.2.12)

Let v ε express v ε  v ε in the sense of distributions. Then using the weak continuity theorem of a 2 × 2 determinant (Theorem 2.1.4), we have Hn (uε , k) = Hn (uε , k),

(3.2.13)

40 CHAPTER 3. CAUCHY PROBLEM FOR SCALAR EQUATION where k is an arbitrary constant and Hn (uε , k) = (Fn (uε ) − Fn (k))(In (uε ) − In (k)) − (fn (uε ) − fn (k))2 , (3.2.14) Hn (uε , k) = (Fn (uε ) − Fn (k))(In (uε ) − In (k)) − (fn (uε ) − fn (k))2 . (3.2.15) So, there exists a set Ω1 of measure zero, such that Hn (uε (x, t), u(y, τ ) = Hn (uε (x, t), u(y, τ )) for any (y, τ ) ∈ Ωc1 , (3.2.16) where Ωc1 is the complement of Ω1 in Ω. Thus Hn (uε (x, t), u(x, t))Hn (uε (x, t), u(x, t)),

(3.2.17)

if choosing (y, τ ) = (x, t) ∈ Ωc1 . Since  |Hn (uε (x, t), u(x, t))|dx Ω

 ≤C+C and

 Ω

Ω

(3.2.18) |uε (x, t)|2(s+1) + |u(x, t)|2(s+1) dx ≤ C1

|Hn (uε (x, t), u(x, t))|dx (3.2.19)

 ≤ C +C

ε

Ω

2(s+1)

|u (x, t)|

2(s+1)

+ |u(x, t)|

dx ≤ C1 ,

we have by Lebesgue’s dominant convergence theorem that lim Hn (uε , u) = −(f (uε ) − f (u))2 .

n→∞

(3.2.20)

So we have from (3.2.14) that  (uε − u)

uε u

(f  (s))2 ds − (f (uε ) − f (u))2 = −(f (uε ) − f (u))2 . (3.2.21)

3.3. RELATED RESULTS

41

To satisfy (uε − u)



uε u

(f  (s))2 ds − (f (uε ) − f (u))2 ≥ 0,

we have f (uε ) = f (u)

(3.2.22)

and  (uε

− u)

uε u

(f  (s))2 ds − (f (uε ) − f (u))2 = 0.

(3.2.23)

The left is exactly analogous to that of Theorem 3.1.1. Thus we get the proof of Lemma 3.2.4. Combining Lemmas 3.2.1-3.2.4, we have the following main theorem in this section: Theorem 3.2.5 Let C1 , C2 hold. If f satisfies meas {u : f  (u) = 0} = 0, then the Cauchy problem (3.1.1) has an L2(s+1) weak solution defined by Definition 1.0.1.

3.3

Related Results

After Tartar’s proof for scalar equation in L∞ space, Schonbek [SC] extended the compensated compactness method to study the strong convergence of the sequence of Lp , 1 < p < ∞ solutions uε,δ for the Cauchy problem of the following Korteweg de Vries equation with viscosity ut + uux + δuxxx = εuxx ,

(3.3.1)

and the generalized BBM-Burger equation with viscosity ut + uux − δuxxt = εuxx .

(3.3.2)

Under suitable order relations between ε and δ, and the strictly convex condition on the nonlinear flux function f , Schonbek succeeded in obtaining an Lp weak solution for the Cauchy problem (3.1.1).

42 CHAPTER 3. CAUCHY PROBLEM FOR SCALAR EQUATION The restriction of the strict convexity on the flux function f in Schonbek’s paper is removed by Lu in [Lu1]. In Lu’s proof, the ideas of Young measures are also avoided. In [Lu3, Lu7], this simplified proof is used to study weak solutions to the special system of two equations,
(u + qz)t + f (u)x = 0, (3.3.3) zt + kg(u)z = 0, which is derived by Majda [Ma] as a chemical reaction model, where q is a positive constant, k denotes the reaction rate of the chemical material. In [Sz], Szepessy extends the concept of measure valued solution, initially introduced by DiPerna [Di4], to the scalar conservation law in the case of several space variables. He obtains the unique Lp , p > 1 weak solution without the condition (3.1.6), i.e., meas {u : f  (u) = 0} = 0 on the flux function f . In his proof, the growth condition (3.2.1) on f is extended to |f (u)| ≤ C|u|q for any 0 < q < p if the initial data is bound in Lp space.

Chapter 4

Preliminaries in 2 × 2 Hyperbolic System Besides the scalar equation studied in Chapter 3, a major part in this book focuses on the applications on hyperbolic conservation systems of two equations. In the next several chapters (Chapters 5-12), we shall apply the compensated compactness method to different types of systems of two equations. Here the types are distinguished by hyperbolicity, linearity and so on.

4.1

Basic Definitions

We consider the pair of conservation laws ut + f (u, v)x = 0,

vt + g(u, v)x = 0,

(4.1.1)

where u and v are in R. We let U = (u, v) and F (U ) = (f, g) so that the equations in (4.1.1) can be written as Ut + dF (U )Ux = 0,

(4.1.2)

where dF (U ) is the Jacobian matrix of F . The following definitions can be found from Smoller’s book [Sm]. Definition 4.1.1 We say that system (4.1.2) is hyperbolic if dF has two real eigenvalues λ1 and λ2 . System (4.1.2) is called strictly hyperbolic if λ1 and λ2 are distinct, i.e., λ1 < λ2 . If λ1 and λ2 coincide at 43

44

CHAPTER 4. PRELIMINARIES IN HYPERBOLIC SYSTEM

some points or domains, system (4.1.2) is called nonstrictly hyperbolic or hyperbolically degenerate. Let lλ1 , lλ2 , rλ1 and rλ2 denote the corresponding left and right eigenvectors. Definition 4.1.2 We say that (4.1.2) is genuinely nonlinear in the λ1 characteristic field if ∇λ1 · rλ1 = 0

(4.1.3)

and genuinely nonlinear in the λ2 characteristic field if ∇λ2 · rλ2 = 0.

(4.1.4)

If ∇λ1 ·rλ1 = 0 or ∇λ2 ·rλ2 = 0 at some domain D, then system (4.1.2) is called linearly degenerate in D in the λ1 characteristic field or in the λ2 characteristic field. Definition 4.1.3 The functions w = w(u, v), z = z(u, v) are called Riemann invariants of system (4.1.2) corresponding to λ1 and λ2 if they satisfy the equations ∇w · rλ1 = 0,

∇z · rλ2 = 0.

(4.1.5)

Definition 4.1.4 A pair of functions (η(u), q(u)) is called a pair of entropy-entropy flux of system (4.1.2) if (η(u), q(u)) satisfies ∇q(u) = ∇η(u)∇f (u).

(4.1.6)

If n ≤ 2, we can always resolve system (4.1.6) and obtain a class of entropies and the corresponding entropy fluxes. However, if n > 2, system (4.1.6) is over-determined and could be resolved only for some very special cases. As we have seen in Chapter 3, the applications of the compensated compactness method on conservation laws depend strongly on the constructions of entropies of hyperbolic systems we considered. This is why, until now, almost all the results obtained by this method are only for systems of two equations.

4.2. L∞ ESTIMATE OF VISCOSITY SOLUTIONS

4.2

45

L∞ Estimate of Viscosity Solutions

Similar to the scalar equation, for a given hyperbolic system, we first need to construct its approximation solutions, for instance, the sequence of viscosity solutions given by the Cauchy problem
ut + f (u, v)x = εuxx , (4.2.1) vt + g(u, v)x = εvxx , with bounded measurable initial data (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)).

(4.2.2)

To obtain an a priori uniform L∞ estimate of (uε , v ε ) with respect to the viscosity parameter ε, in general, the unique framework we could use is found by Chueh, Conley and Smoller [CCS], and is called the Invariant Region Method. We summarize the main result of [CCS] about the solutions of the Cauchy problem (4.2.1), (4.2.2) in the following theorem: Theorem 4.2.1 Let w, z be two Riemann invariants of system (4.1.1). If the curve w = M1 (or z = M2 ) for a constant M1 (or M2 ) in the (u, v)-plane is upper-convex, i.e., wuu a2 + 2wuv ab + wvv b2 ≥ 0 or zuu a2 + 2zuv ab + zvv b2 ≥ 0 for a vector (a, b) ∈ R2 , then the solutions (uε (x, t), v ε (x, t)) satisfy w(uε , v ε ) ≤ M1

(4.2.3)

z(uε , v ε ) ≤ M2

(4.2.4)

or

if the initial data (u0 (x), v0 (x)) satisfy the same estimate w(u0 , v0 ) ≤ M1 or z(u0 , v0 ) ≤ M2 . If the curves w = M1 and z = M2 both are upper-convex, then both estimates (4.2.3) and (4.2.4) are true if the initial data satisfy the same estimates.

Chapter 5

A Symmetry System In this chapter, we are concerned with a hyperbolic system of two equations with symmetry
ut + (uφ(r))x = 0, (5.0.1) vt + (vφ(r))x = 0 with bounded measurable initial data (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)),

(5.0.2)

where φ(r) is a nonlinear symmetric function of u, v with r = u2 + v 2 . System (5.0.1) is of interest because it arises in such areas as elasticity theory, magnetohydrodynamics, and enhanced oil recovery (cf. [KK, LW]). Let F be the mapping from R2 into R2 defined by F : (u, v) → (uφ(r), vφ(r)). Then two eigenvalues of dF are λ1 = φ(r),

λ2 = φ + 2rφ
(r)

(5.0.3)

with corresponding right eigenvectors r1 = (−v, u)T ,

r2 = (u, v)T .

(5.0.4)

By simple calculations, ∇λ1 · r1 = 0,

∇λ2 · r2 = 6rφ
(r) + 4r 2 φ

(r). 47

(5.0.5)

CHAPTER 5. A SYMMETRY SYSTEM

48

Therefore, from (5.0.3) the strict hyperbolicity of system (5.0.1) fails on the points where rφ
(r) = 0, and from (5.0.5) the first characteristic field is always linearly degenerate and the second characteristic field is either genuinely nonlinear or linearly degenerate, depending on the behavior of φ. In this chapter, we assume that φ ∈ C 2 (R+ ),

meas {r : 3φ
(r) + 2rφ

(r) = 0} = 0,

(5.0.6)

which is similar to the condition in (3.1.6). Therefore the second characteristic field could be linearly degenerate on a set of Lebesgue measure zero. Consider the Cauchy problem for the related parabolic system
ut + (uφ(r))x = εuxx , (5.0.7) vt + (vφ(r))x = εvxx with the initial data (5.0.2). We have the main result in this chapter: Theorem 5.0.1 (1) Let the initial data (u0 (x), v0 (x)) be bounded measurable. Then for fixed ε > 0, the viscosity solution (uε (x, t), v ε (x, t)) of the Cauchy problem (5.0.7), (5.0.2) exists and is uniformly bounded with respect to the viscosity parameter ε. (2) Moreover, if condition (5.0.6) holds, then there exists a subsequence of r ε = (uε )2 + (v ε )2 (still labelled r ε ) which converges pointwisely to a function l(x, t). (3) If v0 (x) ≥ c0 > 0 for a constant c0 and the total variation of u0 (x) is bounded in (−∞, ∞) or equivalently v0 (x)  ∞ u0 (x) )x |dx ≤ M, |( (5.0.8) v0 (x) ∞ then there exists a subsequence of (uε , v ε ) (still denoted by (uε , v ε )) which converges pointwisely to a pair of functions (u(x, t), v(x, t)) satisfying l(x, t) = u2 (x, t) + v 2 (x, t), which combining with (2) implies that the limit (u, v) is a weak solution of the hyperbolic system (5.0.1) with the initial data (5.0.2).

5.1. VISCOSITY SOLUTIONS

49

Remark 5.0.2 Since (uε , v ε ) is uniformly bounded with respect to ε, its weak-star limit (u, v) always exists. However, the strong limit l(x, t) of (uε )2 + (v ε )2 needs not equal u(x, t)2 + v(x, t)2 . If this equality is true, then, at least, (u, v) is a weak solution of (5.0.1), (5.0.2) without any more condition such as given in part (3). Remark 5.0.3 Since one characteristic field of system (5.0.1) is always linearly degenerate, the further condition (5.0.8) in Theorem 5.0.1 about the derivative of the initial data seems necessary to ensure the strong convergence of the viscosity solutions (uε , v ε ). This fact also can be seen by the following simple example: Consider the initial value problem for the scalar linear equation
ut = εuxx , (5.0.9) u(x, 0) = uε0 (x). The solutions of (5.0.9) are uε (x, t) =



∞

−∞

uε0 (y)G(x − y, t)dy,

where G(x−y, t) is the Green function. Then clearly uε (x, t) is compact only with some extra compact conditions on the initial data. The proof of Theorem 5.0.1 will be given in the next sections.

5.1

Viscosity Solutions

To prove the existence of the viscosity solution in Theorem 5.0.1, from Theorem 1.0.2, it is sufficient to get the uniform L∞ bound. Multiplying the first and second equations of the parabolic system (5.0.7) by 2u and 2v, respectively, then adding the result, we have rt + rx φ(r) + 2r(φ(r))x = εrxx − 2ε(u2x + vx2 )

(5.1.1)

or rt + f (r)x ≤ εrxx ,

(5.1.2)

CHAPTER 5. A SYMMETRY SYSTEM

50 where

 f (r) =

r 0

φ(s) + 2sφ
(s)ds.

(5.1.3)

Since the initial data is bounded, then r(0, t) is bounded from above. Using the maximum principle to (5.1.2), we get r = (uε )2 + (v ε )2 ≤ M , which implies the uniform boundedness of (uε , v ε ) and hence, the existence of the viscosity solutions. To prove the second part of Theorem 5.0.1, namely the strong convergence of r ε , we multiply Equation (5.1.1) by a test function φ, where φ ∈ C0∞ (R × R+ ) satisfies φK = 1, 0 ≤ φ ≤ 1 and S = supp φ for an arbitrary compact set K ⊂ S ⊂ R × R+ . Then, similar to the proof of (3.1.25), we have that  ∞ ∞   2ε (uεx )2 + (vxε )2 φdxdt 0

−∞

∞ ∞

 = 0

−∞ ∞ ∞

 = 0

−∞



 εrxx − rt − f (r)x φdxdt

(5.1.4)

εrφxx + rφt + f (r)φx dxdt ≤ M (φ)

and hence ε(uεx )2 and ε(vxε )2 are bounded in L1loc (R × R+ ).

(5.1.5)

Let (η(r), q(r)) be any pair of entropy-entropy flux of the scalar equation   r  φ(s) + 2sφ
(s)ds x = 0 (5.1.6) rt + 0

and multiply (5.1.1) by η
(r). Then η(r)t + q(r)x = ε(η
(r)rx )x − εη

(r)rx2 − 2εη
(r)(u2x + vx2 )

(5.1.7)

= I1 − I2 − I3 , −1,2 and I2 + I3 are bounded in L1loc (R × R+ ), where I1 is compact in Wloc −1,α and hence compact in Wloc for α ∈ (1, 2), by (5.1.5). Noticing that

5.1. VISCOSITY SOLUTIONS

51

η(r)t + q(r)x is bounded in W −1,∞ , and using Theorem 2.3.2, we get the proof that −1,2 (R × R+ ), ηi (r ε (x, t))t + qi (r ε (x, t))x are compact in Wloc

(5.1.8)

for i = 1, 2, where (η1 (r), q1 (r)) = (r − k, f (r) − f (k)) and

 (η2 (r), q2 (r)) = (f (r) − f (k),

r k

(f
(s))2 ds),

(5.1.9)

(5.1.10)

and k is an arbitrary constant. Thus, if we consider that r is an independent variable, noticing the condition (5.0.6) on f , which is exactly the same as the proof of Theorem 3.1.1, we get the proof of r ε (x, t) → l(x, t), almost everywhere. Now we are going to prove the third part of Theorem 5.0.1, namely the strong convergence of (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)). Since v0 (x) ≥ c0 , then using the last part of Theorem 1.0.2, we have that vε ≥ c(t, c0 , ε) > 0. By simple calculations, we have from system (5.0.7) that u u 2u 2 u ( )t + λ1 ( )x = ε( )xx − ε( 3 vx2 − 2 ux vx ) v v v v v u 2ε ux u = ε( )xx + vx ( − 2 vx ) v v v v 2ε u u = ε( )xx + vx ( )x . v v v

(5.1.11)

uε is uniformly bounded with respect to ε when we use the vε maximum principle to (5.1.11).

Therefore

Differentiating Equation (5.1.11) with respect to x and then multiplying the function sign θx to the result, where θ = uv , we have |θx |t + (λ1 |θx |)x ≤ ε|θx |xx + (

2ε vx |θx |)x . v

Integrating (5.1.12) in R × [0, t], we have  ∞  ∞ |θx |(x, t)dx ≤ |θx |(x, 0)dx ≤ M, −∞

−∞

(5.1.12)

(5.1.13)

52

CHAPTER 5. A SYMMETRY SYSTEM

uε which implies the pointwise convergence of a subsequence of ε . Comv bining this with the result in the second part of Theorem 5.0.1, we get the pointwise convergence of a subsequence of (uε , v ε ) → (u, v), where the limit (u, v) is clearly a weak solution of the Cauchy problem (5.0.1), (5.0.2). Thus we complete the proof of Theorem 5.0.1.

5.2

Related Results

The proof of Theorem 5.0.1 is from [Lu11]. The study of the Cauchy problem (5.0.1), (5.0.2) by using the compensated compactness method started from [Ch3], where Chen first considered the propagation and cancellation of oscillations for the weak solution. Along the second genuinely nonlinear characteristic field, the initial oscillations cannot propagate and are killed instantaneously as time evolves, but along the first linearly degenerate field, the initial oscillations can propagate. These behaviors about weak solutions of (5.0.1), (5.0.2) are coincided with that for viscosity solutions we studied in this chapter. The first characteristic field of system (5.0.1) is of Temple type [Te], i.e., the characteristic curve in the (u, v)-plane determined by the equation z = c is a straight line, where c is a constant and z = u/v is the Riemann invariant corresponding to the characteristic value λ1 . In Chapter 7, we shall study another system of Temple type, called the Le Roux system.

Chapter 6

A System of Quadratic Flux In this chapter, we consider the existence of global weak solutions for the nonlinear hyperbolic conservation laws of quadratic flux:
1 ut + (3u2 + v 2 )x = 0 (6.0.1) 2 vt + (uv)x = 0 with initial data (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)) (v0 (x) ≥ 0).

(6.0.2)

System (6.0.1) is the special one of the following more general systems of quadratic flux:  ut + (a1 u2 + a2 uv + a3 v 2 )x = 0 (6.0.3) vt + (b1 u2 + b2 uv + b3 v 2 )x = 0, where ai , bi , i = 1, 2, 3 are all constants. System (6.0.3) can be used to approximate any given nonlinear system of two equations near the original point if we represent the nonlinear flux functions by Taylor series first, and then neglect the linear terms and the higher-order small terms. Let F be the mapping from R2 into R2 defined by 1 F : (u, v) → ( (3u2 + v 2 ), uv). 2 53

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

54 Then

  3u v dF = , v u

(6.0.4)

and the eigenvalues of system (6.0.1) are solutions of the following characteristic equation: λ2 − 4uλ + 3u2 − v 2 = 0.

(6.0.5)

Thus two eigenvalues of system (6.0.1) are 1

λ1 = 2u − s 2 ,

1

λ2 = 2u + s 2

(6.0.6)

with corresponding right eigenvectors 1

r1 = (s 2 − u, −v)T ,

1

r2 = (s 2 + u, v)T ,

(6.0.7)

where s = u2 + v 2 . The Riemann invariants of (6.0.1) are functions w(u, v) and z(u, v) satisfying the equations 1

1

wu (s 2 − u) − vwv = 0 and zu (s 2 + u) + vwv = 0.

(6.0.8)

One solution of (6.0.8) is 1

1

w(u, v) = u + s 2 and z(u, v) = u − s 2 .

(6.0.9)

By simple calculations, 1

∇λ1 · r1 = 3(s 2 − u),

1

∇λ2 · r2 = 3(s 2 + u).

(6.0.10)

Therefore it follows from (6.0.6) that λ1 = λ2 at point (0,0), at which the strict hyperbolicity fails to hold, and from (6.0.10), the first characteristic field is linearly degenerate on v = 0, u ≥ 0 and the second characteristic field is linearly degenerate on v = 0, u ≤ 0. Consider the Cauchy problem for the related parabolic system  ut + 12 (3u2 + v 2 )x = εuxx , (6.0.11) vt + (uv)x = εvxx with initial data (uε (x, 0), v ε (x, 0)) = (uε0 (x), v0ε (x)),

(6.0.12)

55 where (uε0 (x), v0ε (x)) = (u0 (x), v0 (x) + ε) ∗ Gε

(6.0.13)

and Gε is a mollifier. Then (uε0 (x), v0ε (x)) ∈ C ∞ × C ∞ , (uε0 (x), v0ε (x)) → (u0 (x), v0 (x)) a.e., as ε → 0,

(6.0.14) (6.0.15)

and uε0 (x) ≤ M1 ,

ε ≤ v0ε (x) ≤ M1 ,

(6.0.16)

for a suitable large constant M1 , which depends only on the L∞ bound of (u0 (x), v0 (x)), but is independent of ε. The main result in this chapter is given in the following theorem: Theorem 6.0.1 Let the initial data (u0 (x), v0 (x)) be bounded measurable and v0 (x) ≥ 0. Then for fixed ε > 0, the viscosity solution (uε (x, t), v ε (x, t)) of the Cauchy problem (6.0.11), (6.0.12) exists and satisfies uε (x, t) ≤ M2 ,

0 < c(ε, t) ≤ v ε (x, t) ≤ M2 ,

(6.0.17)

where M2 is a positive constant independent of ε, c(ε, t) is a positive function, which could tend to zero as ε tends to zero or t tends to infinity. Moreover, there exists a subsequence (still labelled) (uε (x, t), v ε (x, t)) such that (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)), a.e. on Ω,

(6.0.18)

where Ω ⊂ R × R+ is any bounded open set, and (u(x, t), v(x, t)) is a weak solution of the Cauchy problem (6.0.1), (6.0.2). The above theorem includes two parts: one is the existence of viscosity solutions and related estimates (6.0.17), whose proof is given in Section 6.1; the other part is about the strong convergence (6.0.18) of a subsequence of (uε (x, t), v ε (x, t)), whose proof is given in Sections 6.2-6.4.

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

56

6.1

Existence of Viscosity Solutions

To prove the existence of the viscosity solutions in Theorem 6.0.1, it is sufficient to get the a priori L∞ estimate of (uε (x, t), v ε (x, t)) if we use the general framework given in Theorem 1.0.2. By simple calculations, wuu =

v2 , s3

wuv = −

uv , s3

wvv =

u2 , s3

and v2 uv u2 , zuv = 3 , zvv = − 3 . 3 s s s Then w(u, v) and −z(u, v) both are convex functions. Using Theorem 4.2.1, we have zuu = −

Σ1 = {(u, v) : w(u, v) ≤ M, z(u, v) ≥ −M }

(6.1.1)

is an invariant region for a suitable large constant M (see Figure 6.1).

v

Σ1

w = −M

u w=M

FIGURE 6.1 Thus we have the uniform L∞ estimates uε (x, t) ≤ M2 ,

v ε (x, t) ≤ M2

and hence the existence of the viscosity solutions. The positive lower bound of v ε ≥ c(ε, t) > 0 follows from the last part of Theorem 1.0.2.

6.2. ENTROPY-ENTROPY FLUX PAIRS OF LAX TYPE

6.2

57

Entropy-Entropy Flux Pairs of Lax Type

A pair (¯ η (u, v), q¯(u, v)) of real-valued functions is a pair of entropyentropy flux of system (6.0.1) if they satisfy the following system of two equations: ∇¯ η (u, v) · dF (u, v) = ∇¯ q (u, v)

(6.2.1)

or equivalently ηu + v η¯v , q¯u = 3u¯

q¯v = v η¯u + u¯ ηv .

(6.2.2)

Eliminating q¯ from (6.2.2), we get ηuv = 0. v(¯ ηvv − η¯uu ) + 2u¯

(6.2.3)

By the definition of Riemann invariants in Chapter 4, we can easily prove that w, z satisfy ∇w(u, v) · dF (u, v) = λ2 ∇w(u, v), ∇z(u, v) · dF (u, v) = λ1 ∇z(u, v). (6.2.4) If we consider that the entropy-entropy flux pair of system (6.0.1) are functions of variables w, z: (¯ η , q¯) = (¯ η (w, z), q¯(w, z)), then q¯(w, z)w = λ2 η¯(w, z)w ,

q¯(w, z)z = λ1 η¯(w, z)z .

(6.2.5)

Eliminating q¯ from (6.2.4), we have η (w, z)wz + λ2z η¯(w, z)w − λ1w η¯(w, z)z = 0. (λ2 − λ1 )¯

(6.2.6)

Noticing that 1

λ2 = 2u + s 2 = w + z +

3w + z w−z = 2 2

and 1

λ1 = 2u − s 2 = w + z −

w + 3z w−z = , 2 2

we get the other entropy equation of system (6.0.1) in the following form: η¯(w, z)wz +

1 (¯ η (w, z)w − η¯(w, z)z ) = 0. 2(w − z)

(6.2.7)

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

58

Now we make a transformation of variables from (u, v) to (u, s), that is η¯(u, v) = η(u, s),

q¯(u, v) = q(u, s).

By simple calculations, we have η¯u = ηu + 2uηs ,

η¯v = 2vηs ,

η¯vv = 4v 2 ηss + 2ηs

and η¯uu = 4u2 ηss + 2ηs + 4uηus + ηuu .

η¯uv = 4uvηss + 2vηus ,

Then the entropy equation (6.2.7) is changed to the following simple equation: ηss =

1 ηuu . 4s

(6.2.8)

It follows from (6.2.2) that  2uqs + qu = 3u(2uηs + ηu ) + 2v 2 ηs , 2vqs = v(2uηs + ηu ) + 2uvηs

(6.2.9)

and hence the entropy flux q corresponding to the entropy η satisfies qu = 2uηu + 2sηs .

(6.2.10)

If k denotes a constant, then the function η = h(s)eku solves (6.2.8) provided that h (s) − 1

k2 h(s) = 0. 4s

(6.2.11)

1

Let a(s) = s 4 , r = ks 2 , h(s) = a(s)φ(r). Then a simple calculation shows that φ (r) − (1 +

3 )φ(r) = 0, 4r 2

(6.2.12)

which is the standard Fuchsian equation. We can look for a series solution of (6.2.12) with the following form: φ1 (r) = r

3 2

∞  n=0

3

cn r 2n = r 2 g(r).

(6.2.13)

6.2. ENTROPY-ENTROPY FLUX PAIRS OF LAX TYPE

59

Then the coefficients cn must satisfy cn =

( 32

cn−1 , for n ≥ 1 + 2n)( 12 + 2n) − 34

(6.2.14)

and c0 could be any positive constant. Let another independent solution φ2 of (6.2.12) satisfy φ2 = φ1 P . Then P solves P  φ1 + 2φ1 P  = 0.

(6.2.15)

Thus P  = −(φ1 )−2 = −(r 3 g2 (r))−1 , and one function P is given by  ∞ (r 3 g2 (r))−1 dr. P =

(6.2.16)

r

Therefore 3 2



φ2 (r) = r g(r)

∞ r

(r 3 g2 (r))−1 dr.

(6.2.17)

If ηk = a(s)φ(r)eku , then we have from (6.2.10) that φ (r) 1 )ηk (qk )u = 2kuηk + ( + r 2 φ(r)

(6.2.18)

and hence, one entropy flux qk corresponding to ηk is 1 3 r φ (r) − 1) − ). qk = ηk (2u + s 2 + ( k φ(r) 2k

(6.2.19)

Let η−k = a(s)φ(r)e−ku , then by (6.2.10), one entropy flux q−k corresponding to η−k is 1 3 r φ (r) − 1) + ). q−k = η−k (2u − s 2 − ( k φ(r) 2k

(6.2.20)

About the solutions φ1 , φ2 of the Fuchsian equation φ − (1 +

c )φ = 0, r2

where c is a constant, we have the following lemma:

(6.2.21)

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

60

Lemma 6.2.1 If φ1 (r) > 0, φ1 (r) > 0 for r > 0, then 1 φ1 (r) = 1 + O( 2 ), φ1 (r) r

1 c1 φ1 (r)e−r = 1 + O( ) r

(6.2.22)

as r approaches infinity; if φ2 (r) > 0, φ2 (r) < 0 for r > 0, then 1 φ2 (r) = −1 + O( 2 ), φ2 (r) r

1 c2 φ2 (r)er = 1 + O( ) r

(6.2.23)

as r approaches infinity, where c1 , c2 are two suitable, positive constants. Proof. Since φ = (1 + rc2 )φ, if φ1 (r) > 0, φ1 (r) > 0, then φ1 (r), φ1 (r) both tend to infinite as r approaches infinity; if φ2 (r) > 0, φ2 (r) < 0, φ (r) has then φ2 (r), φ2 (r) both go to zero as r goes to infinity. Thus 1 φ1 (r)  φ (r) 0 ∞ and 2 has the form as r goes to infinity. Therefore the form ∞ φ2 (r) 0 we have by the Vol’pert theorem, (1 + rc2 )φ φ (r) φ (r) lim  lim . r→∞ φ(r) r→∞ φ (r) r→∞ φ (r) lim

(6.2.24)

Then lim (

r→∞

φ (r) 2 ) = 1. φ(r)

(6.2.25)

So we have φ1 (r) = 1, r→∞ φ1 (r)

φ2 (r) = −1. r→∞ φ2 (r) lim

lim

(6.2.26)

Using (6.2.21), we have ( Let y =

φ (r) 2 c φ (r)  ) +( ) = 1 + 2. φ(r) φ(r) r

(6.2.27)

φ (r) − 1. Then y satisfies φ(r) y  + (1 +

c φ (r) )y = 2 . φ(r) r

(6.2.28)

6.2. ENTROPY-ENTROPY FLUX PAIRS OF LAX TYPE

61

Integrating Equation (6.2.28) from d to r (d is a positive constant), we have  r ! τ φ
(t) ! r φ
1 (t) ( 1 +1)dt c d ( φ1 (t) +1)dt y(r) − y(d) = e d φ1 (t) dτ, (6.2.29) e τ2 d which implies that φ1 (r) r−d e y(r) − y(d) = φ1 (d)



r

d

φ1 (τ ) τ −d c e dτ, φ1 (d) τ2

and hence φ1 (d) d−r e y(d) + y(r) = φ1 (r)



r d

φ1 (τ ) τ −r c e dτ. φ1 (r) τ2

(6.2.30)

Let d = 12 r. Then clearly φ1 (d) < φ1 (r), φ1 (τ ) < φ1 (r) since φ1 > 0. Since y is bounded from (6.2.26), we have from (6.2.30) that 1

|y(r)| ≤ M e− 2 r +

1 4|c| (1 − e− 2 r ), 2 r

(6.2.31)

where M is the bound of y. Therefore y(r) = O(

1 ) r2

(6.2.32)

as r approaches infinity. The first part in (6.2.22) is proved. Similarly let z =

φ
2 (r) φ2 (r)

+ 1. We have from (6.2.27) that

z + (

c φ (r) − 1)z = 2 . φ(r) r

(6.2.33)

Integrating (6.2.33) from r to d, we have φ2 (d) r−d e z(d) − z(r) = φ2 (r)



d r

φ2 (τ ) r−τ c e dτ. φ2 (r) τ2

(6.2.34)

Let d = 2r in (6.2.34). Then clearly φ2 (d) < φ2 (r), φ2 (τ ) < φ2 (r) since φ2 < 0. Thus we have |z(r)| ≤ M e−r +

|c| (1 − e−r ), r2

(6.2.35)

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

62

which implies that z(r) = O(

1 ) r2

(6.2.36)

as r approaches infinity. The first part in (6.2.23) is proved. Now we prove the remaining estimates in (6.2.22) and (6.2.23). Let φ1 (r) = a(r)er . Then 1 a (r) = y(r) = O( 2 ) a(r) r

(6.2.37)

for r large. Thus !r

a(r) = a(r1 )e

r1

y(r)dr

,

(6.2.38)

for any fixed r1 > 0. Since y(r) = O( r12 ) as r approaches infinity, the limit of lim a(r) r→∞

exists. Then (6.2.38) gives a(r) = a(∞)e−

!∞ r

y(r)dr −

= a(∞) + a(∞)(e 

∞

!∞ r

(6.2.39) y(r)dr

− 1).

1 y(r)dr = O( ) as r approaches infinity, applying the Taylor r r !∞ expansion to e− r y(r)dr − 1 gives

Since

1 a(r) = a(∞) + O( ) r

(6.2.40)

as r approaches infinity. Similarly let φ2 (r) = b(r)e−r . Then b (r) = z(r). b(r)

(6.2.41)

Since z(r) = O( r12 ) as r approaches infinity, we have 1 b(r) = b(∞) + O( ) r as r approaches infinity. Therefore Lemma 6.2.1 is proved.

(6.2.42)

6.2. ENTROPY-ENTROPY FLUX PAIRS OF LAX TYPE

63

It is clear that φ1 (r), φ2 (r) given in (6.2.13) and (6.2.17) satisfy that φ1 (r) > 0, φ1 (r) > 0 and φ2 (r) > 0 for all s > 0. The strict positivity of φ2 (r) gives φ2 (r) < 0 as s > 0 because lim φ2 (r) = 0, lim φ2 (r) = 0. r→∞

r→∞

Applying the estimates in (6.2.22)-(6.2.23) to φ1 (r), φ2 (r), we have ηk1 = a(s)φ1 (r)eku = ekw (a(s) + O( 1r )) = ekw (a(s) + O( k1 ))

(6.2.43)

1

on any compact subset of s > 0 since r = ks 2 ; 1 3 r φ (r) − 1) − ) qk1 = ηk1 (2u + s 2 + ( 1 k φ1 (r) 2k

(6.2.44)

1 = ηk1 (λ2 + O( )) k φ
(r)

on s ≥ 0 by the fact that factor r( φ11 (r) − 1) is uniformly bounded from (6.2.31). Furthermore qk1 = ηk1 (λ2 −

1 3 + O( 2 )) 2k k

(6.2.45)

on any compact subset of s > 0; 1 1 = a(s)φ1 (r)e−ku = e−kz (a(s) + O( )) η−k k

(6.2.46)

on any compact subset of s > 0;  1 1 (λ − r ( φ1 (r) − 1) + 3 ) = η−k q−k 1 k φ1 (r) 2k

(6.2.47)

1 (λ + O( 1 )) = η−k 1 k

on s ≥ 0 and 1 1 = η−k (λ1 + q−k

1 3 + O( 2 )) 2k k

(6.2.48)

on any compact subset of s > 0; 1 ηk2 = a(s)φ2 (r)eku = ekz (a(s) + O( )) k

(6.2.49)

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

64

on any compact subset of s > 0; 3 r φ (r) + 1) − ) qk2 = ηk2 (λ1 + ( 2 k φ2 (r) 2k

(6.2.50)

1 = ηk2 (λ1 + O( )) k φ
(r)

on s ≥ 0 by the fact that factor r( φ11 (r) + 1) is uniformly bounded from (6.2.35). Furthermore qk2 = ηk2 (λ1 −

1 3 + O( 2 )) 2k k

(6.2.51)

on any compact subset of s > 0; 1 2 = a(s)φ2 (r)e−ku = e−kw (a(s) + O( )) η−k k on any compact subset of s > 0; φ
(r)

2 2 (λ − r ( 2 = η−k q−k 2 k φ2 (r) + 1) +

=

2 (λ η−k 2

+

3 2k )

(6.2.52)

(6.2.53)

O( k1 ))

on s ≥ 0 and 2 2 = η−k (λ2 + q−k

1 3 + O( 2 )) 2k k

(6.2.54)

on any compact subset of s > 0. The above estimates about the entropy-entropy flux pairs will be used to prove the Young measure ν, corresponding to the sequence of viscosity solutions, is a Dirac measure in Section 6.4.

6.3

−1 Compactness of ηt + qx in Hloc

In this section, we mainly prove the following theorem: Theorem 6.3.1 For the entropy-entropy flux pairs (η, q) of Lax type constructed in Section 6.2, η(uε , v ε )t + q(uε , v ε )x −1 is compact in Hloc (R × R+ ) with respect to the approximated solutions ε ε (u , v ) constructed by the viscosity method.

−1 6.3. COMPACTNESS OF ηT + QX IN HLOC

65

Proof. For simplicity, we drop the superscript ε in the viscosity solutions (uε , v ε ). It is obvious that system (6.0.1) has a strictly convex entropy η  = u2 +v2 and the corresponding entropy flux q  = u3 + uv 2 . 2 Multiplying the first equation in (6.0.11) by u and the second by v, then adding the result, we have     2  − ε(ηuu u2x + 2ηuv ux vx + ηvv vx ) = εηxx − ε(u2x + vx2 ). ηt + qx = εηxx (6.3.1)

Using the same technique as given in obtaining (5.1.5), we have ε(uεx )2 and ε(vxε )2 are bounded in L1loc (R × R+ ).

(6.3.2)

The first class of entropy-entropy flux pair of Lax type related to the 1 , are clearly function φ1 constructed in Section 6.2, denoted by η±k smooth functions of (u, v). In fact 1 η±k

3 2

=k s

∞ 

cn (k2 s)n e±ku .

(6.3.3)

n=0

So Theorem 6.3.1 can be easily proved for the first class of entropyentropy flux pair of Lax type. However the second order derivatives of the second class of entropyentropy flux pair of Lax type related to the function φ2 constructed in 2 , are singular at the point (u, v) = (0, 0). Section 6.2, denoted by η±k In fact  ∞ 2 − 12 ±ku 2 r g(r) (r 3 g2 (r))−1 dr, (6.3.4) η±k = k e r

where

 r

∞

(r 3 g2 (r))−1 dr = O(

1 ), r2

(6.3.5)

2 and q 2 are as r approaches zero, and hence for any fixed k > 0, η±k ±k uniformly bounded from (6.2.49)-(6.2.53). Moreover,  ∞ − 12 ±ku 2 2 r g(r) (r 3 g2 (r))−1 dr η±k = k e r

=k

− 12

 ∞    1 g (r) ±ku 2 − r g(r) dr , e 2g(r) r 2 g3 (r) r

(6.3.6)

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

66 where g (r) =

∞ 

2ncn r 2n−1 ≤

n=1

thus 2

∞ 

cn−1 r 2n−1 = rg(r),

(6.3.7)

n=1



r g(r)

∞ r

g (r) dr = O(r 2 log r) r 2 g3 (r)

(6.3.8)

as r approaches zero. This implies that for any fixed k > 0, the first 2 are uniformly bounded. It is clear that the first order derivatives of η±k 1

1 2 are smooth; its second order derivatives in η±k part I1 = k− 2 e±ku 2g(r) 2 can be written as r 2 I , where are bounded. But the second part in η±k 2  ∞  1 g (r) dr, (6.3.9) I2 = −k− 2 e±ku g(r) 2 r g3 (r) r

its second order derivatives are singular at the point (0, 0). In fact, from 2 (6.3.8), all derivatives

order of function r I2 are bounded 2 of second 2 except the term (r )uu + (r )vv I2 = 2I2 , which is positive. 2 , we have Therefore, multiplying system (6.0.11) by ∇η±k 2 ) + (q 2 ) (η±k t ±k x

2

2 ) 2 2 2 2 ε(η±k xx − ε (η±k )uu ux + 2(η±k )uv ux vx + (η±k )vv vx =

2 ) ε(η±k xx



− ε A(u, v)u2x + B(u, v)ux vx + C(u, v)vx2

(6.3.10)

−2εI2 (u2x + vx2 ), where A(u, v), B(u, v) and C(u, v) are the regular derivatives of second 2 . order of η±k Let K ⊂ R × R+ be an arbitrary compact set and choose φ ∈ ∞ C0 (R × R+ ) such that φK = 1, 0 ≤ φ ≤ 1 and S = supp φ. Multiplying Equation (6.3.10) by φ and integrating over R × R+ , we obtain  ∞ ∞ 2εI2 (u2x + vx2 )φdxdt ε 0

−∞

= −ε A(u, v)u2x + B(u, v)ux vx + C(u, v)vx2 φ 2 φ + q 2 φ εη 2 φ +η±k t ±k x ±k xx ≤ M (φ),

(6.3.11)

6.4. REDUCTION OF ν

67

where the last inequality follows from the boundedness of viscosity solutions, the local boundedness in L1loc in (6.3.2) of the regular part A(u, v)u2x + B(u, v)ux vx + C(u, v)vx2 . Considering (6.3.10) again, we see that the part

2 2 2 )uu u2x + 2(η±k )uv ux vx + (η±k )vv vx2 ε (η±k −1,α for a constant α ∈ is bounded in L1loc and hence, compact in Wloc −1,2 2 because the (1, 2). The part ε(η±k )xx is clearly compact in Wloc 2 , and the L1 estiboundedness of derivatives of the first order of η±k loc mates (6.3.2) for u2x and vx2 . Noticing the boundedness of 2 2 )t + (q±k )x (η±k

in W −1,∞ , we get the proof of Theorem 6.3.1 by Theorem 2.3.2.

6.4

Reduction of ν

In this section, we shall prove that the family of positive measures νx,t , determined by the sequence of viscosity solutions (uε (x, t), v ε (x, t)) of the Cauchy problem (6.0.11), (6.0.12), must be Dirac measures. Then using Theorem 2.2.3, we get the proof of (6.0.18), which implies that the function (u(x, t), v(x, t)) of support set points of these Dirac measures is a weak solution of the Cauchy problem (6.0.1), (6.0.2). Since the viscosity solutions (uε (x, t), v ε (x, t)) of the Cauchy problem (6.0.11), (6.0.12) are uniformly bounded in L∞ space, by Theorem 2.2.1, we consider the family of compactly supported probability measures νx,t . Without any loss of generality we may fix (x, t) ∈ R × R+ and consider only one measure ν. For any entropy-entropy flux pairs (ηi , qi ), i = 1, 2, of system (6.0.1), −1 (R × R+ ), satisfying the compactness of η(uε , v ε )t + q(uε , v ε )x in Hloc we have from Theorem 2.1.4 that η1 (uε , v ε ) · q2 (uε , v ε ) − η2 (uε , v ε ) · q1 (uε , v ε ) (6.4.1) = η1

(uε , v ε )q

2

(uε , v ε ) −

η2

(uε , v ε )q

1

(uε , v ε ).

Then using Theorem 2.2.1, we have the following measure equation: < ν, η1 >< ν, q2 > − < ν, η2 >< ν, q1 >< ν, η1 q2 − η2 q1 > .

(6.4.2)

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

68

Let Q denote the smallest characteristic rectangle: Q = {(u, v) : w− ≤ w ≤ w+ , z− ≤ z ≤ z+ , v ≥ 0}. We now prove that the support set of ν is either contained in the point (0, 0) or in another point (w , z  ). We assume that supp ν is not the unique point (0, 0). Then < 1 2 > > 0, where η 1 , η 2 are given by (6.2.43) ν, ηk > > 0 and < ν, η−k k −k and (6.2.52). − We introduce two new probability measures µ+ k and µk on Q, defined by 1 1 < µ+ k , h >=< ν, hηk > / < ν, ηk >

(6.4.3)

2 2 < µ− k , h >=< ν, hη−k > / < ν, η−k >,

(6.4.4)

and

where h = h(u, v) denotes an arbitrary continuous function. Clearly − µ+ k and µk both are uniformly bounded with respect to k. Then as a consequence of weak-star compactness, there exist probability measures µ± on Q such that < µ± , h >= lim < µ± k ,h > k→∞

(6.4.5)

after the selection of an appropriate subsequence. Moreover, the measures µ+ , µ− are respectively concentrated on the boundary sections of Q associated with w, i.e.  (6.4.6) supp µ+ = Q {(u, v) : w = w+ } and supp µ− = Q



{(u, v) : w = w− }.

(6.4.7)

In fact, for any function h(w, z) ∈ C0 (Q), satisfying  supp h(w, z) ⊂ Q {w ≤ w0 }, where w0 < w+ is any number, as k → ∞, we have | < ν, hηk1 > | | < ν, ηk1 > |

=

≤

| < ν, hekw (a(s) + O( k1 )) > | | < ν, ekw (a(s) + O( k1 )) > | ek(wo +δ)

c1 c1 = ek(w0 +2δ−w+ ) → 0, k(w −δ) + c2 c2 e

(6.4.8)

6.4. REDUCTION OF ν

69

where c1 , c2 are two suitable positive constants and δ > 0 satisfies 2δ < w+ − w0 , since Q is the smallest characteristic rectangle of ν. Thus we get the proof of (6.4.6). Similarly we can prove (6.4.7). Let (η1 , q1 ) = (ηk1 , qk1 ) in (6.4.2). We have < ν, q2 > − < ν, η2 >

< ν, ηk1 q2 − η2 qk1 > < ν, qk1 > = . < ν, ηk1 > < ν, ηk1 >

(6.4.9)

Noticing the estimate (6.2.45) between ηk1 and qk1 , and letting k → ∞ in (6.4.9), we have < ν, q2 > − < ν, η2 >< µ+ , λ2 >=< µ+ , q2 − λ2 η2 > .

(6.4.10)

2 , q 2 ) in (6.4.2) and use the estimate Similarly, let (η1 , q1 ) = (η−k −k 2 1 (6.2.54) between η−k and q−k . We have

< ν, q2 > − < ν, η2 >< µ− , λ2 >=< µ− , q2 − λ2 η2 > .

(6.4.11)

2 , q 2 ) in (6.4.2). We have Let (η1 , q1 ) = (ηk1 , qk1 ) and (η2 , q2 ) = (η−k −k 2 > 2 − η2 q1 > < ν, ηk1 q−k < ν, q−k < ν, qk1 > −k k − = 2 > 2 >< ν, η 1 > . < ν, η−k < ν, ηk1 > < ν, η−k k

(6.4.12)

We now prove that w− = w+ . If not, choose δ0 > 0 such that 2δ0 < w+ − w− . Then 2 > ≥ c1 e−k(w− +δ0 ) , < ν, η−k

< ν, ηk1 > ≥ c2 ek(w+ −δ0 )

(6.4.13)

for two suitable positive constants c1 , c2 and hence, the right-hand side of (6.4.12) satisfies 2 − η2 q1 > < ν, ηk1 q−k 1 −k k = O( )e−k(w+ −w− −2δ0 ) → 0, 2 1 k < ν, η−k >< ν, ηk >

as k → ∞, (6.4.14)

resulting from the estimates given by (6.2.43), (6.2.44), (6.2.52) and (6.2.53). Letting k → ∞ in (6.4.12), we have < ν + , λ2 >=< ν − , λ2 >. Combining this with (6.4.10)-(6.4.11) gives the following relation: < µ+ , q − λ2 η >< µ− , q − λ2 η >

(6.4.15)

70

CHAPTER 6. A SYSTEM OF QUADRATIC FLUX

−1 (R × R+ ). for any (η, q) satisfying that ηt + qx is compact in Hloc 2 , q 2 ). If w − w > 2δ , we get from Let (η, q) in (6.4.15) be (η−k + − 0 −k the left-hand side of (6.4.15) that c1 (6.4.16) | < µ+ , q − λ2 η > | ≥ ek(w+ −δ0 ) k

and from the right-hand side c2 −k(w− +δ0 ) e (6.4.17) k for two positive constants c1 , c2 . This is impossible and hence, w+ must equal w− . Similar to the above proof, we can use entropy-entropy flux pairs 2 1 , q 1 ) constructed in Section 6.2 to prove z = z . Thus (ηk , qk2 ), (η−k + − −k the support set of ν is either (0, 0) or another point (w , z  ). This completes the proof of Theorem 6.0.1. | < µ− , q − λ2 η > | ≤

6.5

Related Results

The Riemann solutions for general nonstrictly hyperbolic conservation laws of quadratic flux (6.0.3) were constructed by Issacson, Marchesin, Paes-Leme, Plohr, Schaeffer, Shearer, Temple and others (cf. [IMPT, IT, SSMP]). The interactions of elementary waves for systems  ut + 12 (au2 + v 2 )x = 0 (6.5.1) vt + (uv)x = 0 were studied for the case of a > 2 by Lu and Wang [LuW]. The global existence of weak solution to the Cauchy problem (6.0.1), (6.0.2) was obtained in [Lu4]. The proof in this chapter is from [Lu4] and [Lu8]. The entropies constructed in Section 6.2 touch the singular point (0,0) and hence, refine the proof. In [Ka], a different proof was given by Kan independently to the Cauchy problem (6.0.1), (6.0.2). Kan’s proof is based on constructing entropies η(u, v) for system (6.0.1) away from the singular point (u, v) = (0, 0). The ideas in [Ka] were applied to study L∞ solutions for more general systems of quadratic flux in the form (6.0.3) by Chen and Kan (cf. [CK]).

Chapter 7

Le Roux System In this chapter, we shall use the method given in Chapter 6 to study the existence of global weak solutions for the following nonlinear hyperbolic conservation laws:
ut + (u2 + v)x = 0, (7.0.1) vt + (uv)x = 0 with initial data (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)) (v0 (x) ≥ 0).

(7.0.2)

System (7.0.1) was first derived by Le Roux in [Le] as a mathematical model, and so is called the Le Roux system. Let F be the mapping from R2 into R2 defined by F : (u, v) → (u2 + v, uv). Then

 dF =

 2u 1 , v u

(7.0.3)

and the eigenvalues of system (7.0.1) are solutions of the following characteristic equation: λ2 − 3uλ + 2u2 − v = 0.

(7.0.4)

Two roots of Equation (7.0.4) are λ1 =

3u D − , 2 2

λ2 = 71

3u D + , 2 2

(7.0.5)

CHAPTER 7. LE ROUX SYSTEM

72 1

where D = (u2 + 4v) 2 , with corresponding right eigenvectors r1 = (−1, and

u+D T ) , 2

r2 = (1,

−u + D T ) 2

 u 1 u+D T 3     ∇λ1 · r1 ( 2 − 2D , − D )(−1, 2 ) = −2,     ∇λ2 · r2 ( 3 + u , 1 )(1, −u + D )T = 2. 2 2D D 2

(7.0.6)

(7.0.7)

Therefore if we consider the bounded solution in the upper (u, v)-plane (v ≥ 0), it follows from (7.0.5) that λ1 = λ2 at point (0,0), at which strict hyperbolicity fails to hold. Moreover by (7.0.7), both characteristic fields are genuinely nonlinear by the definitions given in Chapter 4. The Riemann invariants of (7.0.1) are functions w(u, v) and z(u, v) satisfying the equations wu − wv

u+D = 0, 2

zu − zv

u−D = 0. 2

(7.0.8)

One solution of (7.0.8) is w(u, v) = u + D,

z(u, v) = u − D.

(7.0.9)

Consider the Cauchy problem for the related parabolic system
ut + (u2 + v)x = εuxx , (7.0.10) vt + (uv)x = εvxx with the initial data (uε (x, 0), v ε (x, 0)) = (uε0 (x), v0ε (x)),

(7.0.11)

(uε0 (x), v0ε (x)) = (u0 (x), v0 (x) + ε) ∗ Gε

(7.0.12)

where

and Gε is a mollifier. Then (uε0 (x), v0ε (x)) ∈ C ∞ × C ∞ ,

(7.0.13)

7.1. EXISTENCE OF VISCOSITY SOLUTIONS (uε0 (x), v0ε (x)) → (u0 (x), v0 (x)) a.e., as ε → 0,

73 (7.0.14)

and uε0 (x) ≤ M1 ,

ε ≤ v0ε (x) ≤ M1 ,

(7.0.15)

for a suitable large constant M1 , which depends only on the L∞ bound of (u0 (x), v0 (x)), but is independent of ε. We have the main result in this chapter: Theorem 7.0.1 Let the initial data (u0 (x), v0 (x)) be bounded measurable and v0 (x) ≥ 0. Then for fixed ε > 0, the viscosity solution (uε (x, t), v ε (x, t)) of the Cauchy problem (7.0.10), (7.0.11) exists and satisfies uε (x, t) ≤ M2 ,

0 < c(ε, t) ≤ v ε (x, t) ≤ M2 ,

(7.0.16)

where M2 is a positive constant, being independent of ε and c(ε, t) is a positive function, which could tend to zero as ε tends to zero or t tends to infinity. Moreover, there exists a subsequence (still labelled) (uε (x, t), v ε (x, t)) such that (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)), a.e. on Ω,

(7.0.17)

where Ω ⊂ R × R+ is any bounded and open set and the pair of limit functions (u(x, t), v(x, t)) is a weak solution of the Cauchy problem (7.0.1), (7.0.2). The first part in Theorem 7.0.1 about the existence of viscosity solutions is proved in Section 7.1; and the second part about the strong convergence (7.0.17) of a subsequence of (uε (x, t), v ε (x, t)) will be proved in Sections 7.2-7.4.

7.1

Existence of Viscosity Solutions

In this section, we shall give the proof of the existence of viscosity solutions for the Cauchy problem (7.0.10), (7.0.11) and related estimates stated in (7.0.16). By simple calculations, we have wu = 1 +

u , D

wv =

2 , D

wuu =

4v , D3

wuv = −

2u , D3

wvv = −

4 , D3

CHAPTER 7. LE ROUX SYSTEM

74 and

u 2 4v 2u 4 , zv = − , zuu = − 3 , zuv = 3 , zvv = 3 . D D D D D Then multiplying system (7.0.10) by (wu , wv ) and (zu , zv ), respectively, we have zu = 1 −

w(u, v)t + λ2 w(u, v)x = ε(wu uxx + wv vxx ) = εw(u, v)xx − ε(wuu u2x + 2wuv ux vx + wvv vx2 ) 4v 4u 4 = εw(u, v)xx − ε( 3 u2x − 3 ux vx − 3 vx2 ) D D D ε = εw(u, v)xx − 3 ((D + u)ux + 2vx )((D − u)ux − 2vx ) D ε = εw(u, v)xx − w(u, v)x z(u, v)x D (7.1.1) and z(u, v)t + λ1 z(u, v)x = ε(zu uxx + zv vxx ) = εz(u, v)xx − ε(zuu u2x + 2zuv ux vx + zvv vx2 ) 4v 4u 4 = εz(u, v)xx + ε( 3 u2x − 3 ux vx − 3 vx2 ) D D D ε = εz(u, v)xx + 3 ((D + u)ux + 2vx )((D − u)ux − 2vx ) D ε = εz(u, v)xx + w(u, v)x z(u, v)x . D (7.1.2) Now consider (7.1.1) as an equation of the variable w, and (7.1.2) as an equation of the variable z. Applying the maximum principle to (7.1.1) and (7.1.2) respectively, we have that w(uε , v ε ) ≤ M, z(uε , v ε ) ≥ −M if the initial data satisfy w(uε0 (x), v0ε (x)) ≤ M, z(uε0 (x), v0ε (x)) ≥ −M . Thus Σ2 = {(u, v) : w(u, v) ≤ M, z(u, v) ≥ −M, v ≥ 0}

(7.1.3)

is an invariant region for a suitable large constant M (see Figure 7.1). The positive lower bound estimate about v ε in (7.0.16) is a direct corollary of the last part of Theorem 1.0.2. Thus we get the proof of the existence of viscosity solutions of the Cauchy problem (7.0.10), (7.0.11) and the estimates in (7.0.16).

7.2. ENTROPY-ENTROPY FLUX PAIRS OF LAX TYPE v

75

Σ2 u z = −M

w=N

FIGURE 7.1

7.2

Entropy-Entropy Flux Pairs of Lax Type

In this section, four classes of entropy-entropy flux pair of Lax type of the following special forms are constructed: ηk1 = ekw (a1 (D) +

b1 (D, k) ), k

qk1 = ekw (c1 (D) +

d1 (D, k) ); k

2 = e−kw (a2 (D) + η−k

b2 (D, k) ), k

2 q−k = e−kw (c2 (D) +

d2 (D, k) ); k

1 = e−kz (a3 (D) + η−k

b3 (D, k) ), k

1 q−k = e−kz (c3 (D) +

d3 (D, k) ); k

b4 (D, k) d4 (D, k) ), qk2 = ekz (c4 (D) + ), k k where w, z are the Riemann invariants of system (7.0.1), and the required estimates on ai , bi (i = 1, 2, 3, 4) are obtained by the estimates on the solutions of the Fuchsian equation (6.2.21) given in Lemma 6.2.1. Let ρ = D 3 , θ = 32 u. Then for smooth solutions, (7.0.1) is equivalent to the following system:   ρt + (ρθ)x = 0 (7.2.1) 2  θ + ( θ + 3 ρ 23 ) = 0. t x 2 8 ηk2 = ekz (a4 (D) +

CHAPTER 7. LE ROUX SYSTEM

76

Considering the entropy-entropy flux pair (η, q) of system (7.0.1) as functions of variables (ρ, θ), we have 1 1 (qρ , qθ ) = (θηρ + ρ− 3 ηθ , ρηρ + θηθ ). 4

(7.2.2)

Eliminating the q from (7.2.2), we have 1 4 ηρρ = ρ− 3 ηθθ . 4

(7.2.3)

If k denotes a constant, then the function η = h(ρ)ekθ solves (7.2.3) provided that h (ρ) = 1

1 2 −4 k ρ 3 h(ρ). 4

(7.2.4)

1

Let h(ρ) = ρ 3 φ(s), s = 32 kρ 3 . Then φ solves the Fuchsian equation φ − (1 +

2 )φ = 0. s2

(7.2.5)

Because it is exactly analogous to that in Chapter 6, we may use the method of Frobenius to give a series solution to Equation (7.2.5) as follows: φ1 = s

2

∞ 

c2n s2n = s2 g(s),

(7.2.6)

n=0

where g(s) =

∞ 

c2n s2n ,

c2n =

n=0

c2(n−1) (2 + 2n)(1 + 2n) + 2

(7.2.7)

and c0 is an arbitrary positive constant. Let another independent solution φ2 of (7.2.5) satisfy φ2 = φ1 P . Then P solves P  φ1 + 2φ1 P  = 0.

(7.2.8)

Thus P  = −(φ1 )−2 = −(s4 g2 (s))−1 , and one special function P is given by ∞ (s4 g2 (s))−1 ds. P = s

(7.2.9)

7.2. ENTROPY-ENTROPY FLUX PAIRS OF LAX TYPE Then 2



φ2 = s g(s)

∞ s

(s4 g2 (s))−1 ds.

77

(7.2.10)

It is clear that φ1 , φ2 given in (7.2.6) and (7.2.10) satisfy that φ1 (s) > 0, φ1 (s) > 0 and φ2 (s) > 0 for all s ≥ 0. The strict positivity of φ2 (s) gives φ2 (s) < 0 as s > 0 because lim φ2 (s) = 0, lim φ2 (s) = 0. s→∞

s→∞

By simple calculations, the two eigenvalues of (7.2.1) are 1

ρ3 , λ1 = θ − 2

1

ρ3 λ2 = θ + 2

(7.2.11)

with corresponding two Riemann invariants 3 1 z = θ − ρ3 , 2

3 1 w = θ + ρ3 . 2

(7.2.12)

Similar to (6.2.5), we have qw = λ2 ηw ,

qz = λ1 ηz . 2

(7.2.13) 2

From (7.2.12), we have θw = 12 , θz = 12 , ρw = ρ 3 , ρz = −ρ 3 . Then  2 2 1 1   qw = qθ + ρ 3 qρ , qz = qθ − ρ 3 qρ 2 2   ηw = 1 ηθ + ρ 23 ηρ , ηz = 1 ηθ − ρ 23 ηρ . 2 2

(7.2.14)

From (7.2.14), (7.2.13) and (7.2.11), we have qθ = qw + qz = θηθ + ρηρ . 1

(7.2.15)

1

Letting ηk = ρ 3 φ(s)ekθ , η−k = ρ 3 φ(s)e−kθ , we have from (7.2.15) that  1 

ρ 3 φ (s)  2   + ,  qk = ηk θ − 3k 2φ(s) (7.2.16) 1 

3 φ (s)  ρ 2   − .  q−k = η−k θ + 3k 2φ(s) 1

Let ηk1 = ρ 3 φ1 (s)ekθ . Then Lemma 6.2.1 gives 1 1 1 1 ηk1 = ρ 3 ekw (1 + O( ))ρ 3 ekw (1 + O( )) s k

(7.2.17)

CHAPTER 7. LE ROUX SYSTEM

78 on any compact subset of s > 0. 1

qk1

=

ηk1 (θ

1

ρ 3 φ1 (s) ρ3 2 1 + ( − 1) + ) = ηk1 (λ2 + O( )) − 3k 2 φ1 (s) 2 k

(7.2.18)

φ


on s ≥ 0 by the fact that s( φ11 − 1) is uniformly bounded. Moreover qk1 = ηk1 (λ2 −

s φ (s) 1 2 2 + ( 1 − 1)) = ηk1 (λ2 − + O( 2 )) (7.2.19) 3k 3k φ1 (s) 3k k

on any compact subset of s > 0. Similarly 1 1 1 ηk2 = ρ 3 φ2 (s)ekθ ρ 3 ekz (1 + O( )) k

(7.2.20)

on any compact subset of s > 0, 1 qk2 = ηk2 (λ1 + O( )) k

(7.2.21)

on s ≥ 0 and qk2 = ηk2 (λ1 −

1 2 + O( 2 )) 3k k

(7.2.22)

on any compact subset of s > 0. 1 1 1 1 = ρ 3 φ1 (s)e−kθ ρ 3 e−kz (1 + O( )) η−k k

(7.2.23)

on any compact subset of s > 0, 1 1 1 = η−k (λ1 + O( )) q−k k

(7.2.24)

on s ≥ 0 and 1 1 = η−k (λ1 + q−k

1 2 + O( 2 )) 3k k

(7.2.25)

on any compact subset of s > 0. Finally 1 1 1 2 = ρ 3 φ2 (s)e−kθ ρ 3 e−kw (1 + O( )) η−k k

(7.2.26)

on any compact subset of s > 0, 1 2 2 = η−k (λ2 + O( )) q−k k

(7.2.27)

−1 7.3. COMPACTNESS OF ηT + QX IN HLOC

79

on s ≥ 0 and 2 2 = η−k (λ2 + q−k

1 2 + O( 2 )) 3k k

(7.2.28)

on any compact subset of s > 0. The estimates (7.2.17)-(7.2.28) will be used in Section 7.4 to reduce the Young measures, corresponding to the sequence of viscosity solutions, to be Dirac measures, and hence the proof of the existence of weak solutions in the second part of Theorem 7.0.1. −1 Compactness of ηt + qx in Hloc

7.3

In this section, we mainly prove the following theorem: Theorem 7.3.1 For the entropy-entropy flux pairs (η, q) of Lax type constructed in Section 7.2, η(uε , v ε )t + q(uε , v ε )x −1 is compact in Hloc (R × R+ ) with respect to the viscosity solutions ε ε (u , v ) obtained in Section 7.1.

Proof. η  (u, v) = v It is obvious that system (7.0.1) has a convex entropy u2 2u3  2 + 0 log vdv and the corresponding entropy flux q (u, v) = 3 + uv log v. Similar to the proof of (6.3.2), we have 1

1

ε 2 ∂x uε and ε 2

1 ∂x v ε are uniformly bounded in L2loc (R × R+ ). vε (7.3.1)

We only prove Theorem 7.3.1 for the entropy-entropy flux pairs A similar method may give the proofs for other entropy1 , q 1 ) and (η 2 , q 2 ). entropy flux pairs (ηk1 , qk1 ), (η−k −k −k −k

(ηk2 , qk2 ).

Multiplying system (7.0.10) by (ηu , ηv ), we have ηt + qx = εηxx − ε(ηuu u2x + 2ηuv ux vx + ηvv vx2 ). Because

s

∞

ds s4 g2 (s)

= O(

1 ) s3

(7.3.2)

(7.3.3)

CHAPTER 7. LE ROUX SYSTEM

80

as s approaches zero, for any fixed k > 0 we have that ∞ 2 3 ds 2 s g(s) ekθ ηk = 4 3k s g2 (s) s

(7.3.4)

and qk2 both are uniformly bounded. Thus (ηk2 (u, v))t + (qk2 (u, v))x is bounded in W −1,∞ (R × R+ ). Because ∞  2ekθ 1 g (s)ds 2 3 ( − 2s g(s) ) (7.3.5) ηk = k g(s) s3 g3 (s) s and g (s)/s ≤ g(s), we have ∞ s

1 g (s)ds = O( ) 3 3 s g (s) s

(7.3.6)

as s approaches zero. Thus (ηk2 )s and (ηk2 )θ both are bounded and (ηk2 )s = O(s) as s approaches zero. Since 1 ∂s = 3k(v 2 + 4u)− 2 , ∂u

1 3kv 2 ∂s = (v + 4u)− 2 , ∂θ 2

(7.3.7)

−1 (R×R+ ) then ε∂x ηk2 = O(ε(|ux |+|vx |)). Thus ε∂xx ηk2 is compact in Hloc from (7.3.1). The proof of Theorem 7.3.1 will be completed if we can prove that the second term in the right-hand side of (7.3.2) is bounded in L1loc (R × R+ ). 2 ekθ is uniformly bounded Since ηk2 = I1 − k4 ekθ I, where I1 = kg(s) ∞  g (s)ds , then the proof of Theorem 7.3.1 is in C 2 , and I = s3 g(s) s3 g 3 s concentrated to the boundedness of L in L1loc (R × R+ ), where

L = ε(Iuu u2x + 2Iuv ux vx + Ivv vx2 ). Let L = L1 + L2 , where

L1 = εIss ((su )2 u2x + 2su sv ux vx + (sv )2 vx2 ) L2 = εIs (suu u2x + 2suv ux vx + svv vx2 ).

(7.3.8)

(7.3.9)

Because g (s)/s ≤ g(s), Is = O(s) as s approaches zero and Iss is v2 bounded, we have that L1 and L2 are controlled by ε(O( x ) + O(u2x )) |v| and hence bounded in L1loc (R × R+ ) from (7.3.1). This completes the proof of Theorem 7.3.1.

7.4. EXISTENCE OF WEAK SOLUTIONS

7.4

81

Existence of Weak Solutions

In this section, we shall use the compensated compactness method to prove the existence of weak solutions of the Cauchy problem (7.0.1), (7.0.10) given in the second part of Theorem 7.0.1. Consider a compactly supported probability measure ν on R2 such that < ν, η 1 >< ν, q 2 > − < ν, η 2 >< ν, q 1 >< ν, η 1 q 2 − η 2 q 1 >

(7.4.1)

for entropy-entropy flux pairs (η i , q i )(i = 1, 2), of system (7.0.1), satisfying the compactness of η i (uε , v ε )t + q i (uε , v ε )x in H −1 (Ω). Then the proof of the existence of weak solutions in Theorem 7.0.1 is reduced to proving that ν is a point mass. We shall realize this goal by the i , q i ) constructed in Section 7.2. entropy-entropy flux pairs (η±k ±k The proof is almost the same as that given in Section 6.4. Let Q denote the smallest characteristic rectangle Q = {(u, v) : w− ≤ w ≤ w+ ,

z− ≤ z ≤ z+,

v ≥ 0}.

Then w− , w+ are nonnegative and z − , z + are nonpositive. If the support of ν only consists exactly of the point (0, 0), we are done. Next, consider the other case, where the support of ν is not concentrated at the point (u, v) = (0, 0), so that < ν, ηk1 > > 0 and 2 > > 0. < ν, η−k We introduce probability measures µ± k on Q defined by 1 1 < µ+ k , h >=< ν, hηk > / < ν, ηk >

(7.4.2)

1 1 < µ− k , h >=< ν, hη−k > / < ν, η−k >,

(7.4.3)

and

where h = h(u, v) denotes an arbitrary continuous function. As a consequence of weak-star compactness, there exist probability measures µ± on Q such that < µ± , h >= lim < µ± k ,h > k→∞

(7.4.4)

CHAPTER 7. LE ROUX SYSTEM

82

after the selection of an appropriate subsequence. We observe that the measures µ+ , µ− are respectively concentrated on the boundary sections of Q associated with w, i.e.,   (7.4.5) Q {(u, v) : w = w+ } and Q {(u, v) : w = w− }. Because 1 1 ηk1 = ekw ρ 3 (1 + O( )), k

1 1 2 and η−k = e−kw ρ 3 (1 + O( )), k

(7.4.6)

on any compact subset of s > 0 and by the assumption, the support of ν is not concentrated at s = 0. Similar to the proof of (6.4.15), we have that < µ+ , λ2 η − q >=< µ− , λ2 η − q >

(7.4.7)

−1 (R × R+ ). for any (η, q) satisfying that ηt + qx is compact in Hloc

Substituting (ηk1 , qk1 ) into (7.4.7) yields that

−

< µ− , λ2 ηk1 − qk1 > ≤ c1 ekw /k

(7.4.8)

and +

< µ+ , λ2 ηk1 − qk1 > ≥ c2 ekw /k,

(7.4.9)

where c1 , c2 are two positive constants. Therefore (7.4.7), (7.4.8) and (7.4.9) imply that w− = w+ . In the same fashion we conclude that z − = z + . This completes the proof of Theorem 7.0.1.

7.5

Related Results

Two characteristic fields of system (7.0.1) or the curves determined by the equations w = const, z = const in the (u, v)-plane are clearly straight lines, so system (7.0.1) is of Temple type [Te], whose shock curves and rarefaction curves coincided. The global existence and uniqueness of an L∞ weak solution for a general n×n system of Temple type in strictly hyperbolic regions were established by Heibig in [He].

7.5. RELATED RESULTS

83

In nonstrictly hyperbolic regions, a compact framework for an n × n system of chromatography uit +

ki ui  = 0, 1+D x

i = 1, 2, · · · , n,

(7.5.1)

was established by James, Peng and Perthame [JPP] by using the kinetic formulation coupled with the compensated compactness method, where ki are positive constants satisfying 0 < k1 < k2 < · · · < kn and D = 1 + k1 + k2 + · · · + kn . System (7.5.1) is the unique application of the compensated compactness method on hyperbolic systems of more than two equations. However, it should be a very interesting topic to construct suitable approximated solutions {uli } of system (7.5.1) and then to prove the −1,2 , for the entropy-entropy flux compactness of η(uli )t + q(uli )x in Wloc pairs (η, q) constructed by the kinetic formulation in [JPP], with respect to the sequence {uli }. If this is done, then the existence of weak solutions to system (7.5.1) follows from the compactness framework given in [JPP]. The proof of Theorem 7.0.1 is from the paper [LMR]. The main difficulty in dealing with system (7.0.1) is the singularity of entropies at the nonstrictly hyperbolic domain. System (7.5.1) should be more difficult since it has more equations.

Chapter 8

System of Polytropic Gas Dynamics We consider the Cauchy problem for the system of isentropic gas dynamics in Eulerian coordinates
ρt + (ρu)x = 0 (8.0.1) (ρu)t + (ρu2 + P (ρ))x = 0, with bounded measurable initial data (ρ(x, 0), u(x, 0)) = (ρ0 (x), u0 (x)),

ρ0 (x) ≥ 0,

(8.0.2)

where ρ is the density of gas, u the velocity, P = P (ρ) the pressure satisfying P
(ρ) ≥ 0. For the polytropic gas, P takes the special form P (ρ) = cργ , where γ > 1 and c is an arbitrary positive constant, for 2 instance, c = k2 = (γ−1) 4γ . System (8.0.1) can be written into  ρt + mx = 0 2 mt + ( mρ + P (ρ))x = 0, where m denotes the mass. Let F be the mapping from R2 into R2 defined by F : (ρ, m) → (m, 85

m2 + P (ρ)). ρ

(8.0.3)

86

CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

Then

 dF =

2

− mρ

0 + P
(ρ)

1 2m ρ

 ,

(8.0.4)

and the eigenvalues of system (8.0.1) are solutions of the following characteristic equation: λ2 −

m2 2m λ+ − P
(ρ) = 0. ρ ρ

(8.0.5)

Thus two eigenvalues of system (8.0.1) are m 
m 
− P (ρ), λ2 = + P (ρ) λ1 = ρ ρ

(8.0.6)

with corresponding right eigenvectors r1 = (1, λ1 )T ,

r2 = (1, λ2 )T .

(8.0.7)

The Riemann invariants of (8.0.1) are functions w(ρ, m) and z(ρ, m) satisfying the equations (wρ , wm ) · dF = λ2 (wρ , wm ) and One solution of (8.0.8) is  ρ
P (s) m + ds, w(u, v) = ρ s 0

(zρ , zm ) · dF = λ1 (zρ , zm ). (8.0.8)

m z(u, v) = − ρ



ρ 0

 P
(s) ds. s (8.0.9)

By simple calculations, m P

(ρ) 1  , )(1, λ1 )T − ρ2 2 P
(ρ) ρ ρP

(ρ) + 2P
(ρ)  , =− 2ρ P
(ρ)

(8.0.10)

m P

(ρ) 1  , )(1, λ2 )T + ρ2 2 P
(ρ) ρ ρP

(ρ) + 2P
(ρ)  . = 2ρ P
(ρ)

(8.0.11)

∇λ1 · r1 = (−

and ∇λ2 · r2 = (−

87 For the case of polytropic gas, we get m m − θρθ , λ2 = + θρθ , λ1 = ρ ρ w(u, v) = where θ =

γ−1 2 ,

∇λ1 · r1 = −

m + ρθ , ρ

z(u, v) =

m − ρθ , ρ

(8.0.12) (8.0.13)

and

(γ − 1)(γ + 1) γ−3 ρ , 4

∇λ2 · r2 =

(γ − 1)(γ + 1) γ−3 ρ . 4 (8.0.14)

Therefore it follows from (8.0.12) that for the case of polytropic gas, system (8.0.1) is strictly hyperbolic in the domain {(x, t) : ρ(x, t) > 0}, while it is nonstrictly hyperbolic in the domain {(x, t) : ρ(x, t) = 0}, since λ1 = λ2 when ρ = 0. From (8.0.14), system (8.0.1) is genuinely nonlinear if the adiabatic exponent γ ∈ (1, 3], while the system is no longer genuinely nonlinear at ρ = 0 if the adiabatic exponent γ > 3. Consider the Cauchy problem for the related parabolic system  ρt + mx = ερxx 2 (8.0.15) mt + ( mρ + P (ρ))x = εmxx , with initial data (ρε (x, 0), mε (x, 0)) = (ρε0 (x), mε0 (x)),

(8.0.16)

(ρε0 (x), mε0 (x)) = (ρ0 (x) + ε, ρ0 (x)u0 (x)) ∗ Gε

(8.0.17)

where

and Gε is a mollifier. Then (ρε0 (x), mε0 (x)) ∈ C ∞ × C ∞ , (ρε0 (x), mε0 (x)) → (ρ0 (x), m0 (x)) a.e., as ε → 0,

(8.0.18) (8.0.19)

and ε ≤ ρε0 (x) ≤ M1 ,

uε0 (x) = 

mε0 (x)  ≤ M1 , ρε0 (x)

(8.0.20)

for a suitable large constant M1 , which depends only on the L∞ bound of (ρ0 (x), u0 (x)), but is independent of ε. We have the main result in this chapter as follows:

88

CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

Theorem 8.0.1 Let the initial data (ρ0 (x), u0 (x)) be bounded measurable and ρ0 (x) ≥ 0, P (ρ) ∈ C 2 (0, ∞), P
(ρ) > 0, 2P
(ρ) + ρP

(ρ) ≥ 0 for ρ > 0; and  c
 ∞
P (ρ) P (ρ) dρ = ∞, dρ < ∞, ∀c > 0. ρ ρ c 0 Then for fixed ε > 0, the smooth viscosity solution (ρε (x, t), mε (x, t)) of the Cauchy problem (8.0.15), (8.0.16) exists and satisfies 0 < c(ε, t) ≤ ρε (x, t) ≤ M2 ,

uε (x, t) = 

mε (x, t)  ≤ M2 , (8.0.21) ρε (x, t)

where M2 is a positive constant, being independent of ε; c(ε, t) is a positive function, which could tend to zero as ε tends to zero or t tends to infinity. Moreover, for the polytropic gas and γ > 1, there exists a subsequence (still labelled) (ρε (x, t), ρε (x, t)uε (x, t)) which converges almost everywhere on any bounded and open set Ω ⊂ R × R+ : (ρε (x, t), ρε (x, t)uε (x, t)) → (ρ(x, t), ρ(x, t)u(x, t)), as ε ↓ 0+ , (8.0.22) where the limit pair of functions (ρ(x, t), ρ(x, t)u(x, t)) is a weak solution of the Cauchy problem (8.0.1), (8.0.2). The existence of viscosity solutions and related estimates (8.0.21) shall be proved in Section 8.1. For the case of adiabatic exponent γ > 3, the strong convergence (8.0.22) of a subsequence of (ρε (x, t), mε (x, t)) is proved in Sections 8.3-8.4. Finally, the proof of the strong convergence subsequence of (ρε (x, t), mε (x, t)) for the case of 1 < γ ≤ 3 is given in Section 8.5, where we introduce a different short proof for this case, but with two more assumptions (A1 ) and (A2 ) on viscosity solutions: (A1 ) The initial data (ρ0 (x), u0 (x)) is small; (A2 ) The viscosity solutions (ρε (x, t), mε (x, t)) of the Cauchy problem (8.0.15)-(8.0.16) have an a priori estimate ρε (x, t) ≥ c(t) > 0, where c(t) is independent of ε, but could tend to zero as t tends to infinity. Remark 8.0.2 The condition (A1 ) ensures that the viscosity solutions are small and so the support sets of the Young measures introduced in Theorem 2.2.1 are also small. The condition (A2 ) is remarked in [LPS], p. 629.

8.1. EXISTENCE OF VISCOSITY SOLUTIONS

8.1

89

Existence of Viscosity Solutions

To prove the existence of smooth viscosity solutions (ρε (x, t), mε (x, t)) for the Cauchy problem (8.0.15), (8.0.16), by Theorem 1.0.2, we only need to prove the a priori estimates given in (8.0.21). m

z = −M Σ3 ρ w=M

FIGURE 8.1 We multiply (8.0.15) by (wρ , wm ) and (zρ , zm ), respectively, to obtain wt + λ2 wx = εwxx +

2ε ε  (2P
+ ρP

)ρ2x , ρx wx − 2 ρ 2ρ P
(ρ)

(8.1.1)

2ε ε  (2P
+ ρP

)ρ2x . ρx zx + 2 ρ 2ρ P
(ρ)

(8.1.2)

and zt + λ1 zx = εzxx +

Then the assumptions on P (ρ) yield wt + λ2 wx ≤ εwxx +

2ε ρx wx ρ

(8.1.3)

CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

90 and

zt + λ1 zx ≥ εzxx +

2ε ρx zx . ρ

(8.1.4)

If we consider (8.1.3) and (8.1.4) as inequalities about the variables w and z, then we can get the estimates w(ρε , mε ) ≤ M, z(ρε , mε ) ≥ −M by applying the maximum principle to (8.1.3) and (8.1.4). This shows that the region (see Figure 8.1) Σ3 = {(ρ, m) : w(ρ, m) ≤ M, z(ρ, m) ≥ −M } ε is an invariant region. Thus we obtain the estimates √ 0
≤ ρ ≤ M2 and  P (ρ) ∞ dρ = ∞ and uε  ≤ M2 for a suitable constant M2 , since c ρ √  c P
(ρ) dρ < ∞ for any constant c > 0. 0 ρ

Since u is uniformly bounded, the last part of Theorem 1.0.2 gives the positive lower bound of ρ. Therefore we get the proof of the existence of smooth viscosity solutions for the Cauchy problem (8.0.15) and (8.0.16).

8.2

−1 Weak Entropies and Hloc Compactness

In this section, we shall first construct the weak entropy-entropy flux pairs (η, q) of system (8.0.1) for the polytropic case P (ρ) = cργ with 2 the exponent γ > 1 and c = k2 = (γ−1) 4γ , and then prove the com−1 (R × R+ ), with respect to pactness of η(ρε , mε )t + q(ρε , mε )x in Hloc the viscosity approximated solutions (ρε (x, t), mε (x, t)) of the Cauchy problem (8.0.15) and (8.0.16). Rewriting the second equation in (8.0.1) as ρt u + ρut + (ρu)x u + ρuux + P (ρ)x = 0

(8.2.1)

and substituting the first equation in (8.0.1) into (8.2.1), we get the following new system:   ρt + (ρu)x =0 ρ 1 P
(s) (8.2.2) ds)x = 0.  ut + ( u2 + 2 s 0

−1 8.2. WEAK ENTROPIES AND HLOC COMPACTNESS

91

If solutions have shock-waves, system (8.0.1) and system (8.2.2) are different. In Chapter 9 and Chapter 10, we shall study the weak solutions for the Cauchy problem (8.2.2) with bounded measurable initial data. However, for smooth solutions, system (8.2.2) is equivalent to system (8.0.1), and particularly, both systems have the same entropyentropy flux pairs. Thus any entropy-entropy flux pair (η(ρ, m), q(ρ, m)) of system (8.0.1) satisfies the additional system   u ρ , (8.2.3) (qρ , qu ) = (ηρ , ηu ) · P
(ρ) u ρ or equivalently qρ = uηρ +

P
(ρ) ηu , ρ

qu = ρηρ + uηu .

(8.2.4)

Eliminating the q from (8.2.4), we have ηρρ =

P
(ρ) ηuu . ρ2

(8.2.5)

An entropy η(ρ, u) of system (8.0.1) is called a weak entropy if η(0, u) = 0, that is, a solution of Equation (8.2.5) with the special initial conditions: η(ρ = 0, u) = 0,

ηρ (ρ = 0, u) = g(u),

(8.2.6)

where g(u) is an arbitrary given function. The solution of (8.2.5)-(8.2.6) is given by the following lemma: Lemma 8.2.1 For ρ ≥ 0, u, w ∈ R, let G(ρ, w) = (ργ−1 − w2 )λ+ ,

λ=

3−γ , 2(γ − 1)

(8.2.7)

where the notation x+ = sup(0, x). Then the solution of (8.2.5)-(8.2.6) is given by the formula  w [(w − s)(s − z)]λ g(s)ds η(w, z) = z

= (w − z)

2 γ−1



1 0

(8.2.8) [τ (1 − τ )]λ g(w − (w − z)τ )dτ

92

CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

or

 η(ρ, u) =

R

=2

g(ξ)G(ρ, ξ − u)dξ

2 γ−1

 ρ

0

1

(8.2.9) [τ (1 − τ )]λ g(u + ρθ − 2ρθ τ )dτ ;

and the weak entropy flux q of system (8.0.1) associated with η is  g(ξ)[θξ + (1 − θ)u]G(ρ, ξ − u)dξ, (8.2.10) q(ρ, u) = R

γ−1 2 .

Moreover, if the initial data g(u) ∈ C 1 (R), then the where θ = weak entropy η satisfies the following estimates: |ηρ (ρ, m)| ≤ M,

|ηm (ρ, m)| ≤ M,

(8.2.11)

where M depends only on the L∞ bound M2 of (ρ, u). Proof. Exactly analogous to (6.2.6), the entropy of system (8.0.1) also satisfies the following equation: ηwz +

λ2z λ1w ηw − ηz = 0, λ2 − λ1 λ2 − λ1

(8.2.12)

where λ1 , λ2 , w, z are given by (8.0.12)-(8.0.13). By simple calculations, we have λ1 =

γ+1 3−γ w+ z, 4 4

λ2z = λ1w =

3−γ , 4

λ2 =

3−γ γ+1 w+ z, 4 4

(8.2.13)

γ−1 (w − z), 2

(8.2.14)

λ2 − λ1 =

and ηρ (ρ, u) = θρθ−1 (η(w, z)w − η(w, z)z ) w−z b ) (η(w, z)w − η(w, z)z ), = θ( 2

(8.2.15)

where b = γ−3 γ−1 . Therefore the entropy equation (8.2.12) is reduced to the following Darboux-Euler-Poisson equation: ηwz +

λ (ηw − ηz ) = 0. w−z

(8.2.16)

−1 8.2. WEAK ENTROPIES AND HLOC COMPACTNESS

93

The weak entropy η satisfies the conditions lim

(w−z)→0

η(w, z) = 0,

(8.2.17)

and lim

(w−z)→0

θ(

w − z γ−3 ) γ−1 (η(w, z)w − η(w, z)z ) = g(w). 2

(8.2.18)

Therefore, by the theory of Darboux-Euler-Poisson equation (cf. [Bi]), we get that the weak entropy is of the representation formula (8.2.8), which is in the form (8.2.9) if we consider it to be the function of (ρ, u). Using the second equation in (8.2.4) and the weak solution formula (8.2.9), we have  1 2 γ−1 θ [τ (1 − τ )]λ g
(u + ρθ − 2ρθ τ )(1 − 2τ )ρθ+1 dτ qu = η + 2 0



2

+2 γ−1 u

1 0

[τ (1 − τ )]λ g
(u + ρθ − 2ρθ τ )dτ. (8.2.19)

Since



u

ug
(u + ρθ − 2ρθ τ )du

ug(u +

ρθ

−

we get from (8.2.19) that   u ηdu + η − q =

+2  = z

w

2 γ−1

 θ

0

1

2ρθ τ )

 −

u

g(u + ρθ − 2ρθ τ )du,

u

ηdu

[τ (1 − τ )]λ g(u + ρθ − 2ρθ τ )(1 − 2τ )ρθ+1 dτ

[(w − s)(s − z)]λ (1 − θ)u + θs)g(s)ds, (8.2.20)

and hence the proof of (8.2.10). Noticing the last part in (8.2.9), we get the boundedness of ∇η and hence, the proof of Lemma 8.2.1.

94

CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

Lemma 8.2.2 For any weak entropy η(ρ, m) of system (8.0.1), ε(ρεx , mεx ) · ∇2 η(ρε , mε ) · (ρεx , mεx )T is bounded in L1loc (R × R+ ), where  ηρρ (ρε , mε ) ηρm (ρε , mε ) 2 ε ε . ∇ η(ρ , m ) = ηmρ (ρε , mε ) ηmm (ρε , mε )

(8.2.21)

(8.2.22)

Proof. For simplicity, we omit the superscript ε in the viscosity solutions (ρε , mε ). It is easy to check that system (8.0.1) has a convex entropy η =

k m2 + ργ , 2ρ γ−1

(8.2.23)

and hence the boundedness of ε(ρx , mx ) · ∇2 η  (ρ, m) · (ρx , mx )T

(8.2.24)

in L1loc (R × R+ ) can be obtained by using the same technique as given in (6.3.2) or (7.3.1). Then it follows that 1 m εkγργ−2 (ρx )2 + ε [ ρx − mx ]2 ρ ρ is bounded in L1loc (R × R+ ). Since 1 m kγργ−2 (ρx )2 + [ ρx − mx ]2 = kγργ−2 (ρx )2 + ρ(ux )2 , ρ ρ

(8.2.25)

(8.2.26)

we get the boundedness of εργ−2 (ρx )2 ,

1 m ε [ ρx − mx ]2 , ρ ρ

ερ(ux )2

(8.2.27)

in L1loc (R × R+ ). By simple calculations, for a weak entropy η represented by (8.2.9), we have that  1 m [τ (1 − τ )]λ g( + ρθ − 2ρθ τ )dτ ηρ (ρ, m) = ρ 0  1 m m [τ (1 − τ )]λ g
( + ρθ − 2ρθ τ )(− + (1 − 2τ )θρθ )dτ, + ρ ρ 0 (8.2.28)

−1 8.2. WEAK ENTROPIES AND HLOC COMPACTNESS

 ηm (ρ, m) =

1

0

[τ (1 − τ )]λ g
(

m + ρθ − 2ρθ τ )dτ, ρ

95 (8.2.29)

ηρρ (ρ, m)  1 m = [τ (1 − τ )]λ g
( + ρθ − 2ρθ τ )[(1 − 2τ )(θ + θ 2 )ρθ−1 dτ ρ 0  + 0

1

[τ (1 − τ )]λ g

(

m m + ρθ − 2ρθ τ )ρ(− 2 + (1 − 2τ )θρθ−1 )2 dτ ρ ρ

= I1 + I2 + I3 + I4 , (8.2.30) where I1 =



m2 ρ3

I2 = θ 2 ρ2θ−1

1 0



1 0

2

[τ (1 − τ )]λ g

(

[τ (1 − τ )]λ g

(

θ−1

I3 = (θ + θ )ρ

I4 = 2θuρ

θ−1

 0

1



1 0

m + ρθ − 2ρθ τ )dτ, ρ

m + ρθ − 2ρθ τ )(1 − 2τ )2 dτ, ρ

[τ (1 − τ )]λ (1 − 2τ )g
(

[τ (1 − τ )]λ (1 − 2τ )g

(

(8.2.31)

(8.2.32)

m + ρθ − 2ρθ τ )dτ, ρ (8.2.33)

m + ρθ − 2ρθ τ )dτ ; ρ

(8.2.34)

ηρm (ρ, m)  1 m m [τ (1 − τ )]λ g

( + ρθ − 2ρθ τ )(− 2 + (1 − 2τ )θρθ−1 )dτ = ρ ρ 0 = I5 + I6 , (8.2.35) where m I5 = − 2 ρ

 0

1

[τ (1 − τ )]λ g

(

m + ρθ − 2ρθ τ )dτ, ρ

(8.2.36)

CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

96

I6 = θρ

θ−1



1 0

[τ (1 − τ )]λ (1 − 2τ )g

(

m + ρθ − 2ρθ τ )dτ ; ρ

(8.2.37)

and ηmm (ρ, m) =

1 ρ

 0

1

[τ (1 − τ )]λ g

(

m + ρθ − 2ρθ τ )dτ = I7 . ρ

(8.2.38)

Since 2θ − 1 = γ − 2, then using (8.2.27), we have that εI2 (ρx )2 is bounded in L1loc (R × R+ ). τ Let l(τ ) = 0 [s(1 − s)]λ (1 − 2s)ds. Then it is easy to see that l(0) = l(1) = 0 and hence  1 m θ−1 [τ (1 − τ )]λ (1 − 2τ )g
( + ρθ − 2ρθ τ )dτ ρ ρ 0 (8.2.39)  1

m θ θ 2θ−1 l(τ )g ( + ρ − 2ρ τ )dτ, = 2ρ ρ 0 ρθ−1



1

m [τ (1 − τ )]λ (1 − 2τ )g

( + ρθ − 2ρθ τ )dτ ρ 0  1 m l(τ )g


( + ρθ − 2ρθ τ )dτ. = 2ρ2θ−1 ρ 0

(8.2.40)

Therefore, εI3 (ρx )2 and εI4 (ρx )2 are bounded in L1loc (R × R+ ). About I6 , we have εI6 ρx mx = εuI6 (ρx )2 + εI6 ρρx ux .

(8.2.41)

From (8.2.27), it is easy to see that the first term in the right-hand side of (8.2.41) is bounded in L1loc (R × R+ ), and the second is also bounded in L1loc (R × R+ ), since |εI6 ρρx ux | = |εθρθ



1 0

[τ (1 − τ )]λ (1 − 2τ )g

(

m + ρθ − 2ρθ τ )dτ ρx ux | ρ

≤ εθ(ρ2θ−1 (ρx )2 + ρ(ux )2 )  ×|

1 0

[τ (1 − τ )]λ (1 − 2τ )g

(

m + ρθ − 2ρθ τ )dτ |. ρ

−1 8.2. WEAK ENTROPIES AND HLOC COMPACTNESS

97

Moreover ε(I1 (ρx )2 + 2I5 ρx mx + I7 (mx )2 ) is clearly bounded in L1loc (R×R+ ) from the second estimate in (8.2.27). Thus we get the proof of Lemma 8.2.2.

Lemma 8.2.3

  K

(ερεx )2 dxdt → 0 as ε → 0,

where K is any bounded set in R × R+ . Proof. We omit the superindex ε. Let Ω1 = {(x, t) ∈ R × R+ : ρ < δ} and Ω2 = {(x, t) ∈ R×R+ : ρ ≥ δ}. Then for a fixed constant δ ∈ (0, 1), Ω1 is an open set in R × R+ . Let K1 be any compact set in Ω1 and choose φ ∈ C02 (Ω1 ) with a compact support set S ⊂ Ω1 and φ = 1 on K1 , 0 ≤ φ ≤ 1. Multiplying the first equation in (8.0.15) by 2ρ, we have (ρ2 )t + 2(ρ2 u)x − 2ρuρx = ε(ρ2 )xx − 2ε(ρx )2 .

(8.2.42)

Multiplying (8.2.42) by the function φ and then integrating in R × R+ , we get  ∞ ∞ 2ε(ρx )2 φdxdt 0

−∞



∞ ∞

= 0

−∞

(ρ2 φt + 2uρ2 φx + ερ2 φxx + 2ρuρx φ)dxdt

≤ Cδ2 + Cδ

 0

∞ ∞ −∞

(ρx )2 φdxdt

(8.2.43)

1 2

,

where C is a suitable positive constant independent of ε, δ. Thus from (8.2.43) we have    ∞ ∞ 2 2 ε (ρx ) φdxdt = ε2 (ρx )2 φdxdt ≤ Cδ2 , 0

−∞

S

(8.2.44)

98

CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

or

  K1

ε2 (ρx )2 dxdt ≤ Cδ2 .

(8.2.45)

Since εργ−2 (ρx )2 s bounded in L1loc (R × R+ ), then for fixed δ, we have   (ερεx )2 dxdt → 0 as ε → 0, (8.2.46) K2

where K2 is any bounded set in Ω2 . Combining (8.2.45) and (8.2.46) yields the proof of Lemma 8.2.3. Theorem 8.2.4 For any weak entropy pair (η, q) of system (8.0.1), −1 (R × R+ ), with respect to the η(ρε , mε )t + q(ρε , mε )x is compact in Hloc viscosity solutions (ρε (x, t), mε (x, t)). Proof. Multiplying system (8.0.15) by (ηρ , ηm ), we have η(ρε , mε )t + q(ρε , mε )x = εη(ρε , mε )xx − ε(ρεx , mεx ) · ∇2 η(ρε , mε ) · (ρεx , mεx )T

(8.2.47)

= I1 + I2 . By the boundedness of the viscosity solutions (ρε , uε ), the left-hand side of (8.2.47) is bounded in W −1,∞ ; by Lemma 8.2.2, I2 is bounded −1 (R × R+ ) in L1loc (R × R+ ) and by Lemma 8.2.3, I1 is compact in Hloc since 1

|η(ρ, m)x | = |ηρ ρx + ηm mx | ≤ C(|ρx | + (ρ) 2 |ux |). Therefore, the proof of Theorem 8.2.4 is completed by Theorem 2.3.2.

8.3

The Case of γ > 3

Theorem 8.3.1 Let γ > 3 and ρε , uε be viscosity solutions given by (8.0.15)-(8.0.16). Then a subsequence of ρε (still labelled ρε ) converges pointwisely to ρ, and (a subsequence of ) uε converges pointwisely to u on the set {ρ(x, t) > 0}. In particular, ρε uε converges pointwisely to ρu.

8.3. THE CASE OF γ > 3

99

Proof. Taking two smooth functions g(ξ1 ), h(ξ2 ) in (8.2.9)-(8.2.10) and using Theorem 2.1.4 and Theorem 2.3.2, we get   g(ξ1 )G(ξ1 )dξ1 h(ξ2 )[θξ2 + (1 − θ)u]G(ξ2 )dξ2 



−

h(ξ2 )G(ξ2 )dξ2

g(ξ1 )[θξ1 + (1 − θ)u]G(ξ1 )dξ1 (8.3.1)

 g(ξ1 )h(ξ2 )G(ξ1 )[θξ2 + (1 − θ)u]G(ξ2 )dξ1 dξ2

=  −

g(ξ1 )h(ξ2 )G(ξ1 )[θξ1 + (1 − θ)u]G(ξ2 )dξ1 dξ2 .

The last equality holds for arbitrary functions g, h, and this yields G(ξ1 ) [θξ2 + (1 − θ)u]G(ξ2 ) − G(ξ2 ) [θξ1 + (1 − θ)u]G(ξ1 ) = G(ξ1 )[θξ2 + (1 − θ)u]G(ξ2 ) − G(ξ2 )[θξ1 + (1 − θ)u]G(ξ1 ) = θ(ξ2 − ξ1 )G(ξ1 )G(ξ2 ). (8.3.2) Here and below we use the overbar to indicate the usual integration with respect to the Young measure; for instance  G(ξ) = G(ρ, u − ξ)dνx,t (ρ, u). We may rewrite (8.3.2) as     1 G(ξ1 )G(ξ2 ) uG(ξ2 ) uG(ξ1 ) θ −1 = − , 1 − θ G(ξ1 ) G(ξ2 ) ξ2 − ξ1 G(ξ2 ) G(ξ1 )

(8.3.3)

for ξ1 , ξ2 ∈ ζ, where ζ is any open connected component in the union of the open intervals (u − ρθ , u + ρθ ), for which (ρ, u) ∈ supp ν. The first step of the proof is to show that for γ > 3, uG(ξ) G(ξ)

is a nonincreasing function of ξ ∈ ζ.

(8.3.4)

100 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS We let f0 (ξ) abbreviate f0 (ξ) = equivalent form

G(ξ)−G(ξ) , G(ξ)

so that (8.3.3) takes the

  1 uG(ξ2 ) uG(ξ1 ) θ − . f0 (ξ1 )f0 (ξ2 ) = 1−θ ξ2 − ξ1 G(ξ2 ) G(ξ1 )

(8.3.5)

Sending ξ2 to ξ1 = ξ in (8.3.5), we should end up with ∂  uG(ξ)  θ , f02 (ξ) = 1−θ ∂ξ G(ξ)

(8.3.6)

and (8.3.4) follows, since γ−1 θ = < 0, 1−θ 3−γ and hence the left-hand side of (8.3.6) is nonpositive for γ > 3. Notice that there is no difficulty passing to the limit on the righthand side of (8.3.5) in the sense of distributions since G(ξ) does not vanish on ζ. In order to pass to the limit on the left-hand side in L2loc (ζ), we require f0 (ξ) and hence G(ξ) ∈ L2 (Rξ ); but since G(ξ)2L2 (Rξ ) = 5−γ  1 ρ 2 −1 (1 − τ 2 )2λ dτ , this requirement of L2 (Rξ )-integrability restricts the range of γ with γ < 5. To extend the statement of (8.3.4) for all γ > 3, we first mollify both sides of (8.3.5) with a unit mass mollifier, ψα (ξ) ≥ 0, and denote fα = f0 ∗ ψα . Then we have   1 uG(ξ2 ) uG(ξ1 ) θ − ∗ ψα (ξ1 ) ∗ ψα (ξ2 ). fα (ξ1 )fα (ξ2 ) = 1−θ ξ2 − ξ1 G(ξ2 ) G(ξ1 ) (8.3.7) Thanks to the boundedness of the left-hand side and the smoothness of the right-hand side, we may now take ξ2 = ξ1 = ξ, to find out that   1 uG(ξ ) uG(ξ ) θ 2 1 − ∗ ψα (ξ1 ) ∗ ψα (ξ2 ) |ξ2 =ξ1 =ξ . f 2 (ξ) = 1−θ α ξ2 − ξ1 G(ξ2 ) G(ξ1 ) (8.3.8) If we now let α tend to zero, then the left-hand side of (8.3.8) yields a negative measure since 1 − θ < 0, whereas the right-hand side tends to ∂  uG(ξ)  , ∂ξ G(ξ)

8.3. THE CASE OF γ > 3

101

which recovers the desired (8.3.4). To complete the proof of Theorem 8.3.1, we need the following two necessary lemmas. The first one is about the construction of the set ζ: Lemma 8.3.2 Let the set ζ be the union of all open intervals (ui − ρθi , ui + ρθi ) for all points (ρi , ui ) ∈ supp ν. Then ζ must be open and connected. The connection in Lemma 8.3.2 is not obvious. For instance, if the support set of ν contains only two points (ρ1 , u1 ) and (ρ2 , u2 ) satisfying u1 + ρθ1 < u2 − ρθ2 , then clearly ζ consists of two disjoint open intervals (u1 − ρθ1 , u1 + ρθ1 ) and (u2 − ρθ2 , u2 + ρθ2 ). Proof of Lemma 8.3.2. To prove Lemma 8.3.2, let the support set S of the Young measure ν determined by the sequence of viscosity solutions to the Cauchy problem (8.0.15), (8.0.16) be contained in the smallest characteristic triangle: Σ4 = {(ρ, u) : w ≤ w0 , z ≥ z0 , ρ ≥ 0}. Then clearly z0 ≤ z ≤ w ≤ w0 (see Figure 8.2). If the support set S of ν is not wholly contained in the vacuum line ρ = 0, then we now show that the intersection point P0 of the lines w = w0 and z = z0 must be in S, i.e., P0 ∈ supp ν.

(8.3.9)

If this is done, then ζ must be the open interval (z0 , w0 ). For the polytropic case P (ρ) = cργ with the exponent γ > 1 and c = γ1 , the entropy equation (8.2.5) is reduced to ηρρ = ργ−3 ηuu .

(8.3.10)

If k denotes a positive constant, then the function η = h(ρ)eku solves (8.3.10) provided that h

(ρ) = k2 ργ−3 h.

(8.3.11)

102 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS

z ρ=0

w Σ4 = 0

(w0 , z0 )

z = z0

w = w0

FIGURE 8.2

1

1

2k 2 (γ−1) ρ . Then h = a(ρ)φ(s) solves (8.3.11) Let a(ρ) = ρ 4 (3−γ) , s = γ−1 if and only if φ satisfies the standard Fuchsian equation

φ

− (1 + µs2 )φ = 0, where µ =

4−(γ−1)2 4(γ−1)2

(8.3.12)

< 0.

The second equation in (8.2.4) is qu = ρηρ + uηu .

(8.3.13)

ηk = h(ρ)eku ,

(8.3.14)

If

then

(qk )u = ρh
(ρ)eku + kuh(ρ)eku

8.3. THE CASE OF γ > 3

103

and hence one entropy flux corresponding to ηk is qk = uh(ρ)eku + (ρh
− h)eku /k  
1 = ηk u + ρ 2 (γ−1) φφ − γ+1 4k .

(8.3.15)

η−k = h(ρ)e−ku ,

(8.3.16)

If

then

(q−k )u = ρh
(ρ)e−ku − kuh(ρ)e−ku

and hence one entropy flux corresponding to η−k is q−k = uh(ρ)e−ku + (h − ρh
)e−ku /k  
1 = η−k u − ρ 2 (γ−1) φφ + γ+1 4k .

(8.3.17)

Then two progressing waves of system (8.0.1) for the case of P (ρ) = are provided by (8.3.14)-(8.3.15) and (8.3.16)-(8.3.17). We may use the method of Frobenius again to obtain a series solution of Equation (8.3.12) as follows: 1 γ γρ

φ = sj

∞ 

en sn ,

(8.3.18)

n=0 γ+1 > 0 is the larger root where e0 can be any positive constant, j = 2(γ−1) of the equation j(j − 1) = µ and en satisfy

en =

en−1 , for n ≥ 1. (2n + j)(2n + j − 1) − µ

(8.3.19)

It is easy to check that φ(s) > 0 and φ
(s) > 0. Let ηk = a(ρ)φ(s)eku . Then using Lemma 6.2.1, we have 1 ηk = a(ρ)φ(s)e−s ekw = ekw (a(ρ) + O( )) k

(8.3.20)

on ρ > 0 as k approaches infinity and the corresponding flux function is of the form

1 1 γ + 1 φ
(s) − 1) − qk = ηk u + ρ 2 (γ−1) + ρ 2 (γ−1) ( φ(s) 4k (8.3.21) 1 γ+1 + O( 2 )) = ηk (λ2 − 4k k

104 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS on ρ > 0. Similarly let η−k = a(ρ)φ(s)e−ku . Then 1 η−k = a(ρ)φ(s)e−s e−kz = e−kz (a(ρ) + O( )) k

(8.3.22)

on ρ > 0 and the corresponding flux function

1 1 γ + 1 φ
(s) − 1) + q−k = η−k u − ρ 2 (γ−1) − ρ 2 (γ−1) ( φ(s) 4k 1 γ+1 + O( 2 )) = η−k (λ1 + 4k k

(8.3.23)

on ρ > 0. It is clear that the entropies η±k given in (8.3.20) and (8.3.22) are weak entropies, since they are zero at the vacuum line ρ = 0. Using these two positive progressing entropy pairs (8.3.20)(8.3.21) and (8.3.22)-(8.3.23), we can complete the proof of (8.3.9). We shall argue by a contradiction. Suppose that the vertex P0 does not lie in supp ν. Using (η±k , q±k ) to the measure equation (6.4.2), we have < ν, ηk q−k − η−k qk > < ν, q−k > < ν, qk > − = . < ν, η−k > < ν, ηk > < ν, η−k >< ν, ηk >

(8.3.24)

Notice that 1 ηk q−k − η−k qk = ek(w−z) {(λ2 − λ1 )a(ρ) + O( )} k

(8.3.25)

/ S. Then we have and (w0 , z0 ) ∈ | < ν, ηk q−k − η−k qk > | ≤ a0 ek(w0 −z0 −δ0 )

(8.3.26)

for appropriate positive constants a0 , δ0 and sufficiently large k. On the other hand, δ0

| < ν, ηk > | ≥ a1 ek(w0 − 4 ) ,

δ0

| < ν, η−k > | ≥ a2 e−k(z0 + 4 )

(8.3.27)

for appropriate positive constants a1 , a2 since S is minimal. Therefore  < ν, η q − η q >  a0 − kδ0   k −k −k k e 2 → 0, k → ∞. (8.3.28) ≤  < ν, η−k >< ν, ηk > a1 a2 Similar to Section 6.4, we introduce two new probability measures µ± k on S defined by < µ± k , h >=< ν, hη±k > / < ν, η±k >

(8.3.29)

8.3. THE CASE OF γ > 3

105

where h = h(u, v) denotes an arbitrary continuous function. Clearly − µ+ k and µk both are uniformly bounded with respect to k. Then as a consequence of weak-star compactness, there exist probability measures µ± on S such that < µ± , h >= lim < µ± k , h >, k→∞

(8.3.30)

after the selection of an appropriate subsequence. Moreover, the measures µ+ , µ− are respectively concentrated on the boundary sections of S associated with w and z, i.e.,  (8.3.31) supp µ+ = S {(ρ, u) : w = w0 } and supp µ− = S



{(ρ, u) : z = z0 }.

(8.3.32)

Then for the left-hand side of (8.3.24), there holds < ν, q−k > < ν, qk > − →< µ− , λ1 > − < µ+ , λ2 > as k → ∞. < ν, η−k > < ν, ηk > (8.3.33) But λ1w = λ2z =

3−γ > 0, 4

which implies that λ2 (w0 , z) ≥ λ2 (w0 , z0 ) > λ1 (w0 , z0 ) ≥ λ1 (w, z0 ) and hence

< µ− , λ1 > − < µ+ , λ2 > < 0,

which is in contradiction with (8.3.24) and (8.3.28). This completes the proof of (8.3.9) and hence that of Lemma 8.3.2. The second lemma is stated as: Lemma 8.3.3 Let ζ = (z0 , w0 ) stand for the open connected component in Lemma 8.3.2, and let u0 = (z0 + w0 )/2. Then lim

ξ→w0

uG(ξ) G(ξ)

≥ u0 ,

lim

ξ→z0

uG(ξ) G(ξ)

≤ u0 .

(8.3.34)

106 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS Proof. According to (8.3.4), uG/G is a monotone function on ζ, and we turn to consider its one-side limits as ξ → w0 and ξ → z0 . The values of (ρ, u) such that G(ξ) > 0 in an interval (w0 − ε, w0 ) satisfy u + ρθ ≥ w0 − ε, and therefore, since w0 ≤ u − ρθ for these (ρ, u) values, we have lim

ξ→w0

uG(ξ) G(ξ)

≥ min{u; (ρ, u) ∈ supp ν, u + ρθ = w0 } w0 + z0 . ≥ 2

(8.3.35)

A similar argument holds for z0 , thus we finish the proof of Lemma 8.3.3. Now we are in the position to complete the proof of Theorem 8.3.1. Combining (8.3.34) with (8.3.35) we obtain that uG(ξ)/G(ξ) is a constant, which in turn tells us, by (8.3.8), that fα2 (ξ) = 0. Hence, fα (ξ) vanishes on the support of ν and in particular, by letting α → 0, so does f0 (ξ): f0 (ξ) =

G(ρ, u − ξ) G(ξ)

− 1 = 0,

(ρ, u) ∈ supp ν.

(8.3.36)

This shows that on the set {ρ > 0}, the Young measure ν is reduced to a Dirac mass and the conclusion holds as usual. This completes the proof of Theorem 8.3.1.

8.4

The Case of 1 < γ ≤ 3

From the entropy-entropy flux equations (8.2.3) of system (8.0.1) and by simple calculations, we have the following four weak pairs of entropyentropy flux to system (8.0.1): (η1 , q1 ) = (ρ, m),

(η2 , q2 ) = (m,

m2 + P (ρ)), ρ

(8.4.1)

(8.4.2)

8.4. THE CASE OF 1 < γ ≤ 3 (η3 , q3 ) =

m2 2ρ



ρ

+ρ

m3

107

P (s) m3 P (ρ) + ds, +( 2 2 s 2ρ ρ





ρ

 P (s) ds)m , s2 (8.4.3)

ρ

P (s) ds, s2  ρ P (ρ) 2 P (s) m4 + 3( 2 + ds)m2 ρ3 ρ ρ s2

(η4 , q4 ) =

ρ2

+ 6m

 +6(P (ρ) Let v = (ρ, m),

ρ

P (s) ds − s2



ρ

(8.4.4)

P 2 (s)  ds) . s2

T  m2 + P (ρ) , f (v) = m, ρ

v¯ = (¯ ρ, m) ¯ = (< ν, ρ >, < ν, m >),

u ¯ = m/¯ ¯ ρ

and 

Qη = η(v) − η(¯ v ) − ∆η(¯ v )(v − v¯) v ) − ∆η(¯ v )(f (v) − f (¯ v )). Q q = q(v) − q(¯

(8.4.5)

Then (Qηi , Q qi )(i = 1, 2, 3, 4) are also the entropy-entropy flux pairs of system (8.0.1) and satisfy the measure equation         Qη Q qi   =  < ν, Qηi > < ν, Q qi > 

ν,  i     Qηj Q qj < ν, Qηj > < ν, Q qj >

(i, j = 1, 2, 3, 4; i = j), (8.4.6)

for the polytropic gas, i.e., if P takes the special form P (ρ) = cργ , where γ > 1. For fixed (x, t), v¯ is a scalar vector, thus it is clear that Qη1 = ρ − ρ¯, ¯ Qη2 = m − m,

Q q1 = m − m, ¯

Q q2 = ρu2 + P (ρ) − ρ¯u ¯2 − P (¯ ρ)

are also entropy-entropy flux pairs of system (8.0.1).

(8.4.7)

(8.4.8)

108 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS By simple calculations from (8.4.5), there hold: Qη3

1 1 2 1 2 ¯ + u ¯ (ρ − ρ¯) − u = ρu2 − ρ¯u ¯(m − m) ¯ +ρ 2 2 2 

P (s) ds − s2  1 2 ¯) + Q(ρ = ρ(u − u 2 ρ¯

−¯ ρ



ρ

ρ¯



ρ

P (s) ds s2

 P (¯ ρ) P (s) (ρ − ρ ¯ ) ds + s2 ρ¯

P (s) ds), s2 (8.4.9)

1 ¯)2 + (u − u ¯)(P (ρ) − P (¯ ρ)) Q q3 = m(u − u 2  ρ P (¯ ρ) P (s) (ρ − ρ¯)), ds − +u(ρ 2 s ρ¯ ρ¯  Qη4 = 6m

Q q

4

=

ρ ρ¯

(8.4.10)

6¯ u P (s) P (¯ ρ)(ρ − ρ¯), ds + ρ(u − u ¯)2 (u + 2¯ u) − s2 ρ¯ (8.4.11) 

ρ



ρ

P 2 (s) ds s2 ρ¯ ρ¯ 6¯ uP (¯ ρ) (m − m). ¯ ¯2 ) + uρ(u − u ¯)2 (u + 2¯ u) − +3P (ρ)(u2 − u ρ¯ (8.4.12)

(6u2 ρ

+ 6P (ρ))

P (s) ds − 6 s2

It follows from (8.4.6) that < ν, Qη1 Q q2 − Qη2 Q q1 > =< ν, (P (ρ) − P (¯ ρ))(ρ − ρ¯) − (u − u ¯)2 ρ¯ρ >

(8.4.13)

=0 and < ν, Qη1 Q q3 − Qη3 Q q1 > =< ν, (u − u ¯)(ρ − ρ¯)P (ρ) − ρ¯ρ(u − u ¯)



ρ ρ¯

1 P (s) ¯)3 ds − ρ¯ρ(u − u s2 2

= 0, (8.4.14)

8.4. THE CASE OF 1 < γ ≤ 3

109

< ν, Qη1 Q q4 − Qη4 Q q1 > ¯2 )P (ρ)) − ρ¯ρ(u − u ¯)3 (u + 2¯ u) =< ν, 3(ρ − ρ¯)(2Q2 + (u2 − u  −6¯ ρρu(u − u ¯)

ρ ρ¯

P (s) ds > s2

= 0, (8.4.15) where

 Q2 = P (ρ)

ρ ρ¯

P (s) ds − s2



ρ ρ¯

P 2 (s) ds. s2

(8.4.16)

Calculate (8.4.15) − 6¯ u × (8.4.14). We have ¯)2 P (ρ) − ρ¯ρ(u − u ¯)4 < ν, 3(ρ − ρ¯)(2Q2 + (u − u  ρ P (s) −6¯ ρρ(u − u ¯)2 ds > s2 ρ¯ = 0.

(8.4.17)

It follows from (8.4.14) that ¯)3 > < ν, 12 ρ¯ρ(u − u



< ν, u − u ¯)(ρ − ρ¯)P (ρ) − ρ¯ρ(u − u ¯)

ρ ρ¯

P (s) ds > . s2

(8.4.18)

Using this and the measure equation < ν, Qη2 Q q3 − Qη3 Q q2 > =< ν, Qη2 >< ν, Q q3 > − < ν, Qη3 >< ν, Q q2 >, we have



(8.4.19)

ρ

P (s) ds > s2 ρ¯ + < ν, P (ρ) >< ν, Q1 > − < ν, P (ρ)Q1 > 1 1 ¯)2 > − < ν, ρP (ρ)(u − u ¯)2 > + < ν, P (ρ) >< ν, ρ(u − u 2 2 ρ)) > + < ν, ρ(u − u ¯)2 (P (ρ) − P (¯

< ν,

ρu2

−

ρ¯u ¯2

>< ν,

1 2 ρ(u

−

u ¯)2

+

= 0, (8.4.20)

110 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS where  Q1 = ρ

ρ ρ¯

P (¯ ρ) P (s) (ρ − ρ¯). ds − 2 s ρ¯

By assumption ρ ≥ ρ0 > 0, then ρ¯ ≥ ρ0 > 0 also. Define: ¯)i (ρ − ρ¯)j ) >, where O denotes an L∞ bound. Oij =< ν, O((u − u By simple calculations, there hold 1 P
(¯ ρ) ¯)2 + (ρ − ρ¯)2 + O21 + O03 , Qη3 = ρ¯(u − u 2 2¯ ρ

(8.4.21)

Qη4 =

3¯ uP
(¯ 6P (¯ ρ) ρ) (u − u ¯)(ρ − ρ¯) + (ρ − ρ¯)2 ρ¯ ρ¯ +3¯ uρ¯(u − u ¯)2 + O21 + O30 + O12 + O03 .

(8.4.22)

u ¯P
(¯ ρ) (ρ − ρ¯)2 ρ)(u − u ¯)(ρ − ρ¯) + Q q3 = P
(¯ 2¯ ρ u ¯ρ¯ ¯)2 + O21 + O12 + O30 + O03 + (u − u 2

(8.4.23)

and
ρ) 3P
(¯ ρ)P (¯ ρ) 2 P (¯ )(ρ − ρ¯)2 + 3¯ u ρ¯2 ρ¯ 6¯ uP (¯ ρ) )(u − u ¯)(ρ − ρ¯) +(6¯ uP
(¯ ρ) + ρ¯ 2 2 ρ))(u − u ¯) + O30 + O21 + O12 + O03 . +3(¯ ρu ¯ + P (¯

Q q 4 = (

(8.4.24)

It follows from (8.4.17) that < ν, (

ρ)P (¯ ρ) + (P
(¯ ρ))2 2P
(¯ ρ)P (¯ ρ) 2P

(¯ − )(ρ − ρ¯)4 2 3 ρ¯ ρ¯

3P
(¯ ρ)P (¯ ρ) (ρ − ρ¯)3 − ρ¯2 (u − u ¯)4 − 3P (¯ ρ)(u − u ¯)2 (ρ − ρ¯) > 2 ρ¯ +O05 + O41 + O23 = 0, (8.4.25) +

8.4. THE CASE OF 1 < γ ≤ 3

111

and from (8.4.20) that (

ρ) ρ¯

P
(¯ ρ)) < ν, (ρ − ρ¯)2 >< ν, (u − u ¯)2 > + P (¯ 2 4 +(

P
(¯ ρ) ρ¯

+ P (¯ ρ)) < ν, (ρ − ρ¯)2 (u − u ¯)2 > 2 4

+

ρ))2 − 5¯ ρP
(¯ ρ)P

(¯ ρ) 2(P
(¯ < ν, (ρ − ρ¯)4 > 2 12¯ ρ

+

P

(¯ ρ)P
(¯ ρ) ρ¯2 (< ν, (u − u ¯)2 >)2 + (< ν, (ρ − ρ¯)2 >)2 2 4¯ ρ

1 (P
(¯ ρ))2 < ν, (ρ − ρ¯)3 > + ρ¯P
(¯ ρ) < ν, (ρ − ρ¯)(u − u ¯)2 > − 2 2¯ ρ +O21 O20 + O20 O03 + O02 O03 + O21 O02 + O05 + O23 = 0. (8.4.26) Calculate (8.4.25) + (8.4.26) ×

6P (¯ ρ) ρ¯P
(¯ ρ) .

Then

 2(P
(¯ ρ))2 − P (¯ ρ)P

(¯ ρ) P
(¯ ρ)P (¯ ρ)  − < ν, (ρ − ρ¯)4 > 2 3 2¯ ρ ρ¯ +

ρ) 3P (¯ ρ)P

(¯ (< ν, (ρ − ρ¯)2 >)2 2 2¯ ρ

+(

ρ)P

(¯ 3P (¯ ρ) 3P (¯ ρ) + )(< ν, (ρ − ρ¯)2 >< ν, (u − u ¯)2 >
ρ¯ 2P (¯ ρ)

¯)2 >) + + < ν, (ρ − ρ¯)2 (u − u

3¯ ρP (¯ ρ) (< ν, (u − u ¯)2 >)2
P (¯ ρ)

= ρ¯2 < ν, (u − u ¯)4 > +O05 + O41 + O23 +O21 O20 + O21 O02 + O20 O03 + O02 O03 . (8.4.27)

112 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS By simple calculations, 1 ¯)4 < ν, Qη3 Q q4 − Qη4 Q q3 >=< ν, ρP (ρ)(u − u 2  ρ 2 P (s) 4 2 +ρP (¯ ρ)(u − u ¯) + 6Q1 Q2 − 3ρ(u − u ¯) ds s2 ρ¯ −6ρ(u −

ρ) u ¯)2 P (¯



ρ ρ¯

P (s) P (¯ ρ) P (ρ) > ds − 3(ρ − ρ¯)(u − u ¯)2 2 s ρ¯

3P (¯ ρ)P
(¯ ρ) 3 ρ)(u − u ¯)4 − (u − u ¯)2 (ρ − ρ¯)2 =< ν, ρ¯P (¯ 2 ρ¯ +

3P (¯ ρ)(P
(¯ ρ))3 (ρ − ρ¯)4 + O41 + O23 + O05 . 2¯ ρ3 (8.4.28)

Then from (8.4.21)-(8.4.24) and the following measure equation: < ν, Qη3 Q q4 − Qη4 Q q3 > =< ν, Qη3 >< ν, Q q4 > − < ν, Qη4 >< ν, Q q3 >, we have 3P (¯ ρ)P
(¯ ρ) 3 ρ)(u − u ¯)4 − (u − u ¯)2 (ρ − ρ¯)2 < ν, ρ¯P (¯ 2 ρ¯ +

3P (¯ ρ)(P
(¯ ρ))2 (ρ − ρ¯)4 > 2¯ ρ3

3(P
(¯ ρ))2 P (¯ ρ) 3 ρ)(< ν, (u − u ¯)2 >)2 + (< ν, (ρ − ρ¯)2 >)2 = ρ¯P (¯ 3 2 2¯ ρ +

3P
(¯ ρ)P (ρ) < ν, (ρ − ρ¯)2 >< ν, (u − u ¯)2 > ρ¯

−

ρ)P (¯ ρ) 6P
(¯ < ν, (ρ − ρ¯)(u − u ¯) > +Error, ρ¯ (8.4.29)

8.4. THE CASE OF 1 < γ ≤ 3

113

where Error is the higher order error given by Error = O41 + O23 + O05 + O20 O30 + O20 O21 + O20 O12 +O20 O03 + O02 O30 + O02 O21 + O02 O12 +O02 O03 + O11 O21 + O11 O12 + O11 O30 + O11 O03 . At this moment, let P (ρ) = k2 ργ for 1 < γ ≤ 3. Then we have from (8.4.27) and (8.4.29) that 3 γ 2 − γ 4 2γ−4 k ρ¯ < ν, (ρ − ρ¯)4 > + (γ 2 − γ)k4 ρ¯2γ−4 (< ν, (ρ − ρ¯)2 >)2 2 2 3 2 γ−1 2 + (γ + 1)k ρ¯ (< ν, (ρ − ρ¯) >< ν, (u − u ¯)2 > 2 3 ¯)2 >) + ρ¯2 (< ν, (u − u ¯)2 >)2 + < ν, (ρ − ρ¯)2 (u − u γ = ρ¯2 < ν, (u − u ¯)4 > +O05 + O41 + O23 +O21 O20 + O21 O02 + O20 O03 + O02 O03 (8.4.30) and 3 ¯)4 − 3γk4 ρ¯2γ−2 (u − u ¯)2 (ρ − ρ¯)2 < ν, k2 ρ¯γ+1 (u − u 2 3 3 = k2 ρ¯γ+1 (< ν, (u − u ¯)2 >)2 + k6 γ 2 ρ¯3γ−5 (< ν, (ρ − ρ¯)2 >)2 2 2 +3γk4 ρ¯2γ−2 < ν, (ρ − ρ¯)2 >< ν, (u − u ¯)2 > 3 ¯) > + k6 γ 2 ρ¯3γ−5 (ρ − ρ¯)4 > −6γk4 ρ¯2γ−2 < ν, (ρ − ρ¯)(u − u 2 +Error. (8.4.31) Calculate (8.4.30) ×

2γk 2 ρ¯γ−1 γ+1

+ (8.4.31). We have

3 − γ 2 γ+1 k ρ¯ < ν, (u − u ¯)4 > 2(γ + 1) 5γ + 1 2 6 3γ−5 γ k ρ¯ < ν, (ρ − ρ¯)4 > + 2(γ + 1) 3(3 − γ) 2 γ+1 k ρ¯ (< ν, (u − u ¯)2 >)2 + 2(γ + 1) 3(γ − 3) 2 6 3γ−5 γ k ρ¯ + (< ν, (ρ − ρ¯)2 >)2 2(γ + 1) +6γk4 ρ¯2γ−2 (< ν, (u − u ¯)(ρ − ρ¯) >)2 + Error = 0.

(8.4.32)

114 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS If 1 < γ < 3, and noticing that (< ν, (ρ − ρ¯)2 >)2 ≤ < ν, (ρ − ρ¯)4 >, we have from (8.4.32) that 3 − γ 2 γ+1 k ρ¯ < ν, (u − u ¯)4 > 2(γ + 1) +

4(γ − 1) 2 6 3γ−5 γ k ρ¯ < ν, (ρ − ρ¯)4 > γ+1

3(3 − γ) 2 γ+1 k ρ¯ (< ν, (u − u ¯)2 >)2 + 2(γ + 1)

(8.4.33)

+6γk4 ρ¯2γ−2 (< ν, (u − u ¯)(ρ − ρ¯) >)2 + Error ≤ 0, which implies that ¯)4 > +C2 < ν, (ρ − ρ¯)4 > ≤ 0, C1 < ν, (u − u

(8.4.34)

for two suitable positive constants C1 and C2 , which depend only on k, ρ¯, γ when the support of the ν is small. Thus ν is a Dirac measure and the support point is (¯ u, ρ¯). If γ = 3, first it follows from (8.4.33) that ν is concentrated on the line ρ = ρ¯ and then it is reduced to the point (¯ u, ρ¯) since we have from (8.4.13) that < ν, (u − u ¯)2 >= γk2 ρ¯γ−3 < ν, (ρ − ρ¯)2 > +O21 + O03 .

(8.4.35)

This completes the reduction of the Young measure to be a Dirac measure in the case of 1 < γ ≤ 3.

8.5

Application on Extended River Flow System

In this section, we shall use the compactness framework introduced in the previous sections to study the river flow equations, a shallow-water

8.5. APPLICATION ON EXTENDED RIVER FLOW SYSTEM 115 model describing the vertical depth ρ and mean velocity u by  ρt + (ρu)x = 0, (ρu)t + (ρu2 + P (ρ))x + a(x)ρ + cρu|u| = 0,

(8.5.1)

with bounded measurable initial data (ρ(x, 0), u(x, 0)) = (ρ0 (x), u0 (x)),

ρ0 (x) ≥ 0,

(8.5.2)

where the function a(x) corresponds physically to the slope of the topography, cρ|u| to a friction term and c is a nonnegative constant. Consider the Cauchy problem for the related parabolic system  ρt + (ρu)x = ερxx , (8.5.3) (ρu)t + (ρu2 + P (ρ))x + a(x)ρ + cρu|u| = ε(ρu)xx , with initial data (ρε (x, 0), (ρε uε )(x, 0)) = (ρε0 (x), ρε0 (x)uε0 (x)),

(8.5.4)

where (ρε0 (x), ρε0 (x)uε0 (x)) are given by (8.0.17), hence satisfy (8.0.18), (8.0.20). Similar to Theorem 8.0.1, we have the main result in this section: Theorem 8.5.1 Let (1) |a(x)| ≤ M and M be a nonnegative constant; (2) the initial data (ρ0 (x), u0 (x)) be bounded measurable and ρ0 (x) ≥ 0; (3) P (ρ) ∈ C 2 (0, ∞), P
(ρ) > 0, 2P
(ρ) + ρP

(ρ) ≥ 0 for ρ > 0 and  d
 ∞
P (ρ) P (ρ) dρ = ∞, dρ < ∞, ∀d > 0. ρ ρ d 0 Then for fixed ε > 0, the smooth viscosity solution (ρε , ρε uε ) of the Cauchy problem (8.5.3), (8.5.4) exists and satisfies 0 < c(ε, t) ≤ ρε (x, t) ≤ M2 (t),

uε (x, t) ≤ M2 (t),

(8.5.5)

where M2 (t) is a positive finite function of t ∈ (0, ∞), being independent of ε, and c(ε, t) is a positive function, which could tend to zero as ε tends to zero or t tends to infinity.

116 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS Moreover, for the case of P (ρ) = k2 ργ , γ > 1, there exists a subsequence (still labelled) (ρε (x, t), ρε (x, t)uε (x, t)) which converges almost everywhere on any bounded and open set Ω ⊂ R × R+ : (ρε (x, t), ρε (x, t)uε (x, t)) → (ρ(x, t), ρ(x, t)u(x, t)), as ε ↓ 0+ , (8.5.6) where the limit pair of functions (ρ(x, t), ρ(x, t)u(x, t)) is a weak solution of the Cauchy problem (8.5.1), (8.5.2). Proof. Exactly analogous to the proof of Theorem 8.0.1 given in Sections 8.1-8.4, we can prove Theorem 8.5.1 if we have the L∞ estimates (8.5.5). zρ , z¯m ), respectively, Multiplying system (8.5.3) by (wρ , wm ) and (¯ where w, z (¯ z = −z) are two Riemann invariants given in (8.0.9), we have c|u| (w − z¯) 2 2ε ε  (2P
+ ρP

)ρ2x = εwxx + ρx wx − 2 ρ 2ρ P
(ρ) 2ε ≤ εwxx + ρx wx ρ

(8.5.7)

c|u| (¯ z − w) 2 2ε ε  (2P
+ ρP

)ρ2x = ε¯ zxx − ρx z¯x − ρ 2ρ2 P
(ρ) 2ε ≤ ε¯ zxx − ρx z¯x . ρ

(8.5.8)

wt + λ2 wx + a(x) +

and z¯t + λ1 z¯x − a(x) +

Making a transformation w = X + M t,

z¯ = Y + M t,

where M is the bound of a(x), we have from (8.5.7)-(8.5.8) that  c|u| 2ε   Xt + λ2 Xx + (X − Y ) ≤ εXxx + ρx Xx , 2 ρ   Yt + λ1 Yx + c|u| (Y − X) ≤ εYxx − 2ε ρx Yx , 2 ρ

(8.5.9)

8.5. APPLICATION ON EXTENDED RIVER FLOW SYSTEM 117 with X|t=0 = w|t=0 ≤ M1 , Y |t=0 = z¯|t=0 ≤ M1 ,

(8.5.10)

where M1 is a positive constant depending only on the bound of the initial data of ρ0 (x), u0 (x). In the following we use the maximum principle to (8.5.9)-(8.5.10) to get the estimates X(x, t) ≤ M1 ,

Y (x, t) ≤ M1 , for (x, t) ∈ R × [0, T ],

(8.5.11)

z ≥ −M1 − M t, for (x, t) ∈ R × [0, T ]

(8.5.12)

which implies w ≤ M1 + M t,

and hence, the upper bound estimates of ρε and uε . To prove (8.5.11), we make the following transformation: 2 t ¯ = X − M1 − N (x + CLe ) , X L2 (8.5.13) 2 N (x + CLet ) ¯ , Y = Y − M1 − L2 where C, L are positive constants and N is the upper bound of X, Y on R × [0, T ] (by local solution, N exists). From (8.5.9), it is easy to ¯ and Y¯ satisfy the inequalities see that functions X

¯ − Y¯ ) + (CLet + 2λ2 x − 2ε − 4ε ρx x) N ¯ x + c|u| (X ¯ t + λ2 X X 2 ρ L2 2ε ¯ xx + ρx X ¯x ≤ εX ρ (8.5.14) and c|u| ¯ ¯ + (CLet + 2λ1 x − 2ε + 4ε ρx x) N (Y − X) Y¯t + λ1 Y¯x + 2 ρ L2 2ε ≤ εY¯xx − ρx Y¯x . ρ (8.5.15) Moreover, ¯ 0 (x) = X0 (x) − M1 − CLN < 0, X L2 CLN <0 Y¯0 (x) = Y0 (x) − M1 − L2

(8.5.16)

118 CHAPTER 8. SYSTEM OF POLYTROPIC GAS DYNAMICS and ¯ X(±L, t) < 0,

Y¯ (±L, t) < 0

(8.5.17)

since N is the upper bound of X, Y . We have from (8.5.14)-(8.5.17) that ¯ X(x, t) < 0,

Y¯ (x, t) < 0, on (−L, L) × (0, T ).

(8.5.18)

¯ (or Y¯ ) at a point We argue by assuming that (8.5.18) is violated for X (x, t) in (−L, L) × (0, T ). Let t¯ be the least upper bound of values of ¯ < 0. Then by the continuity we see that X ¯ = 0, Y¯ ≤ 0 t at which X ¯ x = 0 and ¯ t ≥ 0, X at some points (¯ x, t¯) ∈ (−L, L) × (0, T ). So X ¯ ¯ x, t), i.e., −Xxx ≥ 0 at (¯ ¯ xx − 2ε ρx X ¯ x − εX ¯ x ≥ 0 at (¯ ¯ t + λ2 X x, t¯). X ρ

(8.5.19)

But from the behaviors of local solution ρε , we can choose C large enough so that CLet + 2λ2 x − 2ε −

4ε ρx x > 0, on (−L, L) × (0, T ), ρ

(8.5.20)

CLet + 2λ1 x − 2ε +

4ε ρx x > 0, on (−L, L) × (0, T ). ρ

(8.5.21)

or

(8.5.19) and (8.5.20) yield a conclusion contradicting (8.5.14). So (8.5.18) is proved. Therefore for any point (x0 , t0 ) in (−L, L) × (0, T ), N (x20 + CLet0 ) , L2 N (x20 + CLet0 ) , Y (x0 , t0 ) < M1 + L2 X(x0 , t0 ) < M1 +

(8.5.22)

which yields the desired estimates in (8.5.11) if we let L ↑ ∞ in (8.5.22), and hence completes the proof of Theorem 8.5.1.

8.6. RELATED RESULTS

8.6

119

Related Results

The study of the existence of global weak solutions for the Cauchy problem (8.0.1), (8.0.2) has a long history. For the polytropic gas, the first large data existence theorem with locally finite total variation, for γ = 1 was obtained by Nishida [Ni], and for γ ∈ (1, 1 + δ), δ small, obtained by Smoller and Nishida [NS]. The method used in [Ni, NS] is called the Glimm’s scheme method (cf. [Gl]). Using the compensated compactness ideas developed by Tartar and Murat [Ta, Mu], DiPerna [Di2] established a global existence theorem for γ = 1 + N2 , N ≥ 5 odd, with the aid of the viscosity method. Ding, Chen, and Luo [DCL1] and Chen [Ch1] proved the convergence of the Lax-Friedrichs scheme and the existence of global solutions with L∞ large initial data with adiabatic exponent γ ∈ (1, 53 ]. Lions, Perthame and Tadmor [LPT] proved the global existence of a weak solution for γ ≥ 3 by applying the kinetic formulation coupled with the compensated compactness method. Finally, the method in [LPT] was successfully extended by Lions, Perthame and Souganidis [LPS] to fill up the gap γ ∈ ( 53 , 3) as well as a new proof for whole γ > 1. So the existence of a generalized solution for the Cauchy problem (8.0.1), (8.0.2) was completely resolved for the case of a polytropic gas. The global smooth solution of the Cauchy problem (8.0.1), (8.0.2) for a class of smooth initial data with the vacuum for a general pressure P (ρ) was obtained in [Lu5]. The global existence of L∞ entropy solutions for system (8.0.1) with a special pressure P (ρ) and arbitrarily large L∞ initial data was established in [CL]. In this chapter, the proof for the case of γ > 3 comes from [LPT]. However, to avoid the use of many knotty mathematical formulas, we have not introduced the proofs given in [Di2, DCL1, Ch1, LPS] for the case of γ ∈ (1, 3]. Instead, in Section 8.4 we adopted another proof given by [CL2], although it needs some extra assumptions on the viscosity solutions. The proof of Theorem 8.5.1 is from [KL2]. The related results about system (8.0.1) with inhomogeneous terms can be found in [DCL2, CG].

Chapter 9

Two Special Systems of Euler Equations In this chapter, we consider the existence of global weak solutions for the following nonlinear hyperbolic systems:
ρt + (ρu)x = 0 (9.0.1) ρ
ut + ( 12 u2 + 0 P s(s) ds)x = 0, with bounded measurable initial data (ρ(x, 0), u(x, 0)) = (ρ0 (x), u0 (x)), ρ0 (x) ≥ 0.

(9.0.2)

For smooth solutions, system (9.0.1) is equivalent to the isentropic equations of gas dynamics (8.0.1), but these two systems are different for solutions with shock waves. System (9.0.1) was first derived by S. Earnshaw for isentropic flow (cf. [Ea], [Wh]) and is also called the Euler equations of one-dimensional, compressible fluid flow (cf. [KM]). It is a scaling limit system of a Newtonian dynamics with long-range interaction for a continuous distribution of mass in R (cf. [Oe1, Oe2]) and also a hydrodynamic limit for the Vlasov equation (cf. [CEMP]). Using the method introduced in Chapters 6 and 7, in this chapter, we shall study two special cases for P (ρ):  ρ  ρ 2 s s e ds, and P (ρ) = s2 (s + d)γ−3 ds, (9.0.3) P (ρ) = 0

0

121

122

CHAPTER 9. TWO SYSTEMS OF EULER EQUATIONS

where γ > 3. ρ When P (ρ) = 0 s2 es ds, let F be the mapping from R2 into R2 defined by  ρ   1 2 ses ds . F : (ρ, u) → ρu, u + 2 0 Then two eigenvalues of dF are ρ

λ1 = u − ρe 2 ,

ρ

λ2 = u + ρe 2 ,

(9.0.4)

and the corresponding right eigenvectors are ρ

r1 = (1, −e 2 )T ,

ρ

r2 = (1, e 2 )T .

(9.0.5)

By simple calculations, ρ ρ ρ ∇λ1 · r1 = −2e 2 − e 2 < 0, for ρ ≥ 0, 2

(9.0.6)

ρ ρ ρ ∇λ2 · r2 = 2e 2 + e 2 > 0, for ρ ≥ 0. 2

(9.0.7)

and

Therefore, it follows from (9.0.4) that λ1 = λ2 at the line ρ = 0 at which the strict hyperbolicity fails to hold, and from (9.0.6)-(9.0.7) that both characteristic fields are genuinely nonlinear in the range ρ ≥ 0. ρ Two Riemann invariants of system (9.0.1) for P (ρ) = 0 s2 es ds are ρ

z = u − 2e 2 ,

ρ

w = u + 2e 2 .

(9.0.8)

Then λ1 =

w−z w+z w−z − ln( ), 2 2 4

λ2 =

w−z w+z w−z + ln( ), 2 2 4 (9.0.9)

and w−z 1 ), λ1w = − ln( 2 4

w−z 1 λ2z = − ln( ). 2 4

(9.0.10)

Just as given in (6.2.6) or (8.2.16), the entropies for any 2×2 hyperbolic conservation laws satisfy the equation ηwz +

λ2z λ1w ηw − ηz = 0. λ2 − λ1 λ2 − λ1

(9.0.11)

123 the entropy equation of system (9.0.1) for the case of P (ρ) = Then ρ 2 s s e ds is reduced to 0 ηwz −

1 1 ηw + ηz = 0. 2(w − z) 2(w − z)

(9.0.12)

We recall that the entropies for the system of isentropic gas dynamics (8.0.1) satisfy ηwz +

c c ηw − ηz = 0, w−z w−z

(9.0.13)

ρ 3−γ . So in a sense, the case of P (ρ) = 0 s2 es ds is where c = 2(γ−1) corresponding to the case of γ = ∞. ρ When P (ρ) = 0 s2 (s + d)γ−3 ds, γ > 3, let F be the mapping from R2 into R2 defined by  ρ 1 2 s(s + d)γ−3 ds). F : (ρ, u) → (ρu, u + 2 0 Then two eigenvalues of dF are 1

λ1 = u − ρ(ρ + d) 2 (γ−3) ,

1

λ2 = u + ρ(ρ + d) 2 (γ−3)

(9.0.14)

with corresponding right eigenvectors 1

r1 = (1, −(ρ + d) 2 (γ−3) )T ,

1

r2 = (1, (ρ + d) 2 (γ−3) )T .

(9.0.15)

The two corresponding Riemann invariants for this case are 1 1 w = u + (γ − 1)(ρ + d) 2 (γ−1) , 2

1 1 z = u − (γ − 1)(ρ + d) 2 (γ−1) . 2 (9.0.16)

By simple calculations, 1 1 1 ∇λ1 · r1 = −2(ρ + d) 2 (γ−3) − (γ − 3)ρ(ρ + d) 2 (γ−5) < 0, for ρ ≥ 0, 2 (9.0.17)

and 1 1 1 ∇λ2 · r2 = 2(ρ + d) 2 (γ−3) + (γ − 3)ρ(ρ + d) 2 (γ−5) > 0, for ρ ≥ 0. 2 (9.0.18)

124

CHAPTER 9. TWO SYSTEMS OF EULER EQUATIONS

Therefore, it follows from (9.0.14) that λ1 = λ2 at the line ρ = 0 at which the strict hyperbolicity fails to hold, and from (9.0.17)-(9.0.18) that both characteristic fields are genuinely nonlinear in the range ρ ≥ 0. Consider the Cauchy problem for the related parabolic system


ρt + (ρu)x = ερxx ρ
ut + ( 12 u2 + 0 P s(s) ds)x = εuxx ,

(9.0.19)

with initial data (9.0.2), where P (ρ) is given in (9.0.3). We have the main result in this chapter: Theorem 9.0.1 Let the initial data (ρ0 (x), u0 (x)) be bounded  ρmeasurρ 2 s s e ds or P (ρ) = s2 (s + able and ρ0 (x) ≥ 0. Let P (ρ) = 0

0

d)γ−3 ds, γ > 3. Then for fixed ε > 0, the unique smooth viscosity solution (ρε (x, t), uε (x, t)) of the Cauchy problem (9.0.19), (9.0.2) exists and satisfies 0 ≤ ρε (x, t) ≤ M,

uε (x, t) ≤ M,

(9.0.20)

where M is a positive constant, being independent of ε. Moreover, there exists a subsequence (still labelled) (ρε (x, t), uε (x, t)) which converges almost everywhere on any bounded and open set Ω ⊂ R × R+ : (ρε (x, t), uε (x, t)) → (ρ(x, t), u(x, t)), as ε ↓ 0+ ,

(9.0.21)

where the limit pair of functions (ρ(x, t), u(x, t)) is a weak solution of the Cauchy problem (9.0.1), (9.0.2). In Section 9.1, we shall first prove the L∞ estimates (9.0.20) and hence the existence of viscosity solutions for the Cauchy problem (9.0.19), (9.0.2). From the constructions of entropy-entropy flux pair of Lax type obtained in Sections 9.2-9.3, we can see that all these entropy functions are regular in the range ρ ≥ 0, although they are nonstrictly hyperbolic on the vacuum line ρ = 0. So the pointwise convergence of the viscosity solutions as ε tends to zero follows immediately from the same fashion as in Theorem 6.0.1 or Theorem 7.0.1.

9.1. EXISTENCE OF VISCOSITY SOLUTIONS

9.1

125

Existence of Viscosity Solutions

In this section we consider the existence of viscosity solutions for the Cauchy problem (9.0.19), (9.0.2). By Theorem 1.0.2, it is sufficient to prove the L∞ estimates given in (9.0.20). ρ When P (ρ) = 0 s2 es ds, we multiply system (9.0.19) by the vectors (wρ , wu ) and (zρ , zu ), respectively, where w, z are given by (9.0.8), to obtain ε ρ (9.1.1) wt + λ2 wx = εwxx − e 2 ρ2x ≤ εwxx 2 and ε ρ zt + λ1 zx = εzxx + e 2 ρ2x ≥ εzxx . 2

(9.1.2)

Therefore applying the maximum principle to (9.1.1) and (9.1.2), respectively, we have w(ρε , uε ) ≤ M and z(ρε , uε ) ≥ −M for a suitable large constant depending only on L∞ bound of the initial data. The nonnegativity of ρε ≥ 0 is obvious from ρ0 (x) ≥ 0. Thus we have that Σ5 = {(ρ, u) : w(ρ, u) ≤ M, z(ρ, u) ≥ −M, ρ ≥ 0} ρ

ρ

γ−5 ε(γ − 3) (ρ + d) 2 ρ2x ≤ εwxx , 2

(9.1.3)

is an invariant region (see Figure 9.1), where w = u + 2e 2 , z = u − 2e 2 and hence, the boundedness of (ρε (x, t), uε (x, t)). ρ When P (ρ) = 0 s2 (s + d)γ−3 ds, γ > 3, exactly the same as the above case, we multiply system (9.0.19) by the vectors (wρ , wu ) and (zρ , zu ), respectively, where w, z are given by (9.0.16), to obtain wt + λ2 wx = εwxx − and zt + λ1 zx = εzxx +

γ−5 ε(γ − 3) (ρ + d) 2 ρ2x ≥ εzxx . 2

(9.1.4)

Therefore to apply the maximum principle to (9.1.3) and (9.1.4), respectively, we have that D1 = {(ρ, u) : w(ρ, u) ≤ M, z(ρ, u) ≥ −M, ρ ≥ 0} is an invariant region, which has a similar figure as Σ5 in Figure 9.1, and hence complete the proof of (9.0.20).

CHAPTER 9. TWO SYSTEMS OF EULER EQUATIONS

126

u

w=M Σ5 ρ z = −N

FIGURE 9.1

9.2

Lax Entropy for P (ρ) =

ρ 0

s2 es ds

In this section, we shall the second part of Theorem 9.0.1 for  ρ 2prove s the case of P (ρ) = 0 s e ds, namely the pointwise convergence of the viscosity solutions (ρε (x, t), uε (x, t)) as ε tends to zero. Exactly the same as the proof of Theorem 6.0.1, it is enough to construct four families of Lax entropy-entropy flux pairs and to prove the compactness −1,2 (R×R+ ), for these of η(ρε (x, t), uε (x, t))t +q(ρε (x, t), uε (x, t))x in Wloc entropy-entropy flux pairs, with respect to the sequence of viscosity solutions (ρε (x, t), uε (x, t)). A pair (η, q) of real-valued maps is an entropy-entropy flux pair of system (9.0.1) if (qρ , qu ) = (uηρ + If P (ρ) =

ρ 0

P  (ρ) ηu , ρηρ + uηu ). ρ

(9.2.1)

s2 es ds, the above system of equations is reduced to (qρ , qu ) = (uηρ + ρeρ ηu , ρηρ + uηu ).

(9.2.2)

9.2. LAX ENTROPY FOR P (ρ) =

ρ 0

S 2 E S DS

127

Eliminating the q from (9.2.2), we have ηρρ = eρ ηuu .

(9.2.3)

If k denotes a positive constant, then the function η = h(ρ)eku solves (9.2.3) provided that h (ρ) = k2 eρ h. 1

(9.2.4)

1

Let a(ρ) = e− 4 ρ , r = 2ke 2 ρ and h(ρ) = a(ρ)φ(r). Then φ satisfies a standard Fuchsian equation φ − (1 −

1 )φ = 0. 4r 2

(9.2.5)

Exactly the same as what we did in Chapters 6 and 7, we have a series solution of Equation (9.2.5) as follows: l

φ1 = r (1 +

∞ 

cn r n ),

(9.2.6)

n=1

where 1 l= , 2

c2 =

1 , 22

c2n−1 = 0,

c2n =

c2(n−1) 1 = 2n for n ≥ 2. (2n)2 2 (n!)2 (9.2.7)

Let another independent solution φ2 of (9.2.5) satisfy φ2 = φ1 Q. Then Q solves Q +

2φ1  Q = 0. φ1

(9.2.8)

Thus Q = −(φ1 )−2 = −(rg2 (r))−1 ,

(9.2.9)

where ∞ 

c2n r 2n .

(9.2.10)

(rg2 (r))−1 dr.

(9.2.11)

g(r) = 1 +

n=1

Let

 Q=

r

∞

CHAPTER 9. TWO SYSTEMS OF EULER EQUATIONS

128 Then

1 2



φ2 = r g(r)

∞ r

(rg2 (r))−1 dr.

(9.2.12)

Using (9.2.2), we have qu = ρηρ + uηu

(9.2.13)

and two progressing waves of system (9.0.1) are provided by
ηk = h(ρ)eku qk = uηk + (ρh − h)eku /k and




η−k = h(ρ)e−ku q−k = uη−k + (h − ρh )e−ku /k. 1

(9.2.14)

(9.2.15)

1

Since h(ρ) = a(ρ)φ(r), a(ρ) = e− 4 ρ and r = 2ke 2 ρ , then ρ

qk = (u + ρe 2

φ (r) 4 + ρ − )ηk φ(r) 4k

(9.2.16)

φ (r) 4 + ρ + )η−k . φ(r) 4k

(9.2.17)

and ρ

q−k = (u − ρe 2

Let ηk1 = a(ρ)φ1 (r)eku . Then using Lemma 6.2.1, we have 1 ηk1 = a(ρ)φ1 (r)e−r ekw = ekw (a(ρ) + O( )) k

(9.2.18)

on ρ ≥ 0 as k approaches infinity and the corresponding flux function is of the form   ρ φ
(r) qk1 = ηk1 λ2 + ρe 2 ( φ11 (r) − 1) − 4+ρ 4k (9.2.19)   4+ρ 1 1 = ηk λ2 − 4k + O( k2 ) 1 = a(ρ)φ (r)e−ku . Then on ρ ≥ 0. Similarly let η−k 1

1 1 = a(ρ)φ1 (r)e−r e−kz = e−kz (a(ρ) + O( )) η−k k

(9.2.20)

9.2. LAX ENTROPY FOR P (ρ) =

ρ 0

S 2 E S DS

129

on ρ ≥ 0 and the corresponding flux function   ρ φ (r) 4 + ρ 1 1 λ1 − ρe 2 ( 1 − 1) + = η−k q−k φ1 (r) 4k   1 4 + ρ 1 + O( 2 ) λ1 + = η−k 4k k

(9.2.21)

2 , q 2 ) satisfy on ρ ≥ 0. The entropy-entropy flux pairs (ηk2 , qk2 ), (η−k −k

1 ηk2 = a(ρ)φ2 (r)eku = ekz (a(ρ) + O( )) k

(9.2.22)

on ρ ≥ 0; qk2 = ηk2 (λ1 −

1 4+ρ + O( 2 )) 4k k

(9.2.23)

on ρ ≥ 0; 1 2 = a(ρ)φ2 (r)e−ku = e−kw (a(ρ) + O( )) η−k k

(9.2.24)

on ρ ≥ 0; 2 2 = η−k (λ2 + q−k

1 4+ρ + O( 2 )) 4k k

(9.2.25)

on ρ ≥ 0 respectively. Furthermore, we have from (9.2.18)-(9.2.25) that  4 + ρ −1ρ 1  )e 4 /k + O( 2 )),  qk1 = λ2 ηk1 − ekw ((   4k k     1 = λ η 1 + e−kz (( 4 + ρ )e− 14 ρ /k + O( 1 )),  q−k 1 −k 4k k2 1 4 + ρ 1   )e− 4 ρ /k + O( 2 )), qk2 = λ1 ηk2 − ekz ((    4k k     q 2 = λ2 η 2 + e−kw (( 4 + ρ )e− 14 ρ /k + O( 1 )) −k −k 4k k2 on ρ ≥ 0. It is easy to check that system (9.0.1) for the case P (ρ) = has a strictly convex entropy 1 η  = u2 + eρ 2

(9.2.26)

ρ 0

s2 es ds

CHAPTER 9. TWO SYSTEMS OF EULER EQUATIONS

130

and the corresponding entropy flux 1 q  = u3 + ρueρ . 3 From this strictly entropy-entropy flux pair and the method given in Chapters 6 and 7, we deduce that 1

1

ε 2 ∂x ρε , ε 2 ∂x uε

are uniformly bounded in L2loc (R × R+ ).

(9.2.27)

Noticing that all entropy-entropy flux pairs constructed above are smooth in the range ρ ≥ 0, we have the following lemma, which combining with the compensated compactness method given in Chapters  ρ 26 sand 7 completes the proof of Theorem 9.0.1 for the case P (ρ) = 0 s e ds. Lemma 9.2.1 For any entropy-entropy flux pair (η(ρ, u), q(ρ, u)) given in (9.2.18)-(9.2.25), −1 (R × R+ ) η(ρε , uε )t + q(ρε , uε )x is compact in Hloc

(9.2.28)

with respect to the sequence of viscosity solutions (ρε , uε ).

9.3

Lax Entropy for P (ρ) =

ρ 0

s2 (s + d)γ−3ds

In this section, we construct four families of the entropy-entropy flux pair of Lax type and prove the compactness of η(ρε (x, t), uε(x, t))t + ρ −1,2 (R × R+ ) for the case of P (ρ) = 0 s2 (s + q(ρε (x, t), uε (x, t))x in Wloc d)γ−3 ds in system (9.0.1). ρ If P (ρ) = 0 s2 (s + d)γ−3 ds, γ > 3, system (9.2.1) is reduced to (qρ , qu ) = (uηρ + ρ(ρ + d)γ−3 ηu , ρηρ + uηu ).

(9.3.1)

Eliminating the q from (9.3.1), we have ηρρ = (ρ + d)γ−3 ηuu .

(9.3.2)

If k denotes a positive constant, then the function η = h(ρ)eku solves (9.3.2) provided that h (ρ) = k2 (ρ + d)γ−3 h.

(9.3.3)

9.3. LAX ENTROPY FOR P (ρ) =

ρ 0

S 2 (S + D)γ−3 DS

1

131

1

2k (ρ + d) 2 (γ−1) . Then h = a(ρ)φ(s) Let a(ρ) = (ρ + d) 4 (3−γ) , s = γ−1 solves (9.3.3) if and only if φ satisfies the standard Fuchsian equation

φ − (1 + µs2 )φ = 0, where µ =

4−(γ−1)2 4(γ−1)2

(9.3.4)

> − 14 .

From (9.3.1), we have qu = ρηρ + uηu .

(9.3.5)

ηk = h(ρ)eku ,

(9.3.6)

If

then

(qk )u = ρh (ρ)eku + kuh(ρ)eku

and hence one entropy flux corresponding to ηk is qk = uh(ρ)eku + (ρh − h)eku /k  1 φ (γ + 1)ρ + 4d  . = ηk u + ρ(ρ + d) 2 (γ−3) − φ 4k(ρ + d)

(9.3.7)

If η−k = h(ρ)e−ku , then

(9.3.8)

(q−k )u = ρh (ρ)e−ku − kuh(ρ)e−ku

and hence one entropy flux corresponding to η−k is q−k = uh(ρ)e−ku + (h − ρh )e−ku /k 

= η−k u − ρ(ρ + d)

1 (γ−3) 2

φ (γ + 1)ρ + 4d  + . φ 4k(ρ + d)

(9.3.9)

Then waves of system (9.0.1) for the case of P (ρ) =  ρ 2 two progressing γ−3 ds are provided by (9.3.6)-(9.3.7) and (9.3.8)-(9.3.9). 0 s (s + d) We may use the method of Frobenius again to obtain a series solution of Equation (9.3.4) as follows: φ1 = s

j

∞  n=0

en sn ,

(9.3.10)

CHAPTER 9. TWO SYSTEMS OF EULER EQUATIONS

132

where e0 can be any positive constant, j > 0 is any one root of the equation j(j − 1) = µ, and en satisfy en−1 for n ≥ 1. (9.3.11) en = (2n + j)(2n + j − 1) − µ Let another independent solution φ2 of (9.3.4) satisfy φ2 = φ1 Q. Then Q solves Q +

2φ1  Q = 0. φ1

(9.3.12)

Thus Q = −(φ1 )−2 = −(sj g(s))−2 ,

(9.3.13)

where g(s) =

∞ 

en sn .

(9.3.14)

n=0

Let

 Q=

∞

s

Then j

(sj g(s))−2 ds.



φ2 = s g(s)

∞ s

(sj g(s))−2 ds.

(9.3.15)

(9.3.16)

It is easy to check that φ1 (s) > 0, φ1 (s) > 0, φ2 (s) > 0 and φ2 (s) < 0. Let ηk1 = a(ρ)φ1 (s)eku . Then using Lemma 6.2.1, we have 1 (9.3.17) ηk1 = a(ρ)φ1 (s)e−s ekw = ekw (a(ρ) + O( )) k on ρ ≥ 0 as k approaches infinity and the corresponding flux function is of the form  1 qk1 = ηk1 u + ρ(ρ + d) 2 (γ−3) 1

+ρ(ρ + d) 2 (γ−3) (

(γ + 1)ρ + 4d  φ1 (s) − 1) − φ1 (s) 4k(ρ + d)

 1  (γ + 1)ρ + 4d + O( 2 ) = ηk1 λ2 − 4k(ρ + d) k

(9.3.18)

9.3. LAX ENTROPY FOR P (ρ) =

ρ 0

S 2 (S + D)γ−3 DS

133

1 a(ρ)φ (s)e−ku . Then on ρ ≥ 0. Similarly let η−k 1

1 1 = a(ρ)φ1 (s)e−s e−kz = e−kz (a(ρ) + O( )) η−k k

(9.3.19)

on ρ ≥ 0 and the corresponding flux function  1 1 1 = η−k u − ρ(ρ + d) 2 (γ−3) q−k 1

−ρ(ρ + d) 2 (γ−3) (

(γ + 1)ρ + 4d  φ1 (s) − 1) + φ1 (s) 4k(ρ + d)

(9.3.20)

 1  (γ + 1)ρ + 4d 1 + O( 2 ) λ1 + = η−k 4k(ρ + d) k 2 , q 2 ) satisfy on ρ ≥ 0. The entropy-entropy flux pairs (ηk2 , qk2 ), (η−k −k

1 ηk2 = a(ρ)φ2 (s)eu = a(ρ)φ2 (s)es ekz = ekz (a(ρ) + O( )) k

(9.3.21)

on ρ ≥ 0;  1  (γ + 1)ρ + 4d + O( 2 ) qk2 = ηk2 λ1 − 4k(ρ + d) k

(9.3.22)

on ρ ≥ 0; 1 2 = a(ρ)φ2 (s)e−u = e−kw (a(ρ) + O( )) η−k k

(9.3.23)

 1  (γ + 1)ρ + 4d 2 2 + O( 2 ) = η−k λ2 + q−k 4k(ρ + d) k

(9.3.24)

on ρ ≥ 0;

on ρ ≥ 0 respectively. is easy to check that system (9.0.1) for the case of P (ρ) =  ρ It 2 γ−3 ds has a strictly convex entropy 0 s (s + d) η =

1 1 (ρ + d)γ−1 + u2 . (γ − 2)(γ − 1) 2

Then we have the following lemma:

134

CHAPTER 9. TWO SYSTEMS OF EULER EQUATIONS

Lemma 9.3.1 1

1

ε 2 ∂x uε , ε 2 ∂x ρε are uniformly bounded in L2loc (R × (0, ∞)). (9.3.25) Noticing that all entropy-entropy flux pairs constructed above are smooth in the range ρ ≥ 0, we have the following lemma, which combining with the compensated compactness method given in Chapters 6 and 7 completes the proof of Theorem 9.0.1 for the case P (ρ) = ρ 2 γ−3 ds, γ > 3. 0 s (s + d) Lemma 9.3.2 For any entropy-entropy flux pair (η(ρ, u), q(ρ, u)) given in (9.3.17)-(9.3.24), −1 η(ρε , uε )t + q(ρε , uε )x is compact in Hloc (R × R+ ).

9.4

(9.3.26)

Related Results

u w=M

ρ Σ6

z = −N FIGURE 9.2

The large data existence theorem of global weak solutions with locally finite total variation for the Cauchy problem (9.0.1), (9.0.2) was established by DiPerna [Di1] for general pressure function P (ρ)

9.4. RELATED RESULTS

135

satisfying suitable conditions, for instance, P (ρ) = k2 ργ , γ ∈ (1, 3), such that the invariant region Σ6 = {(ρ, u) : w(ρ, u) ≤ M, z(ρ, u) ≥ M }

(9.4.1)

is away from the vacuum line ρ = 0 as given in Figure 9.2. For this class of functions P (ρ), the Glimm method works well since system (9.0.1) is strictly hyperbolic and two eigenvalues are distant in the range Σ6 . The uniqueness of the weak solution for this case was recently obtained by Bressan [Br]. However, for the case of γ > 3 or more general pressure function P (ρ) given in the next chapter, the Glimm method does not work because the invariant region always includes the nonstrictly  ρ 2 s hyperbolic line graphed in Figure 9.1 for P (ρ) = 0 s e ds or P (ρ) =  ρ 2ρ = 0 as γ−3 ds. s (s + d) 0 proofs in this chapter are from [Lu8] for the case P (ρ) =  ρ The 2 es ds and from [Lu2] for the case of P (ρ) = ρ s2 (s + d)γ−3 ds. s 0 0

Chapter 10

General Euler Equations of Compressible Fluid Flow In this chapter, we consider the existence of global weak solutions for the following nonlinear hyperbolic systems:
ut + ( 12 u2 + f (v))x = 0 (10.0.1) vt + (uv + g(v))x = 0, with bounded measurable initial data (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)).

(10.0.2)

When g(v) = 0, system (10.0.1) is the same as system (9.0.1) with v in (10.0.1) instead of ρ in (9.0.1). In Chapters 6, 7 and 9, we apply the compensated compactness method to the system of quadratic flux, Le Roux system and two special systems of Euler equations. From these applications, we can see that because of the explicit constructions of flux functions in these systems, we can make some suitable transformations of variables to construct explicit entropy-entropy flux pairs of Lax type via the solutions of Fuchsian equations, and hence obtain necessary estimates about these function pairs. Then based on these estimates and using some developed ideas from the theory of compensated compactness, we can prove the large data existence theorem of global weak solutions for these systems. In Chapter 8, the explicit flux function P (ρ) = k2 ργ in the system of gas dynamics also helps much to obtain the explicit 137

138 CHAPTER 10. GENERAL SYSTEM OF EULER EQUATIONS expression of weak entropies, which is clearly the crux to establish the compactness of the sequence of viscosity solutions and hence the existence of weak solutions. However, system (10.0.1) is in a different situation from systems we considered in the above chapters, in which the flux functions are nonlinear and in implicit forms. Entropy-entropy flux pairs to more general strictly hyperbolic systems or systems in the strictly hyperbolic domains were well analyzed by Lax ([La2, La3]). However, to apply the compensated compactness method to some nonstrictly hyperbolic systems just as given in the form of system (10.0.1), some new techniques to construct entropy-entropy flux pairs must be investigated. One new idea in this chapter is to find entropy-entropy flux pairs to system (10.0.1) in the following special form: ηk1 = ekw (a1 (v) +

b1 (v, k) ), k

qk1 = ekw (c1 (v) +

d1 (v, k) ); k

2 = e−kw (a2 (v) + η−k

b2 (v, k) ), k

2 q−k = e−kw (c2 (v) +

d2 (v, k) ); k

1 = e−kz (a3 (v) + η−k

b3 (v, k) ), k

1 q−k = e−kz (c3 (v) +

d3 (v, k) ); k

ηk2 = ekz (a4 (v) +

b4 (v, k) ), k

qk2 = ekz (c4 (v) +

d4 (v, k) ), k

where w, z are the Riemann invariants of system (10.0.1) given by (10.1.1). Notice that all the unknown functions ai , bi (i = 1, 2, 3, 4) are only of a single variable v. This special simple construction yields an ordinary differential equation of second order with a singular coefficient 1/k before the term of the second order derivative. Then the necessary estimates for functions ai (v), bi (v, k) are obtained by the use of the singular perturbation theory of ordinary differential equations. For system (10.0.1), our assumptions about g(v), f (v) and initial data are as follows: (A1 ) f, g ∈ C 3 [0, ∞) and f1 = (f
/v) ∈ C 2 [0, ∞), g1 = (g
/v) ∈ C 2 [0, ∞), satisfy f1 ≥ d and 2f1
+ g1
(s1 + g1 ) ≥ 0, for v ≥ 0,

139 2f1
+ g1
(s1 − g1 ) ≥ 0, for v ≥ 0,  where d is a fixed positive constant, and s1 = g12 + 4f1 . (A2 ) u0 , v0 are bounded measurable and v0 ≥ 0. Example 10.0.1 Besides the general Euler equations of compressible fluid flow (9.0.1), there are many other function pairs (f, g) which satisfy the condition (A1 ). For example, we can specially choose f1 = (v + d)l , g1 = k(v + e)m , where d, e, m, l are positive constants, k is a non-negative constant and e ≥ d, l ≥ m. Then it is easy to check that (A1 ) is satisfied. By simple calculations, two eigenvalues of system (10.0.1) are λ1 =

2u + vg1 (v) − vs1 , 2

λ2 =

2u + vg1 (v) + vs1 2

(10.0.3)

with corresponding right eigenvectors r1 = (−2f1 , s1 − g1 )T ,

r2 = (2f1 , s1 + g1 )T .

(10.0.4)

So, using the assumption (A1 ) we have ∇λ1 · r1 = −2f1 + (s1 − g1 )[

vg
+ g1 − s1 v(g1 g1
+ 2f1
) − ] 2 2s1

1 −4f1 s1 + (s1 − g1 )[s1 (vg1
+ g1 ) − s21 − vg1 g1
− 2vf1
] 2s1 1 [−4f1 s1 + (s1 − g1 )(s1 g1 − s21 )] ≤ 2s1 1 = (−4f1 − (s1 − g1 )2 ) < 0 2 (10.0.5) =

and vg
+ g1 + s1 v(g1 g1
+ 2f1
) + ] 2 2s1 (10.0.6) 1 ≥ (4f1 + (s1 + g1 )2 ) > 0. 2

∇λ2 · r2 = 2f1 + (s1 + g1 )[

Therefore, it follows from (10.0.3) that λ1 = λ2 at the line v = 0, in which the strict hyperbolicity for system (10.0.1) fails to hold. However, both characteristic fields are genuinely nonlinear from (10.0.5) and (10.0.6).

140 CHAPTER 10. GENERAL SYSTEM OF EULER EQUATIONS We add small, positive perturbation terms to system (10.0.1) and consider the Cauchy problem for the following system:
ut + ( 12 u2 + f (v))x = εuxx , (10.0.7) vt + (uv + g(v))x = εvxx , with initial data (10.0.2). Then we have the main result in this chapter as follows: Theorem 10.0.2 Let the assumptions (A1 ), (A2 ) hold. Then, for any fixed ε > 0, the Cauchy problem (10.0.7), (10.0.2) has a unique global smooth solution (uε (x, t), v ε (x, t)) satisfying |uε (x, t)| ≤ M,

0 ≤ v ε (x, t) ≤ M,

(10.0.8)

where M is a positive constant, independent of ε. Moreover, there exists a subsequence (still labelled) (uε (x, t), v ε (x, t)) such that (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)), a.e. on Ω,

(10.0.9)

where Ω ⊂ R × R+ is any bounded and open set, the limit pair of functions (u(x, t), v(x, t)) being a weak solution of the Cauchy problem (10.0.1), (10.0.2). The existence of viscosity solutions for the Cauchy problem (10.0.7), (10.0.2) is given in Section 10.1. In Section 10.2, we shall construct entropy-entropy flux pairs of system (10.0.1) and obtain necessary estimates. In Section 10.3, these estimates will yield the existence of weak solutions for the Cauchy problem (10.0.1)-(10.0.2) when coupled with the method to deal with nonstrictly hyperbolic systems given in Chapters 6, 7 and 9.

10.1

Existence of Viscosity Solutions

In this section, we shall prove the first part of Theorem 10.0.2, namely the existence of smooth viscosity solutions (uε , v ε ) for the Cauchy problem (10.0.7), (10.0.2). By Theorem 1.0.2, the unique thing is to obtain the a priori L∞ estimate (10.0.8).

10.2. LAX ENTROPY AND RELATED ESTIMATES

141

By simple calculations, two Riemann invariants of system (10.0.1) are  w =u+

v 0

g1 + s1 dv, 2

 z =u+

0

v

g1 − s1 dv. 2

(10.1.1)

Along the line of w = M , we have from assumption (A1 ) that g1 + s1 du =− ≤ 0, dv 2

d2 u 2f1
+ g1
(g1 + s1 = − ≤ 0, as v ≥ 0, dv 2 2s1

and along the line of z = −M , we have from assumption (A1 ) that −g1 + s1 du = ≤ 0, dv 2

d2 u 2f1
+ g1
(g1 − s1 = ≥ 0, as v ≥ 0. dv 2 2s1

So by the theory of invariant regions introduced in Theorem 4.2.1, it is easy to get that  D2 = (u, v) : w(u, v) ≤ M,

z(u, v) ≥ −M,

 v≥0

is an invariant region, which has a similar figure as D1 or Σ5 given in Figure 9.1, where M is a suitable large positive constant. This invariant region D2 yields the L∞ bound (10.0.8) and hence the proof of existence of viscosity solutions.

10.2

Lax Entropy and Related Estimates

To prove the existence of weak solutions in the second part of Theorem 10.0.2, in this section, we shall construct the entropy-entropy flux pairs of Lax type for system (10.0.1) and give the required estimates by means of the theory of singular perturbation. We recall that a pair (η, q) of real-valued maps is an entropy-entropy flux pair of (10.0.1) if all smooth solutions satisfy (uηu + vηv , f
ηu + (u + g
)ηv ) = (qu , qv ).

(10.2.1)

Eliminating the q from (10.2.1), we have f
ηuu + g
ηuv − vηvv = 0.

(10.2.2)

142 CHAPTER 10. GENERAL SYSTEM OF EULER EQUATIONS Substituting entropies ηk1 = ekw (a1 (v) + b1 (v, k)/k) into (10.2.2), we obtain that g1
(g1 + s1 ) + 2f1
a1 ] 2s1 g
(g1 + s1 ) + 2f1
b

b1 + 1 +a

1 + s1 b
1 + 1 2s1 k = 0.

k[s1 a
1 +

(10.2.3)

Let s1 a
1 +

g1
(g1 + s1 ) + 2f1
a1 = 0 2s1

(10.2.4)

and a

1 + s1 b
1 + Then

 a1 = exp(−

v 0

g1
(g1 + s1 ) + 2f1
b

b1 + 1 = 0. 2s1 k

g1
(g1 + s1 ) + 2f1
dv) > 0 for v ≥ 0. 2s21

(10.2.5)

(10.2.6)

The existence of b1 and its uniform bound with respect to k can be obtained by the following lemma (cf. [Ka]): Lemma 10.2.1 Let Y (x) ∈ C 2 [0, h] be the solution of the equation F (x, Y, Y
) = 0, and functions f (x, y, z, λ), F (x, y, z) be continuous on the regions 0 ≤ x ≤ h, |y − Y (x)| ≤ l(x), |z − Y
(x)| ≤ m(x) for some positive functions l(x), m(x) and λ0 > λ > 0. In addition, |f (x, y, z, λ) − F (x, y, z)| ≤ ε, |F (x, y2 , z) − F (x, y1 , z)| ≤ M |y2 − y1 |, F (x, y, z2 ) − F (x, y, z1 ) ≥L z2 − z1 for some positive constants ε, M and L.

10.2. LAX ENTROPY AND RELATED ESTIMATES

143

If y(x) = y(x, λ) is a solution of the following ordinary differential equation of second order: λy

+ f (x, y, y
, λ) = 0, with y(0) = Y (0) and y
(0) being arbitrary, then for sufficiently small λ > 0, ε > 0 and P = |y
(0) − Y
(0)|, y(x) exists for all 0 ≤ x ≤ h and satisfies  ε P N  Mx + λ( + ) exp( ), |y(x, λ) − Y (x)| < M L M L where N = max |Y (x)|. 0≤x≤h

Furthermore, we can use Lemma 10.2.1 again to obtain the bound of b
1 with respect to k if we differentiate Equation (10.2.5) with respect to v. Using (10.2.1), we have qu = uηu + vηv ,

qv = f
ηu + (u + g
)ηv .

Then a progressing wave of system (10.0.1) is provided by ηk1 = ekw (a1 (v) +

b1 (v, k) va
− a1 vb
1 − b1 ), qk1 = λ2 ηk1 + ekw ( 1 + ). k k k2 (10.2.7)

In a similar way, we can obtain another entropy-entropy flux pair of Lax type as follows: b2 (v, k) ), k

2 + e−kw ( a2 − va2 + b2 − vb2 ), = λ2 η−k k k2

2 = e−kw (a (v) + η−k 2 2 q−k

(10.2.8)

where a2 (v) = a1 (v) and b2 (v, k) satisfies a

1 − s1 b
2 −

g1
(g1 + s1 ) + 2f1
b

b2 + 2 = 0; 2s1 k

b3 (v, k) ), k va
− a3 vb
3 − b3 + ), qk2 = λ1 ηk2 + ekz ( 3 k k2

ηk2 = ekz (a3 (v) +

(10.2.9)

(10.2.10)

144 CHAPTER 10. GENERAL SYSTEM OF EULER EQUATIONS where s1 a
3 +

g1
(g1 − s1 ) + 2f1
a3 = 0 2s1

(10.2.11)

and a

3 − s1 b
3 −

g1
(g1 − s1 ) + 2f1
b

b3 + 3 = 0; 2s1 k

b4 (v, k) ), k

1 = λ η 1 + e−kz ( a4 − va4 + b4 − vb4 ), q−k 1 −k k k2 where a4 (v) = a3 (v) and b4 (v, k) satisfies 1 = e−kz (a (v) + η−k 4

a

3 + s1 b
4 +

g1
(g1 − s1 ) + 2f1
b

b4 + 4 = 0. 2s1 k

From (10.2.11), we have  v
g1 (g1 − s1 ) + 2f1
dv) > 0 for v ≥ 0. a4 = a3 = exp(− 2s21 0

(10.2.12)

(10.2.13)

(10.2.14)

(10.2.15)

Using the argument in Lemma 10.2.1 in Equation (10.2.14), we can get the existence of b4 and the uniform bounded estimates of b4 , b
4 with respect to k. If making an independent transformation v1 = v − M to Equations (10.2.9) and (10.2.12), where M is the upper bound of v, we also obtain the existence of b2 , b3 and the uniform bounded estimates of b2 , b3 , b
2 and b
3 by Lemma 10.2.1 again. Noticing the assumptions (A1 ), (A2 ), we have that ai − va
i , i = 1, 2, 3, 4 are all positive for v ≥ 0.

10.3

Existence of Weak Solutions

In Section 10.2, four families of entropy-entropy flux pairs of Lax type for system (10.0.1) are constructed. Noticing the estimates about functions ai (v), ai − va
i , i = 1, 2, 3, 4, and the forms of these function pairs given by (10.2.7), (10.2.8), (10.2.10) and (10.2.13), we can use the same method given in Chapters 6, 7 and 9 to deduce that all the Young measures ν determined by the sequence of viscosity solutions of the Cauchy problem (10.0.7), (10.0.2) are Dirac measures if the following lemma is true.

10.4. RELATED RESULTS

145

Lemma 10.3.1 For any entropy-entropy flux pair (η(u, v), q(u, v)) of system (10.0.1) given by (10.2.7), (10.2.8), (10.2.10) and (10.2.13), −1 (R × R+ ) η(uε , v ε )t + q(uε , v ε )x is compact in Hloc

(10.3.1)

with respect to the viscosity solutions (uε , v ε ) of the Cauchy problem (10.0.7), (10.0.2). Proof. It is easy to check that system (10.0.1) has a strictly convex entropy  v y
1 f (s) dsdy. (10.3.2) η  = u2 + 2 s 0 0 Then using this convex entropy, we can prove that 1

1

ε 2 ∂x uε , ε 2 ∂x v ε are uniformly bounded in L2loc (R × R+ ).

(10.3.3)

Notice that all entropy-entropy flux pairs given in (10.2.7), (10.2.8), (10.2.10) and (10.2.13) are smooth in the range v ≥ 0. Thus we complete the proof of Lemma 10.2.2 by using Theorem 2.3.2.

10.4

Related Results

The proof in this chapter is from the paper [Lu6]. The ideas to construct entropy-entropy flux pairs to nonstrictly hyperbolic systems by means of the singular perturbation theory of ordinary differential equations of second order were extended in [KL2] to study some hyperbolic systems with inhomogeneous terms.

Chapter 11

Extended Systems of Elasticity In this chapter, we consider the existence of global weak solutions for the following extended nonlinear hyperbolic systems of elasticity:
ut + (cu + f (v))x = 0 (11.0.1) vt + (u + g(v))x = 0, with bounded measurable initial data (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)),

(11.0.2)

where c is a constant. When g(v) = 0 and c = 0, (11.0.1) is the system of one-dimensional nonlinear elasticity in Lagrangian coordinates which describes the balance of mass and linear momentum, where v denotes the strain, f (v) is the stress and u the velocity. For the more general system (11.0.1), our assumptions about g(v) and f (v) are as follows: (A) f, g ∈ C 3 satisfy f
≥ d for v ∈ R, and 2f

+ g

(s + g
− c) > 0, for v > 0, 2f

+ g

(s + g
− c) < 0, for v < 0, 2f

+ g

(g
− c − s) > 0, for v > 0, 147

148

CHAPTER 11. EXTENDED SYSTEMS OF ELASTICITY 2f

+ g

(g
− c − s) < 0, for v < 0,

where s =



(g
− c)2 + 4f
.

Example 11.0.1 Besides the system of elasticity, there are many other function pairs (f, g) which satisfy the condition (A). For instance, if we choose f
(v) = (v 2 + d)l , g
(v) − c = k(v 2 + e)m , where d, e, m, l are positive constants, k is a non-negative constant and e ≥ d, l ≥ m, then it is easy to check that (A) is satisfied. By simple calculations, the two eigenvalues of system (11.0.1) are λ1 =

c + g
− s2 , 2

λ2 =

c + g
+ s2 2

(11.0.3)

with corresponding right eigenvectors r1 = (g
− c + s2 , −2)T ,

r2 = (g
− c − s2 , −2)T .

(11.0.4)

So 2f

− g

(s2 − (g
− c)) , s2 2f

+ g

(s2 + (g
− c)) . ∇λ2 · r2 = − s2

∇λ1 · r1 =

(11.0.5)

Thus, by the assumption (A), the system (11.0.1) is strictly hyperbolic, but both characteristic fields are linearly degenerate on the line v = 0. Similar to system (10.0.1), the nonlinear flux functions in (11.0.1) are also in implicit forms. We may use the same fashion as given in Chapter 10 to construct the entropy-entropy flux pairs of system (11.0.1) in the following special form: ηk1 = ekw (a1 (v) +

b1 (v, k) ), k

qk1 = ekw (c1 (v) +

d1 (v, k) ); k

2 = e−kw (a2 (v) + η−k

b2 (v, k) ), k

2 q−k = e−kw (c2 (v) +

d2 (v, k) ); k

1 = e−kz (a3 (v) + η−k

b3 (v, k) ), k

1 q−k = e−kz (c3 (v) +

d3 (v, k) ); k

149 ηk2 = ekz (a4 (v) +

b4 (v, k) ), k

qk2 = ekz (c4 (v) +

d4 (v, k) ), k

where w, z are the Riemann invariants of system (11.0.1), i.e.,  v
 v
g − c + s2 g − c − s2 dv, z = u + dv. (11.0.6) w=u+ 2 2 0 0 The necessary estimates for functions ai (v), bi (v, k) can be also obtained by the use of the singular perturbation theory of ordinary differential equations of second order (see Lemma 10.2.1). However, a big difference between system (11.0.1) and system (10.0.1) is that the terms va
i (v) − ai (v), i = 1, 2, 3, 4 could change signs when passing the linearly degenerate line v = 0. From the proof in Section 6.4, we can see that some new difficulties will arise from this linear degeneration. The system of quadratic flux given in (6.0.1) is also linearly degenerate, but the common degenerative domain for two characteristic fields is only at the unique point (u, v) = (0, 0). Using the transformation of variable (su2 + v 2 ), we can deduce the Young measure, determined by the sequence of viscosity solutions of system (6.0.1), to be a Dirac measure with another support point supposing it is not concentrated in s = 0. But to deal with the measure on the line v = 0 is more difficult. Some new ideas are introduced in Section 11.3 to reduce the Young to a Dirac measure, and hence to prove the existence of weak solutions to the Cauchy problem (11.0.1), (11.0.2). We add small, positive perturbation terms to system (11.0.1) and consider the Cauchy problem for the following systems:  ut + (cu + f (v))x = εuxx , (11.0.7) vt + (u + g(v))x = εvxx , with the initial data (11.0.2). We have the main result in this chapter as follows: Theorem 11.0.2 If the initial data (u0 (x), v0 (x)) is bounded measurable and the assumption (A) holds, then for any fixed ε > 0, the Cauchy problem (11.0.7), (11.0.2) has a unique global smooth solution (uε (x, t), v ε (x, t)) satisfying |uε (x, t)| ≤ M,

|v ε (x, t)| ≤ M,

(11.0.8)

150

CHAPTER 11. EXTENDED SYSTEMS OF ELASTICITY

where M is a positive constant, independent of ε. Moreover, there exists a subsequence (still labelled) (uε (x, t), v ε (x, t)) such that (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)), a.e. on Ω,

(11.0.9)

where Ω ⊂ R × R+ is any bounded and open set, (u(x, t), v(x, t)) being a weak solution of the Cauchy problem (11.0.1), (11.0.2). In Section 11.1, we shall prove the existence of viscosity solutions to the Cauchy problem (11.0.7), (11.0.2). In Section 11.2, the method given in Chapter 10 is used to construct the entropy-entropy flux pairs of Lax type, which we shall use in Section 11.2 to prove the existence of weak solutions to the Cauchy problem (11.0.1), (11.0.2).

11.1

Existence of Viscosity Solutions

In this section, we prove the first part in Theorem 11.0.2, namely the existence of global smooth viscosity solutions (uε , v ε ). By Theorem 1.0.2, the unique thing is to get the L∞ estimate (11.0.8). By the expressions of Riemann invariants of system (11.0.1) given in (11.0.6) and the assumption (A), along the curves in the (u, v)-plane, determined by the equations w = N, w = −N, z = N and z = −N , we have the following estimates: On w = N , for v > 0, there hold g
− c + s2 du =− <0 dv 2 and 2f

+ g

(g
− c + s2 ) d2 u = − < 0; dv 2 2s2 On w = −N , for v < 0, there hold g
− c + s2 du =− <0 dv 2 and 2f

+ g

(g
− c + s2 ) d2 u = − > 0; dv 2 2s2

11.1. EXISTENCE OF VISCOSITY SOLUTIONS

151

v z = −N

z=N u Σ7

w=M

w = −M

FIGURE 11.1

On z = N , for v > 0, there hold g
− c − s2 du =− >0 dv 2 and 2f

+ g

(g
− c − s2 ) d2 u = < 0; dv 2 2s2 On z = −N , for v > 0, there hold g
− c − s2 du =− > 0, dv 2 and 2f

+ g

(g
− c − s2 ) d2 u = > 0. 2 dv 2s2 Therefore  Σ7 = (u, v) : w ≤ N,

w ≥ −N,

z ≤ N,

z ≥ −N



CHAPTER 11. EXTENDED SYSTEMS OF ELASTICITY

152

is an invariant region (see Figure 11.1), which implies the L∞ estimate (11.0.8) and hence the existence of the viscosity solutions to the Cauchy problem (11.0.7), (11.0.2).

11.2

Entropy-Entropy Flux Pairs of Lax Type

To prove the second part in Theorem 11.0.2, in this section, we shall construct the entropy-entropy flux pairs of Lax type to system (11.0.1). From the definition about entropy-entropy flux pair given in Chapter 4, a pair (η, q) of real-valued maps is an entropy-entropy flux pair of system (11.0.1) if all smooth solutions satisfy (cηu + ηv , f
ηu + g
ηv ) = (qu , qv ).

(11.2.1)

Eliminating the q from (11.2.1), we have f
ηuu + (g
− c)ηuv − ηvv = 0.

(11.2.2)

Similar to Chapter 10, we can construct four sets of Lax entropy pairs as follows: ηk1 = ekw (a1 (v) +

2 = e−kw (a2 (v) + η−k

ηk2 = ekz (a3 (v) +

1 = e−kz (a4 (v) + η−k

b1 (v, k) ), k

b2 (v, k) ), k

b3 (v, k) ), k

b4 (v, k) ), k

qk1 = λ2 ηk1 + ekw (

a
1 b
1 (v, k) + ); k k2 (11.2.3)

2 2 q−k = λ2 η−k − e−kw (

qk2 = λ1 ηk2 + ekz (

a
2 b
2 (v, k) + ); k k2 (11.2.4)

a
3 b
3 (v, k) + ); k k2 (11.2.5)

1 1 q−k = λ1 η−k − e−kz (

a
4 b
4 (v, k) + ), k k2 (11.2.6)

where ai (v), bi (v, k)(i = 1, 2, 3, 4) satisfy s2 a
1 +

g

(g
− c + s2 ) + 2f

a1 = 0, 2s2

a2 (v) = a1 (v),

(11.2.7)

11.3. EXISTENCE OF WEAK SOLUTIONS s2 a
3 +

g

(g
− c − s2 ) + 2f

a3 = 0, 2s2

a4 (v) = a3 (v),

153 (11.2.8)

and a

1 + s2 b
1 +

g

(g
− c − s2 ) + 2f

b

b1 + 1 = 0; 2s2 k

(11.2.9)

a

2 − s2 b
2 −

g

(g
− c + s2 ) + 2f

b

b2 + 2 = 0; 2s2 k

(11.2.10)

a

3 − s2 b
3 −

g

(g
− c − s2 ) + 2f

b

b3 + 3 = 0; 2s2 k

(11.2.11)

a

4 + s2 b
4 +

g

(g
− c + s2 ) + 2f

b

b4 + 4 = 0. 2s2 k

(11.2.12)

From (11.2.7) and (11.2.8) we have that  v


g (g − c + s2 ) + 2f

dv) > 0 for v ∈ [−M, M ], a1 = a2 = exp (− 2s22 0 (11.2.13) and



a3 = a4 = exp (−

0

v

g

(g
− c − s2 ) + 2f

dv) > 0 for v ∈ [−M, M ]. 2s22 (11.2.14)

Using the arguments in Lemma 10.2.1 in Equations (11.2.9)-(11.2.12), we can get the existence of ai (i = 1, 2, 3, 4) and the uniform bounded estimates of bi , b
i with respect to k. Noticing the assumption (A), a
i (i = 1, 2, 3, 4) all have only one zero point at v = 0.

11.3

Existence of Weak Solutions

In this section, we shall use the entropy-entropy flux pairs of Lax type of system (11.0.1) constructed in Section 11.2 to deduce that the Young measures are Dirac ones, and hence to prove the existence of weak solutions given in the second part of Theorem 11.0.2.

CHAPTER 11. EXTENDED SYSTEMS OF ELASTICITY

154

It is easy to check that system (11.0.1) has a strictly convex entropy  v 1 2  f (s)ds. (11.3.1) η = u + 2 0 Then in a same fashion as Lemma 10.3.1, we can prove the following lemma: Lemma 11.3.1 For any entropy-entropy flux pair (η(u, v), q(u, v)) of system (11.0.1) given by (11.2.3)-(11.3.6), −1 (R × R+ ), η(uε , v ε )t + q(uε , v ε )x is compact in Hloc

(11.3.2)

with respect to the viscosity solutions (uε , v ε ) of the Cauchy problem (11.0.7), (11.0.2). Lemma 11.3.1 guarantees the measure equation to be true, namely < ν, η 1 >< ν, q 2 > − < ν, η 2 >< ν, q 1 >< ν, η 1 q 2 − η 2 q 1 >

(11.3.3)

for any entropy-entropy flux pairs (η i , q i )(i = 1, 2) of system (11.0.1), −1 (R × R+ ). which satisfy that η i (uε , v ε )t + q i (uε , v ε )x is compact in Hloc

1 > > 0 and Since ai > 0 for all v ∈ [−M, M ], then clearly < ν, η±k 2 < ν, η±k > > 0.

Let Q denote the smallest characteristic rectangle: Q = {(u, v) : w− ≤ w ≤ w+ ,

z− ≤ z ≤ z+ }.

− + − We introduce four new probability measures µ+ k , µk , θk and θk on Q defined by 1 1 < µ+ k , h >=< ν, hηk > / < ν, ηk >,

(11.3.4)

2 2 < µ− k , h >=< ν, hη−k > / < ν, η−k >,

(11.3.5)

< θk+ , h >=< ν, hηk2 > / < ν, ηk1 >

(11.3.6)

1 2 > / < ν, η−k >, < θk− , h >=< ν, hη−k

(11.3.7)

and

11.3. EXISTENCE OF WEAK SOLUTIONS

155

where h = h(u, v) denotes an arbitrary continuous function. Clearly − + − µ+ k , µk , θk and θk all are uniformly bounded with respect to k. Then as a consequence of weak-star compactness, there exist probability measures µ± and θ ± on Q such that < µ± , h >= lim < µ± k ,h >

(11.3.8)

< θ ± , h >= lim < θk± , h >

(11.3.9)

k→∞

and k→∞

after the selection of an appropriate subsequence. Moreover, similar to the proof in (6.4.8), we have that the measures µ+ , µ− , θ + , θ − are respectively concentrated on the boundary sections of Q associated with w and z, i.e.,  (11.3.10) supp µ+ = Q {(u, v) : w = w+ } = Iw+ , supp µ− = Q



supp θ + = Q

{(u, v) : w = w− } = Iw− ,



{(u, v) : z = z+ } = Iz+ ,

(11.3.11)

(11.3.12)

and supp θ − = Q



{(u, v) : z = z− } = Iz− .

(11.3.13)

Similar to the proof of (6.4.15), we have < µ+ , q − λ2 η >< µ− , q − λ2 η >

(11.3.14)

< θ + , q − λ1 η >< θ − , q − λ1 η >

(11.3.15)

and

−1 (R × R+ ). for any (η, q) satisfying that ηt + qx is compact in Hloc

Now we are in the position to prove that the Young measure ν is the Dirac measure. Let the line v = 0 in the (u, v)-plane be Γ, i.e, Γ = {(u, v) : v = 0} = {(w, z) : w = z}.

156

CHAPTER 11. EXTENDED SYSTEMS OF ELASTICITY

Step 1. If any one of two points P + = Iw+ ∩ Iz+ and P − = Iw− ∩ Iz− does not lie on Γ, then it is easy to deduce that the Young measure is a Dirac measure. In fact, in this case, at least one of the boundary arcs Iw+ , Iw− , Iz+ , Iz− is contained in the component of Γc , i.e., in an open set where both of the characteristic fields are genuinely nonlinear, or all a
i (v), i = 1, 2, 3, 4 are not zero. For example, if P + lies in Γc , then either Iw+ ⊂ Γc or Iz+ ⊂ Γc .

If Iw+ ⊂ Γc , we may use the same method given in the proof of (6.4.16) and (6.4.17) to deduce that w+ = w− if we go by (ηk1 , qk1 ) instead of (η, q) in (11.3.14). Then the support of ν is reduced to the line I = Iw+ = Iw− , where both of the characteristic fields are genuinely nonlinear. Thus we can apply the entropy-entropy flux pair (ηk2 , qk2 ) instead of (η, q) in (11.3.15) to deduce z + = z − . Step 2. If the points P + = Iw+ ∩ Iz+ and P − = Iw− ∩ Iz− both lie on Γ, we observe that the restriction of a
i (v), i = 1, 2 to Iw+ vanishes at only one point, namely P + , while the restriction of a
i (v), i = 1, 2 to Iw− vanishes at only one point, namely P − . Thus, the support of the boundary measures µ+ and µ− are contained within arcs Iw+ and Iw− along which a
i (v), i = 1, 2 maintain one sign. Then, µ+ and µ− both are Dirac measures and the supports are contained at P + and P − , respectively. In fact, we argue that if besides the point P + , the support of measure µ+ has another point in Iw+ , then we can again apply (ηk1 , qk1 ) instead of (η, q) in (11.3.14) to deduce that w+ = w− , and hence P + = P − . This is a contradiction. Similarly we can prove that θ + and θ − both are also Dirac measures and the supports are concentrated at P + and P − , respectively. Therefore supp µ+ = supp θ + = P + ,

supp µ− = supp θ − = P − . (11.3.16)

Noticing (11.3.14) and (11.3.15), we have q(P + ) − λi (P + )η(P + ) = q(P − ) − λi (P − )η(P − ), for i = 1 and i = 2 and for all pairs (η, q).

(11.3.17)

11.4. RELATED RESULTS

157

Let P + = (u+ , 0), P − = (u− , 0). Then especially choosing (η, q) = (v, u + g(v)), we have u+ + g(0) = u− + g(0) and hence u+ = u− . Thus the Young measure ν is a Dirac measure, which implies the existence of weak solutions to the Cauchy problem (11.0.1), (11.0.2).

11.4

Related Results

The large data existence theorem of global L∞ weak solutions for the Cauchy problem (11.0.1), (11.0.2) was first established by DiPerna [Di3] for the system of elasticity, namely g(v) = c = 0 in system (11.0.1). This is also the first application of the compensated compactness method on hyperbolic systems of two equations. In DiPerna’s original paper, the idea to use the entropy-entropy flux pairs of Lax type to reduce the Young measure to be a Dirac measure was first introduced to this strictly hyperbolic system of elasticity. The proof in this chapter for more general systems (11.0.1) is from [Lu6].

Chapter 12

Lp Case to Systems of Elasticity In the last chapter, we studied the extended systems of one-dimensional nonlinear elasticity in Lagrangian coordinates
ut + f (v)x = 0 (12.0.1) vt + ux = 0, with bounded measurable initial data (u(x, 0), v(x, 0)) = (u0 (x), v0 (x)),

(12.0.2)

where v is the strain, f (v) the stress, and u the velocity. The basic assumptions on the nonlinear function f (v) are as follows: (a) f
(v) ≥ c > 0,

(b) v · f

(v) > 0 ∀ v = 0.

The condition (a) ensures the hyperbolicity of the system, which is essential for all existence results we have had up to now for the system of elasticity; and the condition (b) ensures that the system is linearly degenerate only on the line v = 0 as well as the L∞ estimate of solution, which makes the construction of entropy-entropy flux pairs and the reduction of the Young measure to be a Dirac measure much easier. However, if the condition (b) does not hold, in this case, the simplest situation is if the second derivative has still one zero point, namely v · f

(v) < 0 ∀ v = 0, then system (12.0.1) no longer has the L∞ estimate although it is still linearly degenerate only on the same 159

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY

160

line v = 0. In this case, only the Lp (1 < p < ∞) a priori estimate is available (cf. [Da1]). As shown in Chapter 3, for the scalar equation, we can easily extend the compensated compactness method from L∞ space to Lp space since any smooth function is an entropy for the scalar equation. However, for systems of two or more equations, there are many difficulties in the case of Lp space, such as how to construct suitable entropy-entropy flux pairs (η, q) such that the compactness of ηt (sl (x, t)) + qx (sl (x, t) in −1,2 (R × R+ ) holds with respect to a suitable sequence of approxiWloc mated solutions sl ; how to reduce the corresponding Young measures, having unbounded support sets, to be Dirac measures and so on. It is very important both in mathematics and in physics to prove the existence of Lp solutions for hyperbolic systems. Unfortunately, until now, the existence of Lp solutions was obtained only for the above physical model - the system of elasticity. (See [FS1, FS2] for some mathematical systems.) It is interesting that for the above system of elasticity, at almost the same time, there are two different proofs obtained by Lin [Lin] and Shearer [Sh] independently. In this chapter, we shall introduce these proofs as well as an application of Shearer’s proof on an extended system of elasticity of three equations (12.3.1).

12.1

Lin’s Proof for Artificial Viscosity

In this section, we shall introduce a proof of Lp weak solution existence to the Cauchy problem (12.0.1), (12.0.2), which was obtained by Lin by the compensated compactness method coupled with the artificial viscosity approximation. The artificial viscosity solutions to the Cauchy problem (12.0.1), (12.0.2) satisfy the following singular parabolic systems:
ut + f (v)x = εuxx (12.1.1) vt + ux = εvxx , with the initial data (12.0.2). Our basic assumptions are as follows:

12.1. LIN’S PROOF FOR ARTIFICIAL VISCOSITY

161

(A1 ) There exist positive constants M, M1 such that f (v) ∈ C 4 (R),

|(

d k ) f (v)| ≤ M, ∀v ∈ R, k = 2, 3, 4, dv

1

and (f
(v))− 2 is concave for v ≥ M1 , convex for v ≤ −M1 for a large constant M1 . (A2 ) There is a constant c > 0 such that f
(v) ≥ c, ∀v ∈ R. (A3 ) v · f

(v) < 0, ∀v ∈ R − {0}. ¯, v¯, w0 , z 0 and w0 > z 0 such that (A4 ) There exist real numbers u ¯ ∈ L2 (R), u0 (x) − u w(u0 (x), v0 (x)) ≥ w0 ,

v0 (x) − v¯ ∈ L2 (R),

z(u0 (x), v0 (x)) ≤ z 0 ,

∀x ∈ R,

where w and z are two Riemann invariants of system (12.0.1),  v  v 1 1 (f
(s)) 2 ds, z(u, v) = u − (f
(s)) 2 ds. w(u, v) = u + 0

0

Then we have the following theorem: Theorem 12.1.1 (P.X. Lin) If the assumptions (A1 )−(A4 ) hold, then for any fixed ε > 0, the global smooth solution (uε (x, t), v ε (x, t)) of the Cauchy problem (12.1.1), (12.0.2) exists. Furthermore, there exist a subsequence (still labelled) (uε (x, t), v ε (x, t)) and u(x, t), v(x, t) ∈ L∞ ([0, ∞), L2 (R)) such that (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)), a.e. in Ω,

(12.1.2)

where Ω ∈ R×R+ is any open and bounded set. Therefore the limit pair of functions (u(x, t), v(x, t)) is a weak solution of the Cauchy problem (12.0.1), (12.0.2). Outline of the proof of Theorem 12.1.1. Using the theory of invariant region, we can prove that w(uε (x, t), v ε (x, t)) ≥ w0 > z 0 ≥ z(uε (x, t), v ε (x, t)).

(12.1.3)

Then we may construct four families of entropy-entropy flux pairs of Lax type as follows: η±k (w, z) = e±kz (A0 + A1 (±k)−1 ) + P±k ;

162

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY q±k (w, z) = e±kz (B0 + B1 (±k)−1 ) + Q±k ; η¯±k (w, z) = e±kw (a0 + a1 (±k)−1 ) + p±k ; q¯±k (w, z) = e±kw (b0 + b1 (±k)−1 ) + q±k ,

where A0 (w, z), A1 (w, z), B0 (w, z), B1 (w, z) are functions having compact support sets on z ∈ [z − , z + ], z − and z + are constants, z + < w0 ; a0 (w, z), a1 (w, z), b0 (w, z), b1 (w, z) are functions having compact support sets on w ∈ [w− , w+ ], w− and w+ are constants, w+ > z 0 . Moreover P±k (w, z), Q±k (w, z), p±k (w, z), q±k (w, z) have suitable bounds such that the entropies η and corresponding entropy fluxes q satisfy |∇2 η| ≤ C,

|∇η| ≤ C(1 + |u|α + |v|α )

(12.1.4)

and |η| ≤ C(1 + |u|α + |v|α ),

|q| ≤ C(1 + |u|α + |v|α ),

(12.1.5)

for 0 < α < 1. Noticing that system (12.0.1) has a strictly convex entropy  v 1 2  f (v)dv η = u + 2 0 and a corresponding entropy flux q  = uf (v), we have that 1

ε 2 ∂x uε ,

1

ε 2 ∂x v ε

are uniformly bounded in L2loc (R × R+ ). (12.1.6)

Because of the growth conditions (12.1.4)-(12.1.5) on the entropyentropy flux pairs (η, q), we can apply Theorem 2.3.2 to prove the com−1,2 (R × R+ ) with respect to the pactness of ηt (uε , v ε ) + qx (uε , v ε ) in Wloc ε artificial viscosity approximation (u (x, t), v ε (x, t)). Finally, combining some basic ideas given by DiPerna in the L∞ space (see Section 11.2) with these entropy-entropy flux pairs, through a complicated analysis, we can prove the compactness (12.1.2) and hence the existence of a weak solution of the Cauchy problem (12.0.1), (12.0.2). This completes the proof of Theorem 12.1.1. More details about the proof of Theorem 12.1.1 can be found in [Lin].

12.2. SHEARER’S PROOF FOR PHYSICAL VISCOSITY

12.2

163

Shearer’s Proof for Physical Viscosity

As shown in the last section, in the proof of Theorem 12.1.1, one basic technical restriction is that the artificial viscosity solutions satisfy the invariant region (12.1.3), which forces us to use artificial viscosity exactly as in (12.1.1). However, as a physical model, the following physical viscosity approximation to the system of elasticity (12.0.1) is of more interest:
ut + f (v)x = εuxx (12.2.1) vt + ux = 0, in which the invariant region (12.1.3), generally speaking, does not hold. At almost the same time, independent of Lin’s proof, J.W. Shearer (cf. [Sh]) considered the Lp solution for the same system of elasticity (12.0.1) also with the artificial viscosity approximation (12.1.3), but using different entropy-entropy flux pairs, which were extended late by Serre and Shearer to study the compactness of the physical viscosity approximation (12.2.1). Roughly speaking, Shearer constructed two classes of entropy and entropy flux pairs to system (12.0.1). One class is the Fourier entropy and the other is the half plane supported entropy. Both classes of entropy-entropy flux pairs (η, q) satisfy the following estimates:  1 1  η(u, v) = (f
(v))− 4 O(1), q(u, v) = (f
(v)) 4 O(1),       ηu (u, v) = (f
(v))− 14 O(1), ηv (u, v) = (f
(v)) 14 O(1), (12.2.2) 
(v))− 14 O(1),
(v)) 14 O(1),  (u, v) = (f η (u, v) = (f η uu uv     3  ηvv (u, v) = (f
(v)) 4 O(1), where O(1) denotes a bounded function. Shearer’s result is given in the following theorem: Theorem 12.2.1 (J.W. Shearer) If f
(v) ≥ c > 0, and

f

(v) ∈ L1 ∩ L∞ (R),

f

(v) = 0 f


(v) ∈ L∞ (R),

164

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY

then there exists a subsequence (still labelled) (uε (x, t), v ε (x, t)) of the artificial viscosity solutions (uε (x, t), v ε (x, t)) of the Cauchy problem (12.1.1), (12.0.2) and u(x, t), v(x, t) ∈ L∞ ([0, ∞), L2 (R)) such that (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)), a.e. in Ω,

(12.2.3)

where Ω ∈ R×R+ is any open and bounded set. Therefore the limit pair of functions (u(x, t), v(x, t)) is a weak solution of the Cauchy problem (12.0.1), (12.0.2). Remark 12.2.2 The existence result in Theorem 12.2.1 is almost the same as that in Theorem 12.1.1, but the two classes of entropy pairs constructed by Shearer are more flexible and could be applied to study many different problems as follows: (1) The compactness of physical viscosity solutions of the Cauchy problem (12.2.1) with the initial data (12.0.2); (2) The existence of global Lp weak solutions for the system of adiabatic gas flow through porous media (12.3.1), in which there are three conservation laws; (3) The relaxation problem to hyperbolic systems with more than two equations. The application of Shearer’s proof on (2) in Remark 12.2.2 is given in the next section and the details about the application on (3) can be found in Chapter 16. The following is about the application of Shearer’s proof on (1) in Remark 12.2.2. Consider the physical viscosity approximation, that is, when system (12.0.1) is approximated by its singular perturbation, the Cauchy problem of system (12.2.1) with the initial data (12.0.2). Then we have the following compact result obtained by Serre and Shearer (unpublished, cf. [Sh]): Theorem 12.2.3 (Serre and Shearer) If f
(v) ≥ c > 0, and

v · f

(v) < 0, ∀ v ∈ R − {0}

f

(v) ∈ L1 ∩ L∞ (R),

f


(v) ∈ L∞ (R),

12.3. SYSTEM OF ADIABATIC GAS FLOW

165

then there exists a subsequence (still labelled) (uε (x, t), v ε (x, t)) of the Cauchy problem (12.2.1), (12.0.2) and u, v ∈ L∞ ([0, ∞), L2 (R)) such that (uε (x, t), v ε (x, t)) → (u(x, t), v(x, t)), a.e. in Ω.

(12.2.4)

Therefore the limit pair of functions (u(x, t), v(x, t)) is a weak solution of the Cauchy problem (12.0.1), (12.0.2).

12.3

System of Adiabatic Gas Flow

In this section, we are concerned with the existence of weak solutions of the Cauchy problem for the nonlinear system of three equations:  v − ux = 0,    t (12.3.1) ut − σ(v, s)x + αu = 0,    st = 0, with L2 bounded initial data (v(x, 0), u(x, 0), s(x, 0)) = (v0 (x), u0 (x), s0 (x)),

(12.3.2)

where α ≥ 0 is a constant. System (12.3.1) can be used to model the adiabatic gas flow through porous media, where v is specific volume, u denotes velocity, s stands for entropy, and σ denotes pressure. Its form in Eulerian coordinates is also a model of isothermal unsteady two-phase flow in pipelines (cf. [LLL]). In dealing with the Cauchy problem (12.3.1), (12.3.2), one basic difficulty is the a priori estimate of the viscosity solutions, independent of ε in a suitable Lp space (p > 1), of the following parabolic system:  v − ux = εvxx    t (12.3.3) ut − σ(v, s)x + αu = εuxx    st = εsxx , with the initial data (v0ε (x, 0), uε0 (x, 0), sε0 (x, 0)) = (v0ε (x), uε0 (x), sε0 (x)).

(12.3.4)

166

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY

Since system (12.3.1), in general, cannot be diagonalized by Riemann invariants method, it is not to be expected that viscosity solutions (v ε , uε , sε ) of the Cauchy problem (12.3.3), (12.3.4) will be bounded in L∞ space, uniformly in ε, by using the invariant region principle [CCS]. We have to search for solutions of system (12.3.1) in Lp space. In some sense, the a priori estimate of the solutions of the Cauchy problem (12.3.3), (12.3.4) in L2 space is easy to get, if we can find a strictly convex entropy for system (12.3.1). However, a new difficulty arises when considering the compactness of the viscosity solutions in Lp space by trying to use the compensated compactness method. Shearer’s work given in Section 12.2 provided us with an ideal framework to deal with system (12.3.1) with three conservation laws. In this section, we shall study the global generalized solution for the Cauchy problem (12.3.1), (12.3.2) by the compensated compactness method combined with the entropy pairs constructed by Shearer. We make the assumptions about the nonlinear function σ(v, s) and the initial data as follows: (1) σ(v, s) = σ(v)g(s) − cs, g(s) ∈ C 3 , and σ(v) satisfies (a) σ(v) ∈ C 3 (R), σ(0) = 0, σ
(v) ≥ d > c2 , for a constant d; (b) σ

(0) = 0, and σ

(v) = 0 for v = 0; (c) σ

(v), σ


(v) ∈ L2 ∩ L∞ , (2) (v0 (x), u0 (x), s0 (x)) are all bounded in L2 and tend to zero as |x| → ∞ sufficiently fast such that lim (v0ε (x), uε0 (x), sε0 (x)) = (0, 0, 0),

(12.3.5)

|x|→±∞

 dv ε (x) duε (x) dsε (x)  0 , 0 , 0 = (0, 0, 0), dx dx dx |x|→±∞ lim

(12.3.6)

where (v0ε (x), uε0 (x), sε0 (x)) are smooth and obtained by smoothing the initial data (v0 (x), u0 (x), s0 (x)) with a mollifier, satisfying lim (v0ε (x), uε0 (x), sε0 (x))(v0 (x), u0 (x), s0 (x)),

ε→0

v0ε (x) L2 ≤ v0 (x) L2 ≤ M,

a.e.,

(12.3.7)

v0ε (x) H 1 (R) ≤ M (ε), (12.3.8)

12.3. SYSTEM OF ADIABATIC GAS FLOW

uε0 (x) L2 ≤ u0 (x) L2 ≤ M,

167

uε0 (x) H 1 (R) ≤ M (ε), (12.3.9)

sε0 (x) L2 ≤ s0 (x) L2 ≤ M, (12.3.10)

sε0 (x) H 1 (R) ≤ s0 (x) H 1 (R) ≤ M and

di v0ε (x)

, dxi



di uε0 (x)

, dxi



di sε0 (x)

≤ M (ε), dxi

i = 0, 1, 2, (12.3.11)

where M is a positive constant independent of ε, and M (ε) a positive constant, but dependent on ε. We have the main result in the following theorem: Theorem 12.3.1 Let the conditions in (1) and (2) hold. Then for any fixed ε, there is a global solution (v ε , uε , sε ) of the Cauchy problem (12.3.3), (12.3.4) such that the following estimates hold:

v ε (·, t) L2 (R×R+ ) ,

uε (·, t) L2 (R×R+ ) ,

sε (·, t) H 1 (R×R+ ) ≤ M. (12.3.12)

Moreover, there exists a subsequence (v ε , uε , sε ) (still labelled (v ε , uε , sε )) such that (v ε , uε , sε ) → (v, u, s)

as ε → 0,

(12.3.13)

the limit pair of functions (v, u, s) is bounded in L2 (R × R+ ) and there is a weak solution of the Cauchy problem (12.3.1), (12.3.2). Proof. To prove the existence of viscosity solutions (v ε , uε , sε ), it is enough to get the following L∞ estimate, although the bound M (ε, T ) could tend to infinity as ε tends to zero or the time T tends to infinity:

v ε ≤ M (ε, T ),

uε ≤ M (ε, T ),

sε ≤ M (ε, T ).

(12.3.14)

We multiply the first equation in (12.3.3) by σ(v)g(s) − cs, the second v equation by u, the third equation by g
(s) 0 σ(v)dv − cv + γs, then

168

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY

add the result to obtain

u2 + 2



v

0

σ(v)dvg(s) − csv +

γs2  2 t

+(csu − σ(v)g(s)u)x + αu2

u2 + =ε 2



v 0

σ(v)dvg(s) − csv +

γs2  2 xx

(12.3.15)

−ε σ
(v)g(s)vx2 + 2(σ(v)g
(s) − c)sx vx + u2x 

v

+ 0

 σ(v)dvg

(s)s2x + γs2x ,

where γ is a large positive constant. By the condition

s0 (x) H 1

loc (R)

≤ M,

lim s0 (x) = 0,

x→±∞

we get the uniform boundedness of s0 (x) in L∞ (R). So the functions sε0 (x) satisfy |sε0 (x)|L∞ ≤ M,

1

|ε 2 sε0x (x)|L∞ ≤ M,

|εsε0xx (x)|L∞ ≤ M

and hence the third equation in (12.3.3) yields the estimates    sε (x, t) L∞ ≤ M,   |εsε (x, t)|L∞ ≤ M, xx

1

|ε 2 sεx (x, t)|L∞ ≤ M,

sε (·, t) H 1

loc (R)

(12.3.16) ≤ M.

Thus g(s), g
(s) and g

(s) in (12.3.15) all are bounded. Moreover, since

12.3. SYSTEM OF ADIABATIC GAS FLOW |

v

169

σ(v)dv| ≤ M v 2 , we have by integrating (12.3.15) in R × [0, T ] that,

0





∞

u2 + −∞ 2 

T

v

σ(v)dvg(s) − csv +

0



∞

+ε 0

 ≤

−∞

∞

 u2

−∞

2





T

v0

+ 0

∞

+ 0

c1 vx2 + u2x + c2 s2x + αu2 dxdt 

0

−∞

γs2  dx 2

σ(v)dvg(s0 ) − cs0 v0 + 

2

M v dxdt ≤ M1 + M2

0

T



γs20  2 ∞

−∞

(12.3.17) dx

v 2 dxdt,

for some positive constants c1 , c2 , M1 , M2 and M . Since 

v

σ(v)dvg(s) − csv +

0

γs2 ≥ c3 (v 2 + s2 ) 2

(12.3.18)

for a positive constant c3 , we get by applying the Bellman inequality to (12.3.17) that  ∞  ∞  ∞ 2 2 v dx ≤ M (T ), u dx ≤ M (T ), s2 dx ≤ M (T ), −∞

−∞

−∞

(12.3.19) and  ε

T 0



∞

−∞

vx2 + u2x + s2x dxdt ≤ M (T ).

(12.3.20)

Differentiating the first equation in (12.3.3) with respect to x, then multiplying the result by 2vx , we get 2 = ε(vx2 )xx . (vx )2 − 2(vx ux )x + 2ux vxx + 2εvxx

(12.3.21)

Integrating (12.3.21) in R × [0, T ], we have that 

∞

−∞

vx2 dx



T



∞

+ 0

−∞

2 εvxx dxdt ≤ M (ε, T ).

(12.3.22)

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY

170

Similarly we can get   ∞ 2 2 ux + sx dx + −∞

T 0



∞ −∞

ε(u2xx + s2xx )dxdt ≤ M (ε, T ).

(12.3.23)

Now we have the L∞ estimates given in (12.3.14), and hence the existence of viscosity solutions to the Cauchy problem (12.3.3), (12.3.4). For instance,  x  ∞  ∞ 2 2 2 (v )x dx ≤ v dx + vx2 dx ≤ M (ε, T ). v = −∞

−∞

−∞

To prove the existence of weak solutions in Theorem 12.3.1, we let s be fixed as a constant and construct the entropy-entropy flux pairs for the following system:
vt − ux = 0, (12.3.24) ut − (g(s)σ(v))x = 0. We make the transformation x = g(s)y, t = t,

v = v,

u=



g(s)w,

(12.3.25)

and so system (12.3.24) is rewritten as follows:

vt − wy = 0, wt − σ(v)y = 0.

(12.3.26)

We may apply Shearer’s method to construct the same two classes of entropy-entropy flux pairs (¯ η (v, w), q(v, ¯ w)) for the system (12.3.26), one being the Fourier entropy and the other being the half plane supported entropy. Both classes of entropy-entropy flux pairs satisfy the estimates in (12.2.2). Thus we can get the function pairs (η(v, u, s), q(v, u, s)) = (¯ η (v, w), q¯(v, w)) = (¯ η (v,

u g(s)

), q¯(v,

u

)) g(s) (12.3.27)

to system (12.3.1) satisfying the estimates   η(v, u, s) = (σ
(v))− 14 O(1), q(v, u, s) = (σ
(v)) 14 O(1),  η (v, u, s) = (σ
(v))− 14 O(1), u

1

ηv (v, u, s) = (σ
(v)) 4 O(1), (12.3.28)

12.3. SYSTEM OF ADIABATIC GAS FLOW and

 1  ηuu (v, u, s) = (σ
(v))− 4 O(1),    1 ηvu (v, u, s) = (σ
(v)) 4 O(1),     η (v, u, s) = (σ
(v)) 34 O(1),

171

(12.3.29)

vv

where O(1) denotes a bounded function. Since ηw , −¯ ηv ), (¯ qv , q¯w ) = (−σ
(v)¯ we have (consider here s as a variable) 
(v) g(s)η (v, u, s), (v, u, s) = −σ q  v u     1    ηv (v, u, s), q (v, u, s) = −   u g(s)  1 
ug
(s) ∂w  = η ¯ ηu (v, u, s), (v, u, s)¯ η u = − η  s w w  ∂s g(s)  g(s)  
   q (v, u, s) = − ug (s) q (v, u, s).  s u g(s)

(12.3.30)

(12.3.31)

Multiplying system (12.3.3) by (ηv , ηu , ηs ), and using (12.3.31), we have εηv vxx + εηu uxx + εηs sxx = ηv vt + ηu ut + ηs st −ηv ux − ηu (g(s)σ(v))x + cηu sx + αuηu = ηt + qu g(s)ux + cηu sx + αuηu

 +ηu g(s)σ
(v)vx + σ(v)g
(s)sx = ηt + g(s)(qu ux + qv vx )

(12.3.32)

−σ(v)g
(s)ηu sx + cηu sx + αuηu = ηt + ( g(s)q)x − g(s)qs sx − ( g(s))x q −σ(v)g
(s)ηu sx + cηu sx + αuηu . The left-hand side of (12.3.32), εηv vxx + εηu uxx + εηs sxx = ε(ηv vx )x + ε(ηu ux )x + ε(ηs sx )x

 −ε ηvv vx2 + ηuu u2x + ηss s2x 

−ε 2ηvu vx ux + 2ηvs vx sx + 2ηus ux sx .

(12.3.33)

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY

172

It follows from (12.3.16), (12.3.19) and (12.3.20) that |σ(v)g
(s)ηu sx |L1

+ loc (R×R )

ε|ηss s2x |L1

+ loc (R×R )

≤ M |v|L2

+ loc (R×R )

≤ εM |u2 s2x |L1

+ loc (R×R )

|sx |L2

+ loc (R×R )

≤ M1 |u2 |L1

≤ M (T ), (12.3.34)

+ loc (R×R )

≤ M (T ) (12.3.35)

and ε|ηvs vx sx + ηus ux sx |L1

+ loc (R×R )

≤ εM (|uvx sx | + |usx ux |)L1

+ loc (R×R )

≤ M1 |u2 |L1

+ loc (R×R )

+ ε|vx2 |L1

+ loc (R×R )

+ ε|u2x |L1

+ loc (R×R )



≤ M (T ), (12.3.36) for some positive constants M, M1 , M (T ). Therefore we have ηt (v, u, s) + ( g(s)q(v, u, s))x = I1 + I2 ,

(12.3.37)

where −1 (R × R+ ), I1 = ε(ηv vx )x + ε(ηu ux )x + ε(ηs sx )x is compact in Hloc (12.3.38)

and

I2 = ( g(s))x q + g(s)qs sx + σ(v)g
(s)ηu sx

 −cηu sx − αuηu − ε ηvv vx2 + ηuu u2x + ηss s2x 

−ε 2ηvu vx ux + 2ηvs vx sx + 2ηus ux sx

(12.3.39)

−1,k in Wloc (R × R+ ) is bounded in L1loc (R × R+ ) and hence compact for a constant k ∈ (1, 2). However, ηt (v, u, s) + ( g(s)q(v, u, s))x is −1,p + in (12.3.28), thus bounded in W loc (R×R ) for p > 2 by the estimates −1 ηt (v, u, s) + ( g(s)q(v, u, s))x is compact in Hloc (R × R+ ) by Theorem 2.3.2.

12.4. RELATED RESULTS

173

From (12.3.16), we have the pointwise convergence of the sequence {sε (x, t)}. Then the compact support of Young measures ν(x,t) , determined by the viscosity solutions (v ε (x, t), uε (x, t), sε (x, t)) to the Cauchy problem (12.3.3), (12.3.4) is reduced to the (v, u)-plane, and the following measure equations are satisfied for any entropy-entropy flux pairs (ηi (v, u, s), qi (v, u, s)) constructed above: < ν, η1 (v ε , uε , s) g(s)q2 (v ε , uε , s) − η2 (v ε , uε , s) g(s)q1 (v ε , uε , s) > =< ν, η1 (v ε , uε , s) >< ν, g(s)q2 (v ε , uε , s) > − < ν, η2 (v ε , uε , s) >< ν, g(s)q1 (v ε , uε , s) >, (12.3.40) or equivalently < ν, η1 (v ε , uε , s)q2 (v ε , uε , s) − η2 (v ε , uε , s)q1 (v ε , uε , s) > =< ν, η1 (v ε , uε , s) >< ν, q2 (v ε , uε , s) > − < ν, η2 (v ε , uε , s) >< ν, q1 (v ε , uε , s) > (12.3.41) since g(s) ≥ d > 0. Now consider s to be a constant in (12.3.41). Then the compactness framework in Theorem 12.2.1 or Theorem 12.2.3 deduces that all the Young measures ν are Dirac measures, which implies the proof of Theorem 12.3.1.

12.4

Related Results

Besides the physical model (12.0.1), the existence of Lp weak solutions is also obtained by Frid and Santos [FS1, FS2] for the following mathematical system of two equations: z γ )x = 0, zt − (¯

1 < γ < 2,

(12.4.1)

where z = u + iv ∈ C. System (12.3.1) is more or less similar to system (12.0.1), but it is the unique system of three equations for which we can obtain Lp weak

174

CHAPTER 12. LP CASE TO SYSTEMS OF ELASTICITY

solutions until now. System (12.3.1) and the system of chromatography (7.5.1) are only two applications of the compensated compactness method on hyperbolic systems of more than two equations. The proof of Theorem 12.3.1 is from the paper [LK1].

Chapter 13

Preliminaries in Relaxation Singularity We are concerned with the system of partial differential equation in the form 1 U (x, t)t + F (U )x + R(U ) = εUxx . τ

(13.0.1)

Here U = U (x, t), which takes on value in RN , represents the density vector of basic physical variables over the space variable x, the quantity τ is the relaxation time, which is small in many physical situations. In the kinetic theory it is the mean free path, in elasticity the duration of memory. ε is the artificial viscosity parameter, or the diffusion coefficient. When ε = 0, and the corresponding systems U (x, t)t + F (U )x = 0

(13.0.2)

are hyperbolic, that is, the N ×N matrix ∇F (U ) has N real eigenvalues, the relaxation systems 1 U (x, t)t + F (U )x + R(U ) = 0 τ

(13.0.3)

in the level of hyperbolic equations arise in many physical situations, such as combustion theory [Lu3, Lu7, Ma], multiphase and phase transition [ChL], chromatography [RAA1, Wh], viscoelasticity [CLL, Na], 175

176

CHAPTER 13. PRELIMINARIES IN RELAXATION

kinetic theory [BR, Ca, Ce], river flows [KL2, Wh] and traffic flows [Sc, Wh]. Given a system in the form of (13.0.1), an interesting thing both in physics and mathematics is the limit behavior of the solutions U τ,ε (x, t) of system (13.0.1) as the relaxation time and the viscosity ε go to zero. As an illustrative 2 × 2 model, consider the following system of two equations:
ut − vx = εuxx , (13.0.4) = εv , vt − cux + v−h(u) xx τ where c is a constant and h(u) is the equilibrium value for v. System (13.0.4) could be the simplest model of systems in the form (13.0.1). For system (13.0.4), we shall see that if ε = 0 and the system
ut − vx = 0, (13.0.5) vt − cux = 0, is hyperbolic, i.e., c > 0, the basic condition √ √ − c < h
(u) < c,

(13.0.6)

so called the subcharacteristic condition (cf. [Liu]), is necessary to ensure the stability of solutions (uτ , v τ ) of the following system:
ut − vx = 0, (13.0.7) =0 vt − cux + v−h(u) τ as τ → 0. However if ε = 0 and c < 0, system (13.0.4) or system (13.0.7) is ill posed; if ε > 0 and τ = o(ε), that is, τ is smaller than ε, then the solutions (uτ,ε , v τ,ε ) of system (13.0.4) are always stable. These can be seen through an asymptotic expansion of the ChapmanEnskog type. Let v = h(u) + τ v1 + O(τ 2 ).

(13.0.8)

Then from the second equation in (13.0.4) we obtain v1 = εvxx − vt + cux + O(τ ) = εvxx − h
(u)ut + cux + O(τ ) = εvxx − εh
(u)uxx − h
(u)vx + cux + O(τ ) = εvxx − εh
(u)uxx − (h
(u))2 ux + cux + O(τ ).

(13.0.9)

177 Substituting (13.0.9) into the first equation in (13.0.4), we have ut − h(u)x = uxx + τ v1x + O(τ 2 )    = ε + τ (c − h
(u)2 ) ux + O(τ 2 ) + O(τ ). x

(13.0.10) Thus if ε = 0, c > h
2 (u) or ε > 0 and τ = o(ε), in the latter case, ε > τ (h
2 (u) − c), Equation (13.0.10) is well posed since the coefficient of the diffusion term is positive. Unfortunately if ε = 0 and c < 0, we have the following system:
ut − vx = 0, (13.0.11) vt + ux = 0, and hence system (13.0.7) is ill posed as we proceed to show. By the first equation in (13.0.7) there must exist a function w such that wx = u,

wt = v.

(13.0.12)

Thus the second equation in (13.0.7) can be put in the form 1 wtt − cwxx + (wt − h(wx )) = 0. τ

(13.0.13)

This is an elliptic equation and its Dirichlet problem in the domain t > 0 can be solved with the data  x u0 (s)ds. (13.0.14) w(x, 0) = 0

But then v0 (x) cannot be chosen independently of u0 (x) since in this case we must have v0 (x) = wt (0, x) and w depends on u0 (x). The above analysis illustrates that, besides the subcharacteristic condition (13.0.6) for the stability of solutions of hyperbolic system (13.0.7), viscosity ε in (13.0.1) is not only of mathematical expedience when acting together with relaxation but may also be another necessary stability mechanism. In Chapter 14, we are concerned with singular limits of stiff relaxation and dominant diffusion for general 2 × 2 nonlinear systems of

178

CHAPTER 13. PRELIMINARIES IN RELAXATION

conservation laws, that is, the relaxation time τ tends to zero faster than the diffusion parameter , τ = o(), ε → 0. Some compactness frameworks without the subcharacteristic condition are established. In Chapter 15, we shall study the limiting behavior of some special hyperbolic systems of two equations with stiff relaxation terms, where all compact results are based on the subcharacteristic condition. In Chapter 16, relaxation problems for some special hyperbolic systems of three equations are studied.

Chapter 14

Stiff Relaxation and Dominant Diffusion In this chapter, we are concerned with singular limits of stiff relaxation and dominant diffusion for general 2 × 2 nonlinear systems of conservation laws, that is, the relaxation time τ tends to zero faster than the diffusion parameter , τ = o(),  → 0. We establish the following general framework: If there exists an a priori uniform L∞ bound with respect to  for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system. Our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems, which is not the case for other types of limits in Chapter 15 for relaxation limits without viscosity. This framework applies to some important nonlinear systems with relaxation terms, such as the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates, and the extended models of traffic flows. The singular limits are also considered for some physical models, without L∞ bounded estimates, including the system of isentropic fluid dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms. The convergence of solutions in Lp to the equilibrium solutions of these systems is established, provided that the relaxation time τ tends to zero faster than ε. 179

CHAPTER 14. RELAXATION WITH DIFFUSION

180

14.1

Compactness Results

We are concerned with singular limits of stiff relaxation and dominant diffusion for the Cauchy problem of general 2 × 2 quasilinear conservation laws with relaxation and diffusion:
vt + f (v, u)x = εvxx , (14.1.1) ut + g(v, u)x + τ1 α(v, u)(u − h(v)) = εuxx , with initial data (v, u)|t=0 = (v0 (x), u0 (x)).

(14.1.2)

The second equation in (14.1.1) contains a relaxation mechanism, with h(v) as the equilibrium value for u, τ the relaxation time, α(v, u) > 0, and ε is the diffusion coefficient. The relaxation term serves as a damping in some suitable system coordinates. The singular limit problem for (14.1.1) can be considered as a singular perturbation problem as τ tends to zero. When τ = ε, the relaxation systems have been studied for some typical models (cf. [ES, Fi, RSK] and the references cited therein). The relaxation systems will be studied in Chapter 15 in the case where ε = o(τ ) and the corresponding 2 × 2 systems
vt + f (v, u)x = 0, (14.1.3) ut + g(v, u)x = 0, are hyperbolic. In this section, we consider the case of stiff relaxation and dominant diffusion, that is, τ = o(), as ε → 0. When the solutions of the Cauchy problem (14.1.1), (14.1.2) are uniformly bounded in L∞ , we show that the limit is always stable and no oscillation arises for any C 1 flux functions f and g. Theorem 14.1.1 Let f, g ∈ C 1 (R2 ), h ∈ C 2 (R) and α0 ≤ α ∈ C(R2 ) for a positive constant α0 . Let τ = o(ε) as ε → 0. If the solutions (v ε , uε ) ≡ (v ε,τ (ε) , uε,τ (ε) ) of the Cauchy problem (14.1.1), (14.1.2) have an a priori L∞ bound: |(v ε , uε )(x, t)| ≤ M (T ),

(x, t) ∈ R × [0, T ],

(14.1.4)

14.1. COMPACTNESS RESULTS

181

for any given time T, where M (T ) is independent of ε, then there exists a subsequence (v εk , uεk ) converging strongly to the functions (v, u) as εk → 0, which are the equilibrium states uniquely determined by (E1 )(E2 ): (E1 ) u(x, t) = h(v(x, t)), for almost all (x, t) ∈ R × (0, T ]; (E2 ) v(x, t) is the L∞ entropy solution of the Cauchy problem vt + f (v, h(v))x = 0,

v|t=0 = v0 (x).

(14.1.5)

If the solutions of the Cauchy problem (14.1.1)-(14.1.2) have no a priori L∞ estimate, we can prove that the limit is also stable, provided that the functions f, g, and α satisfy certain growth conditions. Assume ¯) ∈ L∞ ∩ Lp (R), (v0 (x) − v¯, u0 (x) − u

u ¯ = h(¯ v ),

1 < p < ∞. (14.1.6)

Also assume the following conditions: (B1 ) |f (v, u)| + |g(v, u)| ≤ c1 + c2 (|v|q + |u|q ),

q ∈ [1, 3),

|∇v,u f (v, u)| + |∇v,u g(v, u)| ≤ c3 + c4 (|v|q−1 + |u|q−1 ); (B2 ) 0 < c5 +c6 (|v|r +|u|r ) ≤ α(v, u) ≤ c7 +c8 (|v|r +|u|r ), (B3 ) |h(v)| ≤ c9 + c10 |v|k ,

0 ≤ r < 4;

k ≥ 1,

where q, r, k, ci , 1 ≤ i ≤ 10, are positive constants. Theorem 14.1.2 (I) Assume conditions (B1 )-(B3 ) are satisfied, and there exists a strictly convex function p¯(v, u) in the range of solutions of the Cauchy problem (14.1.1), (14.1.2) with initial data (v0 , u0 ) satisfying (14.1.6) such that p¯u (v, u) = p1 (v, u)(u − h(v)),

p¯(v, u) = η(v, u) + p(v, u),

where p1 (v, u) ≥ c0 > 0, p(v, u) is a smooth function whose second order derivatives are bounded, and η(v, u) is an entropy of system (14.1.3). Then, if 2k(q − 1) ≤ r and M1 τ ≤ ε for some large constant M1 , for fixed ε, τ , the Cauchy problem (14.1.1), (14.1.2) with initial data

182

CHAPTER 14. RELAXATION WITH DIFFUSION

(v0 , u0 ) satisfying (14.1.6) has a unique smooth solution (v ε,τ , uε,τ ) satisfying |(v ε,τ , uε,τ )(x, t)| ≤ C(ε, τ ),

(14.1.7)

for some constant C(ε, τ ) > 0 depending on ε and τ . (II) If the conditions of (I) are satisfied and 2k(q − 1) + p − 2 ≤ r, then |v ε,τ − v¯|p ≤ M . (III) If the conditions of (II) are satisfied, p > max{1, q, qk}, and τ = o(ε) as ε → 0, then there exists a subsequence (v εk , uεk ) of (v ε , uε ) ≡ (v ε,τ (ε) , uε,τ (ε) ), converging pointwisely almost everywhere: (v εk , uεk ) → (v, u),

as

εk → 0,

(14.1.8)

where the limit functions (v, u) satisfy (F1 )-(F2 ): (F1 ) u(x, t) = h(v(x, t)), for almost all (x, t) ∈ R × (0, T ]; (F2 ) v(x, t) is the unique Lp entropy solution of the Cauchy problem: vt + f (v, h(v))x = 0,

v|t=0 = v0 (x).

(14.1.9)

Remark 14.1.3 If 2k(q − 1) ≤ r, then from (B1 ) − (B3 ), we have (B4 ) {fu (v, β)2 + gu (v, β)2 + f  (v, h(v))2 + g (v, h(v))2 }/α(v, u) ≤ M, where β takes a value between u and h(v). This inequality will be used to prove Theorem 14.1.2 in Section 14.4. The proof of Theorem 14.1 is given in Section 14.2. Its applications to the system of elasticity, the system of isentropic gas dynamics in Eulerian coordinates, and the extended models of traffic flows with stiff relaxation terms will be given in Section 14.3. The proof of Theorem 14.1.2 is given in Section 13.4, whose applications to the system of isentropic gas dynamics in Lagrangian coordinates and the models of traffic flows with stiff relaxation terms are given in Section 14.5.

14.2. PROOF OF THEOREM 14.1.1

14.2

183

Proof of Theorem 14.1.1

2 = R×R+ = (−∞, ∞)×(0, ∞), Throughout this Chapter, we denote R+ and M (T ) and M are generic constants independent of  and τ , which may be different at each occurrence. Before proving Theorem 14.1.1, we first introduce two lemmas. First, we have the following global existence result about the Cauchy problem (14.1.1), (14.1.2):

Lemma 14.2.1 If the solutions of (14.1.1), (14.1.2) have an a priori L∞ bound (14.1.7), then, for any fixed ε and τ , the Cauchy problem (14.1.1), (14.1.2) has a unique smooth solution (v ε,τ , uε,τ ) on R×(0, T ]. The local existence and regularity of solutions for t > 0 of the Cauchy problem (14.1.1), (14.1.2) can be obtained by applying the Banach contraction mapping theorem to an integral representation of (14.1.1), where the local time depends only on , τ , the L∞ norm of the initial data. The global existence is based on the local existence and the a priori L∞ estimate (14.1.7). Second, we have the following estimates: Lemma 14.2.2 If the solutions of (14.1.1)-(14.1.2) have an a priori L∞ bound (14.1.7) and f, g ∈ C 1 (R2 ), h ∈ C 2 (R), then (εvx2 , εu2x ,

(u − h(v))2 ) L1 (R2+ ) ≤ M, loc τ

(14.2.1)

provided that M1 τ ≤ ε for some large constant M1 > 0. Proof. Since (v, u) is bounded, we can choose a large constant C1 such 2 2 that the function p(v, u) = u2 − h(v)u + C12v satisfies pvv (v, u)vx2 + 2pvu (v, u)vx ux + puu (v, u)u2x ≥ C2 (vx2 + u2x ),

(14.2.2)

for some constant C2 > 0. Multiplying system (14.1.1) by (pv , pu ), we have from (14.2.2) that p(v, u)t + pv (v, u)f (v, u)x + pu (v, u)g(v, u)x +α(v, u) ≤ εpxx (v, u).

(u − h(v))2 + εC2 (vx2 + u2x ) τ

(14.2.3)

CHAPTER 14. RELAXATION WITH DIFFUSION

184 Notice that

pv (v, u)f (v, u)x = (pv (v, u)(f (v, u) − f (v, h(v))))x +pv (v, h(v))f (v, h(v))x −pvx (v, u)(f (v, u) − f (v, h(v))) +(pv (v, u) − pv (v, h(v)))f (v, h(v))x (14.2.4)

= (pv (v, u)(f (v, u) − f (v, h(v))))x  v +( pv (s, h(s))f  (s, h(s))ds)x −(pvu (v, u)ux + pvv (v, u)vx )fu (v, β1 )(u − h(v)) +pvu (v, β2 )(u − h(v))f  (v, h(v))vx and pu (v, u)g(v, u)x = (pu (v, u)(g(v, u) − g(v, h(v))))x  v +( pu (s, h(s))g (s, h(s))ds)x −(puu (v, u)ux + pvu (v, u)vx )gu (v, β3 )(u − h(v))

(14.2.5)

+puu (v, β4 )(u − h(v))g (v, h(v))vx , where pv (v, h(v)) = pv (v, u)|u=h(v) ,

f  (v, h(v)) =

df (v, h(v)) dv

and βi , 1 ≤ i ≤ 4, take values between u and h(v). It follows from (14.2.2)–(14.2.5) that p(v, u)t + q(v, u)x + c0 −τ C3 (vx2 + u2x )

(u − h(v))2 + εC2 (vx2 + u2x ) 2τ

(14.2.6)

≤ εpxx (v, u), for a function q and positive constants c0 and C3 depending on the bounds of second derivatives of p and first derivatives of f and g. Multiplying (14.2.6) by a suitable nonnegative test function and 2 , we get the estimates in (14.2.1) prothen integrating by parts on R+ vided that 2τ C3 ≤ εC2 .

14.2. PROOF OF THEOREM 14.1.1

185

Proof of Theorem 14.1.1. To prove Theorem 14.1.1, we rewrite the first equation in (14.1.1) as follows:   (14.2.7) vt + f (v, h(v))x = εvxx + f (v, h(v)) − f (v, u) x . Let η(v) be any entropy of the scalar equation vt + f (v, h(v))x = 0. Multiplying (14.2.7) by η  (v), we have η(v)t + q(v)x = −η  (v)(f (v, u) − f (v, h(v)))x + εη  (v)vxx = −(η  (v)(f (v, u) − f (v, h(v))))x + εη(v)xx +(f (v, u) − f (v, h(v)))η  (v)vx − εη  (v)vx2

(14.2.8)

= −(η  (v)fu (v, β1 )(u − h(v)))x + εη(v)xx +fu (v, β2 )η  (v)(u − h(v))vx − εη  (v)vx2 , where βi , i = 1, 2, take values between u and h(v). It follows from the estimates in (14.2.1) that, on any compact Ω ⊂ there hold  |fu η  (v)(u − h(v))vx |dxdt

2, R+

Ω

≤M and



(u −

Ω

h(v))2 τ

dxdt

 1  2 Ω

1 τ vx2 dxdt 2 → 0,

     (η  (v)fu (u − h(v)))x φdxdt Ω     =  η  (v)fu (u − h(v))φx dxdt Ω  1 1   (u − h(v))2 2 2 dxdt 2 → 0 τ φx dxdt ≤M τ Ω Ω

(14.2.9)

(14.2.10)

as ε → 0. Moreover, since εη  (v)vx2 is bounded in L1loc and εη(v)xx → 0 in the sense of distributions, the right-hand side of (14.2.7) is compact −1,q , for a constant q ∈ (1, 2). Noticing the left-hand side of in Wloc (14.2.7) is bounded in W −1,∞ , we have that η(v ε )t + q(v ε )x

−1,2 is compact in Wloc

CHAPTER 14. RELAXATION WITH DIFFUSION

186

for any entropy η, with respect to the viscosity solutions v ε . Therefore the strong convergence of v ε follows from the compactness framework about the scalar equation given in Chapter 3, which implies the strong convergence of uε by the third estimate in (14.2.1). This completes the proof of Theorem 14.1.1.

Remark 14.2.3 In Theorem 14.1.1, the condition (3.1.6) on the nonlinear flux function f (v, h(v)) is not assumed to ensure the strong convergence of v ε . In fact, this condition is removed by Szepessy [Sz] as shown in the last part of Section 3.3 (see [Lu9] for the details).

14.3

Applications of Theorem 14.1.1

In this section we apply Theorem 14.1.1 to some important physical models such as the system of elasticity, the isentropic system of gas dynamics in Eulerian coordinates, and the extended models of traffic flows with relaxation terms. 14.3.1. The System of Elasticity The system of elasticity is given by
vt + ux = 0, ut + σ(v)x = 0,

(14.3.1)

which describes the balance of mass and momentum. The existence of L∞ weak solutions for system (14.3.1) with large bounded initial data is obtained in Chapter 11 for the case of σ  (v)v > 0, for all v ∈ R\{0}, and the Lp weak solutions are introduced in Chapter 12 for the case of σ  (v)v < 0, for all v ∈ R\{0}. Consider the viscosity solutions for the system of elasticity with a relaxation term:
vt + ux = εvxx , (14.3.2) = εuxx , ut + σ(v)x + u−h(v) τ with bounded initial data (v0 (x), u0 (x)). The zero relaxation and dissipation limits for system (14.3.1) was first studied by Chen, Levermore and Liu ([CLL]). By applying the

14.3. APPLICATIONS OF THEOREM 14.1.1

187

invariant region arguments, they first obtained the L∞ estimate of solutions (v ε,τ , uε,τ ) for the Cauchy problem (14.3.2) with the initial data (v0 (x), u0 (x)) satisfying following assumptions: (C1 ) σ ∈ C 2 , σ  (v) > 0 for all v ∈ R; vσ  (v) > 0 for all v ∈ R\{0};  (C2 ) σ and h satisfy the stability condition |h (v)| ≤ σ  (v); ¯ (h ¯ a constant), as |v| ≥ M0 for a suitable (C3 ) h ∈ C 1 , h(v) = h large constant M0 . In fact, assumption (C3 ) can be removed by applying the comparison principle (cf. [Na]). And condition (C1 ) can be weakened, by the same method as in the proof of Theorem 7.1 in [Na], to the following: (C¯1 ) σ ∈ C 2 , σ  (v) ≥ 0, but meas{v : σ  (v) = 0} = 0; vσ  (v) > 0

for all v ∈ R\{0}. When considering the convergence of (v ε,τ , uε,τ ) as τ and ε tend to zero, the following two assumptions are used both in [CLL] and [Na]: (C4 ) h ∈ C 1 and there is no interval in which h(v) is affine; (C5 ) |(v0 , u0 )(x)| ≤ N0 , for a suitable constant N0 . However, there is no rate restriction between τ and ε, and oscillatory initial data are allowed. Using Theorem 14.1.1 and the boundedness estimate from assumptions (C¯1 ) and (C2 ), we have the following theorem: Theorem 14.3.1 If h, σ satisfy assumptions (C¯1 ) and (C2 ), then there exists a subsequence (v εk , uεk ) of global smooth solutions (v ε , uε ) for the Cauchy problem (14.3.2) with the bounded initial data (14.1.2), that converges strongly to the equilibrium state functions (v, u), determined uniquely by h(v) and v0 (x) (see (E1 )-(E2 ) of Theorem 14.1.1). 14.3.2. The System of Isentropic Gas Dynamics in Eulerian Coordinates The system of isentropic gas dynamics in Eulerian coordinates is described by
ρt + (ρu)x = 0, (14.3.3) (ρu)t + (ρu2 + P (ρ))x = 0, where ρ, m = ρu and P are the density, the mass and the pressure, respectively. For the case of a polytropic gas P (ρ) = κργ , κ > 0, where

188

CHAPTER 14. RELAXATION WITH DIFFUSION

γ is the adiabatic exponent, the existence of global weak solutions with L∞ large initial data for this system with general adiabatic exponent γ > 1 is given in Chapter 8. Adding a relaxation term to system (14.3.3), we get the following system:
ρt + (ρu)x = 0, (14.3.4) = 0, (ρu)t + (ρu2 + P (ρ))x + ρu−h(ρ) τ which arises in many physical situations such as the flood flows with friction or the river equations (cf. [CLL, Chern, Wh]). Consider the viscosity solutions of system (14.3.4):
ρt + mx = ερxx , (14.3.5) 2 εm , mt + ( mρ + P (ρ))x + m−h(ρ) xx τ with bounded initial data:   (ρε,τ,δ , mε,τ,δ )|t=0 = (ρδ0 (x), mδ0 (x)) ≡ (ρ0 (x) + δ, ρ0 (x)u0 (x)),  0 ≤ ρ0 (x),

0 (x) |m ρ0 (x) | ≤ M0 < ∞.

(14.3.6) Applying the invariant region principle, Lattanzio and Marcati obtained the a priori L∞ estimate of solutions of the Cauchy problem (14.3.5), (14.3.6) for h(ρ) = ρ(1 − ρ) (cf. [LM]). They also considered the zero relaxation limit for system (14.3.4) in the domain of the density away from vacuum. Fortunately, since only L∞ estimate uniformly in the relaxation time and the dissipation parameter is required in Theorem 14.1.1, we can obtain the convergence of solutions, including the vacuum, of the Cauchy problem (14.3.5), (14.3.6) for more general cases. Theorem 14.3.2 Let P (ρ) ∈ C 2 (0, ∞), P  (ρ) > 0, 2P  (ρ) + ρP  (ρ) ≥ 0 for ρ > 0 and  c   ∞  P (ρ) P (ρ) dρ = ∞, dρ < ∞, ∀c > 0. ρ ρ c 0 Suppose that there exists a region Σ8 = {(ρ, m) : w ≤ N, z ≥ −L},

14.3. APPLICATIONS OF THEOREM 14.1.1

189

for some N, L, such that the curve m = h(ρ) and the initial data (ρδ0 (x), mδ0 (x)) are inside the region Σ8 , and m = h(ρ) passes the two intersection points (ρ, m) = (0, 0) and (ρ, m) = (¯ ρ, m), ¯ ρ¯ > 0, of the curves w = N and z = −L (see Figure 14.1). m m = h(ρ) w=N

ρ Σ8 (m, ¯ ρ¯)

z = −L FIGURE 14.1

Then, for any fixed ε, τ and δ, the Cauchy problem (14.3.5), (14.3.6) has a unique smooth solution (ρε,τ,δ , uε,τ,δ ) satisfying 0 < c(t, ε, δ) ≤ ρε,τ,δ ≤ M,

|uε,τ,δ | ≤ M,

(14.3.7)

where c(t, ε, δ) is a positive constant. Moreover, there exists a subsequence, still denoted (ρε,τ,δ , uε,τ,δ ), that converges pointwisely almost everywhere: (ρε,τ,δ , ρε,τ,δ uε,τ,δ ) → (ρ, m) as δ, ε ↓ 0+ with τ = o(ε), where the limit functions (ρ, m) are the equilibrium state functions determined uniquely by h(ρ) and ρ0 (x) (see (E1 )-(E2 ) of Theorem 14.1.1).

190

CHAPTER 14. RELAXATION WITH DIFFUSION

Proof. To prove the estimates in (14.3.7), we multiply (14.3.5) by (wρ , wm ) and (zρ , zm ), respectively, to obtain wt + λ2 wx +

ρu − h(ρ) 2ε = εwxx + ρx wx τρ ρ

ε  (2P  + ρP  )ρ2x , P  (ρ) ρu − h(ρ) 2ε = εzxx + ρx zx zt + λ1 zx + τρ ρ ε  (2P  + ρP  )ρ2x . + 2ρ2 P  (ρ) −

2ρ2

Then the assumptions on P (ρ) yield  ρu − h(ρ) 2ε   wt + λ2 wx + ≤ εwxx + ρx wx , τρ ρ ρu − h(ρ) 2ε   zt + λ1 zx + ≥ εzxx + ρx zx . τρ ρ

(14.3.8)

If the curve m = h(ρ) passes the two intersection points (0, 0), (¯ ρ , m) ¯ of curves w = N, z = −L and is above the curve z = −L and below the curve w = N as 0 ≤ ρ ≤ ρ¯, then it is easy to check that the region Σ8 = {(ρ, m) : w ≤ N, z ≥ −L} is an invariant region. Thus we obtain | ≤ M for a suitable constant the estimates √ 0 ≤ ρε,τ,δ ≤ M and |uε,τ,δ

c √P
(ρ)

∞ P
(ρ) dρ = ∞ and 0 dρ < ∞ for any constant M , since c ρ ρ c > 0. The positive lower bound of ρ in (14.3.7) can be obtained by the last part of Theorem 1.0.2. Thus Theorem 14.3.2 follows from Theorem 14.1.1. 14.3.3. Extended Models of Traffic Flows: L∞ Solutions Consider the viscosity solutions to the extended model of traffic flows:
ρt + (ρu)x = ερxx , (14.3.9) 2 = εu , ut + ( u2 + g(ρ))x + u−h(ρ) xx τ with bounded initial data (ρ, u)|t=0 = (ρ0 (x), u0 (x)),

ρ0 (x) ≥ 0.

(14.3.10)

14.3. APPLICATIONS OF THEOREM 14.1.1

191

The existence of weak solutions to the corresponding hyperbolic system of (14.3.9),
ρt + (ρu)x = 0, (14.3.11) 2 ut + ( u2 + g(ρ))x = 0, is obtained in Chapters 9 and 10. The study of the zero relaxation limit for system (14.3.11) with a singular relaxation term,
ρt + (ρu)x = 0, (14.3.12) 2 = 0, ut + ( u2 + g(ρ))x + u−h(ρ) τ was started by Schochet in [Sc]. System (14.3.12) was derived for car traffic flows (cf. [Wh]) and its existence of classical solutions for all time was obtained in [Sc] for the case f (ρ) = µτ log ρ, provided that τ is sufficiently small and τ ≤ µ3+α , α > 0. Using Theorem 14.1.1, we have the following theorem: Theorem 14.3.3 Let g (ρ) > 0, and g (ρ)/ρ be a nondecreasing function. Suppose that there exist two constants N, L such that the curve u = h(ρ) passes the unique intersection point (¯ ρ, u ¯) of curves w = N, z = −L; the curve u = h(ρ) and the initial data (ρ0 (x), u0 (x)) are in the region Σ9 = {(ρ, u) : w ≤ N, z ≥ −L, ρ ≥ 0} as 0 ≤ ρ ≤ ρ¯ (see Figure 14.2). Then, for any fixed ε and τ , the Cauchy problem (14.3.9), (14.3.10) has a unique smooth solution (ρε,τ , uε,τ ) satisfying 0 ≤ ρε,τ ≤ M,

|uε,τ | ≤ M.

(14.3.13)

Moreover, there exists a subsequence of (ρε,τ , uε,τ ) that converges pointwisely to (ρ, u), as ε tend to zero with τ = o(ε), where (ρ, u) are the equilibrium state functions, determined uniquely by h(ρ) and ρ0 (x) (see (E1 )-(E2 ) of Theorem 14.1.1). Proof. Noticing the conclusions in Theorem 14.1.1, the crux to prove Theorem 14.3.3 is still the boundedness estimates in (14.3.13).

CHAPTER 14. RELAXATION WITH DIFFUSION u w=N

192

m = h(ρ) ρ

(m, ¯ ρ¯)

Σ9

z = −L

FIGURE 14.2

Multiplying (14.3.9) by (wρ , wu ) and (zρ , zu ), respectively, where w and z are Riemann invariants of system (14.3.11), we have  wt + λ2 wx + τ1 (u − h(ρ))εwxx − ( g /ρ) ρ2x ≤ εwxx ,  zt + λ1 zx + τ1 (u − h(ρ))εzxx + ( g /ρ) ρ2x ≥ εzxx . From the assumptions, there exist two constants N, L such that the curve u = h(ρ) passes the unique intersection point (¯ ρ, u ¯) of curves w = N, z = −L, and the curve u = h(ρ) and the initial data (ρ0 (x), u0 (x)) are in the region Σ9 = {(ρ, u) : w ≤ N, z ≥ −L, ρ ≥ 0} as 0 ≤ ρ ≤ ρ¯ (see Figure 14.2). Then Σ9 must be an invariant region. This completes the proof of Theorem 14.3.3.

14.4

Proof of Theorem 14.1.2

In this section we prove Theorem 14.1.2 in several steps.

14.4. PROOF OF THEOREM 14.1.2

193

Proof of (I). The local existence and regularity of solutions for t ∈ (0, t0 ) for the Cauchy problem (14.1.1), (14.1.2) can be obtained by applying the Banach contraction mapping theorem to an integral representation of (14.1.1), where the local time depends only on , τ , and ¯). The global the L1 ∩ L∞ norm of the initial data (v0 − v¯, u0 − u existence is based on the local existence and the global a priori L∞ estimate (14.1.7) proved below. Set ˜¯(v, u) = p¯(v, u) − p¯(¯ v, u ¯)(v − v¯) − p¯u (¯ v , u¯)(u − u ¯), p v , u¯) − p¯v (¯ and similarly η˜(v, u) and p˜(v, u). ˜¯(v, u) is strictly convex, then Since p ˜¯vu (v, u)vx ux + p ˜¯uu (v, u)u2x ≥ C2 (vx2 + u2x ), ˜¯vv (v, u)vx2 + 2p p

(14.4.1)

for some constant C2 > 0. Since η˜(v, u) is an entropy of (14.1.3), we set the corresponding entropy flux by Q(v, u). Observing p¯(v, u) = η(v, u)+p(v, u), we multiply ˜¯v , p˜¯u ) to obtain system (14.1.1) by (p ˜¯t (v, u) + Qx (v, u) + p˜v f (v, u)x + p˜u g(v, u)x p 1 + p1 (v, u)α(v, u)(u − h(v))2 + εC2 (vx2 + u2x ) τ ˜¯xx (v, u). ≤ εp

(14.4.2)

Noticing (14.2.4)-(14.2.5) and the conditions of (B1 )-(B4 ), we have 1 2τ c0 α(v, u)(u C3 τ )(vx2 + u2x )

¯ x (v, u) + ˜¯t (v, u) + Q p +(εC2 −

− h(v))2 (14.4.3)

˜¯xx (v, u), ≤ εp ¯ u) and a constant C3 depending on the for a suitable function Q(v, bounds of second derivatives of p. Therefore, we have the following estimates:  ¯) L2 (R) + (εvx2 , εu2x ) L1 (R2+ ) ≤ M,  (v(·, t) − v¯, u(·, t) − u 2

 α(v, u)(u − h(v)) 1 2 ≤ M, L (R+ ) τ provided that εC2 > 2C3 τ .

(14.4.4)

CHAPTER 14. RELAXATION WITH DIFFUSION

194

Multiplying the first equation in (14.1.1) by vxx and the second by 2 , we have uxx , adding the outcome, and then integrating by parts on R+ 1 2



∞

(vx2

 t

∞

2 (vxx + u2xx )dxdt −∞ 0 −∞  t ∞  1 ∞ 2 2 (v + ux )(x, 0)dx + f (v, u)x vxx = 2 −∞ x 0 −∞  +[g(v, u)x + τ1 α(v, u)(u − h(v))]uxx dxdt.

+

u2x )dx

+ε

(14.4.5)

Then it follows from (14.4.5), the estimates in (14.4.4) and the growth conditions (B1 )-(B2 ) that 

∞ −∞

2(q−1)

+ |u|∞

2(q−1)

+ |u|∞

(vx2 + u2x )dx ≤ c0 (ε, τ )(1 + |v|∞ ≤ c0 (ε, τ )(1 + |v|∞

2(q−1)

+ |α(v, u)|∞ )

2(q−1)

+ |v|r∞ + |u|r∞ ). (14.4.6)

Since v2

+

u2



x

= −∞

(v 2 + u2 )x dx ≤ 2|v|2 |vx |2 + 2|u|2 |ux |2 2(q−1)

≤ c1 (ε, τ )(1 + |v|∞

2(q−1)

+ |u|∞

1

+ |v|r∞ + |u|r∞ ) 2 , (14.4.7)

where 2(q − 1) < 4 and r < 4, then |(v, u)(x, t)| ≤ C(ε, τ ),

t > 0,

for some constant C(ε, τ ) > 0. Thus (I) is proved. Proof of (II). Multiplying the first equation in (14.1.1) by p|v − v¯|p−2 (v − v¯), we have (|v − v¯|p )t + P (v, u)x − p(p − 1)(f (v, u) − f (v, h(v)))|v − v¯|p−2 vx = ε(|v − v¯|p )xx − εp(p − 1)|v − v¯|p−2 vx2 , (14.4.8) where P (v, u) = p|v − v¯|p−2 (v − v¯)(f (v, u) − f (v, h(v))) + v¯|p−2 (v − v¯)f  (v, h(v))dv.



v

p|v −

14.4. PROOF OF THEOREM 14.1.2

195

Since |f (v, u) − f (v, h(v))| = |fu (v, µ)(u − h(v))|, where µ takes a value between u and h(v), it follows from (14.4.8) that (|v − v¯|p )t + P (v, u)x + εp(p − 1)|v − v¯|p−2 vx2 α(v, u)(u − h(v))2 τ τ p(p − 1)|v − v¯|p−2 |fu (v, µ)|2 p(p − 1)|v − v¯|p−2 vx2 + α(v, u) ε α(v, u)(u − h(v))2 + p(p − 1)|v − v¯|p−2 vx2 , ≤ ε(|v − v¯|p )xx + τ 2 (14.4.9) ≤ ε(|v − v¯|p )xx +

provided that 2τ p(p − 1)M ≤ ε, where p(p − 1)|v − v¯|p−2 |fu (v, µ)|2 ≤M α(v, u) from the condition 2k(q − 1) + p − 2 ≤ r. Integrating (14.4.9) by parts on R × [0, t], we have the estimate v − v¯ p ≤ M . Thus (II) is proved. Proof of (III). Since |f (v, h(v))| ≤ c1 + c2 (|v|q + |h(v)|q ) ≤ M (1 + |v|q + |v|qk ) and p > max{1, q, qk}, then (F1 ) and (F2 ) in (III) can be proved directly by (14.1.8) and the estimate v − v¯ p ≤ M . We can use a similar method as in the proof of Lemma 3.2.4 to prove (14.1.8). This completes the proof of Theorem 14.1.2.

Remark 14.4.1 For the simplicity of proof, in Chapter 3, we need the technical condition (3.1.6) and the growth condition |f (v)| ≤ c1 + c2 |v|q ,

2q = p

on the nonlinear flux function f (v) to prove the strong convergence of the viscosity solutions in Lp space. In Theorem 14.1.2, the condition (3.1.6) is removed and the growth condition on nonlinear function f (v, h(v)) is weakened to |f (v, h(v))| ≤ c1 + c2 |v|q , The details can be found in [Sz, Lu9].

q < p.

CHAPTER 14. RELAXATION WITH DIFFUSION

196

14.5

Applications of Theorem 14.1.2

In this section we are concerned with the zero relaxation and dissipation limits for some physical models, without bounded L∞ estimates, such as the system of isentropic gas dynamics in Lagrangian coordinates, the system of elasticity in the case of σ  (v) · v < 0, for all v ∈ R\{0} and the models of traffic flows with stiff relaxation terms. 14.5.1. System of Isentropic Gas Dynamics in Lagrangian Coordinates Consider the viscosity solutions of the system of isentropic gas dynamics with relaxation terms in Lagrangian coordinates:   vt − ux = εvxx ,  ut + g(v)x + 1 α(v, u)(u − h(v)) = εuxx , τ

(14.5.1)

with initial data (v, u)|t=0 = (v0 (x), u0 (x)),

(14.5.2)

where g (v) ≤ 0, g (v) > 0 as v > 0. The two eigenvalues of the corresponding hyperbolic system of (14.5.1) are   λ1 = − −g (v), λ2 = −g (v), and the corresponding two Riemann invariants are  v  v  −g (v)dv, w = u − −g (v)dv, z =u+ v1

v1

for a constant v1 > 0. Let (v0 (x), u0 (x)) be in the open region Σ10 = {(v, u) : z > 0, w < ¯) ∈ L1 ∩ Lp , where v¯ ≥ v1 , u¯ = h(¯ v ). 0} and (v0 (x) − v¯, u0 (x) − u Theorem 14.5.1 (I) If   v  −g (v)dv ≤ h(v) ≤ − v1

v v1



−g (v)dv,

14.5. APPLICATIONS OF THEOREM 14.1.2 



2



2



h (v) + g (v) + |h (v)|



v v0



 −g (v)dv /α(v, u)

197 is bounded,

and 0 < α0 ≤ α(v, u) ≤ c1 (1 + |v|r + |u|r ),

r ∈ [0, 4),

as v ≥ v1 ,

then, for fixed ε and τ satisfying M1 τ ≤ ε for a suitable large constant M1 , the Cauchy problem (14.5.1)-(14.5.2) has a unique smooth solution (v ε,τ , uε,τ ) satisfying v1 ≤ v ε,τ ≤ C(ε, τ ),

|uε,τ | ≤ C(ε, τ ),

(14.5.3)

for some positive constant C(ε, τ ) depending on ε and τ . (II) Let the conditions of (I) be satisfied and |v − v¯|p−2 /α(v, u) be finite as v ≥ v1 . Then v ε,τ − v¯ p ≤ M . (III) Let the conditions of (I) and (II) be satisfied, |h(v)| ≤ c3 (1 + and p > k. Then there exists a subsequence of (v ε,τ , uε,τ ) converging pointwisely almost everywhere to (v, u), as  → 0 with τ = o(ε), where (v, u) are the equilibrium state functions, determined uniquely by h(v) and v0 (x) (see (F1 )-(F2 ) of Theorem 14.1.2).

|v|k )

Proof. We only give the proof of some estimates similar to those in (14.4.4). The remaining can be completed similarly as in the proof of Theorem 14.1.2. Multiplying (14.5.1) by (wv , wu ) and (zv , zu ), respectively, we have    wt + λ2 wx + τ1 α(v, u)(u − h(v)) = εwxx + ε( −g (v)) vx2 ≤ εwxx , 

 zt + λ1 zx + τ1 α(v, u)(u − h(v)) = εzxx − ε( −g (v)) vx2 ≤ εzxx . (14.5.4)

v 

v  If − v1 −g (v)dv ≤ h(v) ≤ v1 −g (v)dv as v ≥ v1 , then the curve u = h(v) is inside the region Σ10 (see Figure 14.3). Thus we can get directly from inequalities (14.5.4) that Σ10 is an invariant region. From this we have the estimates  v −g (v)dv. (14.5.5) v ≥ v1 , |u| ≤ v1

CHAPTER 14. RELAXATION WITH DIFFUSION

198

u w=0

v Σ10 (v1 , 0)

u = h(v)

z=0 FIGURE 14.3

Let p(v, u) =

u2 − h(v)u 2    v s  |h (m)| +4 v1

v1

m v1

 −g (n)dn + h (m)2 + 1 dmds, (14.5.6)

and v, u ¯)(v − v¯) − pu (¯ v , u¯)(u − u ¯). p¯(v, u) = p(v, u) − p(¯ v , u¯) − pv (¯ (14.5.7) Multiplying system (14.5.1) by (¯ pv , p¯u ), we have from (14.2.4)-(14.2.5) that p¯t (v, u) + q¯x (v, u) + (−h (v)ux + p¯vv (v, u)vx )(u − h(v)) +h (v)2 (u − h(v))vx + g (v)(u − h(v))vx + τ1 α(v, u)(u − h(v))2 = ε¯ pxx (v, u) − ε(¯ pvv (v, u)vx2 + 2¯ pvu (v, u)vx ux + p¯uu (v, u)u2x ), (14.5.8)

14.5. APPLICATIONS OF THEOREM 14.1.2 where

199

 v p¯v (s, h(s))h (s)ds q¯(v, u) = −¯ pv (v, u)(u − h(v)) − v1  v  p¯u (s, h(s))g (s)ds. + v1

It follows from (14.5.6)-(14.5.7) and the second estimate in (14.5.5) that pvu (v, u)vx ux + p¯uu (v, u)u2x p¯vv (v, u)vx2 + 2¯  v  ≥ C2 {(|h (v)| −g (s)ds + h (v)2 + 1)vx2 + u2x },

(14.5.9)

v1

for some constant C2 > 0. Thus it follows from (14.5.8) and the conditions in (I) that 1 α(v, u)(u − h(v))2 p¯t (v, u) + q¯x (v, u) + 2τ  

v  +(εC2 − M τ ) (|h (v)| v1 −g (s)ds + h (v)2 + 1)vx2 + u2x

≤ ε¯ pxx (v, u), (14.5.10) which yields that  α(v, u)(u − h(v))2    (v(·, t) − v¯, u(·, t) − u L1 (R2+ ) ≤ M ; ¯) L2 (R) + τ  v   −g (s)ds + h (v)2 + 1)vx2 + u2x } L1 (R2+ ) ≤ M,  ε{(|h (v)| v1

(14.5.11) provided that 2M τ ≤ εC2 . Therefore, the proof can be similarly completed as in the proof of Theorem 14.1.2. 14.5.2. Models of Traffic Flows: Lp Solutions We consider the viscosity solutions of the models of traffic flows with relaxation terms
ρt + (ρu)x = ερxx , (14.5.12) 2 ut + ( u2 + g(ρ))x + τ1 α(ρ, u)(u − h(ρ)) = εuxx ,

CHAPTER 14. RELAXATION WITH DIFFUSION

200 with initial data

(ρ, u)|t=0 = (ρ0 , u0 ),

(14.5.13)

where (ρ0 , u0 ) satisfies ¯ ) ∈ L1 ∩ Lp , (ρ0 (x) − ρ¯, u0 (x) − u

1 ≤ p < ∞,

(14.5.14)

with constants (¯ ρ, u ¯), ρ¯ > 0, and u ¯ = h(¯ ρ). For the model of traffic flow, g(ρ) = log ρ. If g (ρ)/ρ is a nonincreasing function, system (14.5.12) generally does not have an a priori L∞ estimate. We study the compactness of solutions (ρε,τ , uε,τ ) of the Cauchy problem (14.5.12), (14.5.13) in Lp . Two eigenvalues of the corresponding hyperbolic system (14.3.11) of (14.5.12) are   λ1 = u − ρg (ρ), λ2 = u + ρg (ρ) and the corresponding two Riemann invariants are  ρ  ρ  g (s)/sds, w = u + g (s)/sds, z =u− ρ1

ρ1

where 0 < ρ1 ≤ ρ¯ is a constant. Let (D1 ) g (ρ) > 0,

(g (ρ)/ρ) ≤ 0,

as ρ ≥ ρ1 ;

(D2 ) 0 < α0 ≤ α(ρ, u) ≤ c1 + c2 (|ρ|r + |u|r ), (D3 )

0 ≤ r < 4;

ρ  (ρh (ρ))2 + h(ρ)2 + g (ρ)2 + ρ2 + u2 + ρ2 |h (ρ)| ρ1 g (s)/sds

α(ρ, u)

ρ  is bounded when |u| ≤ ρ1 g (s)/sds;

(D4 )

ρp /α(ρ, u) ≤ M ;

(D5 )

|h(ρ)| ≤ c3 + c4 |ρ|k .

Theorem 14.5.2 (I) Let conditions (D1 )-(D3 ) be satisfied, and the initial data (14.5.13) be in the region Σ = {(ρ, u) : w ≥ 0, z ≤ 0}. Suppose  ρ  ρ  g (s)/sds ≤ h(ρ) ≤ g (s)/sds, − ρ1

ρ1

14.5. APPLICATIONS OF THEOREM 14.1.2

201

as ρ ≥ ρ1 . Then, for fixed ε, τ satisfying M1 τ ≤ ε for a suitable large constant M1 , the Cauchy problem (14.5.12), (14.5.13) has a unique smooth solution (ρε,τ , uε,τ ) satisfying ρ1 ≤ ρε,τ ≤ C(ε, τ ),

|uε,τ | ≤ C(ε, τ ),

(14.5.15)

for some constant C(ε, τ ) depending on ε, τ . (II) Let the conditions of (I) and (D4 ) be satisfied. Then ρε,τ − ρ¯ p ≤ M . (III) Let the conditions of (I), (II) and (D5 ) be satisfied and p > 1+ k. Then there exists a subsequence of (ρε,τ , uε,τ ) converging pointwisely almost everywhere to (ρ, u), as  → 0 with τ = o(ε), where the limit functions (ρ, u) are the equilibrium states, determined uniquely by h(ρ) and ρ0 (x) (see (F1 )-(F2 ) of Theorem 14.1.2). Proof. Similar to the proof of Theorem 14.5.1, we can prove that Σ is an invariant region. Hence we have the following estimates:  ρ g (s)/sds. (14.5.16) ρ ≥ ρ1 , |u| ≤ ρ1

Choose p(ρ, u) =

u2 − h(ρ)u 2  ρ m  (|h (m)| g (n)/ndn + (h )2 (m) + 1)dmds, +4 ρ1

ρ1

and let ρ, u ¯)(ρ − ρ¯) − pu (¯ ρ, u ¯)(u − u ¯). p¯(ρ, u) = p(ρ, u) − p(¯ ρ, u ¯) − pρ (¯ Multiplying system (14.5.12) by (¯ pρ , p¯u ), we can obtain the following estimates from (D3 ) and the second estimate in (14.5.16):  α(ρ, u)(u − h(ρ))2   (ρ(·, t) − ρ¯, u(·, t) − u |L1 (R2+ ) ≤ M ; ¯) L2 (R) +  τ  ρ   g (s)/sds + h (ρ)2 + 1)ρ2x + u2x } L1 (R2+ ) ≤ M.  ε{(|h (ρ)| ρ1

(14.5.17)

CHAPTER 14. RELAXATION WITH DIFFUSION

202

Using (D1 )-(D2 ) and the estimates in (14.5.17), we can get the estimates in (14.5.15). Then (I) is proved. Since fu = ρ, the proof of (II) follows from (14.4.8) and (D4 ). Since ρ2 /α(ρ, u) is bounded, similar to the proof of Theorem 14.1.2, we can complete the proof of (III). This completes the proof of Theorem 14.5.2.

14.6

Related Results

The nonlinear stability of weak, smooth travelling waves and rarefaction waves for hyperbolic systems of conservation laws with relaxation were first analyzed by T.-P. Liu (cf. [Liu]), in which it is indicated that the effect of relaxation is closely related to a viscous effect when the solution is near a constant equilibrium state (also see [Chern]). By introducing the theory of the compensated compactness, a simple model of combustion (3.3.3) with infinite reaction rate k (whose reciprocal is related to the zero relaxation) was first studied in [Lu7]. Later, the zero relaxation limit of solutions with large oscillation, for hyperbolic conservation laws with relaxation terms containing both damping and sink mechanisms, was systematically studied in Chen-Liu [ChL] and ChenLevermore-Liu [CLL] also by the compensated compactness method. All the results in this chapter are from [Lu9].

Chapter 15

Hyperbolic Systems with Stiff Relaxation In this chapter, we are concerned with singular limits of relaxation approximated solutions (v τ , uτ ) to the Cauchy problem of general 2 × 2 system of quasilinear conservation laws over a one-dimensional spatial domain in the form
vt + f (v, u)x = 0, (15.0.1) ut + g(v, u)x + τ1 (u − h(v)) = 0, with initial data (v, u)|t=0 = (v0 (x), u0 (x)).

(15.0.2)

The eigenvalues of system (15.0.1) satisfy the following characteristic equation: λ2 − (fv + gu )λ + fv gu − gv fu = 0.

(15.0.3)

System (15.0.1) is assumed to be hyperbolic (it is not necessary to be strictly hyperbolic), that is, the two eigenvalues or characteristic speeds λ1 =

  1 fv + gu − (fv − gu )2 + 4gv fu 2

(15.0.4)

λ2 =

  1 fv + gu + (fv − gu )2 + 4gv fu 2

(15.0.5)

and

203

204

CHAPTER 15. SYSTEMS WITH STIFF RELAXATION

are real. Formally, when the relaxation time τ → 0, we have from the second equation in (15.0.1) that u = h(v), which implies, from the first equation in (15.0.1), the local equilibrium equation just as the following scalar conservation law: vt + f (v, h(v))x = 0.

(15.0.6)

In the Chapman-Enskog expansion one seeks to identify the effective response of the relaxation process as it approaches the line of local equilibria u = h(v). It is postulated that the relaxing variable uτ can be described in an asymptotic expansion that involves only the local macroscopic values v τ and its derivatives, i.e., uτ = h(v τ ) + τ sτ (v τ , uτ , vxτ , uτx , · · · ) + O(τ 2 ).

(15.0.7)

To calculate the form of sτ , we use system (15.0.1) to get (for simplicity, we omit the superscript τ ) 0 = vt + f (v, u)x = vt + f (v, h(v) + τ s + O(τ 2 ))x

(15.0.8)

= vt + f (v, h(v))x + τ (sfu (v, h(v)))x + O(τ 2 ) and 1 0 = ut + g(v, u)x + (u − h(v)) τ = h(v)t + g(v, h(v))x + s + O(τ )  dg(v, h(v))  vx + s + O(τ ). dv Use (15.0.8) to eliminate vt in (15.0.9) and obtain  dg(v, h(v))  vx + O(τ ) −s = h
(v)(−f (v, h(v))x + dv  dg(v, h(v)) df (v, h(v))  − h
(v) vx + O(τ ). = dv dv From (15.0.8) and (15.0.10), on u = h(v), we have

(15.0.9)

= h
(v)vt +

(15.0.10)

vt + f (v, h(v))x = −τ (sfu (v, h(v)))x + O(τ 2 )   dg(v, h(v)) df (v, h(v))  − h
(v) fu (v, h(v))vx =τ dv dv x   = τ fu (gv + (gu − fv )h
(v) − fu (h
(v))2 )vx x . (15.0.11)

205 Let φ(v) = fu (gv + (gu − fv )h
(v) − fu (h
(v))2 ) on u = h(v). Then it is easy to prove that φ(v) = (λ2 (v, h(v)) − λ(v))(λ(v) − λ1 (v, h(v))), where λ(v) = (15.0.6).

df (v,h(v)) dv

(15.0.12)

is the eigenvalue of the equilibrium equation

In fact, on u = h(v), we have (λ2 (v, h(v)) − λ(v))(λ(v) − λ1 (v, h(v)))      = 12 fv + gu + (fv − gu )2 + 4gv fu − fv − fu h
(v)     × fv + fu h
(v) − 12 fv + gu − (fv − gu )2 + 4gv fu     = 12 (fv − gu )2 + 4gv fu − 12 (fv − gu ) + fu h
(v)     × 12 (fv − gu )2 + 4gv fu + 12 (fv − gu ) + fu h
(v) = fu (gv + (gu − fv )h
(v) − fu (h
(v))2 ). (15.0.13) Therefore, Equation (15.0.11) is a stable degenerate parabolic equation if the following subcharacteristic condition is satisfied: λ1 (v, h(v)) ≤ λ(v) ≤ λ2 (v, h(v)).

(15.0.14)

Now we give the definition of entropy-entropy flux pair (η(v, u), q(v, u)) for the relaxation system (15.0.1) as follows: Definition 15.0.1 A pair of functions (η(v, u), q(v, u)) is called an entropy-entropy flux pair of (15.0.1) if (e1 ) (qv , qu )(fv ηv + gv ηu , fu ηv + gu ηu ), (e2 ) ηu (v, h(v)) = 0. An entropy η(v, u) is called convex if (e3 ) ηvv a2 + 2ηvu ab + ηuu b2 ≥ c(v, u)(a2 + b2 ) for any vector (a, b) = (0, 0) in R2 and a nonnegative function c(v, u).

206

CHAPTER 15. SYSTEMS WITH STIFF RELAXATION

(e4 ) If c(v, u) ≥ c0 > 0, for a constant c0 , the entropy is said to be strictly convex. Use the entropy-entropy flux equations in (e1 ) to eliminate q, and obtain the following entropy equation of system (15.0.1): gv ηuu − fu ηvv + (fv − gu )ηvu = 0.

(15.0.15)

If fu , gv and fv − gu have a common zero factor Z(v, u), then we delete this factor from (15.0.15) and consider entropies of system (15.0.1) to be solutions of the following equation: fu fv − gu gv ηuu − ηvv + ηvu = 0. Z(v, u) Z(v, u) Z(v, u)

(15.0.16)

For instance, consider the relaxation problem for the extended model of traffic flows:
ρt + (ρu)x = 0, (15.0.17) 2 = 0, ut + ( u2 + p(ρ))x + u−h(ρ) τ which is nonstrictly hyperbolic on the line ρ = 0. One common zero fact of fu (ρ, u) = ρ, gρ (ρ, u) = p
(ρ) and fρ (ρ, u) − gu (ρ, u) = 0 is ρ. In Section 15.1, we shall prove a general compactness framework to the singular limits of the general 2 × 2 system (15.0.1) with the stiff relaxation term, and in Section 15.2, an application of this compactness framework on the nonstrictly hyperbolic system of extended traffic flow (15.0.17) is obtained.

15.1

Relaxation Limits for 2 × 2 Systems

Before we introduce the compactness framework to the singular limits of the general 2 × 2 relaxation system (15.0.1) (Theorem 15.1.3), we first prove two basic lemmas. Lemma 15.1.1 Let (η, q) be a strictly convex entropy-entropy flux pair of (15.0.1) as defined by (e1 ) − (e4 ) above. Then the local equilibrium equation (15.0.6) has the strictly convex entropy l(v) = η(v, h(v)) with the corresponding entropy flux L(v) = q(v, h(v)).

15.1. RELAXATION LIMITS FOR 2 × 2 SYSTEMS

207

Proof. The function l(v) is clearly an entropy of (15.0.6) since it is a scalar equation. By simple calculations, l

(v) = ηvv (v, h(v)) + 2ηvu (v, h(v))h
(v) + ηuu (v, h(v))(h
(v))2 ≥ c0 (1 + (h
(v))2 ) ≥ c0 > 0, (15.1.1) and hence l(v) is strictly convex. To prove L(v) = q(v, h(v)) to be the corresponding entropy flux, we use the equations in (e1 ) and the condition in (e2 ) to obtain qv (v, h(v)) = ηv (v, h(v))fv (v, h(v)), qu (v, h(v)) = ηv (v, h(v))fu (v, h(v)).

(15.1.2)

Therefore L
(v) = qv (v, h(v)) + qu (v, h(v))h
(v) = ηv (v, h(v))fv (v, h(v)) + ηv (v, h(v))fu (v, h(v))h
(v) = ηv (v, h(v))

df (v, h(v)) , dv

(15.1.3)

and hence L(v) is the entropy flux corresponding to l(v).

Lemma 15.1.2 Let l(v) be a strictly convex entropy for the local equilibrium equation (15.0.6). Assume that the stability criterion (15.0.14) holds on u = h(v). If (λ2 (v, h(v)) − λ(v))(λ(v) − λ1 (v, h(v)) > 0, Z(v, u)2

fu (v, u) = 0, (15.1.4) Z(v, u)

where Z(v, u) is a zero factor as given in (15.0.16), then there exists a strictly convex entropy η(v, u) for system (15.0.1) over an open set Dl ⊂ R2 containing the local equilibria curve u = h(v), along which it satisfies η(v, h(v)) = l(v). Proof. If η(v, u) is a strictly convex entropy of system (15.0.1), then it must satisfy

CHAPTER 15. SYSTEMS WITH STIFF RELAXATION

208 (E1 )

gv fu fv − gu ηuu − ηvv + ηvu = 0, Z(v, u) Z(v, u) Z(v, u)

(E2 ) ηu (v, u)(u − h(v)) ≥ 0, (E3 ) ηuu ≥ c1 ,

ηvv ηuu − (ηvu )2 ≥ c2 , for two positive constants c1 , c2 .

The characteristic curve u = e(v) of Equation (E1 ) satisfies the following characteristic equation: fv − gu
fu gv + e (v) − (e
(v))2 = 0. Z(v, u) Z(v, u) Z(v, u)

(15.1.5)

Therefore, if the conditions (15.1.5) in Lemma 15.1.2 are satisfied, then fv − gu
fu gv + h (v) − (h
(v))2 Z(v, u) Z(v, u) Z(v, u) (λ2 (v, h(v)) − λ(v))(λ(v) − λ1 (v, h(v)) Z(v, u) = 0, = · Z(v, u)2 fu (v, u) (15.1.6) and hence, the curve u = h(v) is not a characteristic curve. Thus the classical Cauchy-Kowalewsky local existence theory ensures that the Cauchy problem for the second-order linear hyperbolic equation (E1 ) with the following initial data: η(v, h(v)) = l(v),

ηu (v, h(v)) = 0

(15.1.7)

has a local solution η(v, u) over an open domain Dl containing the initial curve or the local equilibria curve u = h(v). (E2 ) can be easily obtained by (E3 ). In fact, if (E3 ) is true, then ηu (v, u)(u − h(v)) = ηuu (v, α)(u − h(v))2 ≥ 0, where α(u, h(v)) takes a value between u and h(v). If the strict convexity conditions in (E3 ) are satisfied along the local equilibria curve, then by continuity they will also be satisfied in the open domain Dl (possibly smaller). Differentiating the Cauchy data (15.1.7) with respect to v leads to the identities l
(v) = ηv (v, h(v)) + ηu (v, h(v))h
(v) = ηv (v, h(v)).

(15.1.8)

15.1. RELAXATION LIMITS FOR 2 × 2 SYSTEMS

209

Notice that in the second part of (15.1.7), there hold l

(v) = ηvv (v, h(v)) + ηvu (v, h(v))h
(v),

(15.1.9)

and ηuv (v, h(v)) + ηuu (v, h(v))h
(v) = 0.

(15.1.10)

From (15.1.10) and (15.1.9), we have ηuv (v, h(v)) = −ηuu (v, h(v))h
(v) and

ηvv (v, h(v)) = l

(v) − ηvu (v, h(v))h
(v) = l

(v) + ηuu (v, h(v))(h
(v))2 ,

which combining with the entropy equation (E1 ) yields that on u = h(v), there holds  1 gv ηuu − fu ηvv + (fv − gu )ηvu Z  fu 1 gv − (fv − gu )h
(v) − fu (h
(v))2 ηuu − l

(v), = Z Z

0 =

(15.1.11)

or equivalently   1   fu2

l (v) = η − f η + (f − g )η g ηuu f u v uu u vv v u vu Z2 Z2 (λ2 (v, h(v)) − λ(v))(λ(v) − λ1 (v, h(v)) ηuu (v, h(v)) = Z(v, u)2 (15.1.12) and hence, ηuu (v, h(v)) > 0 from the conditions in (15.1.4). To obtain the second part in (E3 ), from (15.1.10), and (fv − gu )ηvu = fu ηvv − gv ηuu , there holds h
(v) = −

ηvu (v, h(v)) . ηuu (v, h(v))

CHAPTER 15. SYSTEMS WITH STIFF RELAXATION

210

Substituting it into the identity (15.0.12), from the entropy equation (15.0.15), we obtain on u = h(v) that (λ2 (v, h(v)) − λ(v))(λ(v) − λ1 (v, h(v))) = fu (gv + (gu − fv )h
(v) − fu (h
(v))2 )   ηvu 2  ηvu − fu = fu gv − (gu − fv ) ηuu ηuu  2  fu2  . = 2 ηuu ηvv − ηuv ηuu Therefore

 2 ηuu ηvv − ηuv > 0 on u = h(v),

(15.1.13)

(15.1.14)

which completes the proof of Lemma 15.1.2. Now we are in the position to give the main result in this section. Suppose that (v τ,ε , uτ,ε ) ∈ Dl are solutions of the Cauchy problem
vt + f (v, u)x = εvxx , (15.1.15) ut + g(v, u)x + τ1 (u − h(v)) = εuxx , with bounded initial data (v, u)|t=0 = (v0 (x), u0 (x)) ∈ Dl ,

(15.1.16)

where Dl is given in Lemma 15.1.2 and (v τ,ε , uτ,ε ) → (v τ , uτ ) a.e. as ε → 0,

(15.1.17)

and the limit (v τ , uτ ) is a weak solution of the Cauchy problem (15.0.1)(15.0.2) in the domain Dl . Then we have the main result in this section given by the following theorem: Theorem 15.1.3 If the conditions in Lemma 15.1.2 are satisfied and meas {v : λ(v) = 0} = 0, then there exists a subsequence (still denoted) (v τ , uτ ) such that (v τ , uτ ) → (v, u)

a.e.,

and the limit functions (v, u) satisfy (i) u(x, t) = h(v(x, t)) a.e., for t > 0; (ii) v(x, t) is the weak solution of the Cauchy problem (15.0.6) with the initial data v(x, 0) = v0 (x).

15.1. RELAXATION LIMITS FOR 2 × 2 SYSTEMS

211

Proof. Let η(v, u) be the strictly convex entropy of system (15.0.1) constructed in Lemma 15.1.2, and q(v, u) be the corresponding entropy flux. Then ηvv a2 + 2ηvu ab + ηuu b2 ≥ c0 (a2 + b2 )

(15.1.18)

for any vector (a, b) = (0, 0) and a positive constant c0 . Since ηuu (v, u) ≥ d1 > 0 and ηu (v, h(v)) = 0, then in Dl , we have ηu (v, u)(u − h(v)) = ηuu (v, α(u, h(v)))(u − h(v))2 ≥ d1 (u − h(v))2 , (15.1.19) where α(u, h(v)) denotes a value between u and h(v). Multiplying system (15.0.1) by (ηv , ηu ), we obtain ηu (u − h(v)) τ   = εηxx − ε ηvv vx2 + 2ηvu vx ux + ηu u2x .

ηt + qx +

(15.1.20)

Use (15.1.18) and (15.1.19) to obtain ηt + qx +

  d1 (u − h(v))2 + εc0 vx2 + u2x ≤ εηxx . τ

(15.1.21)

Multiplying (15.1.21) by a suitable test function and then integrating in R × R+ , we have that 2  2   (15.1.22) ∈ L1loc (R × R+ ) ε vxτ,ε + uτ,ε x and  2 1  τ,ε u − h v τ,ε τ

∈ L1loc (R × R+ ).

(15.1.23)

Let ε tend to zero in (15.1.23) to obtain  2 1 τ u − h vτ τ

∈ L1loc (R × R+ ).

(15.1.24)

Let l1 (v) = v, l2 (v) = f (v, h(v)) = f (v) and related corresponding fluxes of the equilibrium equation (15.0.6) be L1 (v), L2 (v). Let the entropies of system (15.0.1) with initial data l(v) = l1 (v), l(v) = l2 (v) be η1 (v, u), η2 (v, u) with corresponding entropy fluxes q1 (v, u), q2 (v, u), respectively.

CHAPTER 15. SYSTEMS WITH STIFF RELAXATION

212

Multiplying system (15.0.1) by (ηiv , ηiu ), i = 1, 2 , we obtain       ηiu  τ,ε u − h v τ,ε ηi v τ,ε , uτ,ε t + qi v τ,ε , uτ,ε x + τ   2  τ,ε 2  + η . = εηixx − ε ηivv vxτ,ε + 2ηiuv vxτ,ε uτ,ε x iuu ux (15.1.25) Then li (v τ,ε )t + Li (v τ,ε )x       = li (v τ,ε − ηi v τ,ε , uτ,ε + Li (v τ,ε − q v τ,ε , uτ,ε t x    τ,ε 2 ηiuu v τ,ε , α  τ,ε u −h v − + εηixx  τ 2  τ,ε 2  −ε ηivv vxτ,ε + 2ηiuv vxτ,ε uτ,ε , x + ηiuu ux (15.1.26)   where α takes a value between uτ,ε and h v τ,ε .   From (15.1.22), εηixx v τ,ε , uτ,ε tends to zero in the sense of distributions as ε tends to zero. Then letting ε → 0 in (15.1.26), we have τ τ τ τ + Ii2 + Ii3 + Ii4 li (v τ )t + Li (v τ )x = Ii1

(15.1.27)

τ , I τ are weak limits of in the sense of distributions, where Ii3 i4

   2 ηiuu v τ,ε , α  τ,ε u − h v τ,ε − τ and

  2  τ,ε 2  + η −ε ηivv vxτ,ε + 2ηiuv vxτ,ε uτ,ε iuu ux x

as ε → 0, being bounded in L1loc (R × R+ ) from (15.1.22)-(15.1.23) and −1,p (R × R+ ) for 1 < p < 2. Moreover hence, compact in Wloc τ Ii1

     = supφ∈H01 | li (v τ ) − ηi v τ , uτ φdxdt| t

≤ ≤

Cuτ √

−

h(v τ )L2 φt L2

τ Cφt H 1 → 0, as τ → 0

(15.1.28)

15.2. SYSTEM OF EXTENDED TRAFFIC FLOWS and

     Li (v τ ) − qi v τ , uτ φdxdt| t φ∈H01        qi v τ , h(v τ ) − qi v τ , uτ φdxdt| = sup |

213

τ  = sup | Ii2

t

φ∈H01 Cuτ

(15.1.29)

− h(v τ )L2 φx L2 √ ≤ τ Cφt H 1 → 0, as τ → 0.

≤

τ , I τ are compact in W −1,2 (R × R+ ), and hence These imply that Ii1 i2 loc

li (v τ )t + Li (v τ )x −1,p is compact in Wloc (R × R+ ) for 1 < p < 2. From the boundedness of (v τ , uτ ), we conclude that

li (v τ )t + Li (v τ )x −1,∞ is bounded in Wloc (R × R+ ), and hence

li (v τ )t + Li (v τ )x −1,2 is compact in Wloc (R × R+ ) from Theorem 2.3.2.

Therefore the compactness framework given in Chapter 3 shows the strong convergence v τ → v, a.e., which implies also the strong convergence uτ → u, a.e. from (15.1.24). So we get the proof of Theorem 15.1.3.

15.2

System of Extended Traffic Flows

In this section, we shall introduce an application of Theorem 15.1.3 on the relaxation problem for the nonstrictly hyperbolic system of extended traffic flows (15.0.17) with the initial data (ρ(x, 0), u(x, 0)) = (ρ0 (x), u0 (x)) (ρ0 (x) ≥ 0). Two eigenvalues of system (15.0.17) are   λ1 = u − ρp
(ρ) λ2 = u + ρp
(ρ),

(15.2.1)

CHAPTER 15. SYSTEMS WITH STIFF RELAXATION

214

and the first-order relaxation correction corresponding to (15.0.11) is   (15.2.2) ρt + (ρh(ρ))x = τ φ(ρ)ρx x , where φ(ρ) = ρ2 is reduced to

 p
(ρ) ρ

 − (h
(ρ))2 , and hence the condition in (15.0.14) p
(ρ) − (h
(ρ))2 > 0. ρ

(15.2.3)

The entropy equation of system (15.0.17) is p
(ρ) ηuu − ηρρ = 0 ρ

(15.2.4)

since fu (ρ, u) = ρ, gρ (ρ, u) = p
(ρ) and fρ (ρ, u) − gu (ρ, u) = 0 have
a common zero factor ρ if we assume p ρ(ρ) ≥ d > 0. Therefore combining Theorem 15.1.3, Theorem 10.0.2 and Theorem 14.3.3 yields the following theorem:


Theorem 15.2.1 Let p1 (ρ) = p ρ(ρ) ≥ d > (h
(ρ))2 for a positive constant d and p
1 (ρ) ≥ 0. Suppose that there exist two small constants N, L such that the curve u = h(ρ) passes the unique intersection point (¯ ρ, u¯) of curves w = N, z = −L; the curve u = h(ρ) and the initial data (ρ0 (x), u0 (x)) are in the region Σ9 = {(ρ, u) : w ≤ N, z ≥ −L, ρ ≥ 0} as 0 ≤ ρ ≤ ρ¯ (see Figure 14.2). Then, for any fixed τ , the global weak solution (ρτ , uτ ) of the Cauchy problem (15.0.17), (15.2.1) exists. Moreover, if meas {ρ : (ρh(ρ))

= 0} = 0, then there exists a subsequence (still denoted) (ρτ , uτ ) such that (ρτ , uτ ) → (ρ, u)

a.e.,

and the limit functions (ρ, u) satisfy (i)

u(x, t) = h(ρ(x, t)), a.e., for t > 0;

(ii) ρ(x, t) is the unique weak solution of the Cauchy problem to the scalar equation ρt + (ρh(ρ))x = 0 with the initial data ρ(x, 0) = ρ0 (x).

15.3. RELATED RESULTS

15.3

215

Related Results

The general compactness framework to the singular limits of the general 2 × 2 strictly hyperbolic system (15.0.1) with the stiff relaxation term is established in [CLL], where an application on singular limits of relaxation approximated solutions to the system of elasticity is also obtained. The extension, Theorem 15.1.3 of Chen-Levermore-Liu’s framework to nonstrictly hyperbolic systems of two equations with a relaxation term, and its application on the extended model of traffic flow (15.0.17) both are established by Lu [Lu10].

Chapter 16

Relaxation for 3 × 3 Systems In this chapter, we study the singular limit for the following nonlinear systems of three equations:  vt − ux = 0,     ut − σ(v, s)x = 0, (16.0.1)   s − h(v)   st + c1 sx + β = 0, τ with initial data 

   v, u, s t=0 = v0 , u0 , s0 ,

(16.0.2)

where β, τ and c1 are nonnegative constants. When β = 0, the existence of L2 global weak solution for the Cauchy problem (16.0.1), (16.0.2) is obtained in Section 12.3. In the case of β
= 0, for instance, β = 1, when written in Eulerian coordinates, system (16.0.1) can be used to model the chemically reacting flow (cf. [LLL]). Here v is specific volume, u denotes velocity, s is the mass fraction of one mode of the two-mode gas and h(v) is the given equilibrium distribution in v. In this case, τ denotes the reaction time or relaxation time. System (16.0.1) arises also in many physical situations, such as the system of adiabatic gas flow through porous media, the variant system 217

218

CHAPTER 16. RELAXATION FOR 3 × 3 SYSTEMS

of Broadwell model [LK2] and the system of isothermal motions of a viscoelastic material [LWu, Tz]. Formally, when the relaxation time τ → 0, it yields the system of elasticity or equations of isothermal elastodynamics,  vt − ux = 0, (16.0.3) ut − σ(v, h(v))x = 0. The three eigenvalues of system (16.0.1) are λ1 = − σv (v, s), λ2 = σv (v, s),

λ3 = c1

and two eigenvalues of system (16.0.3) are



dσ(v, h(v)) dσ(v, h(v)) , λ2 = . λ1 = − dv dv

(16.0.4)

(16.0.5)

In Section 16.1, we are interested in combining the zero relaxation with the zero dissipation limit of the Cauchy problem  vt − ux = εvxx ,    ut − σ(v, s)x = εuxx , (16.0.6)    = εsxx , st + c1 sx + β s−h(v) τ with the initial data (16.0.2). When τ ≤ M , with M a suitable large constant which depends only on initial data, and  goes to zero, the convergence of the solutions  ,τthe ,τ v , u , s,τ to the Cauchy problem (16.0.6), (16.0.2) is obtained for very general σ(v, s) even if (16.0.1) is a elliptic-hyperbolic mixed system, that is, σv (v, s)
0. In Section 16.2, we consider the relaxation limit for the following special case of system (16.0.1) without introducing the viscosity:  vt − ux = 0,    ut − (v − cs)x = 0, (16.0.7)    = 0, st + s−h(v) τ with the initial data (16.0.2), where c is a positive constant, but the nonlinear function h(v) must satisfy the subcharacteristic condition 0 < d1 ≤ h
(v) ≤ d2 < for positive constants d1 and d2 .

1 c

(16.0.8)

16.1. DOMINANT DIFFUSION AND STIFF RELAXATION

16.1

219

Dominant Diffusion and Stiff Relaxation

In this section, we shall study the singular limits of stiff relaxation and dominant diffusion for the Cauchy problem of (16.0.6), (16.0.2), that is, the relaxation time τ tends to zero faster than the diffusion parameter , τ = o(),  → 0. We establish the following general compactness framework:   Theorem 16.1.1 A: If the initial data v0 , u0 , s0 are smooth functions satisfying the following condition:   (c1 ) v0 , u0 , s0 L2 ∩L∞ (R) ≤ M1  di v di u di s 0 0 0 lim , , = (0, 0, 0), i = 0, 1; i i i dx dx |x|→±∞ dx (c2 ) h(v) = cv, σ(v, s) satisfies the following condition:   σs (v, s) ≤ M2 , σ ¯
(v) ≥ d > max{0, c2 − c +

2c2 c21 }, (M2 + 1)2

where σ ¯ (v) = σ(v, cv), then for fixed , τ satisfying τ (M2 + 1)2 ≤ , the solutions (u, v, s) ∈ C 2 of the Cauchy problem (16.0.6), (16.0.2) exist in (−∞, ∞) × [0, T ] for any given T > 0 and satisfy |v(x, t)|,

|u(x, t)|,

|v(·, t)|L2 (R) ,

|s(x, t)|

|u(·, t)|L2 (R) ,

≤ M (, τ, T ),

|s(·, t)|L2 (R)

(16.1.1)

≤ M,

(16.1.2)

|(s − cv)2 |L1 (R×R+ ) ≤ τ M, (16.1.3) |εvx2 |L1 (R×R+ ) ,

|εu2x |L1 (R×R+ ) ,

|εs2x |L1 (R×R+ )

≤ M.

¯ (v) = σ(v, cv) satisfies the following condition: B: If σ ¯

(v0 ) = 0 and σ ¯

(v)
= 0 for v
= v0 , σ ¯

, σ ¯


∈ L2 ∩ L∞ , then (c3 ) σ there exist a subsequence (still denoted by) v ,τ , u,τ , s,τ of the solutions to the Cauchy problem (16.0.6), (16.0.2) and L2 bounded functions (v, u, s) such that  ,τ ,τ ,τ  → (v, u, s) a.e.(x, t), (16.1.4) v ,u ,s where (v, u, s) satisfies s = h(v) and (v, u) is an entropy solution  of the  equilibrium system (16.0.3) with the initial data v0 (x), u0 (x) .

CHAPTER 16. RELAXATION FOR 3 × 3 SYSTEMS

220

Remark 16.1.2 In Theorem 16.1.1, the condition h(v) = cv is for avoiding the technical details. We shall see from the proof of Theorem 16.1.1 below that all the steps work just as well for a more general function h(v) which satisfies h
(v) ≥ d1 > 0. In fact, we only need to write the term s − h(v) as a different form: s − h(v) = h
(θ)(h−1 (s) − v)

(16.1.5)

where h−1 is the inverse function of h and θ takes a value between h−1 (s) and v. Proof of Theorem 16.1.1. To prove part A, we use the following local existence lemma and the L∞ estimates given in (16.1.1). Lemma 16.1.3 (Local existence) If the initial data satisfies condition (c1 ) in Theorem 16.1.1, then for any fixed  and τ > 0, the Cauchy problem (16.0.6), (16.0.2) admits a unique smooth local solution (u, v, s) which satisfies |

∂iu ∂is ∂iv | + | | + | | ≤ M (t1 , , τ ) < +∞ ∂xi ∂xi ∂xi

i = 0, 1, 2

(16.1.6)

where M (t1 , , τ ) is a positive constant that depends only on t1 , , τ and t1 depends on |v0 |L∞ , |u0 |L∞ , |s0 |L∞ . Moreover  di v di u di s , , = (0, 0, 0), |x|→±∞ dxi dxi dxi lim

i = 0, 1

(16.1.7)

uniformly in t ∈ [0, t1 ]. Proof. Lemma 16.1.2 can be proved by applying the Banach contraction mapping theorem to an integral representation of (16.0.6). For details see Theorem 1.0.2. To derive the crucial estimates given in (16.1.1), we need the necessary condition τ (M2 + 1)2 ≤  and condition (c2 ) in Theorem 16.1.1.

16.1. DOMINANT DIFFUSION AND STIFF RELAXATION

221

Multiplying the first equation in (16.0.6) by σ ¯ (v) + cv − cs, the second by u and the third by s − cv and adding the result, we have  v s2 u2 − csv + σ ¯ (v) + cvdv + 2 2 t 0  c1 s2 + cus − u(¯ σ (v) + cv) + 2 x    (s − cv)2 −cc1 vsx − u σ(v, s) + s − σ(v, cv) + cv + τ x  v s2 u2 − csv + σ ¯ (v) + cvdv + =ε 2 2 xx  0
 2 −ε σ ¯ (v) + c vx − εu2x − εs2x + 2cεsx vx . (16.1.8) For the third and fourth terms on the left-hand side of (16.1.8), we have the estimate   −cc1 vsx − u σ(v, s) + s − (σ(v, cv) + cv) x  c c2    1 v 2 − cc1 vs − u σ(v, s) + s − (σ(v, cv) + cv) = 2 x x +ux σs (v, α) + 1 (s − cv) + cc1 vx (s − cv) (16.1.9) where α takes a value between s and cv. The last two terms in (16.1.9) have the upper bound τ (M2 + 1)2 u2x 3(s − cv)2 + + τ c2 c21 vx2 (16.1.10) 4τ 2 by the first condition in (c2 ). Combining (16.1.8), (16.1.9) and (16.1.10), we get the following inequality:  v s2 u2 − csv + σ ¯ (v) + cvdv + 2 2 t 0   c1 2 c1 c2 2 v + cus − u(¯ σ (v) + cv) + s − cc1 vs − 2 2 x x    (s − cv)2 − u σ(v, s) + s − (σ(v, cv) + c) + 4τ x

 v 
 s2 u2 − csv + (¯ σ (v) + cv)dv + −ε σ ¯ (v) + c vx2 ≤ε 2 2 xx 0 2  τ (M 2 + 1)  2 ux . −εs2x + 2cεsx vx + c2 c21 τ vx2 − ε − 2 (16.1.11)

CHAPTER 16. RELAXATION FOR 3 × 3 SYSTEMS

222

Noticing the second condition in (c2 ) we know that

v s2 u2 − csv + (¯ σ (v) + cv)dv + 2 2 0 is a strictly convex function. If the condition τ (M2 + 1)2 ≤  is satisfied, noticing (16.1.7), we immediately get the estimates (16.1.2), (16.1.3) by integrating (16.1.11) on R × [0, T ]. Differentiating the first equation in (16.0.6) with respect to x, we get     (16.1.12) vx t − uxx = ε vx xx . Multiplying (16.1.12)by vx yields   vx2    vx2  2 − v u + u v = ε − εvxx . x x x xx x 2 t 2 xx

(16.1.13)

Integrating (16.1.13) in R × [0, T ] and noticing |u2x |L1 (R×R+ ) ≤ M (), we obtain the bound |vx2 (·, t)|L1 (R) ≤ M (), where M () is a constant depending on . Therefore

x

∞

∞ 2 2 2 (v )x dx| ≤ v dx + vx2 dx ≤ M (). v =| −∞

−∞

−∞

|s2x (·, t)|L1 (R)

≤ M () and |u2x (·, t)|L1 (R) ≤ M () Similarly, we can get from the second and third equations in (16.0.6). So we get the estimates in (16.1.1) and hence the proof of (A) in Theorem 16.1.1. From the estimates in (16.1.2) and (16.1.3), it is easy to prove the −1 (R×R+ ), where (v ,τ , u,τ ) are compactness of η(v ,τ )t +q(v ,τ )x in Hloc the solutions of the Cauchy problem (16.0.6), (16.0.2) and (η, q) is any entropy-entropy flux pair of Shearer type constructed in Section 12.2. Then the convergence of (v ,τ , u,τ ) follows. From the first estimate in (16.1.3), we obtain the convergence s,τ → s. So Theorem 16.1.1 is proved.

16.2

A Model System for Reacting Flow

In this section we shall consider the stiff relaxation limit of solutions for the system of chemically reacting flow (16.0.7) with the initial data (16.0.2). We establish the following theorem:

16.2. A MODEL SYSTEM FOR REACTING FLOW

223

Theorem 16.2.1 Let the condition (16.0.8) hold. Then, for any fixed τ , the Cauchy problem (16.0.7), (16.0.2) has a unique global smooth solution (v τ , uτ , sτ ). Furthermore, suppose the initial data satisfy

2 2 2 2 2 v0 + u0 + s0 dx ≤ M, τ v0x + u20x + s20x dx ≤ M, (16.2.1) R

R

and h

(v0 ) = 0, h

(v) ∈ L2 (R) ∩ L∞ (R),

h

(v)
= 0 for v
= v0 , h


(v) ∈ L2 (R) ∩ L∞ (R).

Then, along a subsequence if necessary,  τ τ τ v , u , s → (v, u, s)

a.e. (x, t),

(16.2.2) (16.2.3)

(16.2.4)

where (v, u, s) satisfies s = h(v) and (v, u) is an entropy solution of the equilibrium system  vt − ux = 0, (16.2.5) ut − (v − ch(v))x = 0,   with the initial data v0 (x), u0 (x) . The right side of condition (16.0.8) ensures the equilibrium system = 1−ch
(v) > (16.2.5) is strictly hyperbolic since in this case, d(v−ch(v)) dv 0; and the left side of (16.0.8) is equivalent to the strictly subcharacteristic condition ¯ 1 < λ2 < λ ¯ 2 < λ3 , λ1 < λ

(16.2.6)

where λ1 = −1, λ2 = 0, λ3 = 1 are three eigenvalues of system (16.0.7), and ¯ 2 = 1 − ch
(v) ¯ 1 = − 1 − ch
(v), λ λ are two eigenvalues of equilibrium system (16.2.5). We now show through an asymptotic expansion of the ChapmanEnskog type that condition (16.0.8) is necessary to ensure the stability of the relaxation approximated solutions (v τ , uτ , sτ ) of the Cauchy problem (16.0.7), (16.0.2).

224

CHAPTER 16. RELAXATION FOR 3 × 3 SYSTEMS Let sτ = h(v τ ) + τ g(v τ , uτ , vxτ , uτx , . . .) + O(τ 2 ). To calculate the form of g, we use system (16.0.7),  v τ − uτx = 0,    t uτt − (v τ − ch(v τ ))x = −τ gx + O(τ 2 ),    h(v τ )t + O(τ ) = −g + O(τ ).

(16.2.7)

Substituting g from the third equation of (16.2.7) into the second and noticing that vtτ = uτx from the first equation, we have  τ vt − uτx = 0, (16.2.8) uτt − (v τ − ch(v τ ))x = τ (h
(v τ )uτx )x + O(τ 2 ), which is a stable hyperbolic-parabolic mixed system, or the so-called Navier-Stokes equations if h
(v) ≥ d1 > 0. For fixed τ > 0, the existence of the unique global smooth solution τ (v , uτ , sτ ) in Theorem 16.2.1 is obvious since the growth order of the unique nonlinear function h(v) is given by (16.0.8). We shall prove the compactness (16.2.4) in Theorem 16.2.1 in several steps by several lemmas. Lemma 16.2.2 Under condition (16.0.8), there holds

t

1 2 2 2 (v + u + s )dx + (s − h(v))2 dxdt τ C 0 R R

≤ C (v02 + u20 + s20 )dx,

(16.2.9)

R

for some C independent of τ and t. Proof. Let the inverse function of h be h−1 . Then (h−1 )
= h1 ∈ [ d12 , d11 ] and hence h−1 (s)−v = (h−1 )
(α)(s−h(v)) for a value α between s and h(v). Multiplying the first equation in (16.0.7) by (v − cs), the second by u and the third by c(h−1 (s) − v), we get

s 2 2 h−1 (s)ds)t − 2((v − cs)u)x (v + u − 2cvs + 2c 0 (16.2.10) 2c(h−1 )
(α) 2 (s − h(v)) = 0. + τ

16.2. A MODEL SYSTEM FOR REACTING FLOW

225

s 2 2 2 Since 2c 0 h−1 (s)ds ≥ cs2 min{ h1 } ≥ cs d2 > c s , then integrating (16.2.10) in R × [0, t], we get the proof of Lemma 16.2.2. Using the second and third equations in (16.0.7), we have ut − (v − ch(v))x = −c(s − h(v))x = csxt = (vx − ut )t = τ (uxx − utt ), and hence the following system:  vt − ux = 0, ut − g(v)x = τ (uxx − utt ),

(16.2.11)

(16.2.12)

where g(v) = v − ch(v). Lemma 16.2.3 Suppose the initial data satisfy (16.2.1). Then solutions (v, u, s) of (16.0.7) satisfy the τ independent estimates

t (vx2 + u2x + s2x )dxdt ≤ M. (16.2.13) τ 0

R

Proof. Multiplying the first equation in (16.2.12) by g(v), the second by u, we have

 u2   u2   v u2  2 g(v)dv + − (ug(v)) = τ − τ u − τ + τ u2t , x x 2 t 2 xx 2 tt 0 (16.2.14) or equivalently

  v u2 + τ uut t − (ug(v))x + τ (u2x − u2t ) = τ (uux )x . g(v)dv + 2 0 (16.2.15) The problem is that the term u2x − u2t is not positive definite. To compensate for that, we first multiply the second equation in (16.2.12) by ut to obtain   u2 + u2t  (16.2.16) u2t − g
(v)vx vt = τ (ut ux )x − x t 2 and, in turn   τ 2 u2x + u2t t + 2τ (u2t − g
(v)vx ut ) = 2τ 2 (ut ux )x .

(16.2.17)

CHAPTER 16. RELAXATION FOR 3 × 3 SYSTEMS

226

Using the second equation in (16.2.12) again, we have g
(v)vx2 = vx (u + τ ut )t − τ vx uxx = (v(u + τ ut )t )x − v(u + τ ut )tx − τ vx uxx = (v(u + τ ut ))tx − (vt (u + τ ut ))x − τ vx uxx +vt (u + τ ut )x − (v(u + τ ut )x )t

(16.2.18)

= −(vt (u + τ ut ))x + (vx (u + τ ut ))t  2 +ux (u + τ ut )x − τ v2x t , where ux = vt is used in the last term τ vx uxx . (16.2.18) yields τ2

 vx2 u2  − x t − τ (vx (u + τ ut ))t + τ (g
(v)vx2 − u2x ) 2 2 = −τ (vt (u + τ ut ))x .

(16.2.19)

Adding (16.2.15),(16.2.17) and (16.2.19), we obtain that

v  1 2 + 1 τ 2 (u2 + u2 ) + (u + τ u − τ v ) g(v)dv t t x t x 2 2 0

−(ug(v))x + τ (u2t − 2g
(v)vx ut + g
(v)vx2 )

(16.2.20)

= τ 2 (ut ux )x . Using condition (16.0.8) again, we have g
(v) = 1−ch
(v) ∈ [1−cd2 , 1− cd1 ] and hence u2t − 2g
(v)vx ut + g
(v)vx2 ≥ c1 (u2t + vx2 )

(16.2.21)

for a positive constant c1 . Therefore, integrating (16.2.19) in R × [0, t] and noticing (16.2.21), we have

t 2 2 2 ut + vx dxdt ≤ M, τ u2t + u2x dx ≤ M. (16.2.22) τ 0

R

R

Integrating (16.2.15) in R × [0, t] and using the estimates (16.2.9) and (16.2.22), we have

t u2x dxdt ≤ M. (16.2.23) τ 0

R

16.2. A MODEL SYSTEM FOR REACTING FLOW

227

Rewriting the third equation in (16.0.7), then differentiating it with respect to x, we get τ (st )x sx + (sx − h
(v)vx )sx = 0.

(16.2.24)

Integrating (16.2.24) in R × [0, t] and using the estimate about vx2 in (16.2.22), we get

t

t s2x dx ≤ M, τ s2x dxdt ≤ M, (16.2.25) τ2 0

0

R

and hence, the proof of Lemma 16.2.3. Proof of Theorem 16.2.1. Let (η(v, u), q(v, u)) be any entropyentropy flux pair of Shearer type constructed in Chapter 12 for the equilibrium system (16.2.5). Multiplying system (16.2.12) by (ηv , ηu ) and noticing the first equality in (16.2.11), we have η(v, u)t + q(v, u)x = cηv (h(v) − s)   = c ηv (h(v) − s) x − c(ηvv vx + ηvu ux )(h(v) − s) = I1 + I2 . (16.2.26) Clearly from the estimates in Lemma 16.2.2,    1 (h(v) − s)  I1 = c ηv (h(v) − s) x = c τ 2 ηv 1 x τ2 −1,2 is compact in Wloc and 1

I2 = −c(ηvv vx + ηvu ux )(h(v) − s) = −cτ 2 (ηvv vx + ηvu ux )

(h(v) − s) 1

τ2

−1,α is bounded in L1loc , and hence compact in Wloc for a constant α ∈ −1,2 τ τ τ τ (R× (1, 2). Thus we have that η(v , u )t +q(v , u )x is compact in Wloc + R ). Then Shearer’s compactness framework in Chapter 12 shows the convergence (v τ , uτ ) → (v, u), a.e.,

which implies the convergence sτ → s, a.e. by the estimate given in the second part of the left-hand side of (16.2.9). This completes the proof of Theorem 16.2.1.

228

16.3

CHAPTER 16. RELAXATION FOR 3 × 3 SYSTEMS

Related Results

The proof of Theorem 16.1.1 for general system (16.0.1) is from [Lu12]. Before it, two special systems in the form (16.0.1) are studied by LuKlingenberg in [LK1, LK2]. About the relaxation limits for system (16.0.7), an equivalent system in the following form:  vt − ux = 0,     ut − sx = 0, (16.3.1)   s − h(v)   (s − cv)t + = 0, τ was first studied by I.-S. Liu and Y.-M. Wu [LWu] for smooth relaxation approximated solutions. Later, the general singular limits for Lp solutions were obtained by Tzavaras [Tz] by Shearer’s basic framework coupled some ideas given in [LK1, LK2].

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Index adiabatic exponent, 6, 87, 88, 119, 188 adiabatic gas flow, 7, 164, 217 artificial viscosity solutions, 7, 160, 163 Broadwell model, 217 Chapman-Enskog expansion, 204 characteristic field, 6, 44, 48, 49, 52, 54 chemical reaction, 8, 42 chemically reacting flow, 217, 223 chromatography, 83, 174, 175 combustion theory, 175 compensated compactness method, 43, 44, 52, 83, 119, 130, 134, 137, 138, 157, 160, 166 compressible fluid flow, 7, 121, 139 conservation laws, 1, 2, 8, 43, 44, 179, 180, 202, 203 Darboux-Euler-Poisson equation, 92, 93 DiPerna’s method, 7 Dirac measure, 64, 114, 149, 155– 157, 159 dissipation parameter, 188 dominant diffusion, 8, 177, 179, 180, 219

elasticity, 1, 7, 8, 47, 147, 148, 157, 159, 160, 163, 175, 179, 182, 186, 196, 215, 218 enhanced oil recovery, 47 entropy equation, 6, 57, 58, 92, 101, 123, 209, 210, 214 entropy-entropy flux, 7, 31, 44, 50, 57, 106 entropy-entropy flux pair, 6, 57, 76, 91, 126, 130, 134, 141, 145, 152, 154, 156, 205–207, 227 entropy-entropy flux pair of Lax type, 6, 65, 75, 124, 130, 143 Euler equations, 7 Eulerian coordinates, 8, 85, 165, 182, 186, 187, 217 fluid dynamics, 1 Fourier entropy, 163, 170 Fuchsian equation, 6, 58, 59, 75, 76, 102, 127, 131, 137 gas dynamics, 1, 121, 137, 186 genuine nonlinearity, 6 genuinely nonlinear, 6, 44, 48, 87, 122, 124, 139, 156 Glimm method, 135 Glimm’s scheme, 119

239

240 hydrodynamic limit, 121 hyperbolic conservation laws, 1, 2, 9, 31, 53, 70, 71, 202 hyperbolically degenerate, 44 interpolation inequality, 20, 22 Invariant Region Method, 45 Invariant Region Theory, 6 isentropic flow, 121 isentropic fluid dynamics, 8, 179 isentropic gas dynamics, 85, 123, 182, 187, 196 isothermal motions, 217 kinetic theory, 176 Lagrangian coordinates, 8, 147, 159, 179, 182, 196 Lax entropy, 7 Lax-Friedrichs scheme, 119 Le Roux system, 6, 52, 71, 137 linear degeneration, 6, 149 linearly degenerate, 6, 44, 48, 49, 54, 148, 149, 159 magnetohydrodynamics, 47 measure equation, 67, 104, 107, 109, 112, 154 multiphase and phase transition, 175 Murat theorem, 27

INDEX quadratic flux, 6, 53, 70, 137, 149 rarefaction waves, 202 reaction-diffusion equations, 180 relaxation problems, 7, 178 relaxation singular problem, 8 relaxation time, 175, 176, 178– 180, 188, 204, 217–219 Riemann invariants, 6, 44, 45, 54, 57, 72, 75, 77, 86, 122, 123, 138, 141, 149, 150, 161, 166, 192, 196, 200 river flow equations, 114 river flows, 176 shock waves, 1 singular perturbation theory, 7, 138, 145, 149 special symmetric system, 6 stiff relaxation, 8, 177–180, 182, 206, 215, 219, 223 strict hyperbolicity, 6, 48, 54, 72, 122, 124, 139 strictly hyperbolic, 8, 43, 87, 135, 148, 203, 224 subcharacteristic condition, 176– 178, 218, 224

Newtonian dynamics, 121 nonstrictly hyperbolic, 6, 7, 44, 87, 124, 206

Temple type, 6, 52, 82 theory of compensated compactness, 4, 5, 137 traffic flow, 1, 206, 215 traveling waves, 202

physical viscosity, 7, 163, 164 polytropic gas, 6, 85, 87, 88, 119, 187 polytropic gas dynamics, 6, 7

viscoelastic material, 217 viscoelasticity, 175 viscosity solution, 4, 48, 49, 55, 73, 88, 115, 124

INDEX Vlasov equation, 121 weak continuity theorem, 15, 31, 32, 39 weak entropies, 104, 138 Young measure, 64, 99, 101, 106, 114, 149, 155–157, 159 Young measure representation theorems, 5

241