The Characteristic Method and Its Generalizations for First-Order Nonlinear Partial Differential Equations

πCH… H山C Monographs a.nd Surveys in Pure a-ndApplied Mathematics

101

THE CHARACTERISTIC METHOD AND ITS GENERALIZATIONS FOR FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TRAN DUCVAN MIKIOTSUJI NGUYEN DUYTHAI SON

CHAP问AN

& HALUCRC

Monographs and Surveys in

Pure and Applied Mathematics

Main Editors H. Brezis , Université de Paris R. G. Douglas, Texas A&M University A. Jeffrey, UniversiηI of Newcastle upon 乃ne (Founding Editor)

Editorial Board H. Amann , Universi，η ofZürich R. Aris , University of Minnesota G. I. Barenblatt, Universiη of Cambridge H. Begehr, Freie Universität Berlin P. Bullen , Universiη of British Columbia R .J. Elliott, Universiη ofAlberta R卫 Gilbert， University of Delaware R. Glowinski , Universiη ofHouston D. Jerison , Massachuse t!s Institute ofTechnology K. Kirchgässner, Universität Stuttgart B. Lawson , State Universiη ofNew York B. Moodie, Universiη ofAlberta S. Mori , Kyoto University L. E. Payn巳， Comell University D.B. Pearson , Universi，η ofHull 1. Raeburn , Universiη10fNewcωtle G. F. Roach , Universi，η of Strathclyde 1. Stakgold , Universi吃y of Delaware w.A. S位auss， Brown University J. van der Hoek , University of Adelaide

π CH 队4认附酬帕 州柳 AP 仰棚附 P刊仲 tv…L丛 肌 ω 山/尼 Lυ C

and Surveys 阳 in Pure and Applied Mathematics

f问w叮、》叶1onographs

I0I

THE CHARACTERISTIC METHOD AND ITS GENERALIZATIONS FOR FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TRAN DUCVAN MIKIOTS叫 l

NGUYEN DUYTHAI SON

CHAPMAN & HALUCRC Boca Raton London New York Washington, D. C.

Library of Congr咽s Ca taI oging-in-Publication Data Tran, Duc Van. The characteristic method and its generalizations for 且rst -order nonlinear p缸tia1 differentia1呵uations / Tran Duc Van , Mikio Tsuji, and Nguyen Duy Thai Son. p. cm. -- (Chapman & Ha1 VCRC mongraphs and surveys in pure and applied mathematics ; 101) lncIudes bibliographica1 references and index. ISBN 1-58488-016-3 (a1k. paper) 1. Di fferenti a1 equations , Nonlinear--Numerica1 solutions. I.Tsuji , Mikio. 11. Nguyen , Duy Thai Son. 111. Title. IY. Series QA374.T65 1999 519.5'.353--dc21 99-27321 CIP 四 is book contains information obtained from authentic and high1 y regarded sources. Reprinted materi a1 is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable e仔urts have been made to publish reliable data and information, but the author and the publisher cannot assume responsib且ity for the va1idity of a11 materia1s or for the consequences of their use. Neither 由is book nor any part may be reproduωd or transmitted in any form or by any means, eIectronic or mechanica1, incIuding photocopying, microfilming, and recording , or by any information storage or retrieva1 system, without prior pem咀ssion in writing from 由e publisher. The consent of CRC Press LLC d∞s not extend to copying for genera1 distribution, for promotion, for creating new works , or for res a1e. Specific pen回ssion must be obtained in writing from CRC Press LLC for such copying. Direct a11 inquiries to CRC Press LLC, 2α)() N. W. Corporate Blvd. , Boca Raton , F1 0rida 3343 1.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks , and are used only for identification and explan侃侃， without intent to infringe. 。 20∞ by

Chapman

& Ha1VCRC

No cI aim to origi na1 U.S. Government works Internationa1 Standard Book Number 1-58488-016-3 Library of Congress Card Number 99-27321 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

Contents

Contents Preface Chapter 1. Local Theory on Partial Differential Equations of First-Order 1.1. Characteristic method and existence of solutions 1. 2. A theorem of A. Haar 1. 3. A theorem of T. 巩Tazewski

1 1

7 8

Chapter 2. Life Spans of Classical Solutions of Partial Differential Equations of First-Order 2. 1. Introduction 2.2. Life spans of classical solutions 2.3. Global existence of classical solutions

12 12 13 18

Chapter 3. Behavior of Characteristic Curves and Prolongation of Classical Solutions 3. 1. Introduction 3.2. Examples 3.3. Prolongation of classical solutions 3 .4. Su伍cient conditions for collision of characteristic curves 1 3.5. Su伍cient conditions for collision of characteristic curves 11

22 22 23 24 26 28

Chapter 4. Equations of Hamilton-Jacobi Type in One Space Dimension 4. 1. Nonexistence of classical solutions and historical~arks 4.2. Construction of generalized solutions 4.3. Semi-concavity of generalized solutions 4 .4. Collision of singularities

30 30 34 38 41

Chapter 5. Quasi-linear Partial Differential Equations of First-Order 5. 1. Introduction and problems 5.2. Difference between equations of the conservation law and equations ofHamilton-Jacobi type 5.3. Construction of singularities of weak solutions 5 .4. Entropy condition Chapter 6. Construction of Singularities for HamiltonJacobi Equations in Two Space Dimensions 6. l. Introduction

44 44 47 48 52 55 55

CONTENTS 6.2. Construction of solutions 6.3. Semi-concavity ofthe solution u = u(t , x) 6 .4. Collision of singularities

Chapter 7. Equations of the Conservation Law without Convexity Condition in One Space Dimension 7. 1. Introduction 7.2. Rarefaction waves 创ld contact discontinuity 7.3. An example of an equation of the conservation law 7.4. Behavior ofthe shock 51 7.5. Behavior ofthe shock 52 Chapter 8. Differential Inequalities of Haar Type 8. 1. Introduction 8.2. A differential inequality ofHaar type 8.3. Uniqueness of global classical solutions to the Cauchy problem 8 .4. Generalizations to the case of weakly-coupled systems

56 61 63 67 67 68 70 74 77 82 82 84 91 95

Chapter 9. Hopf's Formulas for Global Solutions of Hamilton-Jacobi Equations 9. 1. Introduction 9.2. The Cauchy problem with convex initial data 9.3. The case of nonconvex initial data 9 .4. Equations with convex Hamiltonians f = f(p)

103 103 105 113 121

Chapter 10. Hopιτ丁ype Formulas for Global Solutions in the case of Concave-Convex Hamiltonians 10. 1. Introduction 10.2. Conjugate concave-convex functions 10.3. Hopf-type fo口丑ulas

127 127 128 136

Chapter 11. Global Semiclassical Solutions ofFir吼-Order Partial Differential Equations 146 11.1. Introduction 146 11. 2. U niqueness of global semiclassical solutions to the Cauchy problem 148 156 11. 3. Existence theorems Chapter 12. Minimax Solutions of Partial Differential Equations with Time-measurable Hamiltonians 12. 1. Introduction 12.2. Definition of minimax solutions 12.3. Relations with semiclassical solutions 12 .4. Invariance of definitions 12.5. U niqueness and existence of minimax solutions 12.6. The case of monotone systems Chapter 13. Mishmash 13. 1. Hopf's formulas and construction of global solutions via characteristics

161 161 165 173 176 179 191 198 198

CONTENTS 13.2. Smoothness of global solutions 13.3. Relationship between minimax and viscosity solutions

205 208

Appendix 1. Global Existence of Characteristic Curves

214

Appendix 11. Convex Functions , Multifunctions , and Diπ'erential Inclusions AI I.1. Convex functions AI I. 2. Mu1t ifunctions 缸ld differential inclusions

217 217 222

References

227

Index

236

Preface One of the main results of the classical theory of first-order partial differential equations (PDEs) is the characteristic method which asserts that under certain assumptions the Cauchy problem can be reduced to the corr臼ponding characteristic system of ordinary differential equations (ODEs). To illustrate tlús , let us consider the Cauchy problem for the nonviscid Burger equation: θuθu 一一 +u 一一= 御街

0,

t

> 0,

u(O ， x)=l巾)，

x

(1)

ξR ，

x

εR

We try to reduce the problem (1)-(2) to an ODE along some curve x precisely, let us find x = x(t) such that

(2)

= x(t).

More

去M By the chain n白， we may simply require dxjdt can be defined by

u , and so the characteristics

z 二 x(t)

生 =u(t， x). dt

(3)

Along each characteristic x = x( t) we have d叫 dt = 0 , i.e. , u = u(t , x(t)) takes a constant value and then the characteristic must be a straight line with slope given by (3). Thus , by the initial data (匀， the characteristic passing through any given point (0 , s) on the x-axis is (4) x = s + h(s 沛， on which u has the constant value:

u

= h(s).

(5)

Hence , if the Cl-norm of h h(s) is bounded , then , by means of the implicit function theorem and (份， we can get

s=s(t , x)

(6)

PREFACE

for small values of t. solution)

Substituting (6) into (5) gives the classical solution (C 1 -

u

=

h(s(t , x))

(7)

to our Cauchy problem (1)-(2). However , in general , this solution exists only locally in time. In fa叽 if h = h(s) is not a nondecreasing function of s , there exist two points (0 , Sl) and (0 , S2) on the x-axis such that

Sl < S2

and

h(Sl) > h(S2).

(8)

Then the characteristic curves beginning from (0 , Sl) and (0 , S2) will intersect at tíme

t-

s2- S ~>O.

h(s t} - h(S2)

Since the solution u u( t , x) is constant along each of the two curves but has different values h(s t} and h(S2) , respectively, at the intersection point , the value of the classical solution cannot be uniquely determined. Hence , in this case the Cauchy problem (1)-(2) never admits a global classical solution on {t 主 O}; in fact , the classical solution will blow up in a finite time no matter how smooth and small the initial data h = h( s) 缸e. On the other hand , if h h( s) is a nondecreasing function of s , then the 巳haracteristics emanating from distinct points (O , Sl) and (0 , S2) on the x-axis will not intersect , and thus the solution u = u( t , x) will exist globally for t 主 O. The previous example shows ，但仅ally speaking , 出 t ha 剖t for (陆如 firsωtιω创卧心 创 O r时 伽削臼叫) non d 血 1 partial dωif宦fe臼ren 时 1曲tia 叫1 equations or systems , classical solutions to the Cauchy problem exist only locally in time , while singularities may occur in a finite time , even if the initial data are sufficiently smooth and small. Therefore , the notions of generalized solutions or weak solutions have been introduced. In fact , the global existence and uniqueness of generalized solutions have been well studied from various kinds of viewpoints. In the 1950s-1970s , the theory and methods for constructing generalized solutions of first-order PDEs were disω叮叮ed by Aizawa , S. [2]-[4], Bakhvalo巾， Benton , S.H. [2 月， Conw町， E.D. [32]-[33], Douglis , A. [44]-[剑， Evan , C. , Fleming , W且 [51]-[52 ], Friedman , A. [54 ], Gelfru吨I. M. ， Godunov , S. K., Hopf, E. [63]-[64], Kuznetso叭 N.N. [95], Lax , P.D. [9η-[98]， Oleinik , O.A. [112], Rozdestvenskii , B. L. [118] , and other mathematicians. Among the investigations of this period we should mention the results of Kruzhko巾， S.N. ([8可 -[92] ， [94]) , which were obtained for Hrunilton-Jacobi equations with convex Hamiltonian. The global existence and uniqueness of generalized solutions for convex Hamilton-Jacobi equations were well studied by several methods: variational method , method of envelopes , vanishing viscosity method , nonlinear semi-group method , etc. Since the early 1980s , the concept of viscosity sol包tions introduced by Crandall and Lions has been used in a large portion of research in a nonclassical theory of first-order nonlinear PDEs as well as in other types of PDEs. The primary virtues of this theoηr are that it allo啊s merely nonsmooth functions to be solutions of nonlinear PDEs , it provides v芭ry general existence and uniquen

PREFACE

yields precise fonnulations of general boundary conditions. Let us mention here the Crandall , M.G. , Lions , Pι. ， Aizawa，丘， Barbu, V. , B町di ， M. , Barles , G. , Barron , E.N. , Cappuzzo-Dolcetta , 1., Dupuis , P. , Evans , L. C. , Ishii , H. , Jensen , R. , Lenhart ，丘， Osher，旦， Perthame , B. , Soravia，卫， Souganidis , P. E., Tataru , D. , Tomita , Y. , Yamada , N. , and many others (s白间， [10]-[20 ], [28 ], [35]-[39] , [47]-[50 ], [67]-[72 ], [79] , [99]-[10 月， [122]-[123 ], [131] , and the references therein) , whose ∞n­ tributions make great progress in nonlinear PDEs , and where the global existence and uniqueness of viscosity solutions have been established almost completely. The concept of viscosity solutions is motivated by the classical maximum principle which distinguishes it from other definitions of generalized solutions. Another direction in the theory of generalized solutions is motivated by differential game theory as suggested by A. 1. Subbotin. This leads to the notion of mzmmαx solutions of first-order nonlinear PDEs. As the tenninology "minimax solutions" indicates , the definition of such global solutions is closely connected with the minimax operations. This definition is b田ed ， to some extent , on the so-called "characteristic inclusions" (a generalization of the classical characteristic system in this 此uation). Subbotin 矶d his coworkers ([1 ], [124]-[127 ], [129]-[130]) developed an effective theory of minimax solutions to first-order single PDEs 缸ld gave nice applications to control problems and differential games. The research of minimax solutions employs methods of nonsmooth analysis , Lyapunov functions , dynamical optimization, and the theory of differential games. At the same time , the research contributes to the development of these branches of mathematics. A review of the results on minimax solutions and their applications to control problems and differential games is given in [124]-[125]. We also want to mention the investigations on PDEs based on the idempotent analysis and Cole-Hopftransformation , which have been discovered by V.P. Maslov and his coworkérs. Indeed , V.P. Maslov , V.N. Kolokol'tsov , S.N. Samborskii , and others developed a nonclassical approach to define the weak global solutions to first order nonlinear PDEs , in which by a suitable structure of new function serr让modules ， nonlinear operators become "linear" ones. In this direction , based on the methods and results of the well n创丑es:

PREFACE

As for the theory of partial differential inequalities , the first achievements were obtained by Haar [6 月， Nagumo [107] , and then by Wazewski [154]. Up to now the the。可 has attracted a great deal of attention. (The reader is referred to Deimling [40] , Lakshmikantham and Leela [96 ], Szarski [128 ], and Walter [153] , for the complete bibliography.) It must be pointed out that the characteristic method gives us the local existence and uniqueness of classical solutions to first-order nonlinear PDEs. We would like to use tms method as an important basis for setting the global generalized solutions. This book is devoted to some developments of the characteristic method and mainly represents our results on first-order nonlinear PDEs. Our aim in the first seven chapters is to fill a gap between the local theory obtained by the characteristic method and the global theory which principally depends on vanishing viscosity method. This is to say, we try to extend the smooth solutions obtained by the characteristic method. Our first problem then is to deter mine the life spα ns of the smooth solutions. Next , we want to obtain the generalized solutions or weak solutions by explicitly constructing their singularities. In Chapter 1, we present the classical results wmch are necessary for our follow ing discussions: the characteristic method , existence of local solutions , and Theorems of Haar and Wazewski on the uniqueness of solutions to the Cauchy problem in C 1 -space. Chapter 2 is devoted to the life spans of classical solutions of the noncharacteristic Cauchy problem. Our method depends on the analysis of the smooth mapping obtained by the family of characteristic curves. Even if the Jacobian of the mapping may vanish at some point , we can sometimes extend the classical solution beyond the point where the Jacobian vanishes. Therefore , we are obliged to consider very often some properties of the inverse of the mapping in a neighborhood of a singular point. This is the subject of Chapter 3. In Chapters 4 and 5, we consider the extension of solutions beyond the singularities of solutions in the case where the dimension of space is equal to one. Then our principal problem is to construct the singularities of generalized solutions or of weak solutions. The theme of constructing the singularities of solutions is picked up again in Chapter 6 for convex Hamilton-Jacobi equations in two space dimensions. The difference between Chap

PREFACE

"characteristic bundles" are invoked instead of characteristic differential equations and characteristic curves. Chapters 9-10 are devoted to the study of Hopf-type formulαs for global solutions to the Cauchy problem in the case of non-convex, non-concave Hamiltonians or initial data. In Chapter 9 , we first consider the case where the initial data can be represented as the minimum of a family of convex functions , and next the case where it is a d.c. function (i.e. , it can be represented as the difference oftwo convex functions). In Chapter 10 , the Hamiltonians are concα ve-convex functions. The method of Chapters 8-10 allows us to deal with global solutions , the condition on whose smoothness is relaxed significantly. In Chapter 11 , we propose the notion of globα 1 semiclassical solutions , which need only be absolutely continuous in the time variable , and investigate their uniqueness and existence. By the way, an answer to an open uniqueness problem of S.N. Kruzhkov [93] is given. In Chapter 12 , we extend the notion of Subbotin 's 口IÏnimax solutions to the case of first-order nonlinear PDEs with time-measurable Hamiltonian. The uniqueness and existence of such solutions are investigated by the theory of multifunctions and differential inclusions. Our road here is devious (by some 平 erturbation technique" on sets of Lebesgue measure 的， and proceeds via an implicit version of Gronwall's inequality and via a sharpening of a well-known theorem on the Lebesgue sets for functions with par臼neters. The results are new even when restricted to the case of continuous Hamiltonians. Generalizations for monotone systems w i11 also be considered. Finally, in Chapter 13 we examine Hopf's formulas in relations with the construction of global solutions via characteristics and the smoothness of the solutions. In this chapter , the relationship between minimax and viscosity solutions is also investigated. We have to say that this book is not designed as an introduction to , or a guidebook on , the general theory of 如st-order nonlinear PDEs. Our goal is not to try to cover as many subjects as possible , but rather to concentrate on some basic facts and ideas of the generalized characteristic methods for studying global solutions. Suitable as a text , the book is self-contained and assumes as prerequisites only calculus , linear algebra, topology, ODEs , and basic measure theory. In the appendices at the end of the book we collect nec

Chapter 1 Local Theory on Partial Differential Equations of First-Order ~1.1.

Characteristic method and existence of solutions

Partial differential equations of first-order have been studied from various points of view: for example , classical mechanics , variational method , geometrical optics , etc. In this chapter we will always suppose that the equations and solutions are realvalued. The classical method to solve the equations is the characteristic method. As this is the fundamental tool in our following discussions , we will give here a brief explanation of the method. For more detailed results and geometrical meanings , refer to , for example , R. Courant and D. Hilbert [34] and F. John [80]. First we consider a quasi-linear partial differential equation of first-order as follows:

内，事内

去+罢αi(川)23 斗。(川 u(O , X)

= 4> (x)

on

def

of {(O , x ,4> (x)) and

4>

=

x E Uo} in

4> (x) are of class

C1

U,

Uo~' {x ε IR n : (O ， x) ε U} ，

where U is an open neighborhood of (t , x)

IR n +2.

in

= (0 , 0).

(1.1) (1. 2)

Let V be an open neighborhood

Assume that aí = 咐，吼叫 (i=O ， l ， … ， n)

in V and Uo, respectively. A function is said to be

of class C k if it is k-times continuously di晶rentiable ， and Ck(U) is the fan均 of functions being of class C k in U. A Ck-function means that it is a function of class

Ck. Characteristic curves of (1. 1)-(1. 2) 缸e deßned by solution curves of the following system of ordinary differential equations:

t (仔 ιdtι= 坠 可… G

生 = ao(巾， υ) . dt

(1. 3)

1. LOCAL THEORY

2

In accordance with (1 功， the initial condition for (1. 3) is given by 引 (0) =的

(i=1 ， 2 ，...， n) ，

v(O)= φ(y).

The ordinary differential equations in (1. 3) are called the

(1. 4)

"chαracteristic

equations"

for (1. 1) , where we use v

= v(t , y) instead ofu = u(t , x) to avoid confusion. In the following discussions , u = u( t , x) is a solution of (1. 1) and v = υ (t ， y) is a solution of (1. 3)-( 1. 4) which is equal to the value of u = u(t , x) restricted on the corresponding solution curve x = x(t , y) of (1.3)-( 1.4). As ai = ai(t , x , V) (i = O ， l ，...， n) 皿d cþ= φ( x) are of class

C 1 in V and 吨， respectively, the Cauchy problem (1. 3)-( 1. 4)

has a system of solutions xi class

C1

=

Xi(t , y) (i

in a neighborhood of {(O , y)

= 1, 2,... , n)

and v = υ (t ， y) which are of

y ε Uo }.

Let us fix our notations on derivatives of functions. i.e. , x

=

t(XllX2'...'Xn). Therefore dxjdt

=

A vector x is vertical ,

t(dxddt , dxddt ,..., dxnjdt). On

the other hand , given any real-valued functionφ = cþ(叫， we write gradcþ( x ) 御用x

x

=

= cþ'(x) =

(御用xll âcþjθX2 ， …，御用x n ).

For an n-vector valued function

x(y) of an n-vector y , we defi.ne its Jacobi matrix and Jacobian , respectively,

by

θXl

θXl

θXl

θ'yl

θy2

θyn

(ZL=13.Y and

去(y) 哩叫去儿，j=1， 2 ，. .， n

We will sometimes write the Jacobi matrix simply by we see that (DxjDy)(t , y) U ε Uo } ，

=

1 for t

= 0, Y

t.\ I((予X :,-. J.

\θyj}

Since x(O , y)

= y,

E Uo . In a neighborhood of {(O ， ν)

:

as (DxjDy)(t , y) does not vanish, we can uniquely solve the equation

= x(t , y) with respect to y and write the solution by y = ν (t， x). Puttingu(t， x) 哇f v(t , y(t , x)) , we will prove that u = u(t , x) satisfies (1. 1)-( 1. 2) in a neighborhood of x

the origin. Theorem 1.1. The Cauchy problem (1. 1)- (1 功 hαs un叩tely α solution in a neighborhood

01 the

origin.

01 class Cl

~l.l.

CHARACTERISTIC METHOD AND EXISTENCE OF SOLUTIONS

3

Proof. We use the notations introduced in the above. The following discussions are true only in the definition domain of y

= ν (t ，

x). This domain is a neighborhood of

the origin , where the Jacobian (Dx/Dy)(t , y) does not vanish. As x

= x(t , y(t , x)) ,

we have

(~二儿，j=1， 2 ，0 0 ， n (缸，户口， JI(叫

( 1. 5)

0

。x . (θXi 、

θy

a\θyj ) i ，j=l ι.. ，n.

ðt

(1. 6)

As u(t , x) = υ (t ， ν (t ， x)) , we have θuθuθv

一=一 ， y) ðt ðt (t '0'07/

åu

+ åu ;，~ . ~ ðt (t , x).

(1. 7)

•

By (1. 7) , using (1. 5) and (1. 6) , we get 百 (t ，

x)

= αo(巾 ， u(t ， x)) 一

θv

(åUi \

. (一三二 l

τ

θν\θXj ) 1 三'，}三 n å

=α o(t， x , u(t , x)) 生 (t， x) . 生 ðt θx'-'-/

=句川一去川去 As u(O , x) = v(O ， ν (O ， x))

rþ(x) , we see that u

u(t , x) satisfies the Cauchy

problem (1. 1)-( 1. 2) in the above neighborhood of the origin. We will show the uniqueness of solutions. Let u =

u 衍，

C10f(11)-(12) , and put Z(tJ)qzf u(t， zO， ν)) where x the solutions of (1. 3)-( 1. 4). Then the

x) be any solution of class

= x(t， ν) and v = v(t， ν) are def

~

differenceω (t ， ν) ~. u(t ， ν)

- v(t , y) satisfies

the following Cauchy problem:

(d一 ω(川) = L: (αj(t ， x ， v) 一 α (t， x 叫)一+(咐， x ， u) n/~\θu } = 1 \ } \ ", W'~!)θ Xj

ω (O ， y)

I

飞

咐 ， x ， v)) ,

= O.

As the right-hand side of this differential equation can be estimated by

Mlu -

vl =

MI叫， we get ω( t ，的三 0 ， i.e. u(t , y) ==咐， ν) for any (t ， 的 in a neighborhood of

the origin. This rneans that the solution of C 1-class is unique a10ng the curves

x

= x(t ， 的.

That is to

say，出 10吨 as

the Jacobian

the solution of (1. 1)-( 1. 2) is unique in the

(Dx/Dy)(t ， ν)

C 1 -space.

does not vanish , 口

1. LOCAL THEORY

4

Next we consider the Cauchy problem for general partial differential equations of first-order as follows:

去 +f(t， x， u， 去) = 0 u(O , x) =
)

Mud

一一

‘飞

..,,-Ttw

，，a‘飞

noxo U-v

δXl

δXl

θν1θν2

δνn

)

Hg

(1.12) δX n

θX n

δν1δy2

θ yn

θX n

仙 ere

5

p(t , y) = (Pl (t , y) , P2(t , y) , . .. , Pn( t , y)).

Proof. We put θXl

δXl

θXl

δyl

。y2

θyn

。马二

0年二

。土二

θyl

θ'y2

θνπ

defθu

z(t , y) ~θu (t ， y) - p( t , y)

Using (1. 10) , we have

(dθf dtz(t , y) = 一再 (t ， X(t , y) , V(t , y) , p(t , y)) z(式的， z(O ， y) 二 o.

As this is a linear ordinary di fIerential equation concerning z z(t ， y) 三 o.

口

Remark. Fix any y E UO, and let J solutions X

Z(t , y) , we get

X(t , y) ,V

c

V(t , y) and P

IR be an interval around 0 on which the =

p(t , y) of the characteristic equations

(1.1 0)-( 1. 11) exist. Then (1.12) is true for each t vanish at (t , y). Recall that in W we have (Dx / Dν )(t ， y)

手 0，

ε J even if the Jacobian m町

and we can uniquely solve the

= x(t ， ν) wi th respect to y. The sol u tion has been denoted by y = y (t , x) and used to de鱼ne u(t , x) 乍fu(tJ(tJ)).

equation x

Corollary 1. 3. In the definition domain of y = ν (t ， x) ， we have

主 (t， x) = pi(t， y(t， x)) 问 2，...

, n)

Using Lerruna 1. 2 and its corollary, we get the following:

(1.13)

1. LOCAL THEORY

6

Theorem 1. 4. Suppose f εC 2α nd ~q队 ν) :~ (t , z , w , q) , |ωιθf dt 孚Jθ'Pi

i=l

dqi θfθf 一=一一 (t ， z , w , q) - qi(t , y) 一(毛巾 ， q) dt

with

ω(0 ， ν)=φ(ν)

θ的

and q(O , y)

= φ， (ν).

(z ， ω ， q)

=

(i=1 , 2 ,..., n) ,

Hence

(z(t ， 的， ω ( t ， 的， q(t ， ν))

~1. 2.

A THEOREM OF A. HAAR

7

satisfies the Cauchy prob1em (1. 10)-( 1. 11). By the uniqueness of solutions of ordinary differentia1 equations , we have v(t ， ν)=ω (t ，

x(t , y) = z(t , y) ,

y) ,

p(t ， ν)

=

q(t ， ν) ,

where (叭叭 p) = (x(t ， y) ， v(t ， ν ) ， p(t ， y)) is the solution of (1.1 0)-( 1.1 1) which has a1ready appeared in Lemma 1. 2. This says that the solution of class C 2 is unique a10ng the curves x = x (t , y). Therefore , as

10吨 as

the Jacobian

(Dx/Dy)(t ， ν)

not vanish , the 叫ution of class C 2 of (1. 8)-( 1. 9) is unique.

does 口

31. 2. A theorem of A. Haar We have seen by Theorem 1. 4 that the Cauchy prob1em (1. 8)-( 1. 9) has a solution of class C 2 in a neighborhood ofthe origin. But , as the equation (1. 8) is of first-order , we wou1d 1ike to discuss the existence and uniqueness of solutions in C 1 -space. In the following , we write a solution of class C k as a Ck-so1ution , C 1 -s01utions will be sometimes called

吐出sica1

solutions."

One of our aims is to consider what kinds of phenomena may appear when we extend the classica1 solutions of (1. 8)-( 1. 9). In this procedure , we need the uniqueness of solutions in C 1 -space. This subject had been well studied by A. Haar [61] and T. Wazewski [154]-[155]. Moreover , their results are indispensab1e to deve10p our discussions. But , as it seems to us that they are not well-known , we would like to introduce them. In this section , we report a theorem of A. Haar [61]. We consider the Cauchy prob1em (1.创刊1. 9) in one space dimension , that is x E R 1 . Let ß be a triang1e defiued by ß={(t ， x):O 三 t 三 α ，

where α> 0 , L ~三 0 ， c {(u ， p)


0 and

α >

M , this contradicts (1. 14). Therefore z(t ， x) 三 o for

all (t ， x) εß. Since we can similarly prove that z(t ， 叫主 0 ， we obtain Ul(t ， X) 三

U2(t , X) in

ß

口

31. 3. A theorem of T. Wazewski In this section we report a theorem of T. Wazewski [154] which is a generalization of Haar's theorem to arbitrary space dimensions. Let us consider the Cauchy problem (1. 8)-( 1. 9) in a pyramid ß which is defined by

~1. 3.

where α> 0 , Li ?: 0 , ci

A THEOREM OF T. WAZEWSKI

< di

and (2L i )α 三(命 - Ci) for all i = 1, 2 ,.. . , n. A set 筑

is a comp配t set in {(u , p)

U 巳Iæ， p ε Iæ n }.

Theorem 1. 6. (T. W aZewski [154]) Suppose that f follo仇ng

9

f(t , x , u , p) satisβes the

Lipschitz condition: If(t , x , u , p) - f(t ， x ， v ， q)1 三汇 Lilpi - qil

for (t ， x ， u ， p) αnd (t , x , v , q) in ß x .fì. Let Uj

=

+ Mlu 一叫

Uj(t , x) (j

= 1, 2)

be in C 1 (ß)

αnd (Uj(t ， x) ，( θUjj θx)(t ， x)) ε .fì for all (t , x) 巳 ß.lfUj=Uj(t ， x) (j =1 ， 2) αre

solutions of (1. 8) in the domain ß and Ul(t , X)

=

U2(t , X) on ß

n {t = O}, then

Ul(t ， X) 三句 (t ， x)inß.

As preparation for the proof of this theorem , we give a fundamental lemma which plays an important role in his many works. Lemma 1. 7. Let

~

be the

pνra αm 旧iω d defin 附ed 切 in 伪 t he α abo ω 盹 v y 唱

def

(t , x) εß}. Let U = u(t , x) be in C 1 (ß) αnd put ω (t) ~. max{u(t , x)

x ξ ßt }

for each t 巳 [O ， a]. Then ω=ω (t) is differentiable from the right on [0 ， α) ， αnd

叫(t) = 尝(们)一全Li I~~. (t , x)1 …… for some x ε ß t ωith ω (t)

= u(t , x).

Proof. As the first step , consider the case where Li = 0 , ci for all i

-1 and d i

1

1, 2,… , n. By the definition of ω=ω (t ， x) ， we easily see that it is

continuous. Let U (t) ~f {x 巳 ß t

ω (t) = u(t , x)} for t ε[0 ， α). Then the sets

U(t) (0 三 t 三 α) are closed , bounded , and non-empty. As U = u(t , x) is in C 1 (ß) ,

for any fixed t O ε [O ， a) ， we.can pick up a point x O ξ U(t O ) C

生 (t O ， xO )

=

ßto

such that

max 些 (tO ， x).

"'EU(t O ) θt

We will prove thatω=ω (t) is differentiable from the right at t O withω+ (tO) = (θuj街 )(t O ， x O ). For any x ε U( t O ) , we have

ω (t O ) = U内) and 去川三言川)

LOCAL THEORY

1.

10

Define z(t) ~f [ω (t) 一 ω (tO)l/ (t - t勺， and put

ß~flimsupz(t)

α 乞f limi~fz(t).

and

t一斗ftO

t-+t t>t O U

t>tO

As ω (t) 一 ω (t O ) ::::: u(t , x O) - u(tO , x O), we see that α 主 (θu/θt)(tO ， X O ). By the definition of ß, we can pick up a sequence {tm}m

c (t O, a)

with lim t m = t O and ，π-+00

JlyiJ(tm)=F.For each tm , t k arbitrarily zmε U(t m ) ， so that u(tm , xm) ω (t m ).

8ince the set

{(t m ， 俨

=

= 1, 2 ,...} is bounded , we can assume that

m

the sequence{(tm , zm)}m is convergent to a point(to , ZO)in A.As the functions u = u(t , x) and ω=ω (t) are continuous , this implies that u(t口， 20)=ω (t O ) ， i.e. , ε U( t O), hence that (θu/θt)(tO ， ~。 x-) 三 (θu/ât)(tO ， X O ).

Z

ω=ω (t) ， we have ω (t O ) 主 u(t O ，

By the de负rútion of

xm) for all m = 1, 2,.... Using this inequality, we

get

z(t m )

=

ω (t m ) 一 ω (t O ) 。

tm

-

tU

ω ( t O)

u (tm ， xm)

tm

-

tO

u (tm ， xm)-u(俨 ， x m )θ U .......m 。 =τ (t , x m ) , t m - tU ät ......m

ith )i m (t

,..

......0θu

, xm) = (t O,X--)./" It_follows that ..._-

→∞，-

~

.~..~.. ~

ß::::: r---'

一(tO ， ;;U)I ---' :::::一(tO ât \ _ ,8t ， xO). There-

fore ,

去川)三 α 三比去川) 80 ω=ω (t) is differentiable from the right at t O, and ω't (t O) = (θu/δt)(t O ， x O). Next , we consider the same problem in a general pyramid ð. To do so , we take the transform of coordinates a.s follows:

{ :二 S! 向=三 [Ci + di + Yi(di -

Ci -

2Lρ) 1

Then the set ð is mapped to the set {怡 ， y)

(i

=

1, 2, . .. , n).

0 三 s 主 α ，一 1 三 ω 三 1

(i =

1, 2,... , n)}. Combining the result obtained in the first step and the above transform of coordinates , we complete the proof of the

lemma.

口

~1. 3.

A THEOREM OF T. WAZEWSKI def

Proof of Theorem 1. 6. We put u(t ， x)~' Ul(t ， X)

max{u(t , x)

x 巳 ßtl.

and

问 (t ， x) ，

11

and defineω (t) ~

For t ε [0 ， a) , let x(t) ε ß t be such thatω (t) = u(t , x(t))

叫 (t) = … 尝(t， x(t)) 一天Li I 芒。， x(叫 …

(cf. Lemma 1. 7). Then (户tw(t))~

_, IθuJ、

= e- at

|θu ，

,,,

1

,,,

1

~页(机 (t))- ~Lilð;i(t ， x(t))1 一 αu(t ， x(t)) ~.

(1.1 5)

On the other hand , it follows from the hypotheses of the theorem that

1: 川) 1 兰叫去川)1 + Mlu(t , x(t))I. If we choose

α >

(1.1 6)

M , then from (1.1 5)-( 1.1 6) we get

(

川ω (t))~ 俨 ργ? 三 俨 e 严〔一叮 叫 d 时t气 a 叫(MIμω 叫州(仅仲 t

0, from which it may be concluded that ω (t) 三 0 ， i.e. , Ul (t, x) 三句 (t ， x ), for all ω(仰 0) =

(t , x)

巳 ß.

句 (t ， x)

Since we can simi1a r1y prove U2(t , x) 三 Ul (t , x) , we finally get Ul (t , x) 三

on ß.

口

We will rewrite Theorem 1.6 in a general form. A function f

= f(t , x , u , p)

is

said to be locally Lipschitz continuous with respect to (u , p) if, for any comp肌t set .Iì in lR x lRn x lR x lR n , there exist constants Ll' L2' … , Ln and M such that

If(t , x , u , p) - f(t , x , v , q)1 三汇 Lilpi 一创 + Mlu -vl i=l for all (t , x ， 飞 p) and (t ， 叭叭 q) in .Iì. Theorem 1.8. Suppose 陀esψ T :pect 归 to 仙 ( u ， 时p) .扩 u

伪 t.hαωt

f

= f(t ， 凯 x， 叽 u， 叫 p)

u(t ， x)andv

neighborhood of the origin satisfying u(O , x)

is

loc ωα l闯 l仿 yL μzp 严schi必tz cω o时 n timω包旧s ω t伪 h

υ (t ， x) αre 三 v(O ， x) ，

neighborhood n of the origin, such that u(t , x)

三 v(t ， x)

C 1 -s01utions of (1. 8)

in α

then there exists an open in

n.

We willleave the proof to the readers. Remark. The uniqueness of C 1 -s01utions is assured by the Lipschitz continuity of

f = f( t , x , u , p) with respect to (u , p). But this condition is not sufficient to get the solvability of the Cauchy problem (1. 8)-( 1. 9). Wazewski [156] has given necessary and sufficient conditions which guarantee the local existence and uniqueness of classical solutions of (1. 8)-( 1. 9).

Chapter 2 Life Spans of Classical Solutions of Partial Differential Equations of First-Order 32. 1. Introduction As we have shown in Chapter 1, the Cauchy problems (1.1)-(1. 2) and (1. 8)-( 1. 9) have locally classical solutions. It is well-known that the solutions may generally have singularities in finite time even for smooth initial data. For some equations of the conservation law , the life spans of classical solutions have been exactly calculated , for example by E. Hopf [63] , P.D. Lax [97]-[98 ], E.D. Conway [32 ], etc. The principal aim of this chapter is to determine the life spans of classical solutions of general partial differential equations of first-order. Moreover , using the results on the life spans of classical solutions , we will give necessary and

su面cient

conditions

which guarantee the global existence of classical solutions for (1. 1)-( 1. 2) 皿d (1. 8)-

(1. 9). As an example , we consider a simple equation of the conservation law as follows:

生+于α巾)生 =0

{t>O， 叫町，

in

…一….

u(O , x)= 4> (x) where ai

αi(U)

on

{t=O , x

(2.1)

巳]R"}，

(2.2)

(i = 1, 2 ,... , n) and 4> = 4> (ν) are of class C 1 in ]R and ]R",

respectively. The equation treated in E. Hopf [63] is the case where n = 1 and E.D. Conway considered the above equation

(2.1)-(2功 in

α l(U)

= u.

[32]. 1n this case , the

characteristic curves which are the solutions of (1. 3)-( 1. 4) are written by x= ν +tα(4) (ν) )，

υ (t ， y) =4>(ν)

(2.3)

where α(u) 哲 (α1 (u) ， α2(U) ，... , a,, (u)). Then the Jacobian of the mapping x =

x(t , y) is given by

(Dx/D训， ν)=1+ 训ν)

with

Obviously, the Jacobian (Dx/ Dy)(t , y) does not

>.(y) 哩艺 αH4>(州θ4>/8Yi) (ν)

var山h

in a neighborhood of (t ， ν)=

~2.2.

,

13

LIFE SPANS OF CLASSICAL SOLUTIONS

,

,

(0 0). Therefore as we have shown in 31. 1 the Cauchy problem (2.1)-(2.2) has a unique C 1 -so1ution u

u(t , x) in a neighborhood of the origin. Since u(t , x)

4> (y( t x)) where y

x) is the solution of the equation x =

,

= ν (t ，

x(t ， ν) ，

we have

(茬，却=品而(茬，... ，茬儿(t，.， ) Then the Jacobian (DxjDy)(t O ， 的= 0 for t O ~f 1 j ( →λ(俨)). We see by (2 .4) that when (t x) goes to (to x O) along the curve x = x(t ， 俨) where Assume λ(俨)

< O.

,

,

,

,

zotf z(tO ， ν勺， at least one of the first derivatives of u = u( t x) tends to infinity. Therefore , if the Jacobian vanishes somewhere , then the Cauchy problem (2.1)(2.2) can not admit a global solution of class C 1 • In 32.2 and 32.3 , we will give similar results on the life spans of classical solutions for general partial differential equations of first-orde r. What we would like to remark here is to point out that there exist some differences between quasi-linear equations and general partial differential equations. See Theorem 2.1 and Theorem 2.2. In 32 .4, we will consider the global existence of classical solutions. These existence results are corollaries of theorems given in 32.2 and 32.3.

32.2. Life spans of classical solutions Let us consider a quasi-linear partial differential equation of first-order as follows:

2 十也川)去= aO川 ln 书> 0, x E ~n}，

,

u(O x)= 4> (x)

on

,

{t=o x

, ,... ,n) and 4> =

where 向=句 (t ， x ， u) (i = 0 1

ξ ~n} ，

4>(ν) are of class

(2.6) C 1 in

~ x ~n X ~

and ~n ， respectively. The characteristic equations for (2.5)-(2.6) are written by (L dt A t u ) ( = 1 2 n )

去 =αO(t， x ,v) ,

(2.7)

with the initial conditions

, ,...

Xi(O) = 衍。= 1 2

，叫，叫 0)

= φ (y).

(2.8)

2. LIFE SPANS OF CLASSICAL SOLUTIONS

14

We write the solutions of (2.7)-(2.8) by x = 仰 ， y) and v = 咐， ν). Then v = 咐， ν) means the value of a solution of (2.5) restricted on the curve

x 二 x(t ， ν).

Here we

assume the following condition: (A .I) The Cauchy problem (2.7)-(2.8) has uniquely a global

v = v(t , y) on {t

~

O} lor

sol1巾 on

x(t ， 的 F

x

a叼 yε Jæ.n.

It is not easy to write down sufficient conditions which guarantee the assumption (A .I). Co配erru吨 this subject , see for example B. Doubnov [43] and M.S Krasnosel'skii [83]. We wi11 consider this in Appendix 1. When we assume (A .I), we

= x( t ， 的

get a smooth mapping x

from

Rn to Jæ.n for each t

~

O. The 1ife span of

classical solutions of (2.5)-(2.6) is determined by the followi吨: Theorem 2. 1.

Un 叫 de旷r

Condition (A .I), s包叩 ，ppose

山阳/川 均νω D 州州川)(附队 圳札阳 (tt式， 川阳 y 0川10川 T川t 01 the x

solution u

0, we can see from

Lemma 2.5 that H t is a diffeomorphism from Jæn to Jæn for any t inverse function y = ν (t ， x) of x

= Ht (ν) is a function of class

C1

主 O.

Hence the

defined on D and

u = u(t , x) ~f v(t ， ν (t ， x)) is the unique global C 1 -so1ution of (2.5)-(2.6). (For the

uniqueness , follow the proof of Theorem 1. 1 or 1.6.) Next , for the necessity, suppose that (Dx / Dy)( t ， 俨)

= 0 for some t

ε(0 ，∞)

= 1 for all νε Jæn ， we can get a unique C 1 -so1ution of (2.5)-(2.6) in a neighborho6d of {t = O}. Then we see by Theorem 2.1 that this classical solution blows up at time t O 乞f inf{t > 0 (Dx/ Dy)(t ， 俨) = O}. (Notice O that 0 < t 手 t.) 口 and yOε Jæn. As (Dx/Dy)(O , y)

Next we consider the Cauchy problem (2.13)-(2.14) for general partial differential equations of first order. We assume the following hypotheses: (A .I)' For

α叼 νε Jæ飞 the Cαuchν problem

(2.15)-(2.16) has

alwaνsα unique

global

C 1 -s01ution x = x(t ， ν) ， v = v(t ， ν) ， p=p(t ， ν) on the half space {t 主 O}. (A.II)' On the solution curves of (2.15)-(2.16) , it holds that

|去 (t， x(t , y) , v(t , y) , for

a叼 t

E [0 , T ], y

ε Jæn

and i

1, 2 , ..., n

叫例 如 constant

M depends only

on T.

For any t H巾).

~

0, we define a C 1 -mapping H t from Jæn to Jæn by x

= x(t , y)

def

~

Then Conditions (A .I)' and (A.II)' guarantee that the mapping H t is proper.

Therefore , Lemma 2.5 is also true for this H t . Theorem 2.7.

Under the

αssumptions

(2.13)-(2.14) has uniq包ely a global

(A.I)' and (A.II)' , the

C 2 -solution

C.αuchy

problem

on the domain D 哲 {t ~ 0 , x 巳Jæn}

~2.3.

GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS

21

if and only if the Jα cobian of the mapping x = Ht (ν) does not vanish for αny t 主 O and y

ζ ]Rn.

Proof. When

(Dx/Dy)(t ， y) 笋 o

the equation x

= Ht (ν) with respect to y for any (t , x) ι D.

for any t

~

0 and y

ζ ]Rn ，

we can uniquely solve Let y

= y(t , x) be the

ínverse functíon. A global C与olutíon of (2.13)-(2.14) ís gíven by u 咐 ， y(t ， x)). The uníqueness ofthís

C 2 -so1utíon

= u(t , x) 彗

follows from Theorem 1.6. (It can

a1so be deduced from the method of proof used ín Theorem 1.4.) The necessíty of the above condítíon comes from Theorem

2.2.

口

Chapter 3 Behavior of Characteristic Curyes and Prolongation of Classical Solutio-ns 33. 1. Introduction This chapter is continued from Theorem 2.2 in 32.2. Let us rewrite the equation which we will again consider here:

去 + f(t , X， u， 去) = 0 in 书> 0, u(O , x)= 4> (x) where f

= f(t , x , u , p)

and 4>

=

on

4> (ν) are of class

respectively. The characteristic equations (2.15)-(2.16). Let x

{t=O , x

(1) (DxjDy)(t O ， 俨 )=0 ，

and

(3.1)

ξ ]R.n} ,

C 2 in

]R.

x ]R. n

(3.2) X ]R.

x ]R. n and ]R. n ,

correspondi吨 to (3.1)-(3.2) 町e

= x(t , y) , v = v(t , y) , and p =

(2.16). We consider the Cauchy problem

x E ]R. n} ,

p(t ， ν)

(3.1)-(3 勾 in

given by

be solutions of (2.15)-

the following situation:

(II)(DxjDy)(t ， yO) 笋 o for

t < t O.

We put x O 乞:f x(tO , yO). Theorem 2.2 says that , when (t , x) goes to (t o, X O) along the curve x = x(t , y勺， one of the second derivatives of the solution u = u(t , x) of (3.1 )-(3.2) tends to infinity. But this does not prevent the existence of C 1 -so1ution in a neighborhood of the point (t O, x O ). Our problem is to see whether or not we can extend the classical solution u

u(t , x) beyond the time t O. On the other

hand , we will show later that , if the characteristic curves meet in a neighborhood of (t O ， 的， then the Cauchy problem (3.1)-(3.2) cannot admit a classical solution there. Therefore , it is necessary for us to consider whether or not the characteristic curves meet in a neighborhood of (t O , X O ) , i.e. , whether or not there exist two points

yl and y2

(yl 乒 y2)

satisfying x(t , yI) =

x(t ， 的)

for some t. In 33.2 , we will give two

examples in which characteristic curves do not meet though the Jacobian vanishes. In 33.3 , we wi1l consider the case where we can extend classical solutions of (3.1) (3.2) beyond a point where the Jacobian vanishes. In 33 .4 and 33.5 , we wiU give

~3.2.

EXAMPLES

23

sufficient conditions so that the characteristic curves meet in a neighborhood of the point (t口， X O ).

~3.2.

Examples

Example 1. We consider the Cauchy problem for a quasi-linear partial differential equation as follows:

( 2仨卜卜←忖叫叫J( α叫伊(呻巾 川 t， u ux

u(O , x)=x where a(t , u) ~rα'(t)e-tu

on

(3.3)

{t=O , x

+ ß'(t) ε -3t u 3

ε ]Rl },

and the two functions α=α(叶， β = ß(t)

satisfy the following conditions:

ß = ß(t) are in C1 ( 1R1 ). 三 o for each t ， α(0) = 1, andα( t) = 0 for all 主 o for each t , ß(O) = O. + β (t) 并 o for all t ε 1R 1

1)α=α (t) and 2)α (t)

3)

ß(t)

4)α (t)

t 三 K

= constant

> O.

Then the characteristic curves for (3.3) are written by

x = x(t , y) = α ( t) ν + ß(t)y3

and

v = v(t ， ν) = ety ,

(3 叫

from which we easily see that 吗

瓦 (t ， y) =α (t)+3ß(t)y~ ，

and

Dx

一 (t ， O)=O forall

Dy

t?K.

But we can also see from (3 .4) that the characteristic curves x = x(t , y) do not meet for all t 三 O. In this case the solution u = u(t , x) is represented as

u(t , x) = ß(t)-1/3 et x 1/ 3 for

t 主 K.

This representation says that the solution contains algebraic singularity at x = 0 , and that the singularity of shock type does not appe缸 though the Jacobian vanishes.

3. PROLONGATION OF CLASSICAL SOLUTIONS

24

Example 2. We consider the following Cauchy problem:

(去忏叶= 0 in 书> 0, x E R 咐叫 =jz2on

1

}

(3.5)

{t=0, zd1} ,

where

f贝(阳川川 ， 叫p) 彗 ;←护扣扒 α旷叽印，气勺(t)扩 ♂叶 f γ泸p2+叶咐呐 t与 肉州，气勺 βr 例 (tt叶) 产 and the functions

α=α(吟， β=β (t)

are the same functions that we introduced

in Example 1. This example is not of quasi-linear type , and it satisfies Conditions (A .I)' and (A.II)' given in

~2.3.

The characteristic curves for (3.5) are written as

x= 巾) =州 +4β(旷 and u=ty(M)=jα(t)e t y2 + 切(呐 Therefore the J acobian (Dx / Dy)( t , y) = α(t) + 12ß(t)y2 vanishes on L 哇f {(t ， ν)

t ;:: K and y = O}. But x = of y = 0 for each

t 主 K.

x(t ， ν)

is a bijective mapping defi.ned in a neighborhood

In a neighborhood of L , the solution u = u(t , x) is written

by

u(t , x) = const.β (t)-1/3 e t X 4 / 3

for

t 主 K.

This says that the solution u 二 u(t ， x) is of class C I, but not of class C 2, in a neighborhood of L.

~3.3.

Prolongation of classical solutions

As we have shown in

~3.2 ，

there exists the case where the characteristic curves do

not meet in a neighborhood of points where the Jacobian vanishes. In this case we can uniquely extend classical solutions even if the Jacobian may vanish. This is the problem which we would like to prove in this section. Now let us make clear the situation under which we consider the Cauchy problem (3.1)-(3.2). We always assume Condition (A .I)' (Chapter 2) which assures the global existence of characteristic curves. Let x

x (t , y) ,

v

υ (t ， y)

and p = p(t , y) be

the solutions of (2.15)-(2.16 ), and define a mapping H from Rn+l to Rn+1 by

H(t ， 的乞f (t , x(t ， ν)). Suppose: (1)

(DxjDy)(t O ， 俨) = 0 ,

33.3. PROLONGATION OF CLASSICAL SOLUTIONS

25

皿d

(II) (Dx/ Dy)(t ， 俨)手 o for t

< t O.

In trus section we consider the case where the characteristic curves do not meet. Therefore , furthermore , we assume the following condition:

(B) The mapping H is bijective (tO ， X O ) ωhere x O~f x(tO , yO).

from α neighborhood

of

(t O ， 俨)

to

αnother

one of

By Condition (B) , we can uniquely solve the equation x = x(t , y) with respect to y , and denote it by y = ν (t ， x) (for (t , x) in a neighborhood of (沪 ， x勺， (t , y) in a neighborhood of (t O ， 俨)). The function y = 仰， x) is obviou句 continuous ， though it may not be differentiable. Then we get the following. Theorem 3. 1. Under the hypothesis (A .I)飞 suppose (I)-(II)α nd (B). Then the

solution u = u( t , x) 乞f v(t , y(t , x)) remains a C 1 -s01ution of(3.1) in α neighborhood of (t O, x勺 ， though it is not of class C 2 • Proof. Let V and U be open neighborhoods of (to, yO) a叫(tO ， xO) , respectively, such that the mapping H is bijective from V to U. Consider S ~f {(t ， ν)ε V

(Dx/ Dy)(t ， ν) = O} and H(S) = {H(t , y) (t ， ν)ε S}. By Sard's theorem , the Lebesgue measure of H(S) is zero. Therefore U\ H(S) is dense in U. We see that u = u( t , x) is of class C 2 in the domain U \ H(S). The reason is as follows. For any (t ， x) ε U \ H( 町， there exists uniquely a point (t , y) ε V satisfying

(Dx/ Dy) (i, y) 并 o and

x= x(i , y). The i盯erse functionν = y(t , x) is of class C

1

in a neighborhood of (t , x) wruch is contained in U \ H(S). Here we recall Lemma 1. 2 and Corollary 1. 3 in ~1. 1 ， and we see that u 二 u(t ， x) is a function of class

C 2 satisfyi吨 the equation (3.1) in the neighborhood of (t , x). Next we show that

u(t , x) is continuously differentiable in the domain U. We pick up arbitrarily ~O _n a point (t , i) in H(S). Then we can choose a sequence {( t m , xm)}m of points in u

=

~O

n

U\ H(S) such that , when m tends toinfinity, (tm , xm) is co盯ergent to (t ， 王). As the mapping H is bijective from V to U , there exists a unique point (t m satisfying H (t m ， νm)

= (tm , xm) for each m. Since y

= ν (t ，

, y Tn)

εV\s

x) is continuous in U ,

3. PROLONGATION OF CLASSICAL SOLUTIONS

26

= y(tm , Xm)

it follows that ym

is convergent to

~O def

y

~.

infinity. As u = u( t , x) is continuously differentiable at

~O ~O

y(t , x-) when m tends to (t飞 xm) ，

we have by (1. 13)

p( Because p

= p( t ， ν)

is continuous on the whole half space {t 主 0 ， y ε IRn} ， we get

h 生 (t m ， xm) = ......c白 ux

The derivative

θu/θx

of u

a way that (θu/θx) (t , x) C1

= u(t , x)

liIIl p(川m)

m......o。

~O ~。

= p( t

， ν)

can thereby be continuously extended (in such

= p( t ， ν (t ， x))) over U.

That is to say, u

= u (t , x) is of class

in the domain U. Moreover , notice that the uniqueness of the above (extended)

C 1 -s01ution

is assured by Theorem 1. 6.

口

Remark. In Theorem 3.1 , the assumption (A .I)' is a crucial point. It will be proved in Chapter 5 that , for quasi-linear equations of first-order , the assumption (A .I)'

i日

not compatible with the property that the Jacobian vanishes somewhere.

33 .4. Suflìcient conditions for collision of characteristic curves 1 For quasi-linear equations

of 岳rst-order，

Theorem 2.1 says that , if the Jacobian

vanishes somewhere , the classical solutions blow up there. Therefore we cannot extend the classical solutions beyond the time t O when the Jacobian vanishes. This obliges us to treat weak solutions for t

>

t O • The typical singularity of weak

solutions is "shock." As it wi11 be shown in Chapter 5, the shock appears by the collision of characteristic curves. Therefore we try here to give sufficient conditions so that the characteristic

cu凹es

meet after the Jacobian vanishes.

In this section we consider quasi-linear equations of first-order in one space dimension as follows:

主 +αl(t， x， U)去 =α巾 ， u) in 川， u(O , x) = ~(x)

0 , 1) and ~ respectively. The characteristic curves for where ai

αi(t ， x , u) (i

on

x E IR 1 } ,

{t = 0, x εIR 1 } ，

(~.6) (3.7)

~(y) are of class C 1 in IR3 and IR\ (3.6)-(3.7) 缸e defined 田 solution

curves

~3 .4.

of

(2.7)-(2.8)

for n

COLLISION OF CHARACTERISTIC CURVES 1

=

2, we

1. Here , as in Chapter

which assures the global existence of characteristic v= υ (t ， y)

be the solutions of

(2.7)-(2.8) for

Theorem 3.2. Under Condition

=

n

27

again assume Condition (A .I) cu凹es.

x(t , y) and

Let x

1.

(A .I)， ωEαssume:

吟川=斗0， α7叫 (11咋 I l厅f 仰 ( θα叫dθu叫)(t俨O ， 泸 xO ， 俨 v 0) 手 0 ω 叫her何e ♂ x o
= x(t , y) with

respect to y is violated. First we prove that (θxjθy)( t , yO) is negative for t

> tO

(2.7) with respect to y , we get the system

with t-t O small. Actually, differentiating

of ordinary differential equations (2.10) for n

=

1. Then (2.10) is linear with respect

to 8xj句 andθ旷句. As the initial data are not zero , we get θzθu

(一 (t ， y) , :" (t , y)) 并 (0 ， 0) θY \ """ ， θy\"'"'' T \~，

for any

t 川， yE Jæ1 .

Hence we have (θυ/θy)( t O ， 俨)并 O. By the assumption , we get d / θz\l 月a， 一(一)|=」 (t勺 O ， V O ) 一(t0 ， yO) 笋 O dt \δν J I(t ，y)=(沪 ， yO)

Moreover , as (δxj句 )(t O ，俨) d

= 0 and

/ θz 飞

(

~..

)1

dt 飞 θν J

θu

l(t ,y)=(tO ,yO)

Therefore we get (θxjθν)( t ， ν。)

(θxj句) (t , yO)

剑

.....:~，


0 for t < t O, we get

d / θz\|

~(~..

11

dt 飞 θy J I(t ， y)=(沪，苗。)

< O.

> t O with t - t O small.

That is to say,

(θxj句) (t ， 俨) changes its sign at t = t O. We now define a function h = h(y) by

setting h(ν) 哲 inf{t

θxj句 )(t ， y) =

Cl in a neighborhood of y

(δ2xj街句)(t0 ，俨)

< 0, we

O}

for each y. Then h

= h(y)

俨. In f:缸t ， as we have (θxj句 )(t O ， 俨)

can uniquely solve the equation (θxj句 )(t ， y)

is of class

= 0 and = 0 with

respect to y in a neighborhood of (t o, yO). Let us restrict our following discussions into a small neighborhood of (t O ， 沪，俨) only. We see by the same reasoning as in the above that , as a function oft , (θxj句 )(t ， y) changes its sign at t

= h(y)

for each

y.

Next , we will show that x a point y

并 y

o

. If h(y)

= x(t , y)

is not monotone with respect to y. Pick up

< t U , then we get

字 (t， y) < 0 and 生 (t， yO) > 0 8y θy

3. PROLONGATION OF CLASSICAL SOLUTIONS

28

for t ε (h( 剖 ， t O ). As trus means that x

=

x(t ， νis not monotone with respect

to y , the characteristic curves meet in a neighborhood of (t O , X O ). When h = 州的 沪， it follows that (θx/句 )(t O ， y) 三 o in

is constant in a neighborhood of y a neighborhood of y

俨

i.e. ，

x(tO , y) is constant there. This says that

x

the characteristic curves meet at the point (沪 ， X O ). When there exists a point

y=

Y such that h(y)

> 沪， we can similarly see that x

= x(t , y)

is not monotone

with respect to y in a neighborhood of y = 俨 for each t ε (t O ， h( y) ). Hence the characteristic curves x = x(t , y) meet in a neighborhood ofthe point (tO ， x O ).

口

33.5. Sufficient conditions for collision of characteristic curves 11 In this section we consider the same problem as in equations of first-order in one space

dimension 出 follows:

些主 +f(t， x， u 旦旦 )=0 θr

J \ '"

....."

'"""θz

u(O ， x) =φ(x)

where f

= f(t , x , u , p)

33 .4 for general partial differentia1

on

in

{t>O， 川剧，

(3.8)

{t = 0 , x ε Iæn ，

(3.9)

and O ~fυ (tO ， yO) and pO p(t O ， υ勺 ， then the characteristic c包rves meet in α neighborhood 0/ (t O, XO). Proof.

V

As in the proof of Theorem 3.2 , we will show that x

monotone with respect to y. We repeat the same discussions in

x(t ， νis

31.1.

def

not

When the

Jacobian (θx/句 )(t ， y) 笋 0 ， we can solve the equation x = x(t , y) with respect to

53.5. COLLISION OF CHARACTERISTIC CURVES 11

y , and denote the solution by y

= ν (t ，

29

x). Then the solution of (3.8)-(3.9) is given

by u(t ， x) 乞f u(tJ(t , z))aad

叫树)=去 (tJ)=2川峰0 =

芋(t， y(阳呐)). (~宇己(t川 t ， ω) y)川)-1 dy

飞 dy'

-, I

百 =y(t ，:z:)

When a point (t , x) goes to (t o, x O) along the curve ((t , x) x = x(t , yO)} , 但'x/句 )(t ， y(t ， x)) tends to zero and p(t， ν (t ， x)) remains bounded. Therefore it must be true that (θυ/θy)(tO ， yO) = O. Differentiating the system (2.15) with respect to y , we get a system of linear ordinary differential equations for ðx /δy ， θυ/θν andθ'p/句 just

like (2.10). As the initial data (θx/θy， θ。/θ'y ， ðp/δy)(O，

y) = (1 ，4>'(ν) ，4>"(ν))

are not zero , it follows that (δ'x/θν ， θ旷θy ， θIp/θν)( t , y) 并 (0 ， 0 ， 0) for any t 2:

o.

Hence (θ'p/θν )(tO ， yO) 并 O. (θx/句) (t , y)

We can see that

satisfies the following differential equation:

d ( θz 飞伊 1

ðx . δ21 θυIθ21 θp

一.一.一一一一二­

dt 飞 θyJ

δzθpθν 『 δuδpθ'y

，

θ'p2 θν-

U sing the assumptions and the above results , we get d ( θz 飞 |δ21 1 , 0U

I :,-

J 1.

dt 飞 θy J l(t ，y)=(tO ，yO)

Moreover , as (θx/句 )(tO ， yO)

0

0

0\

ðp I , 0

0

一τ (t ， XU, VU , pU) 一 (tU ， yU) 并 O. δpθν

= 0 and

(θx/句 )(t ， 俨)

> 0 for t < t O, it

must be true

that d f δz 、 一{一)

1. , ,. ., ~ 0 , i.e

dt 飞 θν/I(t ，y)百 (tO ，y勺..~.，

d

θx ， 1

一(一)1.， dt δ'y

'1 (t ，y)=(沪，宙。)

< O.

= t O. Here we introduce a function h = h(y) , just as in the proof of Theorem 3.2 , by setting h(的管 inf{t : (θ'x/句)(t ， y) = Hence (θx/句 )(t ， 俨) changes its sign at t

O} for each y. We consider the problem in a small neighborhood of 仰，沪，沪 ， pO)

δ2//θ'p2 乒 O. We pick up a point y (y 并 J)·By the same reasoning as in the above , we see that (δx/θy)( t , y) changes its sign at t = h(y). If h(y) > t O, we where

have

生 (t， y)>O and 生 (t， yO) t O with t - t Osmall). By the assumption (4.5) , we treat here the

case where

θ2 f

f10

0

0

0

苟言 (tU ， XU ， VU ， pU)

> O.

(4.9)

Then we get the following. Lemma 4.3. i) For αny t

> t O with t - t O

smαII

and for

αny

x

ε (Xl(t) ， X2(t)) ， 切t

hαve

Ul(t , X) -U2(t , X) < 0αnd U2(t , X) - U3(t , X) > O. ii) There exists uniquely x

Ul (t ， γ (t)) =

= γ (t) ε (Xl(吟 ， X2(t)) sα tisfyin

U3(t ， γ (t)).

Proof. i) By (4.8) and (4.9) , we have (θIp/θν )(t O ， 的< 0, i.e. , (θ'p/句 )(t ， y) p(t , g2(t , X)) > p(t , g3(t , X)). Using these inequalities , we get

￡川 x) 一州

(4.10)

4. EQUATIONS OF HAMILTON-JACOBI TYPE

38

with

(咐，仆地(机))|Fh(t)=0 < 0 for 川队 (t) ， X2(t)). In the we obtain the inequality U2(t , X) - U3(t , X) > 0 for X 巳 (Xl(t) ， X2(t)]. Hence it holds that Ul(t , X) - U2(t , X)

same w町，

ii) Using the results of i , we have

￡MZ)- 州 with

(咐， x) 一句 (t， x))1叫。=(咐， x) 一句(机))1一例 >0 and

(Ul(t , x)

- U3(t , x))1

,.

I "'="'1 (t)

Therefore there exists uniquely

= (Ul(t , x) - U2(t , x))1

" I",=",t{ t)

γ (t) ε (Xl (t) , X2( t))

x

< O.

satisfying Ul (t ， γ (t))

U3(t ， γ(t)).

= 口

We see by Lemma 1.2 that the three functions

U

= Ui (t , x)

(i

= 1, 2, 3) satisfy the

> tO(t-t Osmall) and X2(t) 主 Z 主 Xl(t). As single-valued and continuous solution , we extend the solution

equation (4.3) in their domain where t we are looking for a U

=

u(t , x) into the above domain by defining if x 三 γ (t ), if x> γ (t).

(4.11)

This extended solution is obviously Lipschitz continuous. It satisfies the equation

(4.3) except on the curve {( t , x)

t 主 tO ，

x

= γ (t)}. The next problem is to prove

the uniqueness of generalized solutions. This is the subject of the following section.

34.3. Semi-concavity of generalized solutions First we give an example of non-uniqueness of Lipschitz continuous solutions which is due to Y. Tomita (see also Remark 2 of Theorem 11. 5 in Chapter 11). Example. Consider the Cauchy problem:

(生+(去)二。 ln … E ~1} U(O , x)

=

0

on

{t = 0 , x E

~1}.

~4.3.

SEMI-CONCAVITY OF GENERALIZED SOLUTIONS

Then it has a trivial solution u =

def

UO(t ， x)~'

39

0 , and also

u= 咐， x) 纠。

0:::: t 三

if

l -t + IXI

if t>

Ixl ，

IXI.

The above example shows the non-uniqueness of generalized solutions in the space of Lipschitz continuous functions. Therefore we must introduce a supplementary condition on generalized solutions which guarantees the uniqueness of solutions.

If f

f( t , x , u , p) is convex with respect to p , the supplementary condition would

=

be the semÏ-concavity condition (4.2). When f

=

f(t ， x ， 矶 p)

is concave , then it

becomes the semi-convexity condition as fo l1ows:

u = u( t , x) is said to be semÏ-convex if it satisfies the inequality u(t , x + ν) for 缸ly X

+ u(t , x -

y) - 2u(t , x) ~ KI ν1 2

(4.12)

, Y and t where K is a constant locally independent of x , y and t.

The supplementary condition (4.2) [or (4.12)] is veηimporlant ， because it is related to the stability of solutions of the Cauchy problem (4.3)-(4 .4). What we would like to claim here is that the solution

constn川ed

by (4.11) in 34.2 satis-

fies automatically the above supplementary condition; that is to say, we get the fo l1owing: θ2 f 1.0 _0 _.0 _0θ2f Theorem 4 .4. Assume 一一 (tU ， xU ， VU ， pU) > 0 (resp 一τ (tO ， XO ， VO ， pO) θIp2'"' ,..., ,..... 'r 1"" ..... '...........r. θp

Then the CωO η 俐 ve口z 功 叫)

in

solt出on

u

u(t , x) extended by (4.11) is


0 (y

is small).

Then

u(t , x ={u(t ， x+ ν)

+ ν)

- u(t , x

+ u(t , x -

y) - 2u(t , x)

+ O)} 一 {u(t ， x -

0) - u(t , x - y)}

={去(机+句)一去(川的)叫卖(仙一 0) 一去(t， x-øy)}ν +{去 (t， x+O)- 去(机一 0) }y , where 0

< (}, (} < 1.

4. EQUATIONS OF HAMILTON-JACOBI TYPE

40

To estimate the first and second terms , we must calculate the second derivative of u = u(t , x):

jL(tJ)= ￡(叩斗(机)=￡POJ(tl))

=(去(t， y)/::(t， y))ly=y(t，，，) By (4.8)-(4. 时，但p/ 句 )(t ， ν) is negative in a neighborhood of (t O, yO). From one of the assumptions , it follows that 伽/句)(tO ， yO) = 0 and (θx/句)( t , yO) > 0 for t < t O. Hence , when (t , x) goes to (t O, x O) along the curve x = x( t , Y勺， (θ2U/θx 2 )(t ， x) tends to 一∞， i.e. ， θ2U/θX 2 )( t , x) is bounded from above in a neighborhood of (tO , X O). By (4.10) and the definition of u(t ， 叫， the third term is estimated as follows:

ZM)+0)ZM)-0)zpMM-PM 川

ti}(i 二 1 ， 2) be si吨ula副es of the solution u 口 u(t ， x).

Assume that W 1 and W 2 meet at a time t

= T;

that is to say, let γ1 (t) < γ2(t) (t

T) andγl(T) = γ2(T). By (4.10) and (4.13) ，阴阳 that θu ZMt)-0)> 轨 γl(t) + 0) and 一川t) θz

for t

< T.

Taking the limit as t

•

0) >

~ - (t, , (川) 2

T - 0 in the above inequalities , we get

θuθu 2川(川 T盯)一0创) >坐川川阳+刊0)仨=一 (σ川 T， γ% 划甘响 '22纠J 川 川 (σ T盯)-0创) >一川阳0创)归 θz'θz

As f = f(t , x , u , p) is convex with respect to p, we have by (4.14) dγ

d"Y

云 (T)>Ef(T) ，

ie ， γ1 (t) > γ2(t) for t > T


T

x l nD

E

tJ np 『J

马(t)~---

O

Y

Figure 4.2 First , see Figure 4.2 which expresses the behavior ofthe curve x =

x(t ， ν)

for t

>

T. We use the notations indicated in Figure 4.2. Each part AiBi (i = 1, 2 , 3) ofthe curve x = x(t , y) can be

u叫uely

solved with respect to y , and we write y = gi(t , X)

def

(i = 1, 2 , 3). Put Ui(t ， X)~' V(t , gi(t , X)) (i = 1, 2 , 3). As γ1 (t) > γ2(t) for t > T , the solution

U

= u(t , x)defined by (4.11) takes two values for x

εb2(t) ， γl(t)J.

define I(t , x) ~ Ul(t ， 叫一句 (t， x). By Lemma 4.3 , we get I(t ， γ2(t))

I(t ， γl(t))

= Ul(t , ì2(t)) -

U3(t门2 (t))

=Ul(t ， γ'2(t))

U2(t ， γ2(t))

-

< Ul(t , ìl(t)) - U2(t , ìl(t)) = 0 ,

= Ul(t ， γl(t)) - U3(t ， γ1 (t))

= U2(t ， γ1 (t)) -

U3(t门l(t)) > U2(t ， γ2(t))

-

U3(t ， γ2(t)) = O.

Moreover , it fol1ows from (4.10) that δI

否E(t ， z)=p(t， g10， z))-p(t ， gs(t ， z))>0.

We

~4 .4.

COLLISION OF SINGULARITIES

Hence there exists uniquely x

43

γ (t) ε[γ2(t) ， γ1 (t)] satisfyi吨 I(t ， γ (t))

= o. As

we are looking for a continuous and single-valued solution , we define the solution

u = u(t, x) for x

ε[γ2 (t) ， γl(t)] as follows: -O

(( )) ZZ uu if b ，，

唱A

，?心

rEE、 eEE 、

Z

、‘，，，，

ATtu

一一

，，.‘、

u

if

x 主 γ (t) ，

x < γ (t).

u(t , x) is Lipschitz continuous and semi concave. Taking the derivative of I(t ， γ (t)) = 0 with r臼pect to t , we also get the Then we can similarly show that u

same jump condition (4.14). The collision of a finite number of singularities can be treated by the above method. For example , let {x

= ìi(t) , t > td

(i

= 1, 2,... , k)

be singularities of

the solution u = u(t , x). Suppose (1) 悦。) 但 dt (T) \ - > …>但 dt (T). l'

l'

...,

Therefore it follows that γ1 (t) > γ2(t)>...> γk(t)

for

t > T.

Next , we can prove that there exists uniquely x = γ (t) εbk(吟， γ1 (t) 1satisfying Ul(t ， γ(t))

=

Uk(t ， γ (t)).

A new generalized solution may be defined in the same

way as before. Except the case where an infinite number of singul町ities collide simultaneously, we can repeat the above discussions forever. 1f (δ211δIp2)(t ， x , u , p) changes its sign, the situation may generally become complicated. The phenomena are locally almost the same as the above , but the global behavior of singularities is generally quite different. We wi11 consider this subject in Chapter 7.

Chapter 5 Quasi-linear Partial Differential Equations of First-Order 95. 1. Introduction and problems In this chapter we consider the Cauchy problem for quasi-linear equations of firstorder in one space dimension as follows:

2+忖α向1 毗 叫(O ， x) u

= cþ(x)

on

{杠t

= 0, x

ε Jæl} ，

(5.2)

where 向 =αi(t ， x ， U) (i = 0, 1) and cþ = cþ(y) are of class C∞ in Jæ l

X

Jæ l

X

Jæl

and Jæl , respectively. Our principal interest is on the construction of singularities of weak solutions of (5.1). Therefore we put a little strong assumptions on the regulari ty of 向 =αi(t ， x ， U) (i = 0 , 1) and cþ = cþ(ν). On the other hand , there 缸e

many works on the global existence and uniqueness of weak solutions for quasi-

linear equations of first-order , especially for equations of the conservation law. For example , refer to P.D. Lax [98], J. Smoller [121] , A. Majda [104] , A. Jeffrey [77]. Though the difference method is very important from the point of view of numerical analysis , the best result on the global existence of weak solutions was obtained by the vanishing viscosity method (see O.A. Oleinik [112 ], S.N. Kruzhkov [88] , [92]). By the same reason stated in Chapter 4 , it seems to us that it would be

di缸cult

to get the informations on the singularities of solutions by the vanishing viscosity method. Therefore , in this chapter as we l1, we construct the singularities , especially of "shock" type , by the analysis of characteristic curves. The characteristic curves for (5.1)-(5.2) are the solution curves x

v = v(t , y) of (2.7)-(2.8) for

n 且1.

= x(t , y)

and

Here , we assume (A .I) (stated in 92.2) , which

assures the global existence of characteristic curves. We have seen by Theorem 2.1 that , if the Jacobian of the mappi吨 x = x ( t , y) vanishes at a point (t o , y勺， then the classical solutions blow up at a point (沪，的， where X O 哲 x(tO ， yO). Therefore we

~5. 1.

45

INTRODUCTION AND PROBLEMS

must introduce weak solutions for equation (5.1). To define it , we rewrite equation (5.1) 出 follows:

θuθ

一 + ~f(t ， x ， u) = g(巾 ， u)

(5.1')

θtθz

where (θf/θu)(t ， x ， u) = α1 (t ， 吼叫 and g(t ， x ， u) 一 (θf/θx)(t ， x ， u) = α。 (t ， x ， u). If g(t ， 吼叫三 0 ，

Equation (5.1') is of conservation law. Let

ω =

w(t , x) be locally

integrable in lR.2. The function ω=ω (t ， x) is called a weαk solution of (5.1 )-( 5功 if, for any k = k(t , x) in CO" (JR勺， it holds that

fω|θkθk 叫一 + f(川)一

__1 dtdx + 户(x )k(O , x )dx = 0, + g(川kl

|挠'加|

~

(5.3)

~

where JR~ 乞f{(t， z):t >0 ,

Z

εJRl}. Let ω =

w ( t , x) be a piecewise smooth

weak solution of (5.1) which has jump discontinuity along a curve x

= γ (t).

Then ,

by the definition of weak solutions , we get the following Rankine-Hugoniot jump

U :l:

问u h-

where

I'一 d

-a '归

condition:

I'd-u

(5 .4)

(t) = 吨， γ(t) 土 0) 誓 :EU(的(批 ε)

From now on , we will consider the Cauchy problem (5.1)-(5.2) in a neighborhood of (t o, x O) (x O = x(tO , yO)) under the following two assumptions:

(1) (Dx/Dy)(t O ， 俨)

=

O.

(11) (Dx/Dy)(t ， ν) 手 o for t

< t O and y

εIωhere 1

is an 叩en neighbourhood of

y= νo

Our problem is to see what kinds of phenomena may appe町 for t

>

t O• As

Example 1 in 33.2 shows , we have to consider two cases as follows:

(I1I) Though the Jαcobian of x

x(t ， ν) vanishes at α point (t O ， ν。)， the chα 旷r、-

创 α cter 付i臼 巾sti 时化 C CU1 旷 阿 r、

XO ~f x(tO , yO).

(IV) For t > t O, 叫 ere (Dx/Dy)(tO , yO) = 0, the chαracteristic CU1 旷附r. meet in α neighborhood of (沪 ， x O).

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

46

In the case (I II) , we can uniquely solve the equation x = x( t , y) with respect to

y and denote it by y = ν (t ， x) which becomes a continuous function. Put u(t , x) ~f v(t , y(t , x)). Then Theorem 2.1 means that u

=

u(t , x) is not a classical solution of

(5.1) , however , we can see that it is a continuous weak solution. The proof follows. The continuity of u = u( t , x) comes from the continuity of v

y = y(t , x). Put S ~f {(t , y)

= υ (t ，

is 出

y) and

(Dx/Dy)(t , y) = O} and H(S) 哇 {(t ， x)

x =

x (t , y) for (t ， υ)ε S}. Then , by Sard's theorem , the Lebesgue measure of H(S)

is zero. As a corollary of Theorem 2.1 , we equation (5.1) outside H(S). Hence u

ca且 see

= u(t , x)

that u = u(t , x)

satis丑es

the

is a continuous weak solution of

(5.1)-(5.2). The important problem remaining is to characteristic curves meet , that is to

s叮，

co日sider

x(t , yd

=

the case (IV). Suppose that the

x(t , y2) where

yl 并 ν2.

By the

uniqueness of solutions of (2.7)-(2.8) , we have (x( t , y仆， υ ( t ， 的))并 (x(t ， y2) ， υ (t ， 的)) . As x(t ， 的) = X(t , Y2) , it follows v(t ， yd 并咐， y2). This means that we will not be able to get continuous weak solutions. Therefore we willlook for piecewise smooth def

weak solutions. As we will see a little later , the solution u = u(t , x) ~υ( t , Y(t , x)) in the case (IV) takes several values after the Jacobian vanishes. As we are

looki吨 for

a single-valued solution , our problem is how to choose one appropriate value from these so that the solution becomes a may jump from a branch of

si吨le-valued

solu川川 tio ∞ I丑1 tωoa 皿 I且10 时the 盯r

weak solution of (5.1). If

0丑

one , the jump discontinuity must

satisfy Rankine-Hugoniot's equation (5 .4). Solving the Cauchy problem for (5 .4), we can get a curve of jump discontinuity. J. Guckenheimer [59] and G. Jennings [78] took this approach. But they forgot to pay attention on the uniqueness of solutions for the Cauchy problem to (5 .4). In fact , the right-hand side of (5 .4) is not Lipschitz continuous at an initial point (see Lemma 5.3). The aim of 35.3 is to prove the

of solutions for the Cauchy problem to (5 .4)

u叫ueness

The generic property of singularities was studied in D.G. Schaeffer [120] and applyi吨 Thorr山 "Catastrophe

T. Debeneix [41] by

theory." The construction of

singularities in two space dimensions for Hamilton-Jacobi equations was studied by M.

Ts毗 [137].

By a similar method , S.

Naka时 [109]-[110]

singularities of shock type in several space dimensions.

constructed the

But , as he treated the

singularities of fold and cusp types only, his results are essentially those in the case of two space di

~5.2.

EQUATIONS OF HAMILTON-JACOBI TYPE AND CONSERVATION LAW

47

example , [74]-[76]) give the generic classifications for the bifurcations of singularities of geometric solutions.

!ì 5.2.

Difference between equations of the conservation law and equations of Hamilton-Jacobi type In this section we wí11 give some property of equation (5.1). This will characterize a difference between Equation (5.1) and equations satisfying (A .I)\ A difference between characteristic Equations (2.7) and (2.15) is that (2.7) does not contain an equation concerning p = p(t , y). Let x = x(t , y) and v = v(t , y) be the solutions of (2.7)-(2.8) for n (δuj 8x )(t , x(t , y)) ,

1.

As p

p(t , y) is corresponding to

an ordinary differential equation 岛r p

= p(t , y) is written as

follows:

dp -r dt

8al 'l c 8ao 8α1 一」 (t ， z ， u)pz+{ 」 (t ， z ， υ) 一一位， υ )}p+ ~，，-U(t ， 川 1.

\

I •

θuθU

\-1 -

1 - /

8x

θz

p(O) = 矿 (y).

(5.6)

The equation (5.5) corresponds to the last one of (2.15). see by Corollary 1. 3 that , if u

(5.5)

More concretely, we

u(t , x) is of class C 2 , p

p(t , y) is equal to

(θujθx)(t ， y(t ， x)).

for t < t O. Then the solution p = p(t ， 俨 ) of (5.5)-(5.6) tends to infinity when t goes to t O- o.

Proposition 5. 1. Assume (δxjθy)( t O ， 俨)

= 0 and

( δxjδ的 (t ， 俨)并 o

Proof. Remember thatθxjθyand θ旷句 satisfy the system of ordinary differential 1.

equations (2.10) for n

Si且ce

this system (2.10) is linear with respect to

{δxjθy， θ旷θy} ， we get (θxjθy ， θ旷 θν) 并 (0 ， 0) for any (θxjδy)(tO ， yO)

= 0 , then

(t , y) εIR 2 . Therefore , if

(θυ/θν )(t O ， 俨)手 O. Moreover , we have by Lemma 1. 2

p(t , y) 生 (t， y) θu

= 些( t , y)

for all (t , y)

Hence we can easily get this proposition.

ε Il~.2 . 口

This proposition means that , if the J acobian vanishes somewhere , (A .I)' cannot be satisfied for quasi-linear equations

of 岛的 -order.

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

48

35.3. Construction of singularities of weak solutions In this section we will consider the case (IV) and construct the singularities of weak solutions which are called "shock" or "shock wave." A sufficient condition which guarante臼 the

situation (IV) is

37(t川 V O ) 并 0，

(5.7)

where v O 哇f v(t O ， 俨). This is the result of Theorem 3.2. If (5.7) is violated , we can construct an example in which characteristic curves do not meet though the Jacobian vanishes (see Example 1 in 33.2). Therefore , the solution of this case remains to be continuous , though it is not differentiable. First we solve the equation x = x( t , y) with respect to νfor t

> t Oin a

neighbor-

hood of (t O ， 的. This procedure is almost the same as in 34.2. By the assumptions

(1) and (11) , we have

(θ2xj句 2)(t O ，

yO) = O. We assume

(θ3 x j句 3)(户，俨)并 O. This

assumption is natural from the generic point of view. In this case also , we get , by the same way as the proof of (4.6) ,

去川 >0 Then the graph of the curve x = x(t , y) for t

(5.8)

> t O is drawn just as in Figure 4.1

(see 34.2). Therefore we use the same notations given in Figure 4. 1. H盯 the functions y = 的 (t) and y = Y2(t) ( ν1 >的) are the solutions of (θxj句 )(t ， y) = 0 and Xi(t) 乞f X(t , Yi(t)) (i

Yi(t) (i Z

1, 2). Then Lemma 4.2 is also true for these ν=

1, 2). When we solve the equation x

ε (Xl (功 ， X2(t)) ， we get three solutions y =

x(t , y) with respect to y for

9i(t , X) (91(t , X) < 92(t , X) < 93(t , X))

and define Ui( t , x) ~f v(t , 9i(t , x)) (i = 1, 2, 3). As we are looking for a single valued solution , we must choose only one

value 仕om

{Ui(t , X)

i = 1, 2 , 3} so that we can

get a weak solution of (5.1). As we have written in 35.1 , we can not get continuous weak solutions under the condition (IV). Therefore we look for a weak solution which is piecewise smooth. If U = u( t , x) is a weak solution of (5.1) which has jump discontinuity along a curve condition

{(t , x , u)

(5叫.

u

x

γ( 抖，

it must satisfy the Rankine-Hugoniot jump

This suggests us that a nice weak solution jumps from a branch

uI( t , x)} to the other ((t , x , u)

u = U3(t , X)} along a curve

35.3. CONSTRUCTION OF SINGULARITIES OF WEAK SOLUTIONS x= γ (t)

on which condition (5 .4) is

satis直ed.

49

Therefore our problem is to solve the

Cauchy problem for Rankine-Hugoniot's equation as follows: (生

f(t ， x 叫 (t ， x)) - f(t ， x 问 (t 川

dt x(tO) = XO.

> t O.

where Xl(t) 三 x(t) 三 X2(t) for t def

r

t > t O,

~' j (t , x) ,

Ul(t , X) - us(t , x)

(5.9)

j(t , x) is obviously O t > t and Xl(t) < x < X2(t)} and

The function j .n

differentiable in a domain U ~' {(t , x) 1.

is continuous on U (the closure of U). As we will show in Lemma 5.3 , it is not Lipschitz continuous at the end point (t O ， 俨). As j

= j(t , x)

is continuous in U , we

can see the existence of solutions by the classical theory. But we can not get the uniqueness of solutions. In J.

G旧ke出eimer

[59] and G. Jennings

[7时，

they did not

pay attention to this point. This is one of our problems which we will consider in this chapter. As we put the hypothesis (5.7) , we assume here more concretely

:?川 ， V O ) > 0 Though j is in

=

C 1 (U).

(5.10)

j(t , x) is not Lipschitz continuous at the point (t o, x勺， j

=

j(t , x)

Therefore , if the solution could enter into the interior of the domain

U , we can easily extend it further. Hence we restrict our discussions in a small

neighborhood of (t O , X O ). Lemma 5.2. i) (θvj句 )(t O ， 俨)

< o.

ii) For (t , x) E U , we get θUl

θU2

Ul(t , X) > U2(t , X) > us(t 'θX ， x) α nd -",-' (t , x) < 0 一 (t ， x) > 0 旦旦 (t ， x) < O. \....,....., ............， θz'θz iii) When (t , x) goes to (tO , XO) in U , then 阴阳/θx)(t ， x) (i

tend to

= 1, 2 , 3)

infiη ìty.

Proof. i) As (θxjθν) (t ， ν。)

d

I θz\

一(一川

> 0 for t < t O and

(θxjθν )(tO ， yO) = 0 , we have

Oa ，

二二土 (t勺。 ， V O ) τ (tO ，

dt 飞 θν) l(t ， y)=(tO ， yO)θx

uy

yO).

Since ((θxjθy)(t ， ν) ， (θvjθy)(t ， 的)并 (0 ， 0) for all (t ， ν) ， i t holds that (θvjθν )(tO ， yO) 笋 o.

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

50

Hence we get i) by (5.10).

9i(t , X) (i 1, 2, 3) , we have 91(t , X) < 92(t , X) < 93(t , X) and Ui(t , X) = V(t , 9i(t , X)) (i = 1, 2, 3). Using the property i , we get the first half of ii. As 91(t , X) < 归 (t) < 92(t , X) < 仇 (t) < 93(t , X) for z ε (Xl(t) ， X2(t)) ， we have ii) By the definition of Y

生 (t， 91(t， X)) > 0

些(t， 92(t， X)) < 0

θν'θν

As (δu;/âx)(t ， x)

，::J ~\U ，..v IJ

..............，

生 (t， 93(t， x)) > o. θν

= (âv/句 )(t ， 9队 x ))/(θx/θν )(t ， 9i(t ， X)) ，

we get the second part

of ii , and also iii.

口

= j(t , x) is continuoωly di.fferentiable in the domain __ def " . U and is continωω on U 叫 ere U~' {(t , x) t > t O , X2(t) > x > Xl(t)}. But it O is not Lipschitz continuous at the point (to , x ). ii) For t > t O, j = j (t , x) is decreasing with respect to x.

Lemma 5.3. i) The function j 一-;-

Proof. The first part of i is obvious. Taking the derivative of j (t , x) with respect to x , we have 1θfθf

:; j(t , x) = 一一一{一 (t ， x , ud 一 1 - U3 θ'x ，-，-，

θz

(5.11)

(t , x , U3)}

十二石[去川1)-t♂川d-f川 r 1δf I~ J âu +二-二|二一-{f(t ， 吼叫一 f(t ， x ， U3)} _ :'J (巾 ， U3) Iτ主 (t， x). Ul - U3

I Ul

- U3

au

I

ax

When (t , x) goes to (tO , x O) in U , the first term of (5.11) is convergent to (伊 f/δzδu) (t o, x O, UO) where U O 乞r u(tO , x O). Therefore it is bounded in a neigh borhood of 川， X O ). When (t , x) (t O, x O) in U , the coeffi. cient of (θUl 月x)(t ， x)

•

tends to (θ2f/θ2U)(t O ， 沪， u O )/2 (δU3/âx)(t ， X)

Lemma 5.2 to

=

(θα1 月u )(t O ，沪， u O )/2

has the same property as

(仇 dâx)(t ， x).

> O. The

coe伍cient of

Here we apply ii and iii of

(5.1 月， 出 t hen

tends to 一∞. This means that j = j(t , x) is not Lipschitz continuous at (tO , x O) , and that it is monotonically decreasing with respect to x in

JJemma 5.4. The functions Xi = Xi(t) properties:

i) 轩仲

def

~. x(t ， 以 t))

U.

(i = 1, 2)

sα tisfy

口

the following

~5.3.

CONSTRUCTION OF SINGULARITIES OF WEAK SOLUTIONS

dx? α叫一三 dt >j(t ， X2(t))

ii) 旦~ dt tO

by

(t , x) =

(Ul(t , X)

if x 主 γ (t) ，

l U3(t , X)

if x> γ (t).

~

(5.12)

Remark. It is impossible to jump from the first branch {u second one {u

U3( t , x)} so

This is corresponding

Ul (t, X)} to the

刷 t hat 创t Ra 础 出山 nl 1垃ki 虫时 n 肘 e-Hu I吨 缸 u19 阴 创m O ∞o

tωo Lem卫皿la

6.2 for Hamilton-Jacobi equations.

35 .4. Entropy condition We first give an example which shows the non-uniqueness of weak solutions of (5.1)(5.2) in the space Lloc (I~.2). Lloc (IR2) is the space of measurable functions which are integrable on any compact set in IR2. Example. Consider the Cauchy problem

(θωθ -+-d=Oin 御街

ω (O ， x)=O

on

{t>0，叫则，

{t=O , x

Then this problem has a trivial solution ω (t ， x)

一-

Z

= 0 and a

01-

、

{(t , x) on {(t , x) on {(t , x) on

E4 唱

iEEtr 、 EEE

r，，‘、

ω

ATb

εIR}.

weak solution as follows:

叫主 t 主 O} ，

t>x>O} , t>-x>O}.

def

We get the above example by putting w(t ， x)~' (仇 jðx)(t ， x) where u = 吨 ， x) is the function given in "Example" of 34.3. As weak solutions of the above example are not unique , we must impose the entropy condition which guarantees the uniqueness ofweak 时utions in the space

Lloc( 1R2). Concerning the entropy condition, we follow

here O.A. Oleinik [112] and P.D. L缸 [9可Consider the equation (5.1') and let u

= u(t , x)

be a weak solution of (5.1')

which has jump discontinuity along the curve x = γ(t). Put U::l: (t) 哲 u(t ， γ(t) 士 0). Then the entropy condition is

expressed 田 follows:

For any value v between u+(t)

and u_(吟， it holds that S[v ， u-l;三 S[u+ ， u-l

where

ef f(υ)

- f(u)

S[v , u] ~υ -u

(5.13)

~5 .4.

ENTROPY CONDITION

53

A jump discontinuity of weak solutions of (5.1) satisfying (5.13) is called a "shock" or "shock

wα ve."

The condition (5.13) is also important from the viewpoint

of stability. For example , B. K. Quinn [115] has proved the following. Theorem 5.6. Assume that g(t , x , u) defined in (5.1') is u(t , x)

αnd

v

υ (t ，

(5. 川 .1'丁) foT' αllx αn 叫 d

x) t

αT'e

>

pzecewzse

identicallν ze T'o. 扩 u

cωOη 时ti饥 ηm ωO包 包 ωs1切引 d 副iffi 万 1e 阿'ff T' η 时且此tiωα b1e 叨tα k s01ωt 包L巾 tio旧

of

0ωωi必th initiωα1 dαtα uo(归 x) αηdυo(x) 1ψvhich αT'e pzece切 ωzse

Cω o时 n ti饥 n1ωωl仿 yd 码i伊 如 ffi 托e 陀 T T'tη tiωα b1扣tαn 叫 d Ll_斗tη 时tε叩 grab1扣e in x , and ifu = u(t ， x) αndv = υ (t ， x)

satisfy the condition (5.13) a10ng

discontin包ity CU T' ves ,

then it h01ds

thαt

Ilu(t) - v(t)llu 三 Iluo - vollu. ConveT'sely， 矿 (5.14)

(5.14)

is t T' ue , then the cond巾on (5.13) is satisfied.

We would like to show that our solution (5.12) is reasonable in the above sense. That is to say: Theorem

5 工 The

solution defined by (5.12) sαtisβes the entropy condition (5.13)

zn α neighboT'hood of the point (t O, x O)

Proof. We consider the case where (5.10) is satisfied. By Lemma 5.2 , we have u 一 (t)

> u+(t).

As the inequality (5.10) means the convexity of

f 口 f(t ， x ， u)

with

respect to u in a neighborhood of (t O, x O, v O ) , we get

f' (t , x , u_) > f' (t , x , u+) along a jump discontinuit

口

Next we extend the weak solutions 如r large t. If (δ2f/θu 2 )(t ， x , u) does not change the sign , we can extend the solutions with singularities by the above method and also treat the collision of shocks just as in

34 .4.

But , if (θ2 f/ 仇 2)(t ， x ， U)

changes its sign , the solutions may sometimes lose the entropy condition. Then we must introduce other types of singularities , for example "contact singularity." This Ís the subject which wi1l be discussed in Chapter 7. Remark 1. From the above discussions , we can say that the essential difference between equations of conservation law and Hamilton-Jacobi equations is the global solvability of ordinary differential equation (5.5)-(5.6) with respect to p

=

p(t , y).

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

54

This property determines whether the singularities of generalized solutions , or of weak solutions , are continuous or not. Remark 2. We solved the ordinary differential equation (5.9) to for quasi-linear differential equations of first order (5.1).

constn时 shocks

But , for equations of

conservation law in one space dimension , we can reduce the construction of shocks to the singularities of solutions for Hamilton-Jacobi equations. Suppose that u

= u(t , x)

satisfies the following equation of conservation law: θuθ

一 +

;;_f(t , x , u) = 0 ðt ' 8x

(5.15)

Put u(t , x) = (θ叫θx )(t , x). Then ω=ω (t ， x) satis直es

苦 +f(t， x，

(5.16)

For Harnilton-Jacobi equation (5.16) , we can con由uct the singularities of generalized solutions , as done in g4.2. In this procedure , we do not need to solve ordinary differential equations. Then we see by (4.9) and (4.13) that u

=

(伽 /θx )(t , x)

is a

weak solution of (5.15) which has jump discontinuity satisfying locally the entropy condition (5.13). B.L. Rozdestvenskii had written this idea a little in [118]. But we cannot apply this idea to quasi-linear equations of first-order which are not of conservation law because the above transform u( t , x) well to

aηive

=

(如 /8x)(t ， x)

does not work

at Hamilton-Jacobi equation. Moreover the equations treated in [59]

and [78] do not depend on (t ， 吟， i.e. , f

= f(u).

By these re出ons ， the discussions

in Chapter 5 are necessary to construct the singularities of shock type for general quasi-linear partial differential equations

of 如st-order.

Chapter 6 Construction of Singularities for Hamilton-Jacobi Equations in T wo Space Dimensions 36. 1. Introduction Consider the Cauchy problem for a Hamilton-Jacobi equation in two space dimen S10口 s

as follows:

。uδu

一 ðt +' f( 一) òx J

\

u(O , x) where f

= f(p)

=0

= φ(x)

is of class C∞ and

4> =

{t

m on

> 0,

x ξ Jæ2} ，

(6.1)

{t 工 0 ， x ζR 2 } ，

4>(ν) is in

(6.2)

S( Jæ 2 ). Here , S( Jæ 2 ) is the space of

rapidly decreasing functions defined in R 2 . We assume that f = f(p) is uniformly convex , that is to say, there exists a constant C edθ2 f

f" (p)~'

1

IiθIPiÒpj "':十一 J1 "5， i ，j 三 2

> 0 such that ?C.I>O

(6.3)

This chapter is continued from Chapter 4. Our aim is to construct the singularities of generalized

soh山 ons

of (6.1)-(6.2) in two space dimensions. The

differe配e

between Chapter 4 and this one is the dimension of spaces. The crucial part in our analysis is to solve the equation x = x(t , y) in a neighborhood of a singular point. To do so , we must see the canonical forms of singularities of smooth mappings in a neighborhood of a singular point. Though the singularities of smooth mappings are simple in the case of one space dimension , this subject is di ffi.cult and complicated in the case of higher space dimensions. Here we apply the well-known results of H. Whitney [159] to get the canonical forms of si吨ular points of the smooth mappings. This is the reason why we restrict our discussions to the case of two space dimensions. First , let us repeat the definition of generalized solutions of (6.1). Deftnition. A Lipschitz continuous function u called a generalized solution of (6.1)-(6.2) if

u(t , x) defined on Jæ 1 x Jæ 2 is

a叫 only

if

56

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

= U(t , x) satisfies the equation (6.1) almost everywhere in JRl x JR2 and the initial condition (6.2) on {t = 0, x ξJR2} ii) u = u(t , x) is semi-concave, i.e. , there exists a constant K such that i) u

u(t , x + y)

+ u(t , x -

y) - 2u(t ， x) 三 Klyl2

Remark. Put Vi(t ， X) 哲 (θu/θ町)(t ， x) (i

x , y εJR2 ， t > O.

for all

= 1, 2).

(6 .4)

Then the equation (6.1) is

written down as a system of conservation law: θθ 一句+ ",,-

ât -.

, åXi

f(v)

=0

(i

= 1, 2).

(6.5)

The inequality (6 .4) turns into the entropy condition for (6.5). See a remark in

36.3. 36.2. Construction of solutions Characteristic

cu凹es

for

(6.1)-(6.2) 町e de负ned 臼 solution

curves of ordinary dif-

ferential equations as follows:

ldzθf dt 瓦 (p) ， I

句 (0)

dv ~;

\

= Yi , v(O)

= - f(p)

= 的)，

+ (p , f' (p)) ,

~. = 0

θ￠

Pi(O) = τ (y)

(i =口)，

(i = 1, 2) ,

~IH

where f' (p)

=

(θf/θ'Pl ， θf/θIp2) and 巾， q)

is the scalar product of two vectors P

and q. Solving these equations , we have

Z 二 ν +tf'(φ'(y)) 乞f Ht(y) , v = v(t , y) = cþ(y)

(6.6)

+ t[- f(矿 (y)) + (矿 (y) ，f' (cþ'(y)))].

(6.7)

Then H t is a smooth mapping from JR2 to JR2 and its Jacobian is given by

玄机 y) = det[I + tf"(内附y)] We write A(ν) ~f 1" (φ， (ν ) )cþ叮y) and denote the eigenvalues of A(ν) by Àl(Y) 主 λ 2(Y).

When the sp配e dimension is one，

λ l(Y)

=

1" (φ'(y))φ"(y).

Since f" 忡， (ν))

>

56.2. CONSTRUCTION OF SOLUTIONS

o and -1 =λ1(ν )

(C.1) The singularities

non-degenerate; i. e. ，

a point y , then the

is regular there.

可 grad>-dν)

= 0 at

(C.2) (θvjθν) (tO , yO) 并 O. Then , for t

>

t O where t - t O is small , I: t becomes a simple closed curve and the

number of elements of

~~

is two.

We denote the restriction of v

υ( t ， ν)

on ~t by vE 工 VE(t ， ν). We see by def

(6.8) that 叩 = VE(t ， ν) takes its extremum on I:~， especially if we put v(t , Yi) ~ ci

(i = 1, 2) and suppose C1

its maximum at y

= 几.

< C2 , then VE(t , y) takes its minimum at y

Denote by Dt the interior of the curve I: t and by Slt the

interior of Ht(~t). Then the curve {νεJR2 ν= 巨 (i

= 旦回d

v(t ， ν) 口 cd is ta吨ent to

Dt at

= 1, 2). Here we apply the results ofH. Whitney [159]. Accordi吨 to his

theorem , the canonical forms of fold and cusp points are expressed respectively as follows: X1

讨

X1 = Y1Y2 - yf ,

X2 =

y2

X2 = y2

in a neighborhood of a fold point ,

(6.9 1)

in a neighborhood of a cusp point.

(6.9 2)

59

96.2. CONSTRUCTION OF SOLUTIONS

This means that the mapping (6 乌)

Ht can be regarded as the mappings of (6 乌) and

in' a neighborhood of a fo1d point and a cusp point , respective1y. Moreover he

proved that any smooth mapping from ]R2 to IR 2 can be approximated by smooth mappings whose singu1arities are on1y fo1d and cusp points. We see by his result that the curve Ht( í:, t) has the cusps at X 1 and X 2 where Xi 哩 Ht( Yi) (i When we solve the equation x = Ht(y) with respect to y for x E

= 1, 2).

nt , the expressions

= y(t , x) takes three values. Write them by Y =9i(t , X) (i = 1, 2 , 3). Here we choose y =92(t , X) as 92(t ， X) ε Dt for x εnt def When we write Ui(t ， X)~' V(t , 9i(t , X)) (i = 1, 2 , 3) , the solution of (6.1)-(6.2) has three va1ues {Ui( t , x) i = 1, 2 , 3)} for x εnt (see Figure 6.1 which shows these (6 乌) a时 (6.9 2 ) S可 that

the solution y

situations).

X

,

H

vCt ,

y)=C

,

H"'

‘

vCt ,

,

y)=C

Figure 6. 1. Curves 马 ， Ht( í:，t) and Ht- 1 (Ht (í:, t)) θ￠

Lemma 6.2. i) 瓦 Ui(t ， X) = 瓦 (9i(t ， X)) for 川岛 (i '…·….、

= 1, 2 , 3).

ii) (去 (t ， x) 一去。 ， x) ) . (ω (t ， x) - 9j(t , x)) t 并 J.


O} and f2 t ,- ~'{x ε f2t U3(t , x) - U1(t , x) < O} , we define

f

U1(t , X) U3(t , X)

(t ， x)~' ~

l

As ft is smooth , it can be

if x ξ f2t汁， if x ξ f2t ，- .

parameterized 出口 = 18凹

{x = x(s)}. Then

8u，. 飞

d/(t , x(s)) = ~舌- ð~~) This means that , though

the 直rst

discontinuity along the curve direction of f

口，

U3(t ， X) 一

dx

石=

= u(t , x) has jump

derivative of the solution u

it is continuous with respect to the tangential

t.

!ì 6.3. Semi-concavity of the solution u = 仰 ， x) Let n(t , x) be a unit normal of ft advancing from f2 t ,- to f2 t ,+, and define at x εft

uu ,.

. _,

def

,.

UU

去 (t ， 让 O)~' li~_ :.= (t, x 土 cn). (J~丁~.....→-U (J~丁

Any C 2 -function satisfies the semi-convexity condition (6 .4). Therefore , for the proof of (6 .4), it we have 吨， x

s咄咄

to treat the

c甜 where x εft

(11 θu ，

+ y) 十 u(t ， x-y)-2u(t ， x)= JoI

( 一 (t ， x

飞 ax'

ax

/

= cn (é > 0). Then

8u

\

+ sy) - -;':(t, x + 0))

飞 θzθx\~'-'-)J

(11 θu ， 8u \/θu ，. 十 I ( τ (t ， x - 0) 一云一 (t ， x - sy) ) . y ds + (了。 ， x In

and y

飞 ax

au

+ 0) 一了。 ，x ax

. yds 飞

0)) . y.

/

62

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

The first and second terms are easily estimated by KI ν1 2 • To get the inequality (6 .4), it must be

(去(机 +0)-2(tJ 一州 n(t， x)::; 0 Contrarily, if (6.11) is true , then we can get

(6 .4).

(6.11)

Hence

(6.11) is equivalent to the

semi-concavity property. On the other hand , we have , by the

de丑 nition

of Dt ，土，

>

if x ξDt汁，

US(t , X)-Ul(t , X)
tO.

65

When it satisfies the condition (8.1)

and (8.2) , we can construct the singularity of solution by just the same way as in

36.2. Remark on Figure 6 .4. Assume that I: 1 ,t and I: 2 ,t meet at y = yO = (a , b) and that the si吨ularities of λ= 人 1(ν) are non-degenerate. As À1 = À1 (y) does not take minimum and maximum at y = yO , we can suppose by Morse's lemma λ l(Y) = λ1 (y O) + (y1 一 α)2 一(的 -b户

1

+ t O λ 1(ν。) = Q.

= 1, 2) have singularities at y = yO. But , for t > t O, the curve 1 + t λ r( y) = O} is smooth in a neighborhood of y = yO.

Therefore I: i ,tO (i {y εR 2

芝 Lt O

(i)

(ii)

Figure 6.4. Change of I: 1 ,t U I: 2 ,t with respect to time 8umming up these results , we get the following. Theorem 6.5. Assume that the assumptions (8.1) αnd (8.2) are alωays satisfied. Then , even if two singularities collide with each other, we cαn uniquely pick up one value from two values of solution so that the solution becomes single-valued and continuous. In this case also , the condition of semi-concavity is satisfied.

66

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

Remark. What we have done until this point is thelocal construction of singularities of generalized solutions or weak solutions. The next problem is to consider the global behavior of singularities. This subject has been considered in several cases. For single conservation laws , see D.G. Schaeffer [120J for n = 1 and B. Gaveau [56J for n

= 2 where

n is the space dimension.

Chapter 7 Equations of Conservation Law without Convexity Condition in One Space Dimension 37. 1. Introduction We consider the Cauchy problem for an equation of the conservation law in one space dimension as follows:

ôu

ô

一 +;;f(u)=O 御街

u(O , x) where f

=

f( u) is of class

=

O ， 川的，

(7.1)

{t=O , x εJRl },

We assume the initial data 0,

υ=UM=(:;;; Assume that f'(b) - 1'(α)

> O.

(7 .4) (7.5)

Then the domain D 乞f {(t , x)

tf'(α )<x
0

and

(i

= 1, 2)

where h(Yi)

f"( ti

where t - ti is small. Our problem is to see what kinds of

phenomena may happen when we extend the shocks

马 (i=1 ， 2)

!ì 7 .4. Behavior of the shock Sl In this section we will extend the shock 51 for large t. To explain the situation , we repeat briefiy how we have constructed the shock Sl. As the graph of x for t

> ti

is drawn as in Figure 4.1 , we use the same notations appeared there.

Solving the equation x three solutions y 咐， ν)

= x(t , y)

= r:þ (y)

x(t ， ν)

= 9i(t , X)

(i 。

with respect to Y for x ε (Xl (t) , X2(t)) , we get

< 92(t , X) < 93(t , X). As r:þ (9i(t , X)) (i = 1, 2 , 3). Then

1, 2 , 3) with 91(t , X) def

for a l1 (t ， ν)ε 腔， we define Ui(t ， X)~'

we have by Lemma 5.2

Ul(t , X) < U2(t , X) < U3(t , X).

~7 .4.

BEHAVIOR OF THE SHOCK Sl

75

Here we must pay attention to the order of this inequality which is contrary to the one given in Lemma 5.2. This is caused by the property that

f" ( 0 , (5.13) says that the graph of f = f(u) lies entirely in the lower side of the chord p+p_. Lemma 7.4. As long as the entropy

is decreasing and

t 叶剑 (t ， γ2 (t))

cond凶on

for S2 is satisfied,

t 叶 91(t ， γ2 (t))

is increasing.

We omit the proof, because it is similarly obtained as Lemma 7. 1. When t gets larger ,

(93(t ， 有 (t)))

tends to 0 and

(91(t ， γ2 (t) ))

tends to the maximum of =

(x). Therefore we assume that , though the entropy condition is satisfied for t < T， 瓦 p_ becomes tangent to the curve {(u , f(u))

follows that

豆豆 dt (T) \- = I

Lemma 7.5.

d2 γ2 ，fT'1\ i) 亏f- (T)

f'((93(t I \ r -,,ì'2( t)))) I JJI I

J

\.:1 õ)

< 0,

I ~ \ -

\

..\a

r1t

u ξ ]Rl} at t = T. Then it

_ = f'(u+(T)) 11\\1

ii) 扩 (u+(t)) I 可 =0

The proof is almost the same as that of Lemma 7.2. By this lemma , we get the following: Proposition 7.6. For t

> T where t - T is small, it holds that J牛

U~2 (t) < f' (u+ (t)). dt This proposition means that the entropy condition is violated for t > T. To overcome this point , we use "contact discontinuity" explained in 37.2.

。

V

Figure 7.5

U

u

57.5. BEHAVIOR OF THE SHOCK 52

79

Consider the equation υ

EJ 一

rJ-u u-)rt1、-

叫:

(-

E' ， 、、‘

ff.飞 1

rJ,

, u > V.

(7.16)

We solve the equation (7.16) with respect to v , and denote the solution by v

h(u). 5ee Figure 7.5. To construct a contact discontinuity starting at the point (t , x)

= (T， γ2(T)) ，

we solve the Cauchy problem as follows:

(22f(州的l(t， x)))) x(T)

(7.17)

= γ2(T) def

We denote a solution of (7.17) by x = γc(t) and put Sc~' ((t , x)

x

= γc(t) ，

t2

T}. Let W be a domain sUITounded by the curve Sc and a characteristic line x= 俨十厅I (4) (yO)) which passes through the point yO must satisfy the equation "1 2(T)

=

yO

(t , x) = (T， η (T)). 50, the point

+ T l' (4)(俨)).

Hence yO

=

g3(T， γ2(T) ).

Figure 7.6 explains this situation.

t x=γ/t)

T

× 。

Figure 7.6 Our problem is how to define a weak solution u

= u( t , x)

in the domain W. As

the domain W is not covered by the family of characteristic lines whose starting

80

7. CONSERVATION LAW WITHOUT CONVE Xl TY CONDITION

points are on the initial line {t

= 吗，

we introduce a family of characteristic lines

which start from the curve Sc , i.e. , we construct the "contact discontinuity." For any point (t , i) ε Sc ， we first define -.. ..

we determine u+ = u+ (t , i) by u+

,,-.... . . . . def u_ by .u_ def ~' 4>(i.J) where y ~. 91( t , i).

Next

def .. ,"""

h(u_) and draw a characteristic line which passes through the point (t ，;;) 臼 follows: x = ;;

~.

+ (t -

i) f'(';;+).

On the characteristic line , we define the value of the solution u

= u( t , x)

by u (t , x) =

u+. As the family of the above characteristic lines covers the domain W , the weak solution u = u( t , x) is completely defined on the domain W. It would be obvious that the entropy condition is satisfied (see Figure 7.7).

t

。

x=y+tF'(υ+(

T) )

O

Figure 7.7 Repeating the above discussions , we can extend further the weak solution of non-convex conservation law (7.1).

What we would like to insist in the above

construction is that we did not use "rarefaction waves." Example. For the Cauchy problem (7.1)-(7.2) , we assume that the initial function

4>

=

4>( x) is of a form as shown in Figure 7.8 , and that it

satisfies 飞皿 4> (x)

> b2

57.5. BEHAVIOR OF THE SHOCK 52

and min a

~(x)


，~ (t ，.，) εp'

We now define a

functionμ=μ( x)

. . (μ vl

v_

μ (x) ~'

+ η1 . [h( t) + 叶， ε 1.

0) for t

On the contrary, suppose

that there exists t' ε (0 ， t O] with ω (t') < ω(0). By the hypothesis of the theorem ,

one 鱼nds

a set G 2 C (0 , T) of Lebesgue

measure 0 such that 。T\ (G2 X ]R n) C nk=lDif(Uk)

(8.29)

ε (0 ， T) \岛 ， x ε ]Rn.

From the above , it follows

and that (8.25) is satisfied for any t def

that (8.15) still holds where G ~' G 1 U G2 ; hence , there is a number ). with max{O ， ω (t')} < λ 0 for al1t ε (t泸l， 俨 t 2]. 叫 It follows that

l W (飞 a contradiction. Therefore , (8.33) must be a comparison equation. Motivated by this fact , we propose the following:

~8 .4.

Proposition

GENERALIZATIONS TO

8 且 Let σ=σ( 叫 be

is α comparison

SYSTEMS

c

lR with Jo+∞ e(t)dt = +∞

equation , then so is the equation ωσ(ω)

Converse旬，

(i i)

99

of class C[O , + ∞)， and e = e(t) 主 o be Lebesgue

integrable on each bounded interval (0 ,,)

(i) 1f (8.33)

WEAKLY回 COUPLED

under the condition essinfe(t)

(8.34)

> 0,

ifmo陀over

(8.34) is a com-

tE(O ，+∞}

pα rison eq1川ion，

then so is (8.33). 二 w 1 (t)

Proof. (i) Let w 1

be a solution of (8.34) on some interval (0 ， γ1) with

li问 ω l(t) = O. Find a number γ2

t -t u

>0

γ1 Setting w 2(t) ~f w 1 (0 ， γ2) with li吨 t -t u

,

w2

u:

=

such that

1'Y叫 r)dr

(8.35)

e( r)的， we see that w 2 = w 2(t) is a solution of (8.33) on

(t) = O. Byassumptio凡 ω 2(t) 三 o on (0 ， γ2). Hence 1川 (t) 三 O

on (0 , 1). This shows that (8.34) is a comparison equation. (ii) Let (0 , +∞ )?d 叶1.( t) be the inverse of (0 , +∞ )?d 叶 J: e( r)击， and ω2 w 2 (t) be a solution of (8.33) on some 时 in 时te 衍rva 叫 al (仰0 ，刁 γ2)with J!坦 r;rtuωv汽 骂 2 y

,

defi肘 a number 1 > 0 by (8.35). Then setting ω10)tf tu2(t(t)) , we also see that ωω1 (t) is a solution of

(8.34) on (0 ， γ1) with ~iIl! W1(t) = 0 (cf. [40 , Proposition t 一+0

3 .4( c)]). The rest of the proof runs as

before.

口

In the sequel , for each function 9 = g(t) defined and continuous in a certain interval (0 , t勺， let 马 denote the open 附 {t ε (0 ， t O )

g(t)

> O}. Here is an

elementary property of comparison equations: Proposition 8.10.

Let (8.32) be a comparison eq1川ton a叫 9

=

g(t) α gwen

function α bso l1山 ly cor阳uo旧 on some mt 例αl(OY)such thdj!59(t) 三 o and that g'(t) 三 ρ (t ， g(t)) αlmost everywhere in 马 . Theng(t) 三 o for all t E (0 , t勺，

Proof. On the contrary, suppose that there exists t 1 ι (0 ， t O ) with w 1 ~f g(t 1) > O. Setting g(O) 哩 !!59(t)and t2 智 sup{tε [0 ， t 1 ) g(t) = O} , we 附 that 0 三 t 2 < 沪 ， g(t 2 ) = 0 and (户 ， t 1 ) C 马. Hence , by assumptio凡 g'(t) 三 ρ (t ， g( t) )

almost everyw here in (沪 ， t 1 ).

(8.36)

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

100

Now take

t2 < t < t O ，

ω 主 max{O ， g(t)} ，

fρ(t ， max{O ， g(t)})

if

1ρ(t ， ω)

if t 2 < t < t O ， 。三 ω< max{O , g(t)}.

þ(t ， ω)~. ~

(8.37)

The above-mentioned Carathéodory conditions (1)-(3) are clearly satisfied for

Þ=

仰， ω) on (户 ， t O ) x [0 , +∞). Let ω=ω (t) be a solution through 肘， ω1) of (8.32)

with Þin place ofρ ， and let (t 3 , t 1 ] C (沪 ， t 1 ] be its left maximal interval of existence. We next claim that Vtε (t3 ， t 1 ].

(0 三 )ω (t) 三 g(t)

(8.38)

Assume (8.38) is false. Then one would find a nonempty interval (沪 ， t 5 ) C

W,t 1 )

such that ω (t)

> g(t)

Vt E (t\ t 5 ) ,

(8.39)

with ω (t 5 )=g(俨) .

It fo l1ows from (8.36)-(8.37) and (8.39) that

(8 .40)

g' 但)三 ρ (t ， g(t))

= þ(t ， ω (t)) = ω'( t)

almost everywhere in (沪，俨). Thus (8 削) implies that g( t) 主 ω( t) for all t E (沪，俨) , which contradicts (8.39). So (8.38) must hold. We proceed to show that t 3 = t 2 • In fact , if (0 三) t2 Nv , thus getting the validity of (E. II).

Remark 3. Condition "gN

=

gN(t) ε L~c"

in the hypothesis (E .I) could not

be replaced by "gN = gN(t) ε Lfoc'" To see this , consider the following Cauchy problem θu/θz θu/街+一一一一τ= 01t - 111/2

u(O , x)=x

on

in ...

、

{t > 0, x E lR} l - , -, - ~ -- J

{t=O , x

ξ lR}.

1. Here. ere. nn ~f 二1.

f(t， p)tf-l一丁 It - 11 11

and

cþ(x) 乞f Z

for x E lR, p ε lR and t > O. Then

旷 (p)

=

~丁∞ if p ::f 1 l U 1I P = 1;

hence (E.II) holds when N(V) 主1. All the assurnptions of Theorem 9.1 with Lfoc in place of L~c for Hypothesis (E .I) are therefore satisfied here. However , in this case , (9.7) gives the function

u=u(t ， x) 乞f X

_

2 - 21t - 111/2sign(t - 1) ,

108

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

which

Lipschitz continuous in any neighbourhood of a point (1 , X O).

is 且ot

For the proof of Theorem 9.1 , we need some preparations. We first rec a11 that a direction a1 derivative is of ç near a point ÇO E

defined 出 follows.

]RTn

Let

e 巳]RTn.

and let

ψ=ψ(.;)

be

a 缸lite-valued

function

Denote

可 ψ(çO) ~f i叫 sup {[ψ(çO + O"ë) 一 ψ(俨 )]ó- 1 } ;>UO< 占(x)

u = u( t , x)

N ow suppose that

"'EI

。 EI

4>

x 巳 IR n .

for all

is thus a global solution of (9.1 )-(9.2). =

口

4>( x) is given in the form

4>(叫苦 LEVa(z)for z εRn， with 4>", = 4>",( x)

a 鱼nite

convex

functωn

for every

αε 1.

(9.18) Combining Theorems

9.1 and 9.6 , we obtain the following first results for the representation of global solutions in the case of nonconvex initial data. Corollary 9.7. Assume (E .I)-(E. II) for eachproblem (9.1)-(9.2 ",), with φ。 = 4>", (x) function， αε 1.

a finite convex

Assume , furthermore , that all the hypotheses of

Th wnm9.6~U~r~ew~~m

u'"

= u"，(t， x) 与古 {(p， x)

problem (9.1 )-(9.2) 仙 ere

t

"'EI

4> = 4> (x) is defined by (9.18).

Corollary 9.8. Let 4> 二功(z)qzf con时x

-l f(T， p)命}

def .

oft伪 hose pro 呻 协 b lem πn 盹 I

some

ω)

min 白 (x) ， with φ

iE{l ,..., k}

and globally Lipschitz continuous

白 (x) ，...， i 但) -

iE{l ,..., k }pE皿

I

Jo

f(T , p)dT}

for

(t， x) ε 否

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

116

Proof. Since 1

def

{1 ,.. . , k} is

~'

Corollary 9.5 and Theorem

a 岳出te

set , the conclusion is straightforward from

9.6.

口

Example 1. Consider the Cauchy prob1em

+ I(δu/δx)2-11=0

δu/θt

{O 0 ,

1+∞

h 2 (p) 乞r ~ 0

l

= 0, < 0,

-pln( -p)

if p if p

+p

= 0, < O.

The solution can a1so be rewritten as u(t ， x) 二 min{m_~{px - p1np p主o

-

+p -

tlp2 一 1 日， m缸 {px p 三。

+ p1n( -p) -

p - tlp2 一 11}}

= m_~{ -plxl- plnp + p - tlp2 一 1 日， P主 U

in which we adopt the convention that p 1n p = 0 if p = Example 2. Let h =

h(α)

o.

be a finite-valued continuous function of αon a given

compact set K C Jæ n. We put

(x) 乞f322忡忡忡 |α1.lxl} for x ε Jæn If

f

=

f(t ,p)

be10ngs to C( 剑， depends on1y on t and Ipl , and is decreasing with

respect to Ipl , then it follows from Corollary 9.7 that a global solution u = u( t , x) of the Cauchy prob1em (9.1)-(9.2) can be found by the formu1a u(t , x) ~f mjg{h(α) aεK 飞

= ~g{h( α) 。E且

+ .~?:X .{(p， x) --- . Ipl 三 lal

+ Iα1.lxl- I

JO

I

Jo

f(r , p)dr}}

f(r , a)dr}

for

(t ， x) ε Ð.

~9.3.

THE CASE OF NONCONVEX INITIAL DATA

117

We now consider the Cauchy problem (9.1)-(9.2) in the main case of this section where 功 = N俨，M.

(9.25)

By (9.20) , it follows that , if (t , x) ε 孔， then ψa (t， x ， p一 α) 三 (p， x)- σ;(p)-rl α| 句­ Therefore , (9.24) and (9.25) imply

Ua(t , x)

主 σ;(α) 十吨但 {(p， x)

-

O'

;(p)} -

rlα 1-

Sr

p t::皿"

=σ;(α)+σl(X)

-

rlα I-sr

主 (r +句+向 +M)'Iα 1- J.l r -rlαI-s俨 >M，

provided (t , x)

ε 几 and 1α 1> N.但.

This means that

lim ua(t , x) =

locally

|α| →+∞

+∞

uniformly in

(t , x)

ε Ð.

Hence (cf. Remark 2 after the formulation of Theorem 9.1) , we may find a positive number N r for each

rε(0 ，+∞)

ig!>a(t , x)

。 ε]Rn

=

such that

皿!Iél_

lal 三 Nr

ua(t , x)

whenever

(t ， x) ε 凡

(It should be noted that U a (t , x) is lower semÍ continuous in αon the whole ]Rn.) Moreover , analysis simi1ar to that in the proof of Theorem 9.9 shows that the solutions Ua = ua(t , x) satisfy a Lipschitz condition on 民 with constants depending on r but independent of

αfor 1α| 三 Nr ;

and that they satisfy (9.1) except on a

cOmmon set of Lebesgue measure O. The proof is thus complete in view of Theorem 9.6.

口

39.4. Equations with convex Hamiltonians f = f(p) We now consider the Cauchy problem θU/θt

+ f( θU/δx) = 0

in

Ð = {t > 0, x εR勺

(9.26)

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

122

u(O , x) = fþ(X)

{t = 0 , x

on

巳]Rn}

(9.27)

under the following two hypotheses.

(F .I) The initial function fþ strictly ωnvex

on

]Rπ with

=

fþ(X) is of class

. .lim f(p)flpl Ipl →+∞

C Oα nd

the

Hamiltoniαnf=f(p)is

= +∞.

(F.II) For every bounded subset V of Ð , there exists a positive number N(V) so thαt

~~..，{ fþ( ω )+t. f* ((x 一 ω )ft)}

|四 I S; N(V)

ωhenever

(t , x)

ε V， IνI

> N(V). Here , f*

=

< fþ( ν) + t. f* ((x - y)ft)

f* (z)

denotes the Fenchel

conj叼αte

function of f = f(p). In the sequel , we use the notation

((t ， x ， y) 乞f fþ(y) where (t , x)

ε Ð，

巳]Rn ，

y

+ t. f* ((x -

y)ft) ,

(9.28)

and shall prove the following theorem.

Theorem 9.12. Assume (F .I )-(F.II). Then the formula

U(t ， X)~f ipL((t , x , y) = ipfJfþ(y)+t. f* ((x-y)ft)} yE lR n

determines α globα1

for

(t ， x) ε Ð (9.29)

solution of the Cauchy problem (9.26)-(9.27).

The next auxiliary lemrna is known ([64] , [117 , Theorems 23.5 , 25.5 , and 26.3]) , but what we would like to insist here is on its simple proofby the use of our Lemmas 9.2-9.3. Lemma 9.13. Let f

Then

f* = f* (z)

= f(p) be strictly

con肥x

on

]Rn

with . .lim f(p)flpl = Ipl →+∞

+∞，

is et

f* (z) = (z ， δf* (z)fδz) -

f(θf* (z)fδz)

Proof of Lemma 9.13. The strict convexity on

]R n

for α11 z ξ ]Rn. ofthe function f

(9.30)

= f(p)

says

that this function is everywhere finite and that f(λpl

+ (1 一 λ )p2)

三 λf(pl)

+ (1 一 λ )f(p2)

for any pl , p2 εRn ， λε[0 ， 1]; the sign of equality holding if and only if pl = 泸 or λε{0 ， 1}.

Accordingly, f

= f(p)

is continuous.

~9.4.

EQUATIONS WITH CONVEX HAMILTONIANS f

= f(p)

123

It will now take a simp1e matter to check thatω=ω (z ， p) ~f (z ， 的一 f(p) satisfies all the conditions of Lemmas 9.2-9.3 where we put E def z , ()~. IRm = IRn and shal1 deal with the function

ψ=ψ (z) 乞f SUp {ω (z ， p) : p 巳 IR n } lndeed , since . lim f(p)/Ipl = Ipl →+∞

+∞，

~.

IRn , m

def

~

def

~π ， 5=

= f* (z).

Condition (i) ofthese Lemmas ho1ds whi1e the

others are almost ready. As f = f(p) is strict1y convex , it can be verified that the multifunction L = L(z) defined by

L(z) 生f {p εIR nω (z ， p) = f气 z)} is actually sing1e-valued on the who1e IRn. Therefore , by Part b of Lemma 9.2 , all the partia1 derivatives 叮叮 z)/θzi exist , and L(z) = {θf气功 /θz}. Property (9.30) is thus

∞ming

from the definitions of f*

= 户 (z)

and L = L(z). Further , Lemma

9.3 and its Remark imp1y the continuity of δf*/θz=

δ尸 (z)/δz.

口

Remark 1. Consider a convex and 10wer semicontinuous function f = f(p) on IRn. Assume domf 弄。 and imf

c

(一∞，+∞J (the function

f = f(p) is then called

proper). It will be shown that 1im f(p)/Ipl = +∞

Ipl →+∞

In fact , if

if and on1y if

dom f* = IR n .

1im f(p)/Ipl = +∞， then for each z εIRn the supremum

Ipl →+∞

f* (z)

= sup {(z , p) - f(p)} pE lR n

c

IRn , and hence finite. Converse1y, 1et there exist an M εIR and nonzero points pl ,p2 , . .. in IRn such that f(pk) 三 Mlpkl for k = 1, 2,... and that Ipkl →+∞出 k →+∞. Since IRn is lS

essential1y taken over al1 e1ements p of just a compact set Kz

10cally compact , we m可 suppose that pk /Ipkl → ZOεIRn. Putting z ~f (M + l)zO , we thus get f* (z) 主 BIP{(z ， ph)-f(ph)} 主 ?{|ph|·[(M+l)(zo ， ph/|ph|)-Ml}

主

lim Ipkl = +∞·

k-+ • -00

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

124

Remark 2. Consider a finite convex function iþ conjugate

Let

旷 二旷 (p).

implies

that

lim

Ipl →+∞

旷 iþb俨**

on

= iþ*(p) is 严 p ro 叩 pe 吭 r，

with the Fenchel

∞ c on 盯.ve 叽 乓， 皿d X 刮10 仰， we 盯 r 优 s e皿血lCO ∞ on 时t 川inuo ∞ u

= iþ. Accordingly, dom 旷* = domiþ =

旷 (p)jlpl

]R n

旷** iþ 口 ￠旷旷旷斗丰中叮"叶.

Then it is known [64J that 旷 O且]Rηand

=功(x)

]Rπhence ，

Remark 1

= +∞.

Proof of Theorem 9.12. By (F .I )-(F.II) and Lemma 9.13 , (9.28) determines a continuous function ( = (( t , x , y) whose derivatives θ((t ， x ， y)j挠， θ((t ， x ， y)jθXl ，... ， 何 (t ，

exist and are continuous on the whole Ð x

]R n;

x, y)j åX n

moreover , one may apply Lemma

9.2 to the function ω=ω (~ ， p) 哩仰，吼叫 where p 哲 ν ， E qzf IRnand mtf n+l , ztf(t , z) , Ctf D.Comeqmntly, the function

U

=u(t , z)dehed by (929)

is loca11y Lipschitz continuous and is directionally differentiable in Ð , withθleu(t ， x) equal to ~n ，((1气 (x-y)jt) 一 ((x - y)jt ， θf 气 (x

ν )jt)jθ斗， θf气 (x - y)jt)j 剑 ， e).

yEL(t ,æ)

(9.31 ) Here ,

...L 1

_

]R n+l '3

e,

'\

L( t , x)

def

~. {νε ]Rn

((t ， x ， ν)

accordi 吨 to Rademacher's theorem , p Ol时 outside

a nu11 set

Qc

= u(t ， x)} 并。 (Lemma 9.3). But ,

u = u( t , x)

is (μto 创t阳 ω叫 a11坊y 川圳 ) differen 时由 tia 由 ble 挝 at 阳 aI且1

Ð. Therefore , suitable choices of e in (9.31) give

θu(t ， x)j战 =uzjfa){f*((z-u)/t) 一 ((x - y)jt ， åf气 (x 一的 jt) jθz)} =田 ~~æ) {J气 (x

ν )jt) - ((x - y)jt ， θf* ((x - y)jt)jθz)}

(9.32)

and θu(t ， x)jθXi =

min θf* ((x - y)jt)jδzi =

YEL(t.æ) -

provided (t , x) E Ð\ Q and Now , given any (t , x)

"

iε{1 ，

ε Ð\ Q ，

maxθf* ((x

yε L(t ， æ)

- y)jt)jθZi ，

(9.33)

2,..., n}.

we pick up some y

ε L(t ， x).

Then it follows from

(9.30) and (9.32)-(9.33) that θu(t ， x)jθt 出广 ((x - y)jt) 一( (x 一 ν ) jt ， δ广 ((x - y)jt)jδz)

= -f(θ尸 ((x

一 ν )jt) jθz) = 一 f(θu(t ， x)j θx ).

The equation (9.26) is thus satisfied almost everywhere in Ð.

= f(p)

39 .4. EQUATIONS WITH CONVEX HAMILTONIANS f

125

As the next step , we claim that

Ð3(t

^, u(t , x)

1im

,.,)• (0 ,., 0)

= (x O )

(9.34)

for each fixed X O 巳 Jæn. Indeed , on the one hand , the definition (9.29) clearly forces u(t ， x) 三 (x)+t.f气。)，

hence 1ims叩 Ð3(t

,.,)• (0 ,., 0)

u(t ， x) 三 (X O ).

(9.35)

On the other hand , 1et us 自rst take a sequence {(沪， x k )} t~

c

D converging to

(O , XO) such that __ )iII?- i~~ ^, u(t , x) = .lim u(沪 ， Xk) and second choose arbitrary Ð3(t ,.,) • (0 ,., 0) " k →+∞ points yk 巳 L(沪 ， Xk) (for k = 1, 2 ,...). Then it will be shown that yk 一-t XO. (k →+∞)

In the contrary case , suppose without 10ss of generality that yk

一→俨 ξRn ，

(k→+∞)

where yO 并 X O . (We emphasize here that the sequence {yk}t~ C Jæn is bounded by Lemma 9.3.) Since

lim j* (z)/Izl = 十∞ (cf. Remark 2 of Lemma 9.13) ,

Izl →+∞

(9.35) and a passage to the 1imit as k →+∞ in the equality

U(tk , Xk) = (泸)十 t k . j* ((x k _ yk)/t k )

(9.36)

wou1d yie1d ( x O )

主

1iminf ^, u(t , x)

Ð3(t

,.,)•

(0 ,., 0)

,

a contradiction. This shows that

= .1im u(沪 ， Xk) = (yO) + (+∞) = +∞， k→+∞

1im

'U k h →+∞v

f* = 户 (z) is bounded from be10w since again

a passage to the 1imit as k

→+∞ also

1iII?- i~~

^, u(t , x)

V3(t ， 叫→ (0 ，.， 0)

Finally,

combini吨 (9.35)

,

X O• ,

But the continuous function

üm f气 z)/Izl

Izl →+∞

=

+∞. Therefore,

in (9.36) imp1ies

=

.lim u(沪 ， Xk) 2: (x O).

k →+∞一

and (9.37) gives (9.34) , which says that

(9.37) u 工 u( t , x)

has a (unique) continuous extension over the who1e 否 satisfying (9.27). The proof is thus

comp1ete.

Remark. Assume (F .I). Then (F.II) is satisfied if in (t , x) on each bounded subset of D.

口

,

lim

Iyl →+∞

仰 ， x ， y) = +∞ uniformly

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

126

In fact , let V C Ð be bounded , Put M ~f r.lf气。)1 + ((t ， x ， 叫 =

1> (x) +

+∞ uniformly

in

日町，

V C (O , r) x B(O , r) for some r

rn ax 1> (x) < +∞It 1"'1 三 r

ξ (0 ，+∞).

follows from (9.28) that 皿in((t， x ， ω) 三 四|三 T

t. f* (0) 三 M whenever (t , x) ε V. Hence , if. lim ((t ， x ， ν)= |纠→+∞

(t , x)

on each such V , then for a suitable number N(V)

~

r we

have mm

仰，吼叫三皿in((t ， x ， ω):::; M

|四|三 N(V).

叫 N(V)j

(t ， x) εV， Iν1

i.e. , (F.II) is satisfied. Corollary 9.14.

U叫 er

Hypothesis (F .I), suppose that

lüninf 1> (x)/lxl >

∞.

1"'1 →+∞

(9.38)

Then (9.29) determines a global sol1山 on of the Cauchy problem (9.26)-(9.27)

Proof. By Remark above , it su ffi. ces to prove that. lim ((t , x , y) = |创→+∞

+∞ uniformly

in (t , x) on each bounded su bset V of Ð. To this end , let V C Ð be bounded , say,

V

c

(0 , r) x B(O , r) for some r E (0 , +∞) and let M ξ(0 ，十∞) be arbitrarily given

Condition (9.38) says that there exist numbers 1>(y) 三

λ|ν1

λ ， N ε(0 ， +∞)

whenever

But we certainly find a positive number

νwith

f* (z)/Izl 主 2(M 十 λ)

such that

Iyl ~ N.

the property that as

Izl 三 ν.

Putting N(V) ~f max{l ， N，衍， r(l + ν)} ， w可 tl即可fore deduce from (9.28) that , if

(t , x)

ε V and 1ν1 ~ N(V) , then \

,

f*((x - y)/t)

Ix - yll

I(x - y)/tl

ν1

仰 ， x ， y) 三|一 λ+'-..--'-'-----'--'-~一一一 I .Iyl

L

'"

J

主 [-λ 十阳+入) . ~] .Iyl 三 M because I(x - y)/tl 兰卡 (1 +ν) - r l/ r =ν， Ix - yl/lyl 三 (Iyl- r)/Iyl ~ 1/2.

If 1> = 1>( x) is globally Lipschitz

continuou日

口

on IR n , then (9.38) clearly holds. The

following result of Hopf [64] can thus be considered as a consequence of Corollary 9.14. Coroll盯Y 9.15. 厅 the

and 矿 the 十∞ ，

initial

function 功=

1>( x)

is globally Lipschitz continuous

Hamiltonian f = f(p) is strictly convex on IR n with

then (9.29) determines a global solution of (9.26)-(9.27).

,

.lim

Ipl →+∞

f(p)/Ipl =

Chapter 10 Hopf-Type Formulas for Global Solutions i n the case of Concave-Convex Hamiltonians !ì 10. 1.

Introduction

Consider the Cauchy problem for the simplest Hamilton-Jacobi equation , namely, θujδt

+ f(θujθx)=O

u(O , x) = O , x

{t=O , x

ε ]Rn} ，

ε ]Rn}.

(10.1) (10.2)

In the previous chapter , a global solution of this problem was given by explicit formulas in some cases a little more general than the following two of Hopf: (a)

f =

f(p) convex (or concave) and 一一一一一→十∞ 一

Ip'什|

f*叫叮 1叫(z'飞， pη

f(O'飞， p"

Ip"l

Ip"

一一一一一 N(V).

The main result of this section reads as follows. Theorem 10.4. Assume (G .I )-(G.III). Then the formula

u(t ， x) 乞f sup

y' εIRnl

，jnfC(t， z ， ν) = ..i!!f__ sup ((t ， 吼叫 for (t , x) ε Ð

y" EIR n 2

霄 "εIR n 2 霄， EIRnl

(10.29) determines a global sol山on of the Cauchy problem (10.26)-(10.27).

For the proof of Theorem 10 .4, we need further generalizations of Lemmas 9.29.3 to the mixed sup-inf c田es. Lemma 10.5. Let 0 be an open subset of Rm and χ=χ( 己 ， y) = χ(ç ， y' ， ν")α continuoω function of ( ιν) = (ινν")εo x Rn l x Rn 2 with the following three properties:

(i) The identity sup

infχ(ιν) = ..ip.L

霄 'ε lR R1y" εIR n 2' - ,_. - •

sup x(已 ν ) holds in O.

y"EIRn 2 y 'EIRnl

(ii) For every bounded subset V ofO , there exists a positive number N(V) such thαt supχ(ç ， ωω勺
3(t

,.,)• ,.,

u(t ， x)

= 4> (x o )

(10.39)

(0 0)

,

for ead axed zoqf(z，oJ，，。 )ε IR n1 x IR n2 • Indeed let us first take a sequence

,x气 X" t~ c 'D converging to (0, , , such that lim u(t k ,x ,k,x"k) and second choose arbitrary points

{(t k

k)}

lim s叩

X O X"O)

u(t ， x) =

1> 3 衍，对→ (0 ，.， 0)

h →+∞

10 def I ,Ie ，，1c品'" UK=(ν' ， ν"~)ε L(t舱， z' ， z")for

Then it will be shown that yk

一→

(k→+∞)

x O in

, ,.

k =1 2

IR n1 x IR n2 • On the contrary, since

the sequence {yk}t~ is bounded (Lemma 10.6) , we can suppose without loss of

~10.3.

generality that yk

→

(k →+∞)

HOPF‘-TYPE FORMULAS

143

yO 哲 (y勺，，0)ε IR n1 x IR n2 with yO 并 X O . It is clear

= L(t , x) that

from (10.28)-(10.29) and the above definition of L

U(tk , x ,k , X"k) = ..i!lL_ ((tk ， x气 z川，沪，旷')三 ((tk ， x气 x'气 jhJ"h) 霄"仨凰 n2

__, k

,

三 φ(川

__， k

、

and

u(沪 ， X， k ， x"k)

= sup ((沪， thJ"KJ'， u"h) 主〈(thJh ， z"hjh ， ν"k) y' ElRn

l

主￠内

(10 .4 1)

We need only consider the following two cases. Case 1: y'。并 x ，O. Then (10 .4*) and (10ω) show that

lim u(沪， z' ， z") 三 φ(u ， z)+|z-u|-hm

f气 (X ， k _y ,k)/tk , O") =一∞. I(X ， k - y ,k)/tkl

So , (10 .41) implies .lim tk f* (O' , (x"k _ y"k)/t k )

一∞， a contraruction with

,0 _ ,,0 \

,

I __

,0

,

_ 0

+∞

I

"_

h→+∞

k 一t+o。、，

(10.3*). Case 2: y"。并 x"O. Analogously, (10.3*) and (10 .4 1) show that

lim

/1k

,k

"k 、自

u(t'".x'-.x"-)=+ ∞-

h→+∞、

hence thatAT∞tkf* ((X， k _ y,k)/tk , O")

= +∞， which contradicts (10 .4*).

The contradictions , which we have got in both Cases 1 and 2 , prove that AT∞yk

x O • Therefore , a passage to the limit as k →+∞ in (10ω)-(10 .4 1) would give (10.39). It may dually be concluded that liminf ~， u(t ， x)

1> 3(t ,.,)

• (0 ,.,0)

= (x'飞， 旷x俨，川，

limsup 一一一一一一∞·

(t ， x)EV 四 11

EIR"2 -

,

J

Of course , ~~~_ .!~LJ(t ， x ， ω 旷') 2':

四'ε]R Rl 四 "EIR n 2

.!~L_((t ， x ， x' ， 旷')主 η+1

一回 "EIR n 2

(lO.4 4)

|恤 'l :S: r

出 (t ， x)

E V. Fu rther , in view of (lO.43) , there exist

numbersλ ， N ε(0 ， +∞)

that 功(ν' ， z勺三 λ|ν'1

whenever

Ix叮主 1"， ly'l 主 N.

such

~10.3.

HOPF-TYPE FORMULAS

Finally, by (10 .4*), we certainly find a positive f* (z'飞， 0"

一二→一三 2 均(归 η一λ 川)

Iz'l

numberνwith

臼

145

the property that

|扫z'l怆主 ν.

Putting N(V) ~f max{l , N， 针，1'(1 + ν)}， we therefore conclude from (10.28) that , if (t , x) εVand 1 旷|三 N(V) ， then

四'" u

id 耳总n叼J爪川〈 JJ2J 仰札 α (t优t式，叫 ω旷 州川勺')三红((阳式阳Z川旷♂川，斗)=仰 J (ων归 μ γγ ，飞〉 ， 旷♂")川+村 x" t. 叫f *气(斗豆 乌句， ι 厅0σ0 ， 仨 t =且

r\

,

f*((x' -y')/t ， O叮

Ix' - y'll

::;1λ+ .:. . '---'0'-:----=-":"":"""，---'-一一一 1 . ly'l l" I (x' - y') / t I 扩1 J

I 三 [λ+ 2(η 一 λ) . ~] . 1ν， 1 三 η

(10 .45)

because I(x' - y')/tl 兰卡 (1 +ν) - 1']/1' = ν， Ix' - y'I/1 旷|主( 1ν'1- 1')/ 切， 1 主 1/2. No啊，

(10 .44)-(10 .45) show that the second condition in (G.III) holds. Analo-

gously for the first one. Thus the corollary follows from Theorem

10 .4.口

Chapter 11 GJobal Semiclassical Solutions of First-Order Partial Differential Equations 31 1.1. Introduction The

prese且t

chapter is in principle a continuation of the previous three. However ,

it was actually originated in the following problem posed by S.N. Kr田hkov [93]. def Let a smooth [i.e. , of class C 1] function ω=ω (t ， x) satisfy in the strip IIT ~

[0 , T]

X

Jæl the inequality |δω (t ， x)jθtl 三 NIθω (t ， x) jθ叫

N

= const. 主 0 ，

(1 1.1)

and the initial condition ω (O ， x) 三 o

Then it is easy to show (cf.

on

{t

Ha町、 Theorem

= 0,

(1 1. 2)

x E Jæ.I}

1. 5) that

ω (t ， x) 三 o

in II T

.

Therefore ,

the Cauchy problem for the fust-order nonlinear equation θujθt

where f

+ f( θujθx) = 0 ,

f(p) is of class C1(IRl) , cannot have more than one solution in IIT ,

say, in the class of smooth functions with bounded derivatives. As Kruzhkov already remarked , the same conclusion may be drawn without appeal to the differentiabili ty of ω=ω (t ， x) (resp. of the solution u = u(t , x)) or the validity of

(1 1. 1) (resp. of the equation) at the points in any given 缸lÌte union of straight lines {t const. , x εIR} C II T . The following question arises naturally: to what extent can the condition on the smoothness of ω=ω (t ， x) and on the va lidity of inequality (11. 1) in the entire strip IIT be weakened? For example , the Cauchy problem for the equation θuj街 +(θujθx) 2 = 0 with the zero initial condition

u(O ， x) 三 o

has a continuum of piecewise smooth global solutions , such as

U 口比 (t ， x) ~f min{O , alxl _α2叶， αconst. 主 O. Note that each function

~11. 1.

INTRODUCTION

147

u'" (t , x) satisfies the correspondi吨 inequality Iθω/街|三 α|δω/θxl almost ev e可 where in IIT. Therefore , it is interesting to find intermediate classes (as wide as possible) between C 1 (II T ) and Lip( (0 , T) X~1) ， in which only the zero function

ω =

can simultaneously satisfy (1 1.1) and (1 1.2). These questions can be generalized to the multi-dimensional case. The study of this problem suggests that we should single out the widest class between the class of continuously differentiable functions and the class of Lipschitz continuous functions in which the Cauchy problem for a first-order nonlinear partial differential equation has a unique global solution. We shall assume 0 T=+∞)

N(V).

As we have mentioned in Chapter 9 , (E .I) implies the t-measurability and

< t < T,

continuity of f = f(t , p) on {O

p ε IR. n }.

In addition , since

4>

=

rr

4>( x)

is

finite on IR. n , this hypothesis allows us to define an upper semicontinuous function ψ=ψ (t ， x ， p) from

[O , T) x IR.n x

IR.πinto [一∞，+∞) by taking

巾，卢 (p， x) 一矿 (p) 一 l t f(r , p)dr

(1 1.1 6)

We shall use the notation E ~f dom 4>*手。 and deal with the function u = u(t , x) and multifunction L = L(t , x) of (t ， x) ε [0 ， T) x IR. n determined by the formulas:

u(t ， x) 乞f supψ (t ， 矶时，

(1 1.1 7)

pEE

L(t， z)qzf{pε E def

Lemma 9.3 , with m~' n ψ(巾，时，

• def ,.

+ 1,

ç~'

shows that L = L( t , x) is a

m山 ifunction

of (t , x)

ε [0 ， T)

ψ (t ， x ， p) = u(t , x)}. /~

def

(1 1.1 8)

[O , T) x IR.n

andω=ω (ç ， p) ~

nonempty刊lued ，由sed

and locally bounded

(t , x) ,

x IR. n. This

()~.

mt山ifunction

is therefore upper semi-

continuous. An analogue of Theorem 9.1 says that (1 1. 17) gives a global solution to the Cauchy problem (1 1.14)-(1 1.15). Next , set

512 乞r ((0 , T) \G) x IR.n = {(t ， x) εnT : t

rf.

G}

for any G C [0 , T]. In this section we have: Theorem 1 1. 12. Let

4> = 4> (x)

be a finite con时 x function on IR.n. Assume (E .I)-

(E.II). Suppose that the m川 Z Qg 旦， G being αωs in (E.I月). Then (1 1.1 7) determines α global semiclassical solt巾on to

the problem (1 1.14)-(1 1.1 5). Proof. According to the maximum theorem [8 , Theorem 1. 4.16 ], the formulas (9.13)-(9.14) show that the derivatives 仇/缸，仇/θXl ，... ， θ时 âX n exist and are continuous in n~ (see also [128 , Corollary 2.2]). Hence , the global solution u u(t , x) is continuously differentiable in n~. Since G is of Lebesgue measure 0 , we conclude that u = u (t , x) belongs to V (n T ). An argument similar to that in the proof of Theorem 9.1 now gives us the validity of (1 1. 14) in n~.

口

11. GLOBAL SEMICLASSICAL SOLUTIONS

158

Remark. It can be proved that in u

=

nc.j. all partial derivatives of the global solution

u(t , x) exist [(11. 14) is then satisfied though the solution may fa.il to be dif-

ferentiable] if and only if the

mt山 ifunction

later for the smoothness of u

= u( t , x).

Corollary 1 1. 13.

(E .I)-(E. II). Then

=

Let cþ

L = L( t , x) is

finite ωnvex

cþ( x) be a

si吨le-valued.

See S13.2

function on Jæ n.

(1口1.1 7η) dete旷 付m r 旧zn 旧E臼 sα globα al semi化 ω cl归 αssz化 Cα al ω s 01沁 t包di ωiω on ω to 伪 t he

Assume problem

(1 1.14 钊)-(υ1 1.1 时 5 )可 on 附e of 伪 t he following t~咀 ωt

(ωi) 旷=旷 (ωωp 叫) is strictly con肘x on its effecti肘 domain E ~f dom cþ*. (ii) f

=

f(t , p) is strictly p-convex; more

to p on E

for αlmost

every fixed t

ε (0 ，

precise旬，

it is strictly

con 肘x

with respect

T).

Proof. Each of (i)-(ii) implies the strict concavity of

ψ=ψ (t ， x ， p)

to p on E for every (t , x)

ε

[0 , T) x Jæ n. It follows that the

mu 世 山 11 ltif1 缸 u 山肌 nlctior丑1L 二 L(归t ， 叫 x)

is indeed single-valued , and hence

with respect

叩 u ppe 缸r 配 s em 血lC ∞ on 毗 l此削.tim∞u

continuous.

口

= 1, an existence result is established as fo l1ows:

In case n

Theorem 1 1. 14. Let cþ

=

cþ( x) be a finite ωnvex function on

and let φ*φ*(p) be of class

C2

Jæ, E ~f dom cþ* ,

in int E. Assume:

(i) The Hamiltonian f = f(t , p) belongs to CO(((O , T) \G) x

E)

with

< +∞

esssup If(t , pO)1 tε(O ， T)

for some pO ε E and some closed set G c Jæ of Lebesgue measure 0; it is , moreover, twice continuously differentiable in

p εintE

with

max {esss叩 sup Iδf(t ， p)jδ'pl ， ess sup sup Iδ2 f(t ， p)jθ泸 I} 0

(1 1.1 9)

~1 1. 3.

159

EXISTENCE THEOREMS

Then (1 1. 17) determines a global semiclassical sol山on to the problem (1 1. 14)-

(1 1.1 5) with n

= 1.

Proof. We first note that E is a nonempty convex set in IR 1 , and is therefore an interval with end-points , say， α ， b ε[ 一∞，十叫， α 三 b. Consequently, [117 , Corollary 7.5.1] implies the continuity of the restriction of 旷=旷 (p) to E. Condition (i) shows that f = f(t , p) is t-measurable and p- continuous on {O

t < T,

p ε E}.




0 be

an arbitrary number. Since t ξ AC(r) C Leb(A) n Leb(町， we have 唱

M~‘ max ~

~

sup

I óε (O ， T-t) ο

"t+ ó

I

"t+ó

f(r)dr ,

Jt

Because t ε A~ ， there must be

sup

~

I

A(r)dr

~

+ 1 < +∞.

(12.29)

óE(O ， T-t)ο Jt

0(1)

> 0 such that

If(t , x' , u' , p') - f(t , x , u , p)1 < ε/3 whenever max{lx - x'l , lu - u'l , Ip - p'l}

(12.30)

< 0(1). Now choose

p' εQn ， z'εQn n Br , u(l) , U(2) εQnE r

so that max{lx - x'l , Ip - p'l}

< o(匀 ，

u - 0(2)

< U(l) < u < u(2) < u + 0(2) , where

6(2)tfmin{6(1) ， ε/[6M(r 十 1月， ε/[6M(lpl Since t εLeb (J(.， x' , u 间 ， p')) (k

=

1 ，匀， we could find

0(3)

+ 1)]}.

(12.31)

> 0 with the following

property:

创k) ,， 汕内阳，竹切怡 p' 阶 )dd古T←←川川… 川叫一→卅 - 贝附(归 f(tt 旷h 川川川 川u(川叫(伪例 ρ (k)川k) ,， 旷，p' I~~ii + 川(仙例(k)州川川))》川叫 t

ó

(12.32) Finally, setting 0(0) 乞r min{o(刻，以3)} > 0, we have only to show that

|ifs 川的dr 一川 p)1 < εh 川 For k

= 1, 2 one writes "t+ ó

8Jt

f(r， 川 p)dr - f(巾 ， u , p)

=iJfU川， 叫 p )一川叫 ， 叽w 旷 u，， p'刀 p' +:Ifff 忡时飞+刊s队归旷州)一川

+[卡: Iff忡+刊飞 s + [f(μt ， 旷x， , u (价例h的) , pj，斗) -

f(μt ， 凯 x， 叽 u ， p)川].

(口 33)

172

12. MINIMAX SOLUTIONS

Let 15 E (0 ， 15r/

still as in (12.16) , we let

Xu(t ， x ， u ， α) 乞r XFu(. ，.，'Uρ)(t ， x) ，

XL(t ， x ， u ， β)qEfXFL(-v向β)(t ， x).

The family of all multifunctions Fu = Fu(t , x , u ,o:) (resp. FL = FL(巾 ， u , ß)) satisfying Conditions (i)-(iii) wi11 be denoted by Fu (f) (resp. FL (f)). Ifthe Hamil-

= f(t , x , u , p) satisfies

tonian f

a)叶， then Fu (f) 并 ø ， FL (f) 弄。. In fact , under

such hypotheses , we can choose any P , Q

c

IRn with the property that

{s. α:αε P， s 主 O}={s.ß:

ß

ε Q ， s 主 O}

= IR n

and then use (12.14) to define a concrete pair of multifunctions Fu = Fu(t , x , u ， α) and FL

= FL(t , x , u , ß). This pair would satisfy (i)-(iii). (See Part 3 of ~12.2 for 0

the case where P 乞r S, Q 乞r S.) Remark 1. By the Hahn-Banach theorem, it follows from (i) and (12.42) that

Fu(t , x , u ， α) n FL(t , x , u ， ß)

并。

(12.44)

for almost all t E (0 , T) and for allαε P， ß ε Q ， (x ， u) εIR n x R Remark 2. Condition (12.43) holds if Fu

= Fu(t , x , u ， α) and FL =

FL(t ， x ， u ， 的

satisfy the inequalities 唱

由εÞ

inf

唱

inf lims叩

βεQ

rt+6

二 l 6、o .，(托X~(t，.，) ð Jt

p li rp. inf

6、o

for almost all t

sup

~

;::: f(t , x ， 矶时，

rt+6

I

"， (.)ε XF(t 叫。 Jt

ε (0 ， T)

国~. . (z , p)dr Zε FU(T卢(T) 冉冉)

.m皿

Zε FL( T，.，(T) ，也，β)

and for all x

(12.45)

(z ， p)dr 三 f(t ， x ， u ， p)

巳 IRn ， u εIR， p 巳 IRn.

Remark 3. In the case where f = f(t , x , u , p) is a continuous function of its arguments , S由botin [124] and Adiatullina and S由botin [1] do not assume (1 2.43)

Fu(t , x , u ， α) ， FL FL( t , x , u , ß); they assume the upper semicontinuity in (t , x) of these multifunctions (instead of the measurability in t and upper

for Fu

12. MINIMAX SOLUTIONS

178

semicontinuity in x as we do) and assume the equalities in (12 .4 2). However , it pro叫 that

can be ε>

0,

(12 .4 3) holds then. In fact , under such assumptions , for all

(t ， x) ε il T ， U 仨lR， p ε lR n

there exists αε P so that . (z , p) ;三 f(t ， x ， u ， p) 一 ε/2.

f9.in

zε Pu(t ， x ，也 ， a)

Given any x(.) ε Xp(t ， 功， the multifunction (0 , T) :3 r 叶 Fu( r , x( 叶 ， u ， α) is upper semicontinuous. From this and the maximum theorem , it follows that the function (0 , T) :3 8

> 0 is

r 叶

min

,

zEPu(r ， x(r) ，也 ， a)

(z ,p) is lower semicontinuous. Hence , when

small enough , one has rt+ð

云 I

ò Jt

唱

__

，mi~ ，

.

rt+ð 萨『

(z ， p)dr 去 ~I

zEPu(r卢 (r) ，毡， α)'

__JIli~ ，

=

(z , p)-c/2I dr .

- .

.

J

. (z ， p) 一 ε/2 三 f(t ， x ， u ， p) 一 ε

min

zEPu(t 冉也 ， a)

ε X p ( t , x)

(i ndependently of the choice of x(.)

,

LzEPu(t ， x( 吟，也，α)

Ò Jt

-.

because the function family Xp( t , x)

is equicontinuous at t). In other words , the first inequality of (12 届) must be true Dual 吨uments

give the second one. Therefore , by Re mark 2 , (12 .4 3) holds.

Remark 4. Let Fu = Fu(t , x , u ， α) ， FL = FL(t , x , u , ß) be Carathéodoryin (t ， x) ε

il T [i.e. , measurable in t ε (0 ， T) and continuous in x εR叮 satisfying Condition (i). Given any x(.) ε Xp(t ， 叫， by [8 , Theorem 8.2.14] (cf. [29 , Theorem II I.1 5]) , the measurability in r of the multifunctions (O ， T) :3 r 叶 Fu(r， x(叶 ， u ， α) ，

(O , T) :3

r 叶 FL(r， x(r) ， u ， β)

(which follows from [8 , Theorem 8.2.8]) implies that of the functions

(O , T) :3 r 叶

mi~.

. (z ， 抖，

zEPu(r ， x(r) ， 也，由)'

(0 , T) :3 r 叶p1a;x:，

.. (z , p).

zEPdr ，x(r) ，也，β)

Therefore , the integrals in (12 .4 5) exist. Moreover , if (12 .4 2) and (12 .45) hold , then one has in fact the equalities in (12 .45) , because by Lemma 12.3 , (12 .4 2) clearly forces rt+ð

inf 二 I 。εp 八o x( 托 X F (阳)8λ

p li:rp. inf

__

，mi~.

. (z ， p)dr 二

zEPu(r，x(r) ，毡，由)

r t +ó 二 I sup _ min . (z , p)dr 6 、o x( 托 :X;'(t ，x) J Jt "'EPzEPu(r ,x(r) ,u ,a)

< liminf

inf

< liminf

inf

rt+ð

s、o x( 托 XF(t 卢)

二 I

J

Jt

f仆， x(叶 ， u ， p)dr =

f(t , x , u , p) ,

~12.5.

UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS

179

r t+ 6

inf lims叩

βEQ

6 、0-

sup

~

"'(.)EX:;"(t ,,,,)()

I

_.(z ， p}dr 主

max

1t

zEFdT ， "'(T) 冉，β)

r t +6

~ limsup

sup

二 I

> lims叩

sup

三 I

8、o "'(托 XF(t♂) 0 ， αε P， O 三 t

0,

(t ， x) ε f!T ， B ε [t ， T].

人 (t ， x) ，

that is ,

λ (t ， x)] ~三

ε

Therefore , by arguments dual to the ones in

the proof of Lemma 12.8 , we conclude that À+ =λ+ (t， x) is a subsolution (relative to FL) of the Cauchy problem

(12.9)-(12.10).

口

The next lemma will play a crucial role in proving our main theorem. Adiatullina and

Subbotir内 method

of proof ([1] and [124]) , based rather directly on Gronwall's

inequality, seems to break down in the case of time-measurable Hamiltonians. Our road to this result here is devious (by some "perturbation technique" on sets of Lebesgue measure 0) , and proceeds via an implicit version of Gronwall's inequality. Lemma 12.12. Let u

= 百(t ， x) and 旦=旦(t ，

x) be super- and sub-solutions (relative to Fu and FL ) 01 the Cauchy problem (12.9)-(12.10) , respectively. Then u(t ， x) 主 旦(t ， x) on f! T.

12. MINIMAX SOLUTIONS

186

Proof of Lemma 12.12. Assume the contrary, that 百(t(O) ， x(O)) < 坐( t(的， x(O))

for some (t(O) ， x(O)) ξ 否T.

(12.63)

Let D ~f Br C Jæn be such that x(t) ε D for all

x(.) ε XF(t(O) ， x(O)) , t ε [0 ， T]; and A = A(t) ~f AD.E(t) where

AD.E

=

AD.E(t)

is

the function (in L 1 (0 , T)) existing in Condition b) corresponding to the bounded sets D and E qfh( 斗，以2)] with

一∞ < m(l) 乞r

,II).ÌI!... _u(t， x)<m(2)~f ，

(t ，，，， )E[O.到 xD

"

，m.a~ _!!.(t ， x) 0 , jf

e>O.

is lower (resp. upper) semicon the upper semicontinuity in x

of the multifunctions (Fu) f(.,., Ui ， α) (resp. (FL)f(.,., Ui ， β)) for 1 三 t 主 m ，向巳 R1 ， α ， ß ε S. And hence , according to the Remark of ~12.2-Part 2 0 , the sets

(Xu) f(儿 ， X. ， 间， α) ~f X(Fu )i(. ，.，u; 叫 (t问 x*) (resp.

(XL) f(儿， hJhmqEfx(FL)了( .川剧，β) (t们 x*))

(12.85)

are all nonempty and compact for (t* ， x*) ε n T . So , we can make the followi吨: Definition 7. A

s 叩 ersolution

JR"'-valued function U = u(t , x)

of Problem (12.80)-( 12.81) is a lower semicontinuous

= (Ul (t , x) , . . . , um (t, x))

on

nT

which satisfies the

condition I!l.a]C sup ,, ___p ,!l}Ín , • JUi(T， X(T))-Ui(t ， X)] 三 0 aES"( .)E(Xu n(t ，." 叫 (t ，.，)， a)

1 至 i 0 ,

(FL )ï( t , x , u , ß) C (FL Mt ， x ， Ul ，...， Ui- 1， 问- ê , Ui +1,"', u m , ß)

if ε> O.

(FuMt ， x ， 包， α) C (FU)i(t ， X ， Ul ，"'， U←1，问 +ε ， uí+ 1，

iii) For

αlmost

=

all t E (0 , T)

αnd

...

αII (x ， u ， p) ε ]Rn X ]R m X ]R n

for

the following

inequalities hold:

sup

min

,

0

由 εÞ zE(Fu) (t''''. 1L .a)

(z , p) :S h(t ， 民 u ， p) :S i~f

max

_0

一 βEQ zE(Fd ， (t. "，.毡，β)

ιhuιμ

PAPA

.， b''ι

>一 O} , the

smal1est v-coordinate of the hypersurface in the following sense. Theorem 13. 1. Let

4>

=φ (x) be of class C 1 in

convex and of class C 1 in Iæn with

,

Iæ n , and let f = f (p) be strictly

lim f(p)/Ipl = +∞ . Assume (F.II). Then the

Ipl →+∞

白alue u(t(的 ， x(O)) at each (t(O) , x(O)) , t(O)

> 0 , ofthe global solution u =

by Hopf' s formula (9.29) of the

problem (13.1 )-( 13.2) can be

the smallest values of v(t(的， ν)

Cαuchy

=

u(t , x) given

determined αs

((t(O) , x(t(O) ， 时， ν); the minimum bei叼 taken over

all y such that the characteristic c旷ves {(t , x(t , y))

t 主 O} starting from

(0 , y)

meet each other α t (t(O) , x(O)). Proof.

By hypotheses ,

f*

f* (z) is the same as the Legendre conjugate of

f = f(p) (see Lemma 9.13 and [117 , Theorem 26.6]); moreover , both f' and one-to-o日e mappings from

from (13.5) that (=

f*'

Iæ onto itself with n

((t ， x ， ν)

f*'

are

= (1 ')-1. Therefore , it follows

is differentiable in y with

θ〈

石。， x， ν)= 矿(y)

-

f*'(斗旦)

It has been shown in the proof of Theorem 9.12 that (9.29)

( 13.6) deterrr山es

a global

solution u = u(t , x) of (13.1)-(13.2) , and that the infimum

叫t叭 z(0))=AC(t 叭 z 叭 y) is attained at some y(O) ι Iæ n .

(13.7)

Clearly, y(O) must be a stationary point of the

function ((t(的，川， .); i.e. ,

手 (t(的， x(O) ， y(O)) uy

= 0

Thus (13.6) implies 扩 (y(O)) = fμ ((x(O) - y(O))/t(O)). 8ince fμ= (1 ')-1 , we have ",

(0) _ ,, (0)

气而~ = f'(4)'( ν(0))) ，盹

x(O) = y(O)

+ t(O忡忡， (ν(0))) =

x(t(O) , Y叫

for every such stationary point y(O). 80 , the characteristic curve {( t , x( t , y(O))) t 主 O} ， which is starting from (O , y(O)) , must p出s through (t(O) ，川的). Because the

minimum in (13.7) is not affected if it is taken not for all y but only for those stationary y , the proof is then

complete.

口

313. 1. CONSTRUCTION OF GLOBAL SOLUTIONS VIA CHARACTERISTICS

We now turn to second case of Hopf's formulas , *) is the relative interior of dom 旷. Therefore , (13.13)-(13.14) yield 仰 ， x) 三 u(t ， x).

and choose

We now prove u(t ， x) 三仰， x). Fix any q εri (domφ*) ， ε> 0 ,

q εdom

4>*

such that

u( t , x) On setting p(叫乞r (1 - l/m)q

三 ψ (t ， x ， q) + ε.

+ (l/m泪，

we see , by (13.15) and [117 , Theorem

6.1 ], that p(m) εri (dom 旷) C E for m = 1, 2, 3,... Since the Fenchel conjugate 伊 =

4>*(p)

Ís a lower semicontinωus proper convex function , [117 , Corollary 7.5.1]

shows that 旷 (q) = lim 旷 (p(m)). Hence , it follows from (13.12) that m---+c白

+

ε

h7ε

阳+

01J 川飞

)ttεPA +zl

目

、叶 J

飞乙

4问」

ψz ET P‘

'忖峰，，，飞、lJ

、PU-uε

/1

ZJ

#川区中

E

Z

、、 ，，，

ATb

Ymspmu 川 =φ(x) be convex and of class C 1 on ]R n. Assume (E .I )-(E.II). It has been proved in Chapter 9 that (9.7) then coincides with (13.14) , and that the supremum is always attained (at some point

p εdom 4>*).

The situation becomes

different for the supremum in (13.13). We show this by the following example. Let n

def ~.

1,

叫 i:一向 and

p> 1, pε[0 ， 1] ，

p

创J(;11，

Then

*(p) =

< 0,

Z 三 0，

> O.

x

pε[0 ， 1 ]，

{士 P

p~[O ， l ]，

and 位 (t ， x) =s叩 {x. e宙一旷 (el')

- tf(e ll )}.

F 三。

It is easily seen that the

value 位 (1 ，一 2)

= sup {-2e ll } = 0 can not

be attained in

F 三。

(13.13) at any point y ε ]Rl. In the remainder of this section , we make the following assumption: (E.V) For each

(t ， x) ε Ð，

the

s叩 remum

in (13.13) is attained at some point

yεRπ.

Remark. Let 4> =仰) be convex and of class C 1 on ]R n. Assume (E .I )-(E.II). Then (E. V) is fulfilled , for example , if dom 旷 is an open set (cf. (13.15) and the Remark above). In

par 叫.

convex function , i.e. when it is strictly convex with . lirp. [4>(入 x)/ λ]= +∞ for all Z

ε ]Rn\{O}.

See [117 , Theorems 26.5-26.6].

λ→+∞

= f(t , p) and 4> = 4> (x) be of class C 2 on {t ~ 0 , p ε ]Rn} r叩ectively， such that 4>" 乞r (θ2 4>/(θ的θXj ))ω=1 ，2 ，...，n 切sαJ灿 ωαν伊s pos必iti肘

Theorem 13.3. Let f

αnd

]Rn ,

deβηite. Assume (E. I1)αnd

(E. V). Then the value u(t{的， x{O)) α t each (t{的， x{O)) ε

13. MISHMASH

204

Ð

0/ the global solution u

=

U(t , x) given

bν Hopf's

/ormula (9.7)

0/ the

Cα也chy

problem (13.8)-(13.9) can be determined as the largest values 0/ u(t(0) ， ν)=ψ (t(O) ， x(t(的， ν) ，矿 (ν) );

the m α xz阳 阳 m ~1包L川mb 川 阮ez饥 叼 n gt归 α ken

0

t 主 0 叶} st切 αr付ting from 仰 ( 0 ， νω) me倪et each other at (t(O) , x(O)).

Proof.

D.

This set is called the affine hull of D and is denoted by aff D. The relative interior of a convex set D in IRn , which we denote by ri D , is defined 出 the

interior which results when D is regarded as a subset of its

a血ne

hull aff D.

In other words , ri D qzfhε affD

(x

+ êB) n (affD) c

D for some

ê

> O} ,

where B stands for the unit ball (centered at the origin) in IR n . Theorem AII. 1. [117 , Theorem 6.1] Let D be a convex 8et in IRn. Let x αnd y ε D. Then (1 一 λ )x + λuεriD for 。三人 f(x ， ν ， u ， θu/ θy) ， Jα;p. J. Math. 15 (1938) , 51-56. [108] N agumo, M. and Simoda ，丘， Note sur l'inégalité différentielle concernant les équations du type paraboliq肘 ， Proc. J，α.pan Acαd. 27 (1951) , 536-539. [109] Naka肘，丘， Formation of shocks for a single conservation law , SIAM J. M，α th. Anα1. 19 (1988) , 1391-1408. [110] -, Fo口nation of singularities for Hamilton-Jacobi equations in several space variables , J. Math. Soc. Japan 43 (1991) , 89-100. [111] Ngoan , H.T. , Hopf's formula for Lipschitz solutions of Hamilton-Jacobi equations with concave-convex Hamiltonian , Acta M，α th. Vietnam. 23 (1998) , 269 294. [112] Oleinik , O.A. , Discontinuous solutions of non-linear differential equations , Uspekhi Mat. Nauk 12 (1957) , 3-73 (in Russian). English trans l. in Amer. M，α th. Soc. Tr，α nsl. 26 (1 963) , 95-172. [113] Pl溢， A. , Characteristics of non-linear partial differential equations , Bull. Acad. Polon. Sci. 2 (1954) , 419-422. [114] Qin Tie-hu , On the existence of global smooth solutions for Cauchy problems to first order quasi-linear partial differential equations , J. of Fudan Univ. 22 (1983) , 41-48. [115] Quinn , B. K., Solutions with shocks , an exar即le of an L 1 -contractive semi-group , •

Comr 阳 m1

Robert , R. P. , Convex /unctions , monotone operators αnd differentiability , Springer , Berlin , 1989. [117] Rockafellar , R. T. , Convex analysis , Princeton Univ. Press , 1970. [118] Rozdestvenskii , B. L., Disc [口116]

REFERENCES

233

[121] Smoller , J. , Shock waves and reaction-diffusion equations , Springer- Verlag , Berlin, 1983. [122] Souganidis , P.E. , Existence of viscosity solutions of Hamilton-Jacobi equations , J. Differential Equations 56 (1985) , 345-390. [123] 一， Approximation schemes for viscosity solutions ofHamilton-Jacobi equations , J. Differential Equations 59 (1985) , 1-43. [124] S由botin ， A. I., Minimax inequalities αnd Hamilton-Jα cobi equations , N auka , Moscow , 199 1. (in Russian) [125] -, Generalized solutions 01 介st-order PDEs: the dynamical optimization perspective , Birkhäuser, Boston , 1995. [126] Subbotin , A. 1. and S1吨alova， L. G. , A piecewise linear solution of the Cauchy problem for the Hamilton-Jacobi equation , Russiαn Acad. Sci. Dokl. Math. 46 (1993) , 144-148. [127] Subbotina, N. 忧， Cauchy's method of characteristics and generalized solutions of Hamilton-Jacobi-Bellman equations , Dokl. Acad. N，α uk SSSR 320 (199 月， 556-56 1. (in Russian) [128] Szarski ，止 ， Differential inequalities , Monografie MatematyczI盹 Warszawa， 1967. (2. , rev. ed.) [129] Tarasyev , A.M. , Approximation schemes for the construction of the generalized 叫ution of the Hamilton-Jacobi (Bellman-Isaacs) equation , Prikl. Mat. Mekh. 58 (1994) , 22剖. (in Russian) [130] Tarasyev , A.M. , Uspenskii , A.A. , and Usl叫ov ， V.凤， Approximation operators and finite-difference schemes for the construction of generalized solutions of Hamilton-Jacobi equations , 1扣. RAN Tek阳 加7加η1. 茹 h Kibernet. 3 (1994) , 173-185. (in Russian) [131] Tataru , D. , Viscosity solutions of Hamilton-Jacobi equations with unbounded non1ine缸 terms， J. Math. Anal. Appl. 163 (1992) , 345-392. [132] Thai Son , N.D. , Liem , N.D. , and Van , T. 且， Minimax solutions of first-order non1inear partial differential equations with time-measurable Hamiltonians , in Dynαmical systems and α.pplicα tions ， pp. 647-668 , World Sci. Ser. App l. Anal., Vol. 4 , World Sci. Publishing , River Edge , NJ , 1995. (Agar~al ， R. P. , Ed.) [133] -, Minimax solutions for monotone systems of first-order nonlinear partial differential equations with time-measurable Hamiltonians , Funkciα laj Ekvα CWJ 40 (1997) , 185-214. [134] Thom , R., The two-fold way of catastrophe theory, Lecture Notes in Math. , Vol. 525 , 1976 , pp. 235-252 (Spri吨er- Verlag). [135] Tomita, Y., On a mixed problem for the multi-dimensional Hamilton-Jacobi equation in a cylindrical domain , Hiroshima Math. J. 8 (1978) , 439-468. [136] Tsuji , M. , Solution globale et propagation des sing

REFERENCES

234

[139] Tsuji , M. and Li Ta-tsien , Gl'O bally classical s'Oluti 'O ns f'Or n 'O nlinear equati 'O ns 'Of first 'Order , Commun. in PDEs 10 (1985) , 1451-1463. [140] 一， Remarks 'O n characteristics 'Of partial differential equati 'O ns 'Of first 'O rder , Funkcial，α~ Ekvacioj 32 (1989) , 157-162. [141] Van , T.D. , Hap , L.V. , and Thai S'On , N.D. , On s'O me differential inequalities and the uniqueness 'Of gl 'Obal semiclassical s'O luti 'Ons t 'O the Cauchy pr'O blem f'Or weakly-c'O upled systems , Jo旷nal 01 Inequaμties αnd Applications 2 (1998) , 357372. [142] Van , T.D. and H 'Oa吨， N. , On the existence 'Of gl'Obal s'Oluti 'O ns 'Of the Cauchy pr'Oblem f'Or Hamilt 'On-Jac 'Obi equati 'Ons , SEA Bu ll. Math. 20 (1996) , 81-88. [143] Van , T.D. , H'Oa吨， N. , and G 'O renfl'O, R., Existence 'Of gl'O bal quasi-classical S'Oluti 'O ns 'Of the Cauchy pr'Oblem f'O r Hamilt 'On-Jac'O bi equati 'O ns , Di.fferentsial'nye Uravneniya 31 (1995) , 672-676. (in Russian) [144] Van , T.D. , H'Oa吨， N. , and Thai S'O n , N.D. , On the explicit representati 'On 'Of gl 'Obal s'O luti 'Ons 'Of the Cauchy pr'Oblem f'O r Hamilt 'O n-Jac'Obi equati 'O ns , Acta Math. Vietnam. 19 (1994) , 111-120. [145] Van , T.D. , H'Oa吨， N. , and Tsuji ，旺， On H 'Op f' s f'O rmula f'O r Lipschitz s'O luti 'O n s 'Of the Cauchy problem f'Or Hamilt 'O n-Jac 'Obi equati 'Ons , Nonlineαr Analysis , Theory , Methods & Applications 29 (1997) , 1145-1159. [146] Van , T.D. , Liem , N.D. , and Thai S'On , N.D. , Minimax s'O luti 'O ns f'O r s'Ome sys t旬 ems 'Off鱼Ìrst-'O创 r由 d er n 'O nlinear partial differential equati 'Ons with time-measurable H皿 aml让 灿 b lt'O ω 'Oma

[147]

[148]

[149] [150]

[151]

[152]

1995 盯/九， pp. 499-511 , W'Orld Sci. Publishing , Ri ver Edge , NJ , 1996. (M 'Orim 'Ot 'O, M. and Kawai , T. , Eds.) Van , T.D. and Thai S'On , N.D. , On the uniqueness 'Of gl 'O bal classical s'Oluti 'O ns 'Of the Cauchy pr 'O blem f'O r Hamilt 'On-Jac'O bi equati 'Ons , Acta Math. Vietnam. 17 (1992) , 161-167. - , On the uniqueness 'Of gl 'Obal cla由 cal s 'O luti 'Ons 'Of the Cauchy 严'Oblems f 'Or n 'Onlinear partial differential equati 'O ns 'Of first 'O rder , Actα Math. Vietnα m. 18 (1993) , 127-136. 一， On a class 'Of Lipschitz c'Ontinu'O us functi 'O ns 'Of several variables , Proc. A mer. Math. Soc. 121 (1994) , 865-870. 一， Uniqueness 'Of gl'O bal semiclassical s'O luti 'Ons t 'O the Cauchy pr'O blem f'Or 鱼rst 'O rder n 'Onlinear partial differential equati'Ons , Di.fferential Equations 30 (1994) , 659-666. Van , T.D. , Thai S'On , N.D. , and Hap , L.V. , Partial differential inequalities 'Of Haar type and their applicati 'Ons t 'O the uniqueness pr'Oblem ，的etnαm J. Math. 26 (1998) , 1-28. Van , T.D. , Than让此】 gl 'Obal s'Ol

REFERENCES

235

[155] -, Sur l'unicité et la limitation des int句rales de certains syst色mes d 毛quations auxd岳rivées partielles du premier ordre , Annα li di Matemαtica pura ed applicata 15 (1 936-37) , 155-158. [156] •, Uber die Bedi吨ungen der Existenz der Integrale partieller Differentialgleichungen erster Ordnu吨， Math. Z. 43 (1938) , 522-532. [157] -, Systèmes des équations et des inégalités différentielles ordinaires aux deuxi品mes membres monotones et leurs applications , Ann. Soc. Polon. Math. 23 (1950) , 112-166. [158] Westphal , H. , Zur Abschätzung der Lösungen nichtlinearer parabolischer Dif二 ferentialgleichungen , Uαth. Z. 51 (1 949) , 690-695. [159] Whitney, H. , On síngularitíes of mappíngs of Euclidean spaces 1, Ann. Math. 62 (1955) , 374-410.

Index

Cantor function (1 adder) , 86 Cantor set , 86 Carathéodory differential equation , 98 Carathéodory's conditions , 97 , 166 , 191 characteristic curves , 1 equations , 2 , 4 strips , 4 system of differential equations , 4 Ck-function , 1 Ck-solution , 7 collision of characteristic curves , 26 , 28 of singularities , 41 , 63 comparision equation , 98 concave-convex function , 128 conJugate (Fenchel) concave-convex , 131 (Fenchel) convex , 105 , 219-222 Legendre , 128, 220-222 contact discontinuity, 70 cusp point , 58 d.c. function , 113 d.c. representation , 117 differential inclusion , 225 differential inequality of Haar type , 85 Dini semiderivative , 108 , 209 Dini semidifferential , 210 directional derivative , 108 effective domain , 105 , 218 entropy condition , 52 epigraph , 218 equation of Hamilton-Jacobi type , 34

equation of the conservation law , 12 , 45 Fi 1ippov's theorem , 180 , 226 fold point , 57 genera1ized solution , 32 , 55 general PDE of first-order , 4 global C 1 -so1ution , 92 global semiclassical solution , 147 , 154 global solution , 104 Haar's differential inequality, 85 Haar's theorem , 7 日 adamard' s lemma , 19 Hopf's formulas , 104 Jacobian , 2 Jacobi matrix , 2 Lebesg肘 's theorem , 170 Legendre transformation , 128 , 220-222 life span of classical solutions , 12

marginal function , 224 maximum theorem , 224 minÌ max solution , 168 , 179 , 194 , 196 monotonÌ city condition , 192 multifunction , 222 closed , 223 closed-valued , 223 continuous , 223 locally bounded , 223 lower semicontinuous , 223 measurable , 223-224 upper semicontinuous , 223

INDEX

non-degenerate singularity, 58 proper, 123 , 218 quasi-linear PDE of first-order , 1 quasi-monotonicity condition , 192

strict convexity, 118 , 122 , 219 subdifferential , 210 subgradient , 210 subsolution , 168 , 179, 194 , 196 superdifferential , 210 supergradie时， 210

supersolution , 167, 179 , 193 , 196 Rankine-Hugoniot's condition , 41 rarefaction wave , 68 relative interior , 217 semi-concavity, 32 , 39 , 56 shock wave , 53

viscosity solution, 212 Wazewski's theorem , 9 weakly-coupled system , 97 weak solution , 45

237