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

CHAPMAN & HALL/CRC

Monographs and Surveys in Pure and Applied Mathematics

I 01

THE CHARACTERISTIC

METHOD AND ITS

GENERALIZATIONS FOR FIRST-ORDER NONLINEAR

PARTIAL DIFFERENTIAL EQUATIONS TRAN DUC VAN MIKIO TSUJI

NGUYEN DUYTHAI SON

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

Main Editors H. Brezis, Universitéde Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board H. Amann, University of ZUrich R. Aris, University of Minnesota G.I. Barenblatt, University of H. Begehr, Freie Universitat Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgassner, Universitat Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta S. Mon. Kyoto University L.E. Payne, Cornell University D.B. Pearson, University of Hull I. Raeburn, University of Newcastle G.F. Roach, University of Strathclyde 1. Stakgold, University of Delaware WA. Strauss, Brown University J. van der Hock, University of Adelaide

IC

CHAPMAN & HALUCRC

Monographs and Surveys in Pure and Applied Mathematics

I0I

THE CHARACTERISTIC

METHOD AND ITS

GENERALIZATIONS FOR FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TRAN DUCVAN MIKIOTSUJI NGUYEN DUY THAI SON

CHAPMAN & HALL/CRC Boca

Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Tran, Due Van. The characteristic method and its generalizations for first-order nonlinear partial differential equations / Tran Due Van, Mikio Tsuji. and Nguyen Duy Thai Son. p cm. -- (Chapman & Hall/CRC mongraphs and surveys in pure and applied mathematics 101) Includes bibliographical references and index. ISBN 1-58488-016-3 (alk. paper) I. Differential equations. Nonlinear--Numerical solutions. 111. Title. I.Tsuji, Mikio. II. Nguyen, Duy Thai Son. IV. Series. QA374.T65 1999 519.5'.353--dc2I 99-27321 CIP

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4567890

Contents

Contents

Preface

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

1

1.1. Characteristic method and existence of solutions

1

1.2. AtheoremofA.Haar

7

1.3. A theorem of T.

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

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. Sufficient conditions for collision of characteristic curves I 3.5. Sufficient conditions for collision of characteristic curves II

12 12 13 18

22 22 23

24 26 28

Chapter 4. Equations of Hamilton-Jacobi Type in One Space Dimension

30

4.1. Nonexistence of classical solutions and historical remarks 4.2. Construction of generalized solutions 4.3. Semi-concavity of generalized solutions 4.4. Collision of singularities

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 of Hamilton-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 6.1. Introduction

44 44

47 48 52

55 55

CONTENTS

6.2. Construction of solutions 6.3. Semi-concavity of the 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 and contact discontinuity

56 61

63

67

7.3. An example of an equation of the conservation law 7.4. Behavior of the shock S1 7.5. Behavior of the shock S2

67 68 70 74 77

Chapter 8. Differential Inequalities of Haar Type

82

8.1. Introduction 8.2. A differential inequality of Haar type

82 84

8.3. Uniqueness of global classical solutions to the Cauchy problem 8.4. Generalizations to the case of weakly-coupled systems

91

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 I = f(p)

95

103 103 105 113 121

Chapter 10. Hopf-Type Formulas for Global Solutions in the case of Concave-Convex Hamiltonians 127 10.1. Introduction 10.2. Conjugate concave-convex functions 10.3. Hopf-type formulas

127 128 136

Chapter 11. Global Semiclassical Solutions of First-Order Partial Differential Equations 146 11.1. Introduction 11.2. Uniqueness of global semiclassical solutions to the Cauchy problem 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. Uniqueness 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

146

148 156

161 161 165 173 176 179 191

198 I 9R

CONTENTS

13.2. Smoothness of global solutions 13.3. Relationship between minirnax and viscosity solutions

205 208

Appendix I. Global Existence of Characteristic Curves Appendix II. Convex Functions, Multifunctions, and Differential Inclusions

214

AII.1. Convex functions AII.2. Multifunctions and differential inclusions

References Index

217 217 222

227 236

Preface

of the main results of the classical theory of first-order partial differential equations (PDEs) is the characteristic method which asserts that under certain One

assumptions the Cauchy problem can be reduced to the corresponding characteristic system of ordinary differential equations (ODEs). To illustrate this, let us consider the Cauchy problem for the nonviscid Burger equation: Ou

,3u

—+u----=O, Ot Ox

t>O,XER,

u(O,x)=h(x),

XER.

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

x(t)) =

x(t)) + u(t,

By the chain rule, we may simply require dx/dt = x = x(t) can be defined by dx

x(t)). u,

and so the characterist2cs

= u(t,x).

Along each characteristic x = x(t) we have du/dt = 0, i.e., ti = u(i,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 (2), the characteristic passing through any given point (0,s) on the x-axis is x = a + h(s)t, on which u has the constant value:

u = h(s).

Hence, if the C1-norm of h = h(s) is bounded, then, by means of the implicit function theorem and (4), we can get S

= .s(t,x)

PREFACE

for small values of t. Substituting (6) into (5) gives the classical solution (C1solution)

u = h(s(t, z))

(7)

to our Cauchy problem (1 )-(2). However, in general, this solution exists only locally

in time. In fact, if h = h(s) is not a nondecreasing function of s, there exist two points

and (0,52) on the x-axis such that

<s2

and

h(s1) > h(s2).

(8)

Then the characteristic curves beginning from (0, Si) and (0, s2) will intersect at time —

the solution u = u(t, x) is constant along each of the two curves but has different values h(si) 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 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) are. On the other hand, if h = h(s) is a nondecreasing function of s, then the characteristics emanating from distinct points (0, and (0,82) on the s-axis wiU not intersect, and thus the solution u = u(t, x) will exist globally for t 0. Since

The previous example shows, generally speaking, that for (first-order) nonlinear partial differential 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 discovered by Aizawa, S. [2]-[4], Bakhvalov, Benton, S.H. 121], Conway, E.D. 1321-1331, Douglis, A. [44]-[45], Evan, C., Fleming, W.H.

[511-1521, Friedman, A. [541, Celfand, I.M., Codunov, S.K., Hopf, E. [63]-[64j, Kuznetsov, N.N. [95], Lax, P.D. [97]-[98], Oleinik, O.A. [112), Rozdestvenskii, B.L. [118], and other mathematicians. Among the investigations of this period we should

mention the results of Kruzhkov, S.N. ([87)-[92], [94)), which were obtained for Hamilton-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 solutions 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 theory are that it allows merely nonsmooth functions to be solutions of nonlinear PDEs, it provides very general existence and uniqueness theorems, and it

PREFACE

yields precise formulations of general boundary conditions. Let us mention here the names: Crandall, M.G., Lions, P.-L., Aizawa, S., Barbu, V., Bardi, M., Barles, G., Barron, E.N., Cappuzzo-Dolcetta, I., Dupuis, P., Evans, L.C., Ishii, H., Jensen, R.,

Lenhart, S., Osher, S., Perthame, B., Soravia, P., Souganidis, P.E., Tataru, D., Toinita, Y., Yamada, N., and many others (see [51, [10]-[20], [28], [351-139], [471-1501, [67]-[72], [79], [99]-[1O1], [122]-[123J, [1311, and the references therein), whose con-

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.!. Subbotin. This leads to the notion of minimax solutions of first-order nonlinear PDEs. As the terminology "minimax solutions" indicates, the definition of such global solutions is closely connected with the minimax operations. This definition is based, to some extent, on the so-called "characteristic inclusions" (a generalization of the classical characteristic system in this situation). Subbotin and his coworkers (11], [124]-[127], [1291-11301) developed

an effective theory of minimax solutions to first-order single PDEs and gave nice applications to control problems and differential games. The research of rninimax 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-Hopf transformation, which have been discovered by V.P. Maslov and his coworkers. Indeed, V.P. Maslov, V.N. Kolokol'tsov, S.N. Sainborskii, and others developed a nonclassical approach to define the weak global solutions to firstorder nonlinear PDEs, in which by a suitable structure of new function semimodules, nonlinear operators become "linear" ones. In this direction, based on the methods

and results of the well-developed linear mathematical physics, investigations of first-order nonlinear PDEs with convex Hasniltonian have been considered (see [82], [105]).

Concerning the theory of differential inequalities, let us mention that the theory of ordinary differential inequalities was originated by Chaplygin [30] and Kaznke 181], and then developed by [157]. Its main applications to the Cauchy problem for (ordinary) differential equations concern questions such as: estimates of solutions and of their existence intervals, estimates of the difference between two solutions, criteria of uniqueness and of continuous dependence on initial data and right sides of equations for solutions, Chaplygin's method and approximation of solutions, etc. Results in this direction were also extended to (absolutely continuous) solutions of the Cauchy problem for countable systems of differential equations satisfying Carathéodory's conditions. We refer to Szarski [128] for a systematic study of such subjects.

PREFACE

As for the theory of partial differential inequalities, the first achievements were obtained by Haar [611, Nagumo [107], and then by [154]. Up to now the

theory has attracted a great deal of attention. (The reader is referred to Deimling [40], Lakshniikanthain and Leela [96], Szarski [1281, 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 this 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 determine the life spans 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 which are necessary for our following discussions: the characteristic method, existence of local solutions, and Theorems of Haar and on the uniqueness of solutions to the Cauchy problem in C1 -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 Chapters 4-5 and this one is the dimension of space. But the problem caused by this difference would become much more complicated. In Chapter 7, we treat the case where equations are not convex and not concave. We will give only typical phenomena appearing in these cases without developing a general theory.

In the last six chapters, the first-order PDEs under consideration are always assumed to satisfy Carathéodory's conditions (or something like them). Chapter 8 introduces the so-called differential ineqtzalities of Haar type. Here we use some new techniques based on the theory of multifunctions and differential inclusions to investigate the uniqueness problem. The idea originates from the generalized characteristic method. Roughly speaking, "characteristic differential inclusions" and

PREFACE

"characteristic bundles" are invoked instead of characteristic differential equations and characteristic curves. Chapters 9-10 are devoted to the study of Hopf-type formulas for global solutzons to the Cauchy problem in the case of non-convex, non-concave Harniltonians 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 of two convex functions). In Chapter 10, the Hazniltonians are concave-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 global 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 193] is given. In Chapter 12, we extend the notion of Subbotin's minimax 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 "perturbation technique" on sets of Lebesgue measure 0), 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 parameters. The results are new even when restricted to the case of continuous Hamiltonians. Generalizations for monotone systems will 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 first-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 necessary facts, mostly of nonsmooth analysis and the theory of differential inclusions. The authors are grateful to Professor H. Begehr for his proposal that we prepare this book for Chapman & Hall / CRC Monographs and Surveys in Pure and Applied Mathematics. The work of the first and third named authors was supported in part by the National Center for Natural Science and Technology and the National Basic Research Program in Natural Science, Vietnam. Some parts of this monogragh were written

and typeset at Ohio University (USA) where the third named author was a guest for two quarters in 1998 by invitation of the Department of Mathematics, College of Arts and Sciences. We record here our gratitude for all the support and help we received from these institutions.

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:

= ao(t,x,u) in

+

U,

tz(0,x)=4)(x) on is an open neighborhood of (i, x) = (0,0). Let V be an open neighborhood Assume that a1 = a,(t,x,u) (1 = 0,1,... ,n) of {(0,x,4)(x)) x U0} in and 4) = 4)(x) are of class C1 in V and U0, respectively. A function is said to be of class Gil if it is k-times continuously differentiable, and Ck(U) is the family of functions being of class C" in U. A C"-function means that it is a function of class U

:

Ck. Characteristic curves of (1.1)-(1.2) are defined by solution curves of the following system of ordinary differential equations:

J

!.a1(i,r,v) = ao(t,x,v).

1. LOCAL THEORY

In accordance with (1.2), the initial condition for (1.3) is given by

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

x,(O) =

(1.4)

v(O) =

The ordinary differential equations in (1.3) are called the "characteristic equations" for (1.1), where we use v = v(t,y) instead of u =u(t,x) to avoid confusion. In the following discussions, u = u(i, x) is a solution of (1.1) and v v=(i, y) is a solution of (1.3)-(1.4) which is equal to the value of ii = u(t,x) restricted on the corresponding solution curve x = x(t,y) of (1.3)-(1.4). As a1 = a1(t,x,v) (i = O,1,...,n) and = are of class C1 in V and (Jo, respectively, the Cauchy problem (1.3)-(1.4) has a system of solutions x1 = x1(i, y) (i = 1,2,..., n) and v = v(t, y) which are of

class C1 inaneighborhoodof{(O,y) yEUo}. Let us fix our notations on derivatives of functions. A vector x is vertical, i.e., x = Therefore dx/dt = On the other hand, given any real-valued function q5 = q5(x), we write grad4(x) = = qV(x) = ,8c6/Ox,1). For an n-vector valued function x = x(y) of an n-vector y, we define its Jacobi matrix and Jacobian, respectively, :

by Ox1

Ox1

Ox1

0111

0112

OYn

df a

III

axn

am 01,1

and

Dx

del

= det

Y

We

01/n

0112

(Ox1

\ Y3 ij=1,2

a

will sometimes write the Jacobi matrix simply by

Since x(O, y) = y,

we see that (Dx/Dy)(t,y) = 1 for t = 0, y E Uo. In a neighborhood of {(O,y) y E Uo}, as (Dx/Dy)(t,y) does not vanish, we can uniquely solve the equation x = x(t,y) with respect toy and write the solution by y = y(t,x). Putting u(t, x) v(t,y(t,x)), we will prove that u = u(t,x) satisfies (1.1)-(1.2) in a neighborhood of the origin.

Theorem 1.1. The Cauchy problem in a neighborhood

of the origin.

(1.1)-(1.2)

has

uniquely a solution of class

C1

§1.1. 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(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

(p!-) Yj

n

= I (identity matrix),

x3

.

Ox

Oy

i,j=1,2 As

(1.5)

i,j=1,2 —

n

u(t,x) = v(t,y(t,x)), we have Ou

=

Ov

+

OvOy

(1.7)

By (1.7), using (1.5) and (1.6), we get Ou

= ao(t,x,u(t,x)) — = ao(i,x,u(t,x)) — = ao(t,x,u)

As u(O,x) = v(O,y(O,x)) =

cS(s),

Ov

Ox

—

\ Ox

Ou

—

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(t, x) be any solution of class

C' of(1.1)-(1.2), and put where x = x(t,y) and v = v(t,y) are the solutions of (1.3)-(1.4). Then the difference w(t,y) — v(t,y) satisfies the following Cauchy problem:

J L.

=

(a,(t,x,v)

—

+

—

ao(i,x,v))

w(O,y) = 0.

As the right-hand side of this differential equation can be estimated by — vi = we get w(t,y) v(i,y) for any (t,y) in a neighborhood of 0, i.e. Miwl, the origin. This means that the solution of C'-class is unique along the curves x = x(t,y). That is to say, as long as the Jacobian (Ds/Dy)(t,y) does not vanish, the solution of (1.1)-(1.2) is unique in the C'-space. 0

1. LOCAL TREORY

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

(1.8)

U

on

is an open neighborhood of the origin. Let V be an open neighborhood of Assume that I = f(t,x,u,p) x E U0} in R x RTh x R x and = are of class C2 in V and U0, respectively. Characteristic strips for (1.8)-( 1.9) are defined as solution curves of the following system of ordinazy differential equations: U

:

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

= dv

at dp1

=

=

"

Of

Of

(1.10)

Of

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

with

x,(0)=y,, v(0)=cb(y),

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

(1.11)

We remark that system (1.10) is called the "characteristic system of differential equations," or simply "characteristic equations," for equation (1.8). As I and are of class C2, the Cauchy problem (1.10)-(1.11) has uniquely the solutions x = x(t,y),v = v(t,y) and p = p(t,y) in a neighborhood oft = 0. Moreover, they are of class C1 with respect to (t, y). As x(0, y) = y, we have (Dx/Dy)(0, y) = 1 for any y E Uo. Therefore, there exists an open neighborhood W of {(0, y) y E Uo} where the Jacobian (Dx/Dy)(t, y) does not vanish and the equation x = x(t, y) can be uniquely solved with respect to y. Denote the solution by y = y(t, x), and put . 2 u(t,x) def ,and that it = v(t,y(t,x)). We will prove that u = u(t,x) is of class C satisfies the Cauchy problem (1.8)-(1.9). For this aim, we prepare some lemmas. :

Lemma 1.2. For all (t,y) in the existence domain of solutions to (1.10)-(1.11), we have

§1.1. CHARACTERISTIC METHOD ANI) EXISTENCE OF SOLUTIONS

5

OX1

81/i

01/2

:::

:::

:::

:::

,

(1.12)

oxn

0Y18Y28Yn where

p(t, y) = (pi(t, y), p2(t, y),... ,

y)).

Proof. We put On OYi deC t9V

z(t,y) =

Ox1 0112

01/n

01/2

01/n

—p(t,y)

Using (1.10), we have

(

y) =

z(0,y)=0.

As this is a linear ordinary differential equation concerning z = z(t,y), we get 0 z(t,y)wO.

Remark. Fix any y E Uo, and let J C R be an interval around 0 on which the solutions x = x(t,y),v = v(t,y) and p = p(t,y) of the characteristic equations (1.10)-(1.11) exist. Then (1.12) is true for each t E J even if the Jacobian may vanish at (i,y).

that in W

have (Dx/Dy)(t, y)

0, and we can uniquely solve the equation x = x(t, y) with respect to y. The solution has been denoted by y = y(t, x) deC and used to define u(t,x) = v(t,y(t,x)). Recall

Corollary

we

1.3. In the definition domain of y = y(t,x),

=

(i =

we have

1,2,... ,n).

Using Lemma 1.2 and its corollary, we get the following:

1. LOCAL THEORY

Theorem 1.4. Suppose f 6 C2 and has

6 C2. Then the Cauchy problem (1.8)-(1.9)

uniquely a solution of class C2 in a neighborhood of the origin.

Proof. We first prove the existence of solutions. Let u(t,x)

v(t,y(t,x)) as in

the above notations. By Lemma 1.2 and Corollary 1.3, using (1.10), we have Ou

x) =

y(t, x)) +

i9v

=

y(t, x)) . —

Oy

x)

f(t,x,v,p)—p(t,y(t,x)) 1O, LO, c 0 (Dx/Dy)(t,y°) = 0}. (Notice :

thatO0, rE u(0,x) = 4(x) on {t =

0,

x E R"},

(3.1) (3.2)

where f = f(t,x,u,p) and

are of class C2 in R x Rx and = respectively. The characteristic equations corresponding to (3.1)-(3.2) are given by (2.15)-(2.16). Let x = x(t,y), v = v(t,y), and p = p(i,y) be solutions of (2.15)(2.16). We consider the Cauchy problem (3.1)-(3.2) in the following situation:

(I) (Dx/Dy)(t°,y°) = 0, and (II) (Dx/Dy)(t,y°)

0 for t < t°.

0def

We put x = x(t o , y o ). Theorem 2.2 says that, when (t, x) goes to (to , x o ) along the curve r = x(t,y°), 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 C1-solution in a neighborhood of the point (t°, x°). Our problem is to see whether or not we

can extend the classical solution u = u(t, x) beyond the time t°. On the other hand, we will show later that, if the characteristic curves meet in a neighborhood of (t°,x°), 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°, x°), i.e., whether or not there exist two points y2) satisfying x(t, yi) = x(t, 112) for some t. In §3.2, we will give two yi and 112 (yi examples in which characteristic curves do not meet though the Jacobian vanishes. In §3.3, we will consider the case where we can extend classical solutions of (3.1)(3.2) beyond a point where the Jacobian vanishes. In §3.4 and §3.5, we will give

EXAMPLES

23

sufficient conditions so that the characteristic curves meet in a neighborhood of the

point (t°,x°).

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

(Ou I.

where o(t,u)

Ou

=u in

{t >0, xE R }, 1

(33)

u(0,x)=x on {t=0, xER'}, + /3l(t)e_SmuS and the two functions

= (t), $ = fl(t)

satisfy the following conditions:

1) o = (t) arid /3 = /3(t) are in C1(R1). 2) ci(t) 0 for each t, ci(0) = 1, and a(i) = 3) i3(t) 0 for each t, /3(0) = 0.

0

for all t K =

constant > 0.

Then the characteristic curves for (3.3) are written by

x = x(t,y) = cl(t)y + /3(t)y3 and v = v(t,y) = ety,

(3.4)

from which we easily see that

= (t) +3/3(t)y2,

=

and

0

for all

t K.

But we can also see from (3.4) that the characteristic curves x = x(t, y) do not meet for all t 0. In this case the solution u = u(t, s) is represented as u(t, x) =

for

t K.

This representation says that the solution contains algebraic singularity at x =

0,

and that the singularity of shock type does not appear though the Jacobian vanishes.

3. PROLONGATION OF CLASSICAL SOLUTIONS

24

Example 2. We consider the following Cauchy problem:

(Ots

Ou

I

(IX

. in

{t>O, xER }, 1

(3.5)

on {t=O,xER'}, where

f(i,u,p)

+ f3'(t)e3tp4 —

u

are the same functions that we introduced and the functions a = a(t), = in Example 1. This example is not of quasi-linear type, and it satisfies Conditions (A.!)' and (A.!!)' given in §2.3. The characteristic curves for (3.5) are written as

x = x(t, y) = ci(t)y + 4/3(t)y3

and v = v(t, y) =

+ 313(t)ety4.

Therefore the Jacobian (Dz/Dy)(t,y) = a(t) + 12i3(t)y2 vanishes on L {(i,y) t K and y = O}. But x = r(t, y) is a bijective mapping defined in a neighborhood of y = 0 for each t K. In a neighborhood of L, the solution u = u(t, z) is written by

u(t, x) =

for

i K.

This says that the solution u = u(t,x) is of class C', but not of class C2, 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.!)' (Chapter 2) which assures the global existence of characteristic curves. Let x = x(t,y), v = v(t,y) and p = p(t,y) be to by the solutions of (2.15)-(2.16), and define a mapping H from

H(t,y) del = (i,x(t,y)). Suppose: (I) (Dx/Dy)(i°,y°) =

0,

§3.3. PROLONGATION OF CLASSICAL SOLUTIONS

25

and

(II) (Dx/Dy)(t,y°)

0 fort 1° with t — 1° small. Actually, differentiating (2.7) with respect to y, we get the system of ordinary differential equations (2.10) for n = 1. Then (2.10) is linear with respect to Ox/Oy and Ov/Oy. As the initial data are not zero, we get Ox

Ot,

(0,0)

d (Ox\

Oaj (tO — Ou ' —

di Oy)

Moreover, as (Ox/Oy)(t°,y°) =

t 0, yE R

1

0. By the assumption, we get

Hence we have (Ov/Oy)(t°,y°)

di Oy

for any

0

(t,y)=(t°,y°)

0

0

Oy'

0

and (Ox/Oy)(t,y°) > 0 for t 0. As p'(y°) = 0, it follows that p'(y) > 0 for y > y° and that p'(y) t° (t—i° small) and x2(t) x we are looking for a single-valued and continuous solution, we extend the solution u = u(t,x) into the above domain by defining

u( ,x) —

J u1(t,z) u3(t,x)

if

x

if

x>

This extended solution is obviously Lipschitz continuous. It satisfies the equation The next problem is to prove (4.3) except on the curve {(t,x) t t°, r = the uniqueness of generalized solutions. This is the subject of the following section. :

§4.3. Semi-concavity of generalized solutions First we give an example of non-uniqueness of Lipschitz continuous solutions which

is due toY. Tomita (see also Remark 2 of Theorem 11.5 in Chapter 11).

Example. Consider the Cauchy problem:

=0 I.

u(0,x)=0

on

in

{t=0, xER1}.

§4.3. SEMI-CONCAVITY OF GENERALIZED SOLUTIONS

39

deC

Then it has a trivial solution u = uo(t,x) = 0, and also

0 0 (y is small). x= neighborhood of {(t,z) :

Then u(t, x + y) + u(t, x — y)

—

2u(t, x)

= {u(t, x + y) — u(t, x + 0)} — {u(t, x —0) — u(t, On

+Oy) —

On

where 0 < 9,9 < 1.

On

On

+0) —

—0)—

On —

0)}y,

x—

y)}

On —

4. EQUATIONS OP HAMILTON-JACOBI TYPE

40

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

= =

-(Ou/Oz)(t,x) = _p(t,y(t,x)) iOp

lOx p=y(t,r)

By (4.8)-(4.9), (Op/Oy)(t,y) is negative in a neighborhood of (t°,y°). From one of

the assumptions, it follows that (Ox/Oy)(t°,y°) = 0 and (Ox/Oy)(t,y°) >0 fort < t°. Hence, when (t,x) goes to (t°,x°) along the curve x = x(t,y°), (82u/8x2)(t,s) tends to —oo, i.e., (02u/0x2)(t, x) is bounded from above in a neighborhood of (t°,x°). By (4.10) and the definition of u(t,x), the third term is estimated as follows: 7(t) —0) = p(t,g3(t, x)) — p(t,g1(t, x)) < 0.

+ 0) —

(4.13)

Summing up the above results, we see that there exists a constant K satisfying

u(t,x + y) + u(i,x — y)

—

2u(t,x)

In the case where (02f/8p2)(t°,x°,v°,p°) is negative, we can similarly prove

0

that the solution is semi-convex in a neighborhood of (t°, x°).

As it is well known that a generalized solution with property (4.2) or (4.12) is unique in the space of Lipschitz continuous functions, we see that our generalized solution (4.11) is reasonable. By (4.11), the singularity of the solution u = u(t,x) lies on the curve {(t,x) x = 1(t)} where u1 (t, 'y(t)) = us(t, -y(t)). Taking the derivative of this equality with respect to t, we get {

'y(t)) —

y(t)) —

By the definition of u = u(t, x), it holds that

=

—

0),

y(t)) =

+ 0).

§4.4. COLLISION OF SINGULARITIES

(i =

As u =

41

1,2) satisfy the equation (4.3) in their definition domain, we

get —

f(t,x,u,

+ 0)) —

f(t,x,u,

—

0))

(• 14 -y(t) + 0) — where

x def =

-y(t), u

7(i) —0)

def

= u(t,-y(t)). The equality (4.14) is correspondmg to the

Rankine-Hugoniot jump condition for equations of conservation law.

Remark. We can extend the singularity of the solution for large t by the same method. But, if (02f/0p2)(t, x, u,p) changes its sign, the solution may generafly lose the above supplementary condition. Then we must introduce other types of singularities. This is the reason why the above construction of singularities is local. This is the subject which will be discussed in Chapter 7.

§4.4. Collision of singularities In this section, we assume that f = f(t,x,u,p) is convex with respect to p in the whole space, and consider the case where two singularities of the solution collide with each other. In this case we see by Theorem 3.3 that, if the Jacobian vanishes at a point, then the characteristic curves meet in a neighborhood of the point. Therefore the singularities of the generalized solution can be expressed in a form of (4.11).

(i = 1,2) be singularities of the solution u = u(t, x). Assume that W1 and W2 meet at a time t = T; that is to say, let 1'1(t)

>

(t) + 0) and

72(t) — 0) >

+ 0)

for t

i.e.,

71(t) > 72(t) for

t > T.

4. EQUATIONS OF HAMILTON-JACOBI TYPE

42

Our problem is how to extend the solution u = u(t,x) for t> T.

x

t

y

Figure 4.2

First, see Figure 4.2 which expresses the behavior of the curve x = x(t, y) for i> T. We use the notations indicated in Figure 4.2. Each part (i = 1,2,3) of the curve x = x(t,y) can be uniquely solved with respect toy, and we write y = g;(t,x) (1 = 1,2,3). Put x) v(t,91(t, x)) (i = 1,2,3). As for t > T, > the solution u = u(t, x) defined by (4.11) takes two values for x E We def define I(t,x) = ui(i,x) — us(t,x). By Lemma 4.3, we get

I(t,

= uj(t, 'y2(t)) — u,(t, 12(i)) = u1(t, 72(t)) — u2(t,

< u1 (t, 71(t)) — u2(t, 11(t)) = 0,

I(t,y1(t)) = u1(i,'71(t)) — = u2(t,y1(t))

—

u,(i,yi(t)) > u2(t,y2(t)) — u3(t,72(t)) = 0.

Moreover, it follows from (4.10) that

a' —(t,x) =p(t,gi(t,x)) —p(t,gs(t,x)) > 0. ax

§4.4. COLLISION OF SINGULARITIES

43

there exists uniquely a, = satisfying E = 0. As we are looking for a continuous and single-valued solution, we define the solution u = u(t,x) for x E [72(t),11(t)] as follows: Hence

u(t,x)

I u3(t,x)

if

x

= 1 uj(t,x)

if

a,

we can similarly show that u = u(t, a,) is Lipschitz continuous and semiconcave. Taking the derivative of I(t, = 0 with respect to i, we also get the Then

same jump condition (4.14).

The collision of a finite number of singularities can be treated by the above method. For example, let {x = -y,(t), t > t} (i = 1,2,... ,k) be singularities of the solution u =u(t,x). Suppose (I) for i

> .. . >

for

t > T.

Next, we can prove that there exists uniquely a, = satisfying E ui(t,-y(t)) = A new generalized solution may be defined in the same way as before. Except the case where an infinite number of singularities collide simultaneously, we can repeat the above discussions forever. If (02f/0p2 )(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 will consider this subject in Chapter 7.

Chapter 5 Quasi-linear Partial Differential Equations of First-Order §5.1.

Introduction and problems

In this chapter we consider the Cauchy problem for quasi-linear equations of firstorder in one space dimension as follows: in

u(0,x) =

on

{t>0, xER },

{t = 0, x E R1},

(5.1) (5.2)

= a1(t,x,u) (i = 0,1) and 4 = are of class in R.' x R1 x and R', 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 regularity of = (i = 0,1) and = 4(y). On the other hand, there are many works on the global existence and uniqueness of weak solutions for quasilinear equations of first-order, especially for equations of the conservation law. For example, refer to P.D. Lax (981, 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 [1121, S.N. Kruzhkov [88), (92)). By the same reason stated in Chapter 4, it seems to us that it would be difficult to get the informations on the singularities of solutions by the vanishing viscosity method. Therefore, in this chapter as well, 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 = x(t,y) and v = v(t,y) of (2.7)-(2.8) for n = 1. Here, we assume (A.!) (stated in §2.2), which assures the global existence of characteristic curves. We have seen by Theorem 2.1 that, if the Jacobian of the mapping x = z(t, y) vanishes at a point (i°, y°), then the classical solutions blow up at a point (i°, x°), where x0 x(t°, po). Therefore we where

INTRODUCTION AND PROBLEMS

45

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

= g(t,x,u)

+

(5.1')

where (Of/Ou)(t,x,u) = aj(t,x,u) and g(t,x,u) — (Of/Ox)(t,z,u) = ao(t,x,u). If g(t,z,u) 0, Equation (5.1') is of conservation law. Let w = w(t,x) be locally integrable in R2. The function w = w(t,x) is called a weak solution of (5.1)-(5.2) if, for any k = k(t,x) in it holds that + f(t, x,

x)

+ g(t, x, w)k] dtdx +

x)dx =

0,

(5.3)

{(t,x) t > 0, x R'}. Let w = w(t,x) be a piecewise smooth weak solution of (5.1) which has jump discontinuity along a curve z = 1(t). Then, by the definition of weak solutions, we get the following Rankine-Hugoniot jump condition: where

:

dy — dt —

= u(t, y(t) ± 0)

where

f(t, x, u+(t)) — f(t,z,u_(t))

54

lirnu(t, 1(t) ± e).

From now on, we will consider the Cauchy problem (5.1)-(5.2) in a neighborhood

of(t°,x°) (x° = x(t°,y°)) under the following two assumptions: (I) (Dz/Dy)(t°,y°) = 0. (II) (Dx/Dy)(t,y)

0 for t < t° and y

I where I is an open neighbourhood of

y = y°.

Our problem is to see what kinds of phenomena may appear for t > t°. As Example 1 in §3.2 shows, we have to consider two cases as follows:

(III) Though the Jacobian of x = x(t,y) vanishes at a point (t°,y°), the characteristic curves x = x(t,y) do not meet in a neighborhood of (t°,x°), where x(t°, y°).

(IV) Fort > t°, where (Dx/Dy)(t°,y°) = meet in a neighborhood of (t°, x°).

0,

the characteristic curves x = x(t,y)

46

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

In the case (III), we can uniquely solve the equation x = x(t, y) with respect to y and denote it by y = y(t,x) which becomes a continuous function. Put u(i,x) v(i,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 is as follows. The continuity of u = u(t, x) comes from the continuity of v = v(t, y) and

r= {(i,y) {(t,x) (Dx/Dy)(i,y) = 0) and H(S) y = y(i,x). Put S z(t,y) for (t,y) E S}. Then, by Sard's theorem, the Lebesgue measure of H(S) is zero. As a corollary of Theorem 2.1, we can see that u = u(t,x) satisfies the equation (5.1) outside H(S). Hence u = u(t,x) is a continuous weak solution of (5.1 )-(5.2).

The important problem remaining is to consider the case (IV). Suppose that the characteristic curves meet, that is to say, x(t,yj) = x(t,y2) where Yl By the uniqueness of solutions of (2.7)-(2.8), we have (x(t, yi), v(i, yi)) (r(t, Y2), v(t, y2)). v(t,y2). This means that we will not be As x(t,yi) = x(t,y2), it follows v(t,yj) able to get continuous weak solutions. Therefore we will look for piecewise smooth weak solutions. As we will see a little later, the solution u = u(t, x) v(t, y(i, z)) in the case (IV) takes several values after the Jacobian vanishes. As we are looking for a single-valued solution, our problem is how to choose one appropriate value from these so that the solution becomes a single-valued weak solution of (5.1). if one may jump from a branch of solution to another 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 §5.3 is to prove the uniqueness of solutions for the Cauchy problem to (5.4). The generic property of singularities was studied in D.G. Schaeffer 1120] and T. Debeneix [41j by applying Thom's "Catastrophe theory." The construction of singularities in two space dimensions for Hamilton-Jacobi equations was studied by M. Tsuji [137]. By a similar method, S. Nakane [109]-[11O] constructed the singularities of shock type in several space dimensions. But, as he treated the

singularities of fold and cusp types only, his results are essentially those in the case of two space dimensions. Recently S. Izumiya and G.T. Kossioris (see for

§5.2. EQUATIONS OF HAMILTON-JACOBI TYPE AND CONSERVATION LAW

47

example, 1741-1761) give the generic classifications for the bifurcations of singularities

of geometric solutions.

§5.2. Difference between equations of the conservation law and equations of Hamilton-Jacobi type In this section we will give some property of equation (5.1). This will characterize a difference between Equation (5.1) and equations satisfying (A.!)'. 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 r = x(t,y) and v = v(t,y) be the solutions of (2.7)-(2.8) for n = 1. As p = p(t,y) is corresponding to (ôu/Ox)(t,x(t,y)), an ordinary differential equation for p = p(t,y) is written as follows:

=

—

x, v)p2 +

x, v) — p(O)

x, v)}p +

=

x, v),

(5.5) (5.6)

The equation (5.5) corresponds to the last one of (2.15). More concretely, we see by Corollary 1.3 that, if u = u(t,z) is of class C2, p = p(t,y) is equal to (ôu/&c)(t, y(t, x)). 0 and (Ox/Oy)(t,y°) 0 fort t°

—

uj (t, x) — us(i, x)

x(t°) = x0.

()

where xi(t) < x(t) < x3(t) for I > t°. The function j = j(t,x) is obviously I > to and xj(t) < x < x2(t)} and {(t,x) differentiable in a domain U :

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 (I°, y°). 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 3. Guckenheimer [59] and (3. Jennings [78], 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

>0.

(5.10)

Though j = j(t,x) is not Lipschitz continuous at the point (t°,x°), j = j(t,x) is in C'(U). 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 (10, x°).

Lemma 5.2. i) (Ov/Oy)(t°,y°) u3(t,x) and

0,

iii) When (t,x) goes to (t°,x°) in U, then (0u1/Ox)(i,x) (i = 1,2,3) tend to infinity.

Proof. i) As (Ox/Oy)(t,y°) > 0 fort < t° and (Ox/Oy)(t°,y°) = =

0>— di '..OyI

we have

°G1(jO Ox

Since ((Ox/Oy)(t,y),(Ov/Oy)(t,y))

0,

'

'

Oy

(0,0) for all (t,y), it holds that

(Ov/Oy)(t°,y°)

0.

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

50

Hence we get i) by (5.10).

ii) By the definition of y = (i = 1,2,3), we have gi(t,x) < g2(t,x) < gs(t,x) and = v(t,g1(t,z)) (i = 1,2,3). Using the property i, we get the first half of ii. As g1(t,x) < y2(t) < gz(t,x) < yi(t) < gs(t,x) for x E (xi(t),x2(t)), we have g2(t, z)) < 0,

(i, a,)) > 0,

g3(t, x)) > 0.

As (0u1/Ox)(t,z) =

we get the second part

0

of ii, and also iii.

Lemma 5.3. i) The function j = j(t, a,) is continuously differentiable in the domain U and is continuous on U where U {(t,x) t > t°, r2(t) > x > x1(t)}. But it is not Lipschitz continuous at the point (t°, a,°). ii) Fort > t°, j = j(t,x) is decreasing with respect to a,.

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

0. +

Of

1

=

—

a,, ui) —

+

U1 —

{f(t, x, u1) —

{f(t,x, u1) — f(t,x, us)}} f(t, x, u3)} —

[

When

(5.11)

—

a,)

a,).

(t,x) goes to (t°,x°) in U, the first term of (5.11) is convergent to

(02f/OxOu) (i°,x°,u°) where u0

u(t°,x°). Therefore it is bounded in a neigh-

borhood of (t°,r°). When (i,x) -4 (i°,x°) in U, the coefficient of (Oui/Ox)(t,x) tends to (02f/02u)(t°,x°,u°)/2 = (Oai/Otz)(t°,x°,u°)/2 > 0. The coefficient of (8u3/Ox)(t,x) has the same property as (Oui/Ox)(t,x). Here we apply ii and iii of Lemma 5.2 to (5.11), then we see that, when (t, x) goes to (t°, a,°) in U, (Oj/Ox)(t, a,)

to —oc. This means that j = j(t,x) is not Lipschitz continuous at (t°,x°), and that it is monotonically decreasing with respect to x in U. 0 tends

Lemma 5.4. The functions x1 = x1(t)

x(t,

(i = 1,2) satisfy the following

properties: i)

=

for t > t°

(i = 1,2).

CONSTRUCTION OF SINGULARITIES OF WEAK SOLUTIONS ii)

<j(t,.r1(t)) and

>

j(t,x2(t)) for

Proof. i) Since the functions y = y1(t) and y2(t) we see that of (Ox/Oy)(t,y) = 0 for t >

51

>

(y1(t) > y2(t)) are

=

+

=

= a1(t,x(t,y1(t)),v(t,y1(t))).

the solutions

ii) By the definition of j(t, x), we have

j(t, x1(t)) = ai(t, xj(t), u3(t, xj(t)) + O(ui(t, x1) — u3(t, x1 ))),

0

< 0 < 1.

As (Oai/Ou)(t°,x°,v°) > 0, ai(t,x,u) is strictly increasing with respect to u in a neighborhood of(t°,x°,u°). Moreover we have ui(t,x) > u3(t,x) for (t, x) U and v(t,y1(t)) us(t,zi(t)). Hence we get j(t, x1 (t))

> a1 (t, x1(t), us(i,

x1 (1))) =

(t).

We can similarly obtain the second inequality. Using the above lemma, we can prove the following.

Proposition 5.5. The Cauchy problem (5.9) has a unique solution in the domain {(t,x) : t > x2(t) > r > xj(t)}. U Proof. We extend the definition domain of j = j(t, x) to the whole space keeping the following two properties: (I) j = j(t,x) is continuous on R2; and (II) j(t,x) is decreasing with respect to x for t > t°. Then it is obvious that the Cauchy problem (5.9) has a solution x = -y(t). From ii of Lemma 5.4 and Lemma 5.3, we get easily x1(t) t°. Next we will prove the uniqueness of solutions. Let x = -11(t) and x = 'V2(t) be two solutions of (5.9). As j = j(t,x) is decreasing with respect to x, we have —

y2(t)]2 = 2(y1(t)

11(t°) — y2(t°) = 0. Hence we get y1(t)

12(t) for t

t°.

—

—j(t,12(t))]

0,

52

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

Now we define the weak solution of(5.1) in the interval (xj(t),x2(t)) fort > t° by

u(t x) =

1

I

ui(t,x) if x < u3(t,x) if x > y(t).

(5.12)

Remark. It is impossible to jump from the first branch {u = u1(t,x)} to the second one {u = us(t,x)} so that Rankine-Hugoniot condition (5.9) is satisfied. This is corresponding to Lemma 6.2 for Hamilton-Jacobi equations.

§5.4. Entropy condition We first give an example which shows the non-uniqueness of weak solutions of (5.1)-

is the space of measurable functions which are (5.2) in the space integrable on any compact set in R2.

Example. Consider the Cauchy problem

(Ow

2

=0 in {t>0,XER},

w(0,x) = 0 on {t =

0,

x E R}.

Then this problem has a trivial solution w(t, x) = 0

w(t,x) =

1

—1

on on on

0

and a weak solution as follows:

{(t,x) ri t 0}, {(t,x) : t > x > 0}, {(t,x) : t> —x > 0}. :

where u = u(t,x) is

We get the above example by putting w(t,x)

the function given in "Example" of §4.3. As weak solutions of the above example are

not unique, we must impose the entropy condition which guarantees the uniqueness Concerning the entropy condition, we follow of weak solutions in the space here O.A. Oleinik 1112] and P.D. Lax [97].

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 = 'y(t). Put Then the entropy condition is expressed as follows: For any value v between u.k. (t) and u_(t), it holds that

S[v,u_] S[u÷,u_]

where

slv,uI

f(v)

1(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 wave." 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 identically zero. If ti = u(t, x) and v = v(t, x) are piecewise continuously differentiable weak solutions of (5.1') for all x and I > 0 with initial data 110(5) and 110(5) which are piecewise continuously differentiable and L'-integrable ins, and if u = u(t,x) and v = v(t,x) satisfy the condition (5.13) along discontinuity curves, then it holds that IIu(t) — v(t)IILI
As the inequality (5.10) means the convexity off = f(t,x,u) with respect to u in a neighborhood of (t0, v°), we get

f'(t,x,u_) > f'(t,x,u+) along a jump discontinuity, from which we can easily obtain (5.13).

0

Next we extend the weak solutions for large I. If 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 §4.4. But, if (52f/0u2)(t, s, 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 is the subject which will be discussed in Chapter 7.

Remark 1.

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).

54

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

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 construct shocks for quasi-linear differential equations of first order (5.1). 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: +

=

0.

(5.15)

Put tt(t,z) = (Ow/Ox)(t,x). Then w = w(t,x) satisfies

+ f(t,x,

= 0.

(5.16)

For Hamilton-Jacobi equation (5.16), we can construct the singularities of generalized solutions, as done in §4.2. In this procedure, we do not need to solve ordinary differential equations. Then we see by (4.9) and (4.13) that u = (Ow/Ox)(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) = (Ow/Ox)(t,x) does not work well to arrive at Hamilton-Jacobi equation. Moreover the equations treated in [59] and [78] do not depend on (t,x), i.e., I = f(u). By these reasons, the discussions

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

Chapter 6 Construction of Singularities for Hamilton-Jacobi Equations in Two Space Dimensions Introduction Consider the Cauchy problem for a Hamilton-Jacobi equation in two space dimensions as follows: in

u(O,x) =

on

{t>O,xER2}, {t = 0, x E R2},

(6.1) (6.2)

is in S(R2). Here, S(R2) is the space of = f(p) is of class C°° and 4i = rapidly decreasing functions defined in R2. We assume that f = f(p) is uniformly convex, that is to say, there exists a constant C > 0 such that where f

ffl()

8

de=f

[

api p,

C. I >

0.

(6.3)

J

This chapter is continued from Chapter 4. Our aim is to construct the singularities of generalized solutions of (6.1)-(6.2) in two space dimensions. The difference between Chapter 4 and this one is the dimension of spaces. The crucial part in our analysis is to solve the equation x = z(i, 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 difficult 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 singular 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).

Definition. A Lipschitz continuous function u = u(t, x) defined on called a generalized solution of (6.1)-(6.2) if and only if

x R2 is

6. CONSTRUCTION OP SINGULARITIES IN TWO SPACE DIMENSIONS

56

I) u = u(t,x) satisfies the equation (6.1) almost everywhere in R' x R2 and the initial condition (6.2) on {t = 0, x E R3} ii) U = u(t, x) is semi-concave, i.e., there exists a constant K such that

u(t,x + y) + u(i,x —

y)

—

2u(t,x)

x,y ER2, t >0.

for all

(6.4)

Remark. Put v1(t,x)

(Ou/0x1)(t,x) (i = 1,2). Then the equation (6.1) is written down as a system of conservation law:

a

+

a

=

0

(z = 1,2).

(6.5)

The inequality (6.4) turns into the entropy condition for (6.5). See a remark in §6.3.

§6.2. Construction of solutions Characteristic curves for (6.1)-(6.2) are defined as solution curves of ordinary differential equations as follows: = —f(p)+ (p,f'(p)),

=

r1(0) =

v(0) =

=0

(1 = 1,2),

(i = 1,2),

=

where f'(p) =

and (p, q) is the scalar product of two vectors p and q. Solving these equations, we have x =p+ v

Then

(6.6)

= v(t, y) = 4'(y) +

(6.7)

+

is a smooth mapping from R2 to R2 and its Jacobian is given by

= det[I + We write A(y)

and denote the eigenvalues of A(y) by

When the space dimension is one, .X1(y) =

Since

>

§6.2. CONSTRUCTION OP SOLUTIONS is in 8(R), it follows that = A1(y) must take negative values at some points. In this case also, we can prove that

0 and 4 =

minA1(y) = Aj(y°) = —M < 0. p

Put t°

1/M. First assume t < t°. As

forany yER2, we can uniquely solve the equation (6.6) with respect to y and write y = y(t,x). . Thenu = u(t,x) def = v(t,y(t,x))asauniqueclassicalsolutionof(6.1)-(6.2)fort t°. Suppose that t — t0 is positive and sufficiently small, and consider the equation (6.6) in a neighborhood of (t°,y°). The Jacobian of vanishes on {y E R2 : 1 + = 0). Assume the condition: .

(S.1) Ai(y)

.

0 on

Az(y), gradAi(y)

and

.

is a simple closed curve.

In this case, is paranieterized as = {y = is a closed interval and = yj(S) is in C°°(I) (i = 1,2). Put {y(s°)

E

a E I}, where!

= 0 at a = s°}.

:

By the definition of H. Whitney [159], a point in mapping

:

is a fold point of the

i.e.,

tx(t,y(s))

for

0

y(s) E

because it holds that

=

=

0.

Lemma 6.1. Assume that the number of elements of E7 is two. Then

=0 for

y(s) E

Proof. Put I + tA(y) =

,

a1(t,y) ER2 (i = 1,2).

(6.8)

58

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

Then a1(t,y) and a2(t,y) are linearly dependent on As they are smooth in the interior of they can not take every direction of R2. That is to say, when (t, y) moves in a small neighborhood of (t°, y°),

a small

neighborhood of a1(t°,y°) (s = 1,2) where ai(t°,y°) and a2(t°,y°)

are linearly

dependent.

(dy/ds)(s) takes all Contrarily, when the point y = y(s) makes a round of the directions. Therefore (d/ds)x(t, y(s)) vanishes at least at two points. But we Hence we get see by (6.8) that the points where it vanishes are contained in

0

this lemma. Assume here the following condition:

(S.2) E =

{Y1,Y2}, i.e., the number of element.s of

is two; andY1 (z = 1,2) are

cusp points of H1, i.e., 0

at y(s) =

(i = 1,2).

Remark. Assume: (C.1) The singularities of A1 = A1(y) are non-degenerate; i.e., ifgradA1(y) = Oat a point y, then the Hessian of A1 = A1 (y) is regular there. (C.2) (Ov/Oy)(t°,y°)

0.

Then, for t > t° where t — number of elements of

t0

is small,

becomes a simple closed curve and the

is two.

We denote the restriction of v = o(t,y) on E1 by yE = VE(t,y). We see by especially if we put = takes its extremum on (6.8) that c, (i = 1,2) and suppose c1 t°. When it satisfies the condition (S.l) and (S.2), we can construct the singularity of solution by just the same way as in §6.2.

Remark on Figure 64. Assume that

meet at y = y° = (a,b) and that the singularities of Aj = Ai(y) are non-degenerate. As X1 = Xi(y) does not and

take minimum and maximum at y = y°, we can suppose by Morse's lemma .X1(y) =

Therefore {y

R2

(i =

+ (yi — a)2

—

(y2

—

b)2,

1 + t°Ai(y°) =

1,2) have singularities at y = y°.

0.

But, for t > t°, the curve

1 + tX1 (y) = 0} is smooth in a neighborhood of y = y°.

>2. t0 (ii)

Figure 6.4. Change of E1,2 U

with respect to time

Summing up these results, we get the following.

Theorem 6.5. Assume that the

(S.1) and (S.2) are always satisfied. Then, even if two s;ngularities collide with each other, we can 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. assumptions

66

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

Remark. What we have done until this point is the local 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 [1201 for n =

for n =

2

where n is the space dimension.

1

and B. Gaveau 156)

Chapter 7 Equations of Conservation Law without Convexity Condition in One Space Dimension §7.1.

Introduction

We consider the Cauchy problem for an equation of the conservation law in one space dimension as follows: in

u(O,z) =

4)(x)

on

{t>O,xER'}, {t = 0, x E R1},

(7.1)

(7.2)

4)(x) to be in Cg°(R'), except in §7.2 where the Riemaun problem will be discussed. Even in the case where f = f(u) is not convex, the global existence of weak solutions of

where f = 1(u) is of class C°°. We assume the initial data 4) =

(7.1)-(7.2) has been well-studied, for example by O.A. Oleinik (112J, S.N. Kruzhkov

[881, [92J. Our interests are in the singularities of weak solutions for general partial differential equations of first-order. In Chapters 4, 5, and 6, we have locally constructed the singularities of generalized solutions or weak solutions. Our next problem is to extend the weak solution in the large. In this chapter, we consider the Cauchy problem (7.1)-(7.2) in one space dimension. if f = 1(u) is convex with respect to u, we can extend the singularities for large t, because the entropy condition is always satisfied. Moreover, we can treat the collision of singularities as in §4.4 and §6.4. But, if f"(u) changes its sign, we may meet a new phenomenon which does not appear in the convex case. The aim of this chapter is to explain this situation. We will not cover here all the results obtained until now on this subject. We present only fundamental notions in non-linear wave propagation and solve explicitly a certain example which shows a new phenomenon caused by the non-convexity of the equation.

7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION

68

§7.2. R.arefaction waves and contact discontinuity In the Cauchy problem (7.1)-(7.2), we assume in this section that the initial data is given by = =

(a xO,

a and b are constants with a b. The Cauchy problem (7.1)-(7.2) with the initial data (7.3) is called "R.iemann problem." In this case, characteristic curves are written as where

(y+tf'(a), y b, and that the entropy condition (5.13) is not satisfied. Sup-

pose that the graph off = f(u) is drawn as in Figure 7.1. We draw tangent lines L passing the points (b,f(b)) and (a,f(a)), respectively. Let L and be tangent to the curve f = 1(u) at (c,f(c)) and (d,f(d)), respectively. See Figure are equal to f'(c) and f'(d), respectively. We 7.1. Then the slopes of L and {(t,x) : x = f'(c)t, t 0}. denote {(t,x) x = f'(d)i, t 0} and C' :

70

7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION

Put ci {(t,x) f'(c)t < x < f'(d)t, t > O} which is contained in R. The boundary äf is equal to comes into the domain ci. U C—. The shock curve x = We assume here that f'(u) is monotonously decreasing on [d, ci. For any (t, x) E ci, f'(v) = s/t. If we use the function we can pick up a value v E [d,c] r = r(p) introduced in the definition of rarefaction wave, we have v = r(x/t). We define a weak solution u = u(i,z) of (7.1) in the domain R as follows: :

f'(a)t > x > f'(d)t, r(x/t), f'(d)t > x > f'(c)t, a, f'(c)t > x > f'(b)i. b,

u(t,x) =

(7.9)

Though the solution u = u(t, x) has jump discontinuity along the curves and C, it satisfies the entropy condition (5.13). As in the above example, a shock whose speed equals the characteristic speed of one side is called a "contact discontinuity."

§7.3. An example of an equation of the conservation law Before giving an example, we explain the reason why we consider "contact discon-

tinuity." The propagation of singularities for equations of conservation law without convexity condition has been studied, for example, by D.P. Ballou [9], J. Guckenheimer [59], and G. Jennings [78]. Their method is to construct locally the singularities of weak solutions for the Cauchy problem (7.1)-(7.2) where the initial data are "piecewise smooth." That is to say, though they started from the "smooth" initial data (especially in [59] and [78]), they solved essentially the Riemann problem for (7.1)-(7.2). Though the initial data are smooth in their discussion, a rarefaction wave appears in the solution of the Cauchy problem (7.1)-(7.2). Let us

recall that the rarefaction wave has been introduced to construct a solution in a region which is not covered by the family of characteristic curves. Therefore our first question is whether or not we need to use rarefaction waves for solving the Cauchy problem (7.1)-(7.2) with the smooth initial data. A typical phenomenon which does not appear for convex equations is "contact discontinuity" introduced in §7.2. See Figure 7.1 which explains the situation. Our second question is whether or not the situation in Figure 7.1 would surely happen even for the Cauchy problem (7.1)-(7.2) with the smooth initial data. In answer to the above two questions,

§7.3. AN EXAMPLE OF AN EQUATION OF THE CONSEftVATION LAW

we will consider the following example: Assume that the graphs of f = 1(u) and = are drawn as in Figure 7.2 (i) and Figure 7.3 (i), respectively. The graphs of their derivatives appear in Figure 7.2 (ii)-(iii) and Figure (ii).

a1

0

U

Figure 7.2 (i) F,

/71

b2 b1

U

Figure 7.2 (ii)

72

7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION

F" F'' =

0

Figure 7.2 (iii)

Figure 7.3 (i)

§7.3. AN EXAMPLE OF AN EQUATION OP THE CONSERVATION LAW

73

45

C

Figure 7.3 (ii)

In the above case, it holds that 1(a1) = = 1,2) and = = 0. Moreover, we assume

0

(i = 1,2), f"(b1) =

0

(1 =

> b2,

because, if not, our problem the

is equivalent to a case

where I =

f(u)

is concave on

whole R1.

For the Cauchy problem (7.1)-(7.2), the characteristic curves are written by x = x(t, y) = y +

v = v(i, y) =

Therefore it follows that Ox y)

=

1

+

set h(y) def = f

(y), and assume that the graph of h = h(y) is drawn as in Figure 7.4. Next, take A1 = (y1,h(y1)) (i = 1,2) where h(y1)

0.

y

A2

A

Figure 7.4

now put t1def = and X, def = x(t;,yj) (: = 1,2). As we have shown in §5.3, we see that a shock appears at each point (t1,X1) (i = 1,2), and we denote We

it by

(z = 1,2). We have proved in §5.4 that each shock S satisfies the entropy

is small. Our problem is to see what kinds of (i = 1,2). phenomena may happen when we extend the shocks condition for t >

where t —

t1

§7.4. Behavior of the shock In this section we will extend the shock S1 for large t. To explain the situation, we repeat briefly how we have constructed the shock Si. As the graph of x =

for t > t1 is drawn as in Figure 4.1, we use the same notations appeared there. Solving the equation x = x(t,y) with respect to y for x E (x1(t),x2(t)), we get < g3(t,x). As three solutions y = g(t,x) (i = 1,2,3) with gj(t,x) < (i = 1,2,3). Then for all (t,y) ER2, we define u1(t,x) v(t,y) = we have by Lemma 5.2

u1(t,x) u1(i, -11(t)) = u_(t).

Then the entropy condition (5.13) gives us

= f'(u_(t))

(7.13)

76

7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION

and

f'(us(t,1'j(t))) = > 0 for i =

Since

1

and i =

3,

we get

and

0

The proof is complete.

The initial function = ç6(x) has the properties as drawn in Figure 7.3. As long as the entropy condition (5.13) is satisfied, we see by Lemma 7.1 that goes to its maximum. We write here = advances toO, and that and P_(t) = (u_(t),f(u_(i))). The shock Sj enjoys the entropy

condition for I > t1 where the graph of I = 1(u) lies entirely in the upper side of the chord joining two points and P_(t). If we extend the shock S1 further, then the chord PTP may be tangent to the curve {(u,f(u)) u E R1} is tangent to the curve in finite time. We will assume so, i.e., assume that is tangent to the curve I = f(u) at I = T. Then we see obviously that but not at the point P_(T). Therefore it holds that f = f(u) at the point

dll(T) = Lemma 7.2.

i)

(P11(T) > 0,

(7.14)

=

ii)

Proof. i) Using the definition (7.12) of (dyi/dt)(i), we have d d12

di

—

I))—

+

di

[f(u+) — f(u_) —

As u÷(t) =

f(u+)—f(u_)] du+(1)

[ f'(u+)

u_)2

—

f'(u_) —

1

u_J di

we get by (7.14) d di

—u÷(t)

t=T

= FOILs i—+ Ous&yi OrOl 101 = —

=

0.

(7.15)

§7.5. BEHAVIOR OF THE SHOCK S2

As u_(t) = u1(t,-11(t)) =

Put Y1 =

77

we

obtain by

(7.13)

= =

—

This leads us to

=

—

—

u_(T)

fl(u_(T))J

Here we recall that the starting point of the shock S1 is the point (t1, X1). Therefore

it follows that qS'(Yj) >0, u4.(t) —u_(t) >0 and (Ogi/ôx)(t,x) >0. Hence we get (&yi/dt2)(T) > 0. Part ii is easily obtained by (7.15). 0 By Lemma 7.2, we have >

t > T.

This implies that the entropy condition is satisfied. Summing up the above results, we have the following.

Proposition 7.3. The shock S1 starting from

the

point (t1, X1) always satisfies

the entropy condition.

§7.5. Behavior of the shock S2 In this section we extend the shock S2 which has appeared at the point (t2,X2).

The graph of x = x(t,y) for t > t2 can be drawn as in Figure 4.1. Though x = x(t,y) in this section is different from x = x(t,y) in §4.2 and §7.4, we use the same notations introduced there. As in §7.4, we solve the Cauchy problem (7.12) with the initial condition x(t2) = X2, and denote the solution by r = '72(t). We write here also = u(t,72(t) ± 0); that is to say, = The shock S2 satisfies and u_(t) = Put = the entropy condition (5.13) for t > t2 where t — t2 is small. In this case, as

78

7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION

> 0, (5.13) says that the graph of f =

f(u) lies entirely in the lower side

of the chord

Lemma 7.4. As long as the entropy condition for S2 is satssfled, t is decreasing and t g,(t,y2(t)) increasing.

We omit the proof, because it is similarly obtained as Lemma 7.1. When t gets larger, tends to 0 and tends to the maximum = Therefore we assume that, though the entropy condition is satisfied for t < T, becomes tangent to the curve {(u,f(u)) u E R'} at t = T. Then it follows that

(T) =

Lemma 7.5.

i)

=

T where t

—T

is small, it holds that

This proposition means that the entropy condition is violated for t > T. overcome this point, we use "contact discontinuity" explained in

F.

V

Figure 7.5

Li

Li

To

§7.5. BEHAVIOR OF THE SHOCK S2

Consider the equation

u>v.

(7.16)

We solve the equation (7.16) with respect to v, and denote the solution by v = h(u). See Figure 7.5. To construct a contact discontinuity starting at the point (t,x) = (T,-yz(T)), we solve the Cauchy problem as follows:

( dx

=

(7.17)

x(T) = 72(T).

and put S, def = {(t,x) x = T}. Let W be a domain surrounded by the curve and a characteristic line We

denote a solution of (7.17) by x

:

which passes through the point (t, x) = (T, So, the point x = y° y° must satisfy the equation -12(T) = y° + Hence y° =

Figure 7.6 explains this situation.

t x

72(t)

T

y

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 CONVEXITY CONDITION

points are on the initial line {t = O}, we introduce a family of characteristic lines which start from the curve i.e., we construct the "contact discontinuity." For any point

we first define

E —

by

Next

where

..-

— deC . = h(u_) and draw a characteristic line which determine u.k. = u+(t,x) by passes through the point (t , as follows:

we

On the characteristic line, we define the value of the solution u = u(t, s) by u(t, x) = 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 T))

(T.y (T))

0

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 = 4S(x) is of a form as shown in Figure 7.8, and that it satisfies max > b2

§7.5. BEHAVIOR OF THE SHOCK 52

Then, after the collision of two shocks, we will arrive at the 4'(x) < situation in Figure 7.1 (see §7.2). and

mm

x

Figure 7.8

Remark. Recently, S. Izumiya [73] and S. Izumiya & G.T. Kossioris [75]-[761 are studying geometric singularities of generalized solutions of general partial differential

equations of first-order in the framework of "Legendrian unfoldings." Concerning the propagation of singularities for non-convex first-order partial differential equations, the results of [76] would be the best at today's point.

Chapter 8 Differential Inequalities of Haar Type §8.1.

Introduction

The theory of ordinary differential inequalities was originated by Chaplygin [30] and Kainke 181] and then developed by [1571. Its main applications to the Cauchy problem for (ordinary) differential equations concern questions such as: estimates of solutions and of their existence intervals, estimates of the difference between two solutions, criteria of uniqueness and of continuous dependence on

initial data and right sides of equations for solutions, Chaplygin's method and approximation of solutions, etc. Results in this direction were also extended to (absolutely continuous) solutions of the Cauchy problem for countable systems of differential equations satisfying Carathéodory's conditions. We refer to Szarski [128] for a systematic study of such subjects. As for the theory of partial differential inequalities, first achievements were obtained by Haar [61], Nagumo [107], and then by Wniewski [154]. Up to now the theory has attracted a great deal of attention. (The reader is referred to Deimling [40], Lakshmikantham and Leela 196], Szarski (128), and Walter [153]. In particular, he is referred to [24]-[27] and the references therein for recent results in functional setting.) We emphasize here that one of its applications to the Cauchy problem for first-order partial differential equations, videlicet the uniqueness criterion formulated in Theorem 1.8, is just for classical solutions and may only be used locally. (For more details, see the introductory comments in the next section.) The present chapter provides a new method, based on the theory of multifunctions and differential inclusions, to investigate the uniqueness problem. This method allows us to deal with global solutions, the condition on whose smoothness is relaxed significantly. As we shall show more concretely in Chapter 11, the equations to be considered satisfy certain conditions somewhat like Carathéodory's, and their global semiclassical solutions need only be absolutely continuous in time variable.

§8.1. INTRODUCTION

83

The structure of the chapter is as follows. In Section §8.2 we introduce a so(See (8.5) later. Note that (1.14) and (1.16) were usually referred to as Haar's differential inequalities.) An estimate via initial values for functions satisfying this differential inequality will be established (cf. (8.6)-(8.7)). As an application, Section §8.3 gives some uniqueness criteria for global classical solutions to the Cauchy problem for first-order nonlinear partial differential equations. In this way, moreover, the continuous dependence on the initial data of solutions can be examined. Finally, Section §8.4 concerns some called differential inequality of Hoar type.

generalizations to the case of weakly-coupled systems. Most of the results presented here were published in [141], [147]-[149J, and [151].

The relevant material on multifunctions and differential inclusions from [8] and [29] may be found in Appendix II given at the end of the book.

Throughout this chapter, 0 < T < +oo, and ciT

The notation 0/Ox will denote the gradient (0/On...

Let

.j and (.,.)

be the Eucidean norm and scalar product in R", respectively. Denote by Lip(ciT) the set of all locally Lipschitz continuous functions u = u(t,x) defined on QT. F'ürther, set Lip([0,T) x x R"). For every function u = u(t, x) defined on cii', we put Dif(u)

{(s,y) E cii'

:

u = u(i,x) is differentiable at (s,y)}.

We shall be concerned with the following class of Lipschitz continuous functions:

E Lip([0,T) x

:

[0,T]

mes(G) =

0,

Dif(u) J ciT\(G x RTh)). (Here, "mes" signifies the Lebesgue measure on R1.) In other words, a function u = u(t, x) in Lip([0, T) x belongs to only if for almost all t, it is differentiable at any point (t, x).

if and

8. DIFFERENTIAL INEQUALITIES OF UAAR TYPE

84

§8.2. A differential inequality of Haar type First, several comments are called for in connection with the classical uniqueness Theorem 1.8, whose proof is essentially based on Theorem 1.6 or Haar's Theorem 1.5. Our comments will concern the Cauchy problem in the large for a general first-order partial differential equation as follows:

Ou/Ot + f(t,x,u,Ou/Ox) = 0 in u(O,x) =

on

{t =

(8.1)

(8.2)

0, x E RTh},

where the Hainiltonian f = f(t,x,u,p) is afunction of(t,x,u,p) E 11r x R1 x and = 4(x) is a given function of x E Assume the function f = 1(1, x, u, p) to be locally Lipschitz continuous with respect to (u,p) in the sense specified in §1.3; i.e., for any bounded set K C x R1 x there exist nonnegative numbers L1,. . , and M such that .

If(t,x,u,p)

—

—

qd

+

— UI

(8.3)

for all (t,x,u,p) and (i,x,v,q) in K. Let u1 = uj(t,x) and u2 = u2(i,x)

be

two C1-solutions on the whole 11T of (8.1)-(8.2). Then Theorem 1.8 assures the equality ui(t,x) = u2(t,z) in a neighborhood of {t = 0, x E in The question arises as to whether this equality can be extended to the entire domain In answer to this question, we must go back to Theorem 1.6. It seems to

us that there is no standard procedure for joining a point (t°, x°) E to the hyperplane {t = 0, x e by consecutively gluing pyramids of the form {(t,x)

:

01 < t

a2, cj +L1t

< d1 — L1t

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

(8.4)

where

0 = Ix(t°)I, there exists a number

E (0,t°) such that

m,7(t) > Ix(t)l whenever t (t° — & t°]. Assume that (8.11) were false, so that there exists t' [0,t°) such that rn,(t) < x(t)I} < t0, we would have: x(t')I. Setting t1 {t E [0,t°) :

Ix(t1 ) I = rn,7(t), Ix(t) I < m,,(t)

and

drn,7(t)/dt =

. (1

+ m,7(t))

0, let

g,(t)

[g(0) +

exp { [C(x°) +

.

[h(t) + itt]

}.

To get (8.13), it suffices to prove that

g(t) 0 Vt E [0,,0). Obviously, w(0) = > 0. We shall show that w(0) Vt E [0,t°). Assume this is false, so there exists t' (0,t°[ such that w(t') < w(O).

It is well-known that there exists a set G1 C (0, T) of Lebesgue measure 0 with the property that dh(t)/dt = £(t) Vt E (O,T)\G1.

By the hypothesis of Theorem 8.1, we find a set G2 C (0, T) also of Lebesgue

measure 0 such that fZT\(G2 x R") C Dif(tt) and that (8.5) holds for all t E (O,T)\G3, x E RTh.

Since the image of a null set under an absolutely continuous mapping is also a null set, Lemma 8.2 implies

mes(w(Gn(9,t0j)) =0 VO€(o,t°), def

where C = C1 U C2. So mes

(C fl [0, tOl)) = limmes (w(G fl [9, tO])) = 0.

(8.15)

From (8.15) and the continuity of w = w(t) on I we conclude that there is a number A with max{0,w(t')}
0, the last inequality together with (8.16) implies

Because

> £(t.)

On the other hand, since Jx,I

+ C(z°) .

[(i + Ix.I) .

.

(8.19)

E Z(t.,t°,x°), Lemma 8.3 yields

(1 + Jx°I)exp / ft.

—1

(1

+ Ix°I)exp I £(r)dr — 1. Jo

the formula (8.7) gives C(x°) lp(x.)l, which shows that (8.19) contradicts (8.5). It follows that there exists no t' E [0,t°] with w(t') < w(0). Thus, w(O) > 0 for all t [0,t°]; the inequality (8.14) is thereby proved. This completes the proof of Theorem 8.1. 0 Therefore,

§8.3. Uniqueness of global classical solutions to the Cauchy problem The advantage of Theorem 8.1, as we have mentioned in the introduction, is that it allows us to discuss the so-called global semiclassical solutions, which are just absolutely continuous in time variable, for first-order nonlinear partial differential equations with time-measurable Hamiltonians. This will be taken up in Chapter 11, where an answer to a problem of S.N. Kruzhkov [931 is given. In the present section we restrict ourselves to the case of C'-solutions, dealing with some applications of Theorem 8.1 to stability questions concerning the Cauchy problem in the large for

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

92

partial differential equations of first-order, namely the problem (8.1 )-(8.2). Even in this "classical case," using the a priori estimate (8.6)-(8.7) of Theorem 8.1, we find some new uniqueness criteria (posed on the Hamiltonian I = f(t,x,u,p)) for global C' -solutions of (8.1)-(8.2). Let us first repeat the definition of solutions to be considered.

Definition 1. A function u = u(t, x) in C' (liT) fl C([O, T) x

is called a global C'-solution to the Cauchy problem (8.1)-(8.2) if it satisfies (8.1) everywhere in liT and (8.2) for all x E RTh.

As was shown in the introductory comments of §8.2, for the uniqueness of global C'-solutions, the following result may be invoked instead of Theorem 1.8.

Theorem 8.4. Suppose I = f(t, x, u, p) satisfies the following condition: there exist nonnegative numbers L, M such that

lf(t,x,u,p) —

L(1 + lxl)ip — qi +

(8.20)

—

If ui = u,(t,x) and u2 = u2(t,r) are

for all (t,x,u,p),(t,x,v,q) E QT x R' x

global C'-solutions to the problem (8.1)-(8.2), then u,(t,x)

u2(t,x) in 11T•

del

Proof. Consider the function u = ts(t, x) = u,(t, x) — u2(t, x). Then u(0, x) Furthermore, by (8.20) and the definition of global C' -solutions, we have =

f(t, x, u,(t, x),

< L(1 + lxi)

x)) —

x)

= L(1 + lxi)

for all (i,x) E

f(t, x, u2(t, x), +

—

0.

x)) z) — u2(t, x)I

+ Mlu(t, x)i

Now it follows from Theorem 8.1 that u(t,x)

0 in

This

0

proves the theorem.

The next sharpening (and its corollary) of Theorem 8.4 will give some useful uniqueness criteria for global C' -solutions with bounded derivatives.

Theorem 8.5. Suppose f = f(t,x,u,p) satisfies the following condition: for any compact sets K, C R', K2 C R" there exist a nonnegative number LK2 and a nonnegative function (8.20) with

=

such that locally bounded on L and M, respectively, holds for all

§8.3. UNIQUENESS OF GLOBAL CLASSICAL SOLUTIONS

93

(t,x,u,p),(t,x,v,q) E IZT x K1 x K2. I/u1 = uj(t,x) and u2 = u2(t,x) are global C1 -solutions to the problem (8.1)-(8.2) with sup then

ui(i,x)

.._L(t,x) 0 temporarily fixed and g,,(t)

[g(0) + ,i] exp { [C(x°) + ti]

.

[h(t) + iii] },

need only claim that c1i(t) w(0) (= rj > 0) for t E I. On the contrary, suppose that there exists t' E (O,t°] with w(t') <w(O). we

By the hypothesis of the theorem, one finds a set G2 C (0, T) of Lebesgue measure 0 such that fZT\(G2 x

(8.29)

C

From the above, it follows and that (8.25) is satisfied for any t E (0, T)\G2, x E that (8.15) still holds where G G1 U G2; hence, there is a number A with

max{0,w(t')} 0, will be called a comparison equation if and p(t, 0) w = w(t) 0 is in every interval the only solution satisfying the condition limw(t) = 0.

t—+o

Remark. Let £ =

£(t) be a nonnegative function Lebesgue integrable on each

bounded interval (0, y) C R, and = a function of class C[0, +m) such that r(0) = 0, > 0 as w > 0, and f(1/c(w))dw = +co for every 5 > 0. Then (cf. [128, Example 14.2])

w' = £(t)u(w)

(8.33)

is a comparison equation. In fact, assume the contrary that (8.33) admits a nonzero solution w = w(t) on some interval (0,7) with limw(t) = 0. Letting w(0) 0, from this we easily find a nonempty subinterval (t1 , t2) of (0, such that w(t') = 0 and w(t) > 0 for all t E (t1,t2]. It follows that I

Jo

dv

a(v)

=I

o(w(t))

dt = I

£(t)dt

< +m,

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

§8.4. GENERALIZATIONS TO WEAKLY-COUPLED SYSTEMS

99

Proposition 8.9. Let c = c(w) be of cla8s C[O, +oo), and £ = 1(t) 0 be Lebesgue +00

integrable on each bounded interval (O,i') C R with f0

£(t)dt = +00.

(i) If (8.33) is a comparison equation, then so is the equation

= a(w).

(8.34)

(ii) Conversely, under the condition essinf 1(t) > 0, if moreover (8.34) is a cornIE (0.+oo)

parison equation, then so is (8.33).

Proof. (i) Let w1 = w'(t) be a solution of (8.34) on some interval (O,.y1) with urn w1 (t) = 1-30

0.

Find a number .72 > 0 such that 72

= JoI w2(J'e(r)dr),

£(r)dr.

(8.35)

see that w2 = w2(t) is a solution of (8.33) on (O,-y2) with lirnw2(t) = 0. By assumption, w2(t) 0 on (O,.y2). Hence w1(t) 0 on (O,-y1). This shows that (8.34) is a comparison equation. (ii) Let (0, +m) t be the inverse of (0, +oo) 3 t j £(r)dr, and w2 = w2(t) be a solution of (8.33) on some interval with limw2(t) = 0. First, Setting w2(t)

we

define a number > 0 by (8.35). Then setting w'(t) w2(1(t)), we also see that w1 = w'(t) is a solution of (8.34) on (O,-y1) with Iimw'(t) = 0 (cf. [40, Proposition

0

3.4(c)J). The rest of the proof runs as before.

In the sequel, for each function g = g(t) defined and continuous in a certain interval (O,t°), let P9 denote the open set (t (O,t°) g(t) > O}. Here is an :

elementary property of comparison equations:

Proposition 8.10. Let (8.32) be a comparison equation and g = g(t) a given function absolutely continuous on some interval (O,t°) such that lLrng(t) < 0 and that g'(t) p(t,g(t)) almost everywhere in P,. Then g(t) < 0 for all t (O,t°). Proof. On the contrary, suppose that there exists t1 (0, t°) with & g(t1) > 0. g(t) = 0}, we see that 0 t2 Setting g(O) limg(t) and t2 sup{t [0, t1) t1, g(t2) = 0 and (t2,t1) C P9. Hence, by assumption, :

g'(t) 0, p E R"}. Moreover, since = us to define an upper senucontinuous function = ço(t,x,p) from x R" into [—oo,+oo) by

(p,x)

—

—

ff(r,P)dr,

which is, for each p E E, actually finite and continuous in (I, x) on The next theorem will be fundamental in this section.

(9.6)

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

106

Theorem 9.1. Let

Assume (E.I)= be a finite convex function on of the Cauchy problem (9.1)-(9.2) is given Then a global solution u = u(t,x) (E.II). by

sup

f(r,p)dr} for (t,x) ED.

{(p,x)

pER"

pER's

0

(9.7)

Remark 1. Requirement (E.II) could in a sense be regarded as a compatible condition between Hamiltonian and initial data for the existence of global solutions of the Cauchy problem (9.1)-(9.2). To see this, we first rewrite (Eli) in an alternative form that is essentially equivalent (by a standard compactness argument) but seemingly more amenable to verification:

(E.II)' For every (t°,x°) so that

x,p)
N(t°, x°) (9.5')

where V(t°,x°) We

{(t,x) E

:

t — 201 + Ix

— x°I
O, xER},

{t=O, xER}.

Then the method of characteristics in Chapter 1 gives the unique classical solution

= u(t, s)

in {O 0, XER},

1]I'2

It

on

{t=0, xER}.

1,

f(t,p) def =

7)

]t

— 1p1/2

deC

= x

and

forxER, p€Randt>0. Then •

1+00 if if

hence

p=l;

1. All the assumptions of Theorem 9.1 with for Hypothesis (E.I) are therefore satisfied here. However, in this

(E.II) holds when N(V)

in place of case, (9.7) gives the function

u=u(t,x) def = x—2—21t—1l 1/2 sign(t-. 1),

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

108

which is not Lipschitz continuous in any neighbourhood of a point (1, x°). For the proof of Theorem 9.1, we need some preparations. We first recall that a be a finite-valued function directional derivative is defined as follows. Let = of

near a point

E Rm and let e E Rm. Denote

inf sup

+

—

inf

+

—

e>0

OO O t° + x°l some index a0 E Jk so that u(t°,r°) = Obviously,

and

u(t,x) — u(t°,x°) uao(t,x) — u0o(t°,x°)

for all (t,x) close enough to (t°,x°). Since u = u(t,x) and UaO = Uao(t,X) are both differentiable at (t°, x°), (9.17) implies

Ou(t°,x°)/Ot But UaO =

Ou(t°,x°)/Ox = OUao(t°,s°)/OX.

and

s9uao(t°,x°)/Ot

Uao(t,X) satisfies (9.1) at (t°,x°); so does u = u(t,z).

On the other hand, it is clear from the hypotheses that

u(O,x) = in!

= inf4,0(x) = 4,(x) aEI

aEJ

for all

x

R".

The function u = u(t,x) is thus a global solution of(9.1)-(9.2). Now suppose that 4, = 4,(x) is given in the form 4,(r)

in! 4,a(X)

aEI

for

R",

z

(9.18)

= a finite convex function for every a E I. 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. with

Corollary 9.7. Assume (E.I)-(E.II) for each problem = a finite convex function, a I. Assume, furthermore, that all the hypotheses of Theorem 9.6 hold for the solutions def

= uQ(t,x) = max{(p,x) — 4,a(p) pER

of those problems. Then u = u(t, x)

—

, f(r,p)dr}

in! ua(t, r) is a global solution of the Cauchy

aEI

problem (9.1)-(9.2) where 4, = 4,(x) is defined by (9.18). with 4'j = 4,j(x),..., 4,k =

Corollary 9.8. Let 4, = 4,(x)

some convex and globally Lipschitz continuous functions. 1ff = f(t,p) is continuous on then a global solution u = u(t,x) of the Cauchy problem (9.1)-(9.2) can be found in the form

u(t,x)

1iE

k}

{(p,z) — g5(p)

—

jf(r,p)dr}

for (t,x) €

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

116

Proof. Since

I

{ 1,.. . , k} is a finite set, the conclusion is straightforward from

0

Corollary 9.5 and Theorem 9.6.

Example 1. Consider the Cauchy problem

Ou/Ot+l(Ou/Ox)2—1l=O

in

{O 0, p E Assume (E.I) and (E.III). Then it can be proved that (E.IV) is satisfied with Q x the set G being as in (E.I), if one of the following two conditions holds:

(E.IV)' f = f(t,p) is strictly p-convex; more precisely, that is to say, it is strictly convex with respect to p on

(E.W)" o =

for almost every fixed t E (0, +oo).

(p) is strictly convex on its effective domain E1

dom o.

9. HOFF'S FORMULAS FOR GLOBAL SOLUTIONS

For strictly convex functions on R", see the proof of Lemma 9.13. The detailed notion of the strict convexity of a function on a convex set may be found in Appendix

Theorem 9.9. Let q4 =

be in the class with a d.c. representation (9.19) such that 02 = a2(x) is globally Lipschitz continuous on Under Hypotheses (E.l), (E.Ill), and (E.IV), the formula u(t,x) def =

on — D,

*

.

aCE2

pER

(9.22)

in which E2 dom o, determines a global solutzon u = u(t, x) of the Cauchy problem (9.1)-(9.2).

Proof.

Then

a

Let

=

are obviously convex functions. For each a E2, consider the Cauchy problem By (E.I) and (E.III), it follows from Theorem 9.1 that the formula def

Ua(t,x) =

— on D

•

PER"

(9.23)

determines a global solution Ua = x) of this problem. Moreover, we may assume that Q D G x the sets Q and G being as in (EIV) and (E.I), and then see that all the solutions Ua = Ua(t,r) satisfy (9.1) at every point of D\Q. (For the smoothness of such u8 = x), see Remark after Proof of Theorem 9.1.) Now, since 02 = r2(x) is globally Lipschitz continuous on R", the (nonempty) set E2 = domo' should be bounded [117, §13.3]. Given any r (O,+oo), denote v,. {(t,x) N(V,.,E2) (cf. (E.III)). For any t + Izi < r} and N,. (t',x'), (t2,x2) in Vr, we can then choose p6 E L6(t1,x1) C and deduce from (9.20)-(9.21) and (9.23) that

u0(t',x')

— u6(t2,x2)

—

_x2)+f —

where

a,. =

+

f(r,p6)dr —

(cf. (E.I)). The solutions Ua = u6(t,z) therefore satisfy CE(O,r)

a

a

Lipschitz condition on V,. E2.

with

constants N,. and a,., which are independent of

§9.3. THE CASE OF NONCONVEX INITIAL DATA

Next, rewrite (9.23) as

= o(a) +

—

pER'

a)

(9.24)

fix temporarily (t,x) E By (9.20) and Hypothesis (E.I), Hence (see 1117]), the right side of (9.24), being the a E supremum of a family of continuous functions, actually determines a lower semiand

continuous function of a from the whole R" into (—oo, +oo] whose effective domain It follows that is precisely the nonempty bounded set E2 C

+00> inf

inf

aEE2

pER's

aEB2

= min{o(a)+ pER's

cx)}

aEE3

= min{u(a) +

—

Finally, since 72(x) = max{(x,a) aEE2

aEE2

—

a)} = rninuQ(t,x)

(>

—oo).

a(a)} (see [64, p. 964]), one has

oEE2

= Ui(X)

—

max{(x,a)

—

a consequence of Theorem 9.6, u = u(t, x)

mm

aCE3

ua(t, x) is therefore a

global solution of the Cauchy problem (9.1)-(9.2).

0

Corollary 9.10. Let I = f(t,p) be of class C° on

and = /(x) have a are globally Lipschitz d.c. representation (9.19) such that = o1(x), = continuous on R". Assume (E.IV). Then the function u = u(t, x) given by (9.22) is a global solution of Problem (9.1)-(9.2).

is bounded [117, §13.3], HypothProof. Since the nonempty set E1 esis (E.III) must hold while (E.I) is trivially satisfied. Hence, the conclusion is

0

immediate from Theorem 9.9.

Example. Consider the Cauchy problem Ou/Ot + f(Ou/Or) =

0

in

{0 M, provided (t,x) E V,. and al > N,.,M. This means that

tim Hence

ua(t,x) = +oo locally uniformly in (t,x) E V.

(cf. Remark 2 after the formulation of Theorem 9.1), we may find a positive

number N,. for each r E (0, +oo) such that jflf

= mm

ua(t,x) whenever (t,x) E Vr.

(It should be noted that Ua(t, x) is lower semicontinuous in a on the whole IR't.) Moreover, analysis similar to that in the proof of Theorem 9.9 shows that the = Ua(t, a,) satisfy a Lipschitz condition on V,. with constants depending solutions

on r but independent of a for al N,.; and that they satisfy (9.1) except on a common set of Lebesgue measure 0. The proof is thus complete in view of Theorem 9.6.

§9.4. Equations with convex Hamiltonians I = 1(p) We now consider the Cauchy problem

au/at + f(Ou/Ox) = 0

in

V = {t > 0, x E

(9.26)

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

122

u(O,.r) =

on

{t

=

(9.27)

xE

0,

under the following two hypotheses.

(F.I) The initial function strictly

=

cb(x)

is of class C° and the Hamiltonian f = f(p)

lim

with

convex on

=

IpI-*+oo

(F.II) For

is

every bounded subset V of V, there exists a positive number N(V) so

that

+ t f((x — y)/t)

+ t . f((x — w)/t)} < > N(V). Here,

whenever (t,x) E V, function off = f(p).

=

f

.

f(z)

denotes the Fenchel conjugate

In the sequel, we use the notation + t f((x — y)/t),

ç(t,z,y) where (t,x)

e

V,

and shall prove the following theorem.

y E

Theorem 9.12. Assume u(t.

x)

inf yEtR

(9.28)

.

((t, x,

(F.I)-(F.II).

y) =

inf

Then

+t

.

the formula

r ((x

y)/t)

—

}

for

(t,

x) E V (9.29)

determines a global solution of the Cauchy problem (9.26)-(9.27).

The next auxiliary lemma is known ([64j,

Theorems 23.5, 25.5, and 26.3]),

but what we would like to insist here is on its simple proof by the use of our Lemmas 9.2-9.3.

Lemma 9.13. Let f = f(p) be strictly convex on Then

f'

with

lim

IpI-++oo

= +00.

= f(z) =

(z,8f(z)/Oz)

—

f(5f*(z)/Oz)

Proof of Lemma 9.13. The strict convexity on that this function is everywhere finite and that f(Ap' + (1— A)p2)

for all

z E W1.

(9.30)

of the function I = f(p) says

+ (1— A)f(p2)

[0, 1]; the sign of equality holding if and only if p1 = E {0, 1}. Accordingly, f = f(p) is continuous.

for any p', p2 E RTh, .\

E

p2

or

§9.4. EQUATIONS WITH CONVEX HAMILTONIANS f = f(p)

It will now take a simple matter to check that w = w(z,p) satisfies z,

def

all the conditions of Lemmas 9.2-9.3 where we put E = R Rm = and shall deal with the function def i,b=i,b(z)=sup{w(z,p)

Indeed, since

,

123

(z,p) — f(p) def def in = Ii, =

• : pER }=f(z).

= +00, Condition (i) of these Lemmas holds while the

lim

IpI-'+oo

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

: w(z,p)=f(z)} is actually single-valued on the whole Therefore, by Part b of Lemma 9.2, all the partial derivatives Of (z)/Oz, exist, and L(z) = {Of'(z)/Oz}. Property (9.30) is thus coming from the definitions of f = (z) and L = L(z). Further, Lemma

I

9.3 and its Remark imply the continuity of Of */Oz = Of (z)/Oz.

0

Remark 1. Consider a convex and lower semicontinuous function I = 1(p) on Assume domf 0 and imf C (—oo,+oo] (the function I = f(p) is then called proper). It will be shown that = +00

lim

In fact, if lim

p1-4+00

if and only if domf* =

= +00, then for each z E

f(z) =

sup

the supremuin

{(z,p) — f(p)}

pER"

and hence is essentially taken over all elements p of just a compact set C finite. Conversely, let there exist an M E R and nonzero points p1, p2,... in such is -+ +00 ask —* +00. Since that f(p") MIp"I fork = 1,2,... and that Putting z z0 E (M + 1)z°, locally compact, we may suppose that we thus get

f'(z) sup{(z,pk) > —

lim

—

f(pk)}

I(M + 1)(zo,pk/Iphhl) — M1}

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

124

Remark 2. Consider a finite convex function = on R" with the Fenchel = conjugate be the Fenchel conjugate of = 4i'(p). Let = (p) is proper, convex, and lower semicontinuous hence, Remark 1 =

Then it is known [641 that çiS =

and that 4" =

on

implies

Accordingly, domqS" =

= +00.

urn

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 x, y)/Ot, OC(t, x, y)/ôxi,..., OC(t, x,

moreover, one may apply Lemma exist and are continuous on the whole V x def del def = —((t,x,y) where p del 9.2 to the function = = y, E = R and m = n + 1, (t, x), 0 V. Consequently, the function u = u(t, x) defined by (9.29) •

is locally Lipschitz continuous and is directionally differentiable in 1), with equal to mm

((f((x — y)/t)

—

((x

—

—

x)

y)/i)/Oz),0f((x — y)/t)/Oz),e). (9.31)

e, L(t,x) ((i,x,y) = u(t,x)} 0 (Lemma 9.3). But, {y according to Radernacher's theorem, u = u(t, x) is (totally) differentiable at any Here,

point outside a null set Q C V. Therefore, suitable choices of e in (9.31) give

Ou,x)/ôt = =

— y)/t)

—

((x — y)/t,0f((x

y)/i)

—

((x

—

—

—

y)/t)/Oz)}

y)/t,0f((x — y)/t)/Oz)}

(9.32)

and

Ou(t,z)/8x1 = nun 0f((x — y)/t)/i3z1 = max 0f((x — y)/t)/0z1, yEL(t.r)

provided (t,x) V\Q and i {1,2,...,n}. Now, given any (t,x) V\Q, we pick up some y (9.30) and (9.32)-(9.33) that

Ou(t,r)/Ot =

f((x

—

y)/t)

—

((x

—

(9.33)

L(i,x). Then it follows from

y)/t,0f((x — y)/t)/Oz)

= —f (i3f((x — y)/t)/c3z) = —f(Ou(t,x)/Ox). The equation (9.26) is thus satisfied almost everywhere in V.

§9.4. EQUATIONS WITH CONVEX HAMILTONIANS f = f(p) As

the next step, we claim that urn

(9.34)

Indeed, on the one hand, the definition (9.29) clearly forces + t . f'(O), hence

for each fixed x0 E

u(t, z)

liinsup

(9.35)

On the other hand, let us first take a sequence {(ik, C V converging to (O,x°) such that liminf u(i,z) = urn u(tk,xk) and second choose arbitrary points

E L(tk,xk) (for k = 1,2,...). Then it will be shown that

In the contrary case, suppose without loss of generality that yk

(k-++ao)

(k—H-cc)

y°

x0.

R",

is bounded x0. (We emphasize here that the sequence C by Lemma 9.3.) Since lim f*(z)/Izp = +00 (cf. Remark 2 of Lemma 9.13), where y°

hi-H-co

(9.35) and a passage to the limit as k —* +oo in the equality

u(tk,xk) =

f((x"

+

—

yk)/jk)

(9.36)

would yield

liminf

u(t,x) = lirn u(tk,xk) =

—

a

+ (+oo) = +00,

k—H-co

contradiction. This shows that lim = x0. But the continuous function k-H-co = is bounded from below since again lim f(z)/IzI = +oo. Therefore, IzI-4+oo

a passage to the limit as k -+ +00 also in (9.36) implies liminf

u(i,x) = lim u(tk,xk) > k—H-co

(9.37)

—

Finally, combining (9.35) and (9.37) gives (9.34), which says that u = u(i,x)

has a (unique) continuous extension over the whole

satisfying (9.27). The proof

0

is thus complete.

Remark. Assume (F.I). Then (Fl!) is satisfied if lim ((t,z, y) = +oo uniformly Ivi-H-oo

in (t, x) on each bounded subset of V.

126

FORMULAS FOR GLOBAL SOLUTIONS

9.

In fact, let V c V be bounded, say, V C (O,r) x B(O,r) for some r E (O,+oo). Put M < +m. It follows from (9.28) that mm ((t, x, w) + 1f(O)I IzIr

((t,x,z) =

+

1. f(O) < M whenever (t,x) E V. Hence, if urn ((t,x,y) = Ii, 1-4+00

+00 uniformly in (t, x) on each such V, then for a suitable number N(V) have

mm ((t,x,w)

IwIN(V)


N(V);

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

Corollary 9.14. Under Hypothesis (F.I), suppose that > —rn.

Ixl-4+oo

(9.38)

Then (9.29) determines a global solution of the Cauchy problem (9.26)-(9.27).

Proof. By Remark above, it suffices to prove that

((t, x, y) = +00 uniformly

urn hi

in (t, x) on each hounded subset V of V. To this end, let V C V be bounded, say, V C (O,r) x B(O,r) for some r E (O,+oo) and let M E (O,+oo) be arbitrarily given. Condition (9.38) says that there exist numbers A, N e (0, +m) such that 2 —AIyI

whenever

2 N.

But we certainly find a positive number j with the property that

f(z)/jz[ 2(M Putting N(V)

(t,x)

E

+ A)

as

IzI

2

ii.

+ v)}, we therefore deduce from (9.28) that, if

V and 1111 2 N(V), then

((t,x,y) [-

because (x —

A

- y)/t)

+

I(x—y)/tI

y)/tI 2 [r(1 + v) — r]/r =

Is

.

Ix Ill

(IyI

.

—

2 1/2.

if = is globally Lipschitz continuous on IR", then (9.38) clearly holds. The following result of Hopf [64] can thus be considered as a consequence of Corollary 9.14.

Corollary 9.15. If the initial funct;on = 4(x) ss globally Lipschztz continuous and if the Ham:ltonzan f = f(p) is strictly convex on IR" with lim =

Chapter 10 Hopf-Type Formulas for Global Solutions in the case ot Concave-Convex Hamiltonians §10.1.

Introduction

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

Ou/Ot+f(Ou/Oz)=0

in

{t>0,

u(0,x) = 4>(x) on {t =

0,

x E R't}.

(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) I = f(p) convex (or concave) and 4> = 4>(x) largely arbitrary; and (b) 4> = 4(x) convex (or concave) and f = 1(p) largely arbitrary. It is unlikely that such restrictions, either on f = f(p) or on 4> = 4>(x), are really vital. A relevant solution is expected to exist under much wider assumptions. According to Hopf [64], that he has been unable to get further is doubtless due to a limitation in his approach: he uses the Legendre transformation globally, and this global theory has been carried through only in the case of convex (or concave) functions [Fenchel's theory of conjugate convex (or concave) functions]. In the present chapter, we propose to examine a class of concave-convex functions as a more general framework where the discussion of the global Legendre transformation still makes sense. Hopf-type formulas for non-concave, non-convex Hamilton-Jacobi equations can thereby be considered. The method here is a development of that in Chapter 9, which involves the use of Lemmas 9.2-9.3 (and their generalizations). It is essentially different from the methods in [64] and [111]. Also, the class of concave-convex functions under our consideration is larger than that in [111] since we do not assume the twice continuous differentiability condition on its functions.

10. HOFF-TYPE FORMULAS FOR GLOBAL SOLUTIONS

128

Let us continue using the notation of Chapters 8-9. We shall often suppose

R" are separated into two as that n n1 + n3 and that the variables x, p Accordingly, the zero(x',x"), p (pS,ph?) with x', p' Rn', x", p" vector in will be 0 = (0', 0"), where 0' and 0" stand for the zero-vectors in and Rn2, respectively.

Definition 1. [117, p. 349] A function f = f(p',p") is called concave-convex if it is a concave function of p'

for each p'

E

for each p" E Rn2 and a convex function of p"

Rfl.

In the next section, conjugate concave-convex functions and their smoothness properties are investigated. Section §10.3 is devoted to the study of Hopf-type formulas in the case of concave-convex Hainiltonians.

§ 10.2. Conjugate concave-convex functions be a differentiable real-valued function on an open nonempty subset A of R". The Legendre conjugate of the pair (A,f) is defined to be the pair (B,g), where B is the image of A under the gradient mapping z = Of (p)/Op, and g = g(z) is the function on B given by the formula

Let f = f(p)

g(z)

—

It is not actually necessary to have z = Of (p)/Op one-to-one on A in order that g = g(z) be well-defined (i.e., single-valued). It suffices if

(z,p')

— 1(p1) = (z,p2) — 1(p2)

whenever Of(p1)/Op = Of(p2)/Op = z. Then the value g(z) can be obtained (z) by any of the unambiguously from the formula by replacing the set (Of vectors it contains. Passing from (A, f) to the Legendre conjugate (B, g), if the latter is well-defined, is called the Legendre transfonnation. The important role played by the Legendre transformation in the classical local theory of nonlinear equations of first-order is well-known. The global Legendre transformation has been studied extensively for convex functions. In the case where f = f(p) and A are convex, we can extend f = f(p) to be a lower semicontinuous convex function on all of R" with A as

§10.2. CONJUGATE CONCAVE-CONVEX FUNCTIONS

129

the interior of its effective domain, if this extended I = f(p) is proper, then the Legendre conjugate (B, g) of (A, is well-defined. Moreover, B is a subset

of domf (namely the range of Of/öp), and g = g(z) is the restriction of the Fenchel conjugate f = f(z) to B. (See 1117, Theorem 26.41; cf. also Lemma 9.13 in the previous chapter.) Chapter 9 has thereby proved an important role of the global Legendre transformation in the global theory of first-order partial differential equations.

For a class of C2-concave-convex functions, H.T. Ngoan [1111 has studied the global Legendre transformation and used it to give an explicit global solution to the Cauchy problem (1O.1)-(1O.2) with I = 1(p) = f(p',p") in this class. He shows that in his class the (Fenchel-type) upper and lower conjugates (117, p. 389],

in symbols f = T(z',z") and f = f(z',z"), are the same as the Legendre conjugate g = g(z',z") off = f(p',p"). Motivated by the above facts, we introduce in this section a wider class of concave-convex functions and investigate regularity properties of their conjugates. (Applications will be taken up in § 10.3.) All concave-convex functions f = f(p',p") under our consideration are assumed to be finite and to satisfy the following two "growth conditions." lim IP"I-4+°°

Ip"I

urn

= —oo

Ip'I-++oo

Let f*2 =

p"

= +00 for each p' E Rn'. for each p" E

(10.3) (10.4)

Ip'I

1s2(ps,zu) [reap. f*i = f'(z',p")] be, for each fixed p'

[resp.

R'2], the Fenchel conjugate of a given p"-convex [resp. p'-concavel function

I = f(p',p"). In other words, z")

sup {(z",p") — f(p',p")}

(10.5)

inf {(z',p') —

(10.6)

p"€R"2

[resp. f'(z',p")

is concaveR" x [reap. (z',p") x R"3]. If f = convex, then the definition (10.5) [resp. (10.6)] actually implies the convexity [resp. concavityl of f*2 = f*2(p',z") [resp. f1 = f'l(z',p")] not only in the variable

for (p',z")

z" E R"2 (reap. z' E x [reap. (z',p") E

but also in the whole variable (p',z")

x

Moreover, under the condition (10.3) (reap. (10.4)1,

_____________

_______________ ________________

10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS

130

the finiteness of f = f(p',p") clearly yields that of f2 = f2(pI, z") (reap.

1' =

I '(z',p")] (cf. Remark 1 of Lemma 9.13) with f•2(p', z")

lim

= +00

Iz"I

z"I-4+oo

f'(z',p")

lim

(resp.

z'(

Iz'I—l+oo

=

(10.7)

—00]

(resp. p" E Rh'2]. To see this, fix any 0
0 is arbitrary, we have lim Ip"l

=

M

(10.19)

§10.2. CONJUGATE CONCAVE-CONVEX FUNCTIONS

133

for any r E (0, +oo). Thus (10.16) holds. Analogously, (10.17) can be deduced from

0

(10.4).

Definition 2. A finite concave-convex function I = f(p', p") on to be strict if its concavity in p' RTh' and convexity in p" E It will then also be called a strictly concave-convex function on

x Rn2 is said

are both strict. x

x Lemma 10.2. Let I = f(p',p") be a strictly concave-convex function on = f*2(ps,zhl) [resp. with (10.3) [resp. (10.4)1 holding. Then its partial conjugate f1 = f*1(ZI,p'l)] defined by (10.5) [resp. (10.6)] is strictly convex ]resp. concave]

in p' E

[resp. p" E RV121 and everywhere differentiable in z"

R'%2 Iresp. z' E

that, the gradient mapping R'hl x (p', z") '-+ Of 2(p', z")/Oz" x (z',p") '-+ 0f1(z',p")/Ozhl is continuous and satisfies the

Besides

[resp.

identity

z")

[resp. f' (z', p")

z")/Oz")

(z",

—

(z', Of" (z', p")/Oz') —

f(p', 0f2(p', z")/Oz")

f(Of" (z', p")/Oz', p")].

(10.20)

Proof. For any finite concave-convex function f = f(p',p") satisfying the property (10.3) [resp. (10.4)], the partial conjugate f'2 = f*2(pS, z") [resp. 1" = is finite and convex [resp. concave] as has previously been proved. x Now, assume that f = f(p', PS') i5 a strictly concave-convex function on

Rn2 with (10.3) holding. Then Lemma 9.13 shows that f'3 = f'3(p',z") must be differentiable in z" and satisfy (10.20). To obtain the continuity of (p', z") i-+ 0f2(p', z")/Oz", let us go back to the gradient mapping R'hl x n2, E m Lemmas 9.2-9.3 and introduce the temporary notations: n +n2, continuous function

=

It follows from (10.16) that the

and

(z",p")

= w(p',z",p")

—

f(p',p")

(10.21)

meets Condition (i) of Lemma 9.2. Therefore, by Lemma 9.3 and its Remark, the nonempty-valued multifunction

L=

= L(p',z")

{p" €

:

=

10. HOPP-TYPE FORMULAS FOR GLOBAL SOLUTIONS

should be upper seinicontinuous. However, since I = f(p', p") is strictly convex in the variable p" E (10.21) implies that L = L(p',z") is single-valued, and hence continuous in RnI x But L(p', z") = {Of*2(p', z")/Oz"}, which may be handled by the same method as in the proof of Lemma 9.13 (we use Lemma 9.2, ignoring the variable p'). The continuity of RnI x (p', z") —* Of 2(pS, z")/Oz" has accordingly been established. of f9 = f3(p', z") is strict. Next, let us claim that the convexity in p' E andp',q' E Rn'. Of course, (10.5) and (10.21) To this end, fixO

max

inf ((t, x, y', w")

*D"ER"3

w'I —m locally uniformly in z E R ',

,,

(10.43)

I

Then (10.29) determines a global solution of the Cauchy problem (10.26)-(10.27).

Proof. Let V C V be bounded, say, V C (O,r) x

for

x

some r (0,+oo). By (10.3*) and (10.42), analysis similar to that in the proof of Corollary 9.14 shows that liixi

hi I-*+oe

((t,x,z',y") =

{ch(x',y") + t. f' (o', S" —

urn

Ii' 1-4+00

11k)

} = +00

uniformly in (t,x) = (t,x',x") E V. Hence, we may deduce that

0>—

inf inf min{0,_1 + (t,a,)EVW"EIU'2

> —m.

Of course, max

inf

w'EIR'i w"EIU'3 1w'

as

((t,x,w',w") > —

inf

w"ER"2

((t,x,x',w") + 1

(10.44)

I

(t, x) E V. Further, in view of (10.43), there exist numbers A, N E (0, +m) such

that
f(t,x,u,p), —

(12.26)

(z,p)dr 0 is small enough, one has nun (z,p)dr ZEFv(r,x(r).ii,a)

—

— =

nun

(z,p) — e/2

dr

J* mm

zEF0 (t,z,U,Q)

(z,p) — e/2

f(t,x,u,p) —

e

(independently of the choice of x(.) E Xp(t, x) because the function family Xp(t, x)

is eqwcontinuous at i). In other words, the first inequality of (12.45) must be true. Dual arguments give the second one. Therefore, by Remark 2, (12.43) holds.

Remark 4. Let Fu =

x, u, a), FL = FL(i, x, u, 13) be Carathéodory in (t, x) E

satisfying Condition QT [i.e., measurable in t E (0, T) and continuous in x E (i). Given any x(.) E Xp(t,x), by [8, Theorem 8.2.14] (cf. [29, Theorem 111.151), the measurability in r of the multifunctions (0,T) 3 1.

Fu(r,z(r),u,a),

(0,T) 3 r

FL(r,x(T),u,[3)

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

(0,T) 3 r '—+

(z,p),

mm

(0,T) 3 r

zEFt

max

(z,p).

Therefore, the integrals in (12.45) exist. Moreover, if (12.42) and (12.45) hold, then one has in fact the equalities in (12.45), because by Lemma 12.3, (12.42) clearly forces

supliminf

aEp ö\O

inf

1 —

r()€Xp(t,z)45

t+6

miii

I

0, this

§13.1. CONSTRUCTION OF GLOBAL SOLUTIONS VIA CHARACTERISTICS

199

hypersurface has a simple projection upon tx-space (the tx-projections x = x(t, y) of the characteristics issuing from different points y at t = 0 do not intersect). Then one can eliminate y from

x = x(t,y),

(13.3)

I. v=v(t,y),

and the hypersurface appears in the form v = u(t, x) with u = u(t, x) single-valued and smooth. This is the classical solution. Outside of the classical tx-domain, however, the hypersurface has, in general, multiple projection upon tx-space or, in other words, v = u(t, x) becomes multivalued. This multi-valued function exists for all t > 0. It is natural to ask [64] whether our single-valued global solution of (13.1)-(13.2) can be determined directly from the several values of v in (13.3) above the same point (t,x). In the first case of Hopf's formulas, I = 1(p) convex (or concave), = largely arbitrary, the answer can be foreseen by (4.11) and Lemma 4.3 as follows: The value at (t, x) of the global solution is either always the smallest or always the largest of the values of the many-valued v = u(t,x), depending on the convexity character off = 1(p). For definiteness, let us suppose I = 1(p) is convex. Recall that the characteristic equations for the Cauchy problem (13.1)-(13.2) can be written by (1.10)-(1.11). Therefore, in this particular case (the case of Hamilton-Jacobi equations), the characteristic strips are defined as

x = x(t,y) = v = v(t,y) = I.

y+i +t.

(13.4)

—

p=p(t,y)=4'(y). df dp

,

dq5

Here and subsequently, we write x, y, p, f = — and 4) = — as horizontal vectors dx

As was mentioned and (.,.) denotes the (Euclidean) inner product in in the above, in general, we could not completely eliminate y from (13.4) to get a single-valued function v = u(t,x) (i.e., in this form, u = u(t,x) might become multi-valued). Anyway, if we try to do so, we may rewrite v = v(t,y) in (13.4) as in

v = ((t,x(t,y),y) where

((t,x,y)

+t.

f((x

—

y)/t).

(13.5)

13. MISHMASH

200

Then the hypersurface (13.4) projects upon the entire half {t 0) of tx-space, and the global solution (9.29) of (13.1)-(13.2) equals, at each point (t, x) of {t >0), the smallest v-coordinate of the hypersurface in the following sense.

Theorem 13.1. Let

be of class C' in R", and let I = f(p) be str*ctly

=

convex and of class C1 in

= +00. Assume (F.lI). Then the

lim

with

value u(t(°), x(°)) at each (t(°), x(°)),

> 0, of the global solution u = u(t, x) given by Hopf's formula (9.29) of the Cauchy problem (13.1)-(13.2) can be determined as the smallest values of v(t(°),y) = ((t(°) ,x(t(°),y),y); the minimum being taken over

all y such that the characteristic curves {(t,x(t,y)) meet each other at (t(°),x(°)).

I

Proof. By hypotheses, f =

I=

:

t 0} starting from (O,y)

(z) is the same as the Legendre conjugate of

.f(p) (see Lemma 9.13 and [117, Theorem 26.61); moreover, both 1' and f' are

onto itself with I = (f') from (13.5) that ( = ((t,x,y) is differentiable in y with one-to-one mappings from

=

Therefore, it follows

(13.6)

—

It has been shown in the proof of Theorem 9.12 that (9.29) determines a global solution u = u(t, x) of (13.1)-(13.2), and that the infimum = inf

(13.7)

pER

is attained at some

Clearly,

E

must be a stationary point of the

function ((t(°>,x(°), .); i.e.,

= 0. Thus (13.6) implies 4,'(y(O)) =

(

=

f'((x(°) or,

for every such stationary point

—

=

y(°))/t(°)). Since f' = +

(f')1,

we have

=

So, the characteristic curve {(t,x(t,y(°))) 0}, which is starting from must pass through (t(°),x(°)). 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. 0

§13.1. CONSTRUCTION OF GLOBAL SOLUTIONS VIA CHARACTERISTICS

We now turn to second case of Hopf's formulas, qS =

201

convex (or concave),

I = f(t, p) largely arbitrary, and continue investigating the construction of global solutions via characteristics. Consider the Cauchy problem

.3u/Ot+f(t,ôu/Ox) = u(0,x) =

0

on

in V {t>0, XE R"},

(13.8)

{t = 0, x

(13.9)

as in Chapter 9, under the following standing hypotheses:

(E.I) The Hamiltonian I = f(t,p) is continuous in {(t,p)

t E (O,+oo)\G, p E R of Lebesgue measure 0. Moreover, to each N E

(0, +oo) there corresponds a function gjq = gri(t) in

sup If(t,p)I

IplN

:

such that

gN(t) for almost all t E (0, +oo).

(E.II) For every bounded subset V of D, there ezzsts a positive number N(V) so that

(p,x)

—

—

whenever (t,x)

ff()d

V, I;'I

—

—

ff(r,q)dr}

> N(V).

It was shown in the proof of Theorem 9.1 that, if = is a finite convex function on a global solution of (13.8)-(13.9) exists and is found by Hopf's formula (9.7). We need to know how this global solution can be constructed by

means of characteristics. For this, suppose the Hamiltonian and initial data are of class C2. The characteristic strips for (13.8)-( 13.9) in this case are defined (cf. (1.10)-(1.11)) as

x =x(t,y) v

=v+f

= v(t,y) =

+

—

f(r,cb'(y)))dr,

(13.10)

p = p(t,y) = In general, we can not completely eliminate y from (13.10) to get a single-valued

function v = u(t,x). Anyway, if we try to do so, we may rewrite v = v(t,y) in (13.10) as

v=

+ (4'(y),x(t,y) — y) —

j f(r,qS'(y))dr.

13. MISHMASH

202

From now on, let us assume = to be convex on 23.5 and 25.11, we can then further rewrite (13.11) as

Using [117, Theorems

v= where

x,p)

(p.x)

—

f f(r,p)dr

ç&(p) —

(t,x,p) E

for

(13.12)

x

It will be proved in Theorem 13.3 that the hypersurface (13.10) projects upon the entire half {t 0) of tx-space, and the global solution (9.7) of (13.8)-(13.9) equals, at each point (t, x) of {t 0}, the largest v-coordinate of the hypersurface. First, let

u(t,x) dcf = u(t,x) def =

(t,x)

for

(y)),

sup

Proof. Let E

:

u(t,x) on V. It follows from Corollary 26.4.1 in [117] that

yE

ri(domq5) c E where ri (dom q5') is the relative interior

and

< u(t,x). We

choose q E dom

now prove u(t,x)

setting p(m)

(1 — 1/rn)q

shows that

of dom

(13.15)

Therefore,

(13. 13)-( 13.14) yield

ü(t,x). Fix any

e

> 0,

x, q) + €.

+

we see, by (13.15) and

C E for

6.11, that

=

c

such that u(t, x)

On

(13.14)

sup pEdom #

We have:

Proposition 13.2. ü(t,x)

ü(i,x)

(13.13)

in

= 1,2,3,...

[117, Theorem

Since the Fenchel conjugate

is a lower semicontinuous proper convex function, [117, Corollary 7.5.1)

= lim

Hence, it follows from (13.12) that

u(i,x)

+e

= lim co(t,x,p(m))+e

0 is arbitrarily chosen, the proof is then complete.

Remark. Let 4) = 4)(x) be convex and of class C' on 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 E dom4)). The situation becomes different for the supremum in (13.13). We show this by the following example. Let

n def =

1

p>l,

(1, def I

1(P) =

p—plnp,

pE[O,1], p0,xER},

S13.2. SMOOTHNESS OF GLOBAL SOLUTIONS

u(0,x)=x2/2 on where

(0, def I p, s 1,

207

xER},

p0, O 0).

.

The sets

are called the superdifferential and the subat the point The elements of these sets = are called supergradients and subgradients, respectively. These sets are also called the upper and lower Dini semidifferentials. If = is differentiable at then in this case, = =

and

differential, respectively, of

= where

—

+ -4 0 as

—

+ o(e

—

According to (13.19)-(13.20),

—+

—

+ for any supergradient -y E one-sided but opposite estimate:

—

+ o(e

—

A subgradient y E

+

—

also

satisfies a

—

+ o(e

We can verify that

(y,e) Ve

= {y E Rm : Rm : ={

In fact, let + as

ej

0

eI/2.

1321 )

Rm}.

Ve

e E Rm\{0}. Consider the function

-y

for

(,e)

0 and

E Rm. Of course,

= =

—

Consequently, (13.18)-(13.20) imply

inf sup

+

e>0 O<S(e Ii—eI<e 3

[mi sup

—

+

i—el<e

=

—

—

mi sup =

—

—

—

—

—

—

§13.3. RELATIONSHIP BETWEEN MINIMAX ANT) VISCOSITY SOLUTIONS

211

Ye E Rm\{O}. This still holds even fore = 0, one would have

from which we get since otherwise,

inf sup

+

S>OOO

=

—

It follows that

0

inf sup

+

—

inf e>O

+

—

e>O O 5(e) > ... > 5(k)

= hence,

0. But, by (12.16) and (12.14), we

f

a)dr

(13.28)

f(r,xo(r), u(i(°>, x(°)),cx)dr,

by (12.13), that

1

J This

together with (12.37) and (13.28) implies (y,b) f(t(°),x(°),u(t(°),x(°)),b).

Therefore, it follows from (13.27)-(13.28) and (13.18)-(13.21) that

0 sup mf { e>00(J