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)) 1
$ 1$

=

—

f(i,x,v,p) —p(t,y).

= —f(t,x,v(t,y(t,x)),p(t,y(t,x))) = —f(t, x, u(t, x),

Ou

x)).

Moreover, by (1.11), u(0,x) = v(0,y(0,x)) = q5(x). It follows that u = u(t,x) is a (i = 1,2,... ,n) are of class C1 in W and = solution of (1.8)-(1.9). As y = y(t,x) is of class C' in its definition domain, we see by (1.13) that u = u(t,x) is of class C2 in the definition domain of y = y(t,x). This domain is actually a neighborhood of the origin.

Finally, we give the sketch of a proof of the uniqueness of solutions. Let u = u(i, be a solution of class C2 of(1.8)-(1.9), and z = z(t, y) = (zi(t, y),... , y)) be a solution of = I.

z1(0)

Of

Ou

(, =

1,2

n),

=

def def We put w(t, y) = u(t, z(t, y)) and q(t, y) = (Ou/Ox)(t, z(t, y)), then we get

= —f(t,z,w,q) + =

with w(0,y) =

çb(y)

—

and q(0,y) =

qV(y). Hence

(z, w, q) = (z(t, y), w(t, y), q(t, !J))

(z = 1,2,...

§1.2. A THEOREM OF A. HAAR

7

satisfies the Cauchy problem (1.1O)-(1.11). By the uniqueness of solutions of ordinary differential equations, we have

r(t, y) = z(t, y),

v(i, y) = w(t, y),

p(i, y) = q(t, y),

where (x,v,p) = (x(t,y),v(t,y),p(t,y)) is the solution of (1.1O)-(1.11) which has already appeared in Lemma 1.2. This says that the solution of class C2 is unique along the curves x = x(i, y). Therefore, as long as the Jacobian (Dx/Dy)(t, y) does not vanish, the solution of class C2 of (1.8)-(1.9) is unique.

0

§ 1.2. A theorem of A. Haar have seen by Theorem 1.4 that the Cauchy problem (1.8)-(1.9) has a solution of class C2 in a neighborhood of the origin. But, as the equation (1.8) is of first-order, we would like to discuss the existence and uniqueness of solutions in C1-space. In the following, we write a solution of class as a C's-solution. C'-solutions will be sometimes called "classical solutions." One of our aims is to consider what kinds of phenomena may appear when we extend the classical solutions of (1.8)-(1.9). In this procedure, we need the uniqueness of solutions in C1-space. This subject had been well studied by A. Haar [611 and T. Walewski [154]-[155]. Moreover, their results are indispensable to develop We

our discussions. But, as it seems to us that they are not well-known, we would like to introduce them. In this section, we report a theorem of A. Haar [61]. We consider the Cauchy problem (1.8)-(1.9) in one space dimension, that is x R'. Let

be a triangle defined by

wherea>O, LO, c
Theorem 1.5. (A. Haar [61]) Suppose that the function I = f(t,x,u,p) satisfies a Lipschitz condition as follows:

If(t,x,u,p) — f(t,x,v,q)I Lip— qI +Miu — vi

1. LOCAL THEORY

for all (t,x,u,p) and (t,x,v,q) in x Let u3 = u,(t,z) (j = 1,2) be sn and (u1(t,x),(Ou,/Ox)(t,x)) E for all (t,z) E If u,(t,x) (j = 1,2) satisfy the equation (1.8) in the domain and ui(O,x) = u2(O,z) for x E [c,d], then uj(i,x) in Proof. According to the Lipschitz continuity off = f(t,x,u,p), the difference z(t,x) — u2(t,z) satisfies the inequality

Assume

(1.14)

+ Mlz(t, x)I.

L

that z = z(t,x) takes some positive values in a > M. As w = w(t,x) is continuous on

for

We

there

put w(t,x) exists a point

P E at which w = w(t, x) attains the maximum. The point P is not on the initial line {t = 0) because w(O,x) 0. Let us consider the derivatives of w = w(t,x) at P with respect to the directions (—1,--L) and (—1,L). As w = w(t,s) takes the maximum at the point P, we have

<0,

—

Hence it holds that

—

<0.

8w

9w

Rewriting this for z = z(t,x), we get Or

az(P) + L

Oz

As z(P) > 0 and a > M, this contradicts (1.14). Therefore z(t,x) < 0 for all (t,x) Since we can similarly prove that z(t,x) 0, we obtain ui(t,x)

0

§ 1.3. A theorem of T. Wazewski In this section we report a theorem of T. Waiewski [154] which is a generalization of Haar's theorem to arbitrary space dimensions. which is defined Let us consider the Cauchy problem (1.8)-(1.9) in a pyramid by

= {(t,x)

O t

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

§1.3. A THEOREM OF T. WAZEWSKI

wherea>O, L1O, is a compact set in {(u,p)

,n. :

u E R, p

R't}.

Theorem 1.6. (T. Waiewski [154J) Suppose that I = f(t,x,u,p) satisfies the following Lipschitz condition:

if(t,x,u,p) —

—

for (t,x,u,p) and (t,x,v,q) in and (u,(t,x),(Ou,/ôx)(t,x))

x

solutions of (1.8) in the domain uj(t,x) u2(t,x) in As

+

MItz — vi

u = u(t,x) (j = 1,2) be in all(t,x) If u = u,(t,x) (j = 1,2) Let

and

ui(t,z) = u2(t,x)

on

are

fl {t = 0), then

preparation for the proof of this theorem, we give a fundamental lemma

which plays an important role in his many works.

Lemma 1.7. Let

be the pyramid defined in the above, and {x E R" Let u = u(t,x) be in and put xE (t,x) E for each t E [0,a]. Then w = w(t) is differentiable from the right on [0,a), and :

n

= for some x E

—

with w(t) = u(t,x).

Proof. As the first step, consider the case where L1 = 0, cj = —1 and d1 = 1 for all i = 1,2,... ,n. By the definition of w = w(t,x), we easily see that it is continuous. Let U(t) w(t) = u(t,x)} for t E [0,a). Then the sets {x U(t) (0 < t < a) are closed, bounded, and non-empty. As u = u(t,x) is in such that for any fixed t0 E [0,a), we can pick up a point x° E U(t°) C 0) Ot

—

max —(t° ZEU(t0)Ot

a,)

We will prove that w = w(t) is differentiable from the right at t° with For any x E U(t°), we have w(t°) = tz(t°,x) and

=

1. LOCAL THEORY

10

Define z(t)

[w(t) —

w(t°)]/(t — t°), and put def.

/3 = hmsupz(t) and

def. Cl = hminfz(t).

g-4t0

u(t,x°) — u(t°,x°), we see that & (Ou/Ot)(t°,x°). By the definition of/3, we can pick up a sequence {tm}m C (t°,a) with urn tm = t° and limz(t"t) = /3. For each tm, take arbitrarily xm E U(tm), so that u(tm,xm) = w(tm). Since the set {(tm,xm) m = 1,2,... } is bounded, we can assume that Xm)}m is the sequence {(tm, convergent to a point (t°, in As the functions u = u(t,x) and w = w(t) are continuous, this implies that u(t0,°) = w(t°), i.e., As w(t) — w(t°)

:

(Ou/&)(t°,x°). By the definition of

E U(t°), hence that

w = w(t), we have w(t°) u(t°, xtm) for all m = 1,2

Using this inequality, we

get

z(tm) = w(tm)

—

w(t°)

ttm — to —

u(tm,xm) —w(t°)

—

tm_to _u(tO,xm)

<

with m-Ioo lim (t ,xm) =

— —

tm_to

—

0

).

It follows that /3 <

Ot' 0

Ott

0

Ot

fore,

)

< —(t°,x°). There-

at


{

—

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

—

Then the set is mapped to the set {(s,y) s < a, —1 < 0 1 (i = 1,2,... , n)}. Combining the result obtained in the first step and the above trans:

form of coordinates, we complete the proof of the lemma.

0

§1.3. A THEOREM OF T. WAZEWSKI

Proof of Theorem 1.6. We put u(t,x) max{u(t,x)

—

For t E [O,a), let r(t) E

xE

=

be

11

u2(t,x), and define w(t)

such that w(t) = u(t,x(t))

—

(cf. Lemma 1.7). Then

=

x(t))

—

—

{

clu(t,

x(t)) }

.

(1.15)

On the other hand, it follows from the hypotheses of the theorem that (1.16)

+ MIu(t, If we choose o > M, then from (1.15)-(1.16) we get (Mlw(t)I f

—

1. w(O) = 0,

from which it may be concluded that w(t) 0, i.e., u1(t,x) < u2(t,x), for all (t, x)

Since

we can similarly prove uz(t, x) u1(t, x), we finally get ui(t, x)

0

u2(t,x) on

We will rewrite Theorem 1.6 in a general form. A function f = f(t,x,u,p) is said to be locally Lipschitz continuous with respect to (u,p) if, for any compact set Rx and M such that in R x there exist constants L1,L2,... —vi

for all (t,x,u,p) and (t,x,v,q) in fi..

Theorem 1.8. Suppose that f = f(t, x, u,p) is locally Lipschitz continuous with respect to (u,p). If u = u(t,x) and v = v(i,x) are C1 -solutions of (1.8) in a neighborhood of the origin satisfying u(0,r) neighborhood of the origin, such that u(t,x)

v(O,x), then there exists an open

v(t,x) in

We will leave the proof to the readers.

Remark. The uniqueness of C'-solutions is assured by the Lipschitz continuity of f = f(t,x,u,p) with respect to (u,p). But this condition is not sufficient to get the 11561 has given necessary solvability of the Cauchy problem (1.8)-(l.9). and sufficient conditions which guarantee the local existence and uniqueness of classical solutions of (1.8)-(1.9).

Chapter 2 Life Spans of Classical Differential Soluti9ns of Equations of I-irst-Order §2.1.

Introduction

As we have shown in Chapter 1, the Cauchy problems (1.1)-(1.2) and (1.8)-(1.9) have locally classical solutions. It is well-known that the solutions may generally have singularities in finite time even for smooth initial data. For some equations of the conservation law, the life spans of classical solutions have been exactly calculated, for example by E. Hopf [63], P.D. Lax [97]-[98], E.D. Conway [32], etc. The principal aim of this chapter is to determine the life spans of classical solutions of general partial differential equations of first-order. Moreover, using the results on the life spans of classical solutions, we will give necessary and sufficient conditions which guarantee the global existence of classical solutions for (1.1)-(1.2) and (1.8)(1.9). As an example, we consider a simple equation of the conservation law as follows:

'I

=

+ u(O,x) =

0

on

where a1 = a(u) (i = 1,2,... ,n) and

in

{t >

{t =

0, x

(2.2)

0, x

=

(2.1)

are of class C1 in Rand RTh,

respectively.

The equation treated in E. Hopf [63] is the case where n = 1 and ai(u) = u. E.D. Conway considered the above equation (2.1)-(2.2) in [32]. In this case, the characteristic curves which are the solutions of (1.3)-(1.4) are written by

x=

y

v(t, y) =

+

wherea(u) def = (ai(u),az(u),...

x(t,y)

is given

by (Dx/Dy)(t,y) =

Then the 1

(2.3)

Jacobian of the mappmgx =

+ tA(y) with A(y)

Obviously, the Jacobian (Dx/Dy)(t, y) does not vanish in a neighborhood of (t, y) =

§2.2. LIFE SPANS OF CLASSICAL SOLUTIONS

13

(0,0). Therefore, as we have shown in §1.1, the Cauchy problem (2.l)-(2.2) has a unique C1-solution u = u(t,r) in a neighborhood of the origin. Since u(t,x) = x)) where y = y(t, z) is the solution of the equation x = x(t, y), we have

f'Ou

1

'

1

24

+

'

Assume A(y°) <0. Then the Jacobian (Dx/Dy)(t°,y°) = 0 for t° l/(—A(y°)). We see by (2.4) that, when (t, x) goes to (t°, x°) along the curve x = x(t, y°) where 0det . z = x(t o ,yo ), at least one of the first derivatives of u = u(i,x) tends to infixuty. Therefore, if the Jacobian vanishes somewhere, then the Cauchy problem (2.1)(2.2) can not admit a global solution of class C1. In §2.2 and §2.3, we will give similar results on the life spans of classical solutions for general partial differential equations of first-order. What we would like to remark here is to point out that there exist some differences between quasi-linear equations and general partial differential equations. See Theorem 2.1 and Theorem 2.2. In §2.4, we will consider the global existence of classical solutions. These existence results are corollaries of theorems given in §2.2 and §2.3.

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

= ao(i,x,u)

+

u(0,x) =

where a = a1(t,x,u) (i = and

on

{t =

in

0,

{t >0, x E R't}, x E W'},

(2.5) (2.6)

n) and 4 = are of class C1 in R x R respectively. The characteristic equations for (2.5)-(2.6) are written by 0,1

= a(t,x,v) (i = 1,2,... ,n),

I

(2.7)

= ao(t, x, v),

with the initial conditions

=

y1

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

v(0) =

(2.8)

2. LIFE SPANS OF CLASSICAL SOLUTIONS

14

We write the solutions of (2.7)-(2.8) by r = x(t, y) and v = v(t, y). Then v = v(t, y) means the value of a solution of (2.5) restricted on the curve x = x(t, y). Here we assume the following condition:

(A.I) The Cauchy problem (2.7)-(2.8) has uniquely a global solution x = x(t,y), v = v(t,y) on {t O} for any y E It is not easy to write down sufficient conditions which guarantee the assumption (A.I). Concerning this subject, see for example B. Doubnov [43J and M.S. Krasnosel'skii [83J. We will consider this in Appendix I. When we assume (A.!), we

get a smooth mapping x = x(t,y) from R" to RTh for each t 0. The life span of classical solutions of (2.5)-(2.6) is determined by the following:

Theorem 2.1. Under Cond;t*on (A.!), suppose that (Dx/Dy)(t°,y°) =

0

and

(Dx/Dy)(t,y°) 0 fort < t°. Then at least one of the first derivatives of the solution u = u(t,x) tends to infinity when t goes to t0 — 0 along the curve x = x(t, y°).

Proof.

: x=x(t,y°),0<

:

t
U

.

on — Ov

,n

On the other hand, from (2.7),

(2.9)

V.

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

and

the following system of linear equations: —

d dt

—

Osk

Lank

+

Ov

Oa0Ov dt

Oy,)

—

Oy. + Ov Oyj

12 10

§2.2. LIFE SPANS OF CLASSICAL SOLUTIONS

15

with the initial data =

11

if

0

if

z=j,

and

Ov

Oçb

=

By the linearity of (2.10), we have

oy.

=rank

rank

=n forall

•..

Oy" Ox,

0

Oy

0x1\

/ Ox1

t0,

(2.11)

0

Ov

/Ov

Ov

and — = 0111 (31Jn 0Y = 0111 , —) Assume that all the components of Ou/Ox remain bounded along the curve C0, then we can pick up a sequence {tm}m C and one c = (ci,c2,... E such that

where we write

01,

1)

lim t'Th = j0, and

2)

lim

Ox

=c.

From (2.9'), we get

=

(2.12)

As (Dx/Dy)(t°,y°) = 0, (2.12) contradicts (2.11). This means that at least one component of (Ou/Ox)(t, x(i, y°)) tends to infinity when t

—0

along C0.

0

Next we consider general partial differential equations of first-order as follows: in

u(0,x) =

where f = f(t,x,u,p) and 4 =

on

{t>O, xER"},

{t = 0, x E

(2.13) (2.14)

Rx and are of class C2 in R x respectively. The characteristic equations for (2.13)-(2.14) are written by

2. LIFE SPANS OF CLASSICAL SOLUTIONS

= — = di dp1

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

j1

=

—f(t,x,v,p),

VP,

Of

Of

(2.15)

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

—

with x1(O) =

y1,

v(O) =

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

=

(2.16)

We assume here the following condition:

(A.!)' The Cauchy problem (2.15)-(2.16) has uniquely a global solution on {i O} for any y E

We denote the solution of (2.15)-(2.16) by x = x(t,y), v = v(t,y), and p = p(t,y). As (Dz/Dy)(O,y) = 1 for all y E RTh, the Jacobian does not vanish in a neighborhood of {i = O}. Therefore we can uniquely solve the equation x = x(t, y) Then with respect toy, and denote it by y = y(t,x). Define u(t,x) u = u(t, x) is a C2-solution of (2.13)-(2.14) in a neighborhood oft = 0. Moreover, we can see by Theorem 1.4 that there does not exist another C'-solution of (2.13)(2.14) in a neighborhood of {t = 0). When we extend this solution for large t, we get the following.

Theorem 2.2. Under Condition (A.!)', suppose that (Dx/Dy)(t°,y°) = 0 and " O2ti (t,x) tends to infinity when i (Dz/Dy)(t,y°) 0 for t < t°. Then E i,j=1 UX;UXJ

goes to

—0 along the curve x = z(i, y°).

Proof. As the proof is similar to that of Theorem 2.1, we use the same notations Next we choose an open introduced there. Put L {(t, y°) t< 0 def neighborhood V of L so that the Jacobian (Dx/Dy)(t, y) does not vanish on V = I
:

S2.2. LIFE SPANS OF CLASSICAL SOLUTIONS

17

Our aim is to show that, when t goes to t0 —0 along the curve x = x(t, y°), at least one of {Ou/0x1, 02u/Ox1Ox,

:

i,j = 1,2,... ,n} tends to infinity.

Differentiating (2.15) with respect to y, (j = 1,2,... ,n), we get a system of linear ordinary differential equations of first-order concerning {0x1/0y1, .3v/Oy,, i,j = 1,2,... ,n} like (2.10). As this system of equations is linear, we have

(0z1/Oy)(t)

(Oxi/Oy)(O)

(Ox,1/Ôy)(t)

(ôXm/Oy)(O)

= rank

rank

(Ov/äy)(0)

(0p1/Oy)(t)

= n for any t 0.

(2.17)

(Opl/Oy)(O) (Opfl/Oy)(0)

Since (2.17) is true at (t,y) = (t°,y°), we can choose n vectors in {8x1/äy,... such that they are linearly independent at (t, y) = (t°, y°). We denote them by a1(t, y) (8/Oy)b1(t, y) (i = 1,2,. . , n), where b1(t, y) is one of {xk(t, y), v(t, y), pk(t, y) : k = 1,2,... , n}. As (Dx/Dy)(t°, y°) = Here we recall 0, we can find v(t,y) or some pk(t,y) in .

Corollary 1.2:

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

in

U

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

(2.18)

On the other hand, we get in U ai(t, y) r o

1

a2(t,y)

=

...

-(t,x)

1

= y)

a neighborhood of

Since {a1 (t, y), a2(i, y),... ,

(t°,y°) and (Dx/Dy)(t°,y°) =

0,

it follows that at least one component of the

2. LIFE SPANS OF CLASSICAL SOLUTIONS

matrix tends to infinity when t goes tot0 —o along the curve x = x(t,y°). But we see by (A.!)' and (2.18) that

y(t, x))} {

=

x(t, y°)) = p,(t,y°)

= 1,2

z-z(*y°)

remain bounded (though the Jacobian vanishes); therefore, some pa(t, y) must be contained in {bi(t,y),b2(t,y),... Hence we get Theorem 2.2. 0

Theorem 2.2 says that, if the Jacobian vanishes at a point (t°, y°), then the second derivatives of classical solutions blow up at time t = t°. But this does not prevent the existence of C1-solutions even if the Jacobian may vanish. To understand this situation, we need to know the behavior of characteristic curves in a neighborhood of the point where the Jacobian vanishes. After having studied this subject, we will again consider the extension of classical solutions in Chapter 3.

§2.3. Global existence of classical solutions As we have shown in §1.1, the Cauchy problem for a partial differential equation of first-order has locally a classical solution. We have also seen in §2.2 that the solution may generally have singularities in finite time even for smooth initial data. But in some cases the Cauchy problem admits a global classical solution. In this section, following Tsuji-Li [1391, we will give necessary and sufficient conditions which guarantee the global existence of classical solutions. First we give some comments on this subject. For the Cauchy problem (2.1)-(2.2), E.D. Conway (32] gave a necessary and sufficient condition for the global existence of C1-solutions on the half space {t O}, using Hadamard's lemma on the diffeomorphism of Euclidean spaces. Another proof

was given by Li Ta-tsien and Chen Shu-xing [102J. Qin Tie-hu [1141 treated the case where the functions a (z = 1,2,... ,n) depend on t and x. Li Ta-tsien and Shi Jia-hong [103] gave certain relations between the global existence of nontrivial smooth solutions in the whole (t, x)-space and the linear dependence of the functions aj = aj(u) = Now we review the results on the diffeomorphism of the Eucidean space Concerning this subject, we must go back to J. Hadamard, but here we follow W.B. Gordon's note [58]. Let H be a C1-mapping from R" to R" defined on the whole

§2.3. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS

space. Then the mapping H is said to be proper if K is compact. We have:

is compact whenever

Lemma 2.3. (J. Hadainard) The mapping H is a diffeomorphssm from R" to if and only if H is proper and the Jacobian of H does not vanish anywhere on the whole space.

First we consider the Cauchy problem for a quasi-linear equation, say Problem (2.5)-(2.6). We assume the following conditions. (Here and subsequently, ri denotes the Euclidean norm of x E (A.I) For any y E the Cauchy problem (2.7)-(2.8) has always a unique global C'-solution x = x(t,y), v = v(t,y) on {t O}. (A.II) On the solution curves of (2.7)-(2.8), it holds that a1(t, x(t, y), v(t, v))i

for any t on T.

[O,T], y E

+

and i = 1,2,... ,n where the constant M depends only

When Conditions (A.1) and (A.Il) are satisfied, we can define, for any t -mapping C from R to R by x = x(t,y) def = Ht(y). Then we get:

0, a

1

Lemma 2.4. Under the assumptions (A.I) and (A.!!), the mapping for any t 0.

is proper

Proof. As we have y)a2(t, x(t, y), v(t,

Ix(t, )I

—nM(1

we get Ix(t,y)i (1 + means that is proper.

— 1.

+ z(t,y)l), Therefore

—* 00 as

00. This

0

Using Lemma 2.3, we have:

is a diffeomorphism from Lemma 2.5. For any fixed t 0, the mapping if and only if its Jacobian does not vanish at any point y €

to

2. LIFE SPANS OF CLASSICAL SOLUTIONS

20

By means of Lemma 2.5, we obtain the following.

Theorem 2.6. Under the assumptions (A.I) and (A.!!), the Cauchy problem (2.5)(2.6) has a global C' -solution uniquely on the domain D {t 0, x E if and only if the Jacobian (Dx/Dy)(t,y) of the mopping does not vanish for any t 0 and

y E RTh.

Proof. When (Dx/Dy)(t,y)

and t 0, we can see from 0 for any y E Lemma 2.5 that H, is a diffeomorphism from R" to R't for any t > 0. Hence the inverse function y = y(t, x) of x = H,(y) is a function of class C1 defined on D and u = u(t,x) v(t,y(t,x)) is the unique global C1-solution of (2.5)-(2.6). (For the uniqueness, follow the proof of Theorem 1.1 or 1.6.)

Next, for the necessity, suppose that (Dx/Dy)(t,y°) =

0

for some t E (0,oo)

As (Dx/Dy)(O, y) = 1 for all y E R't, we can get a unique C1-solution and y° E of (2.5)-(2.6) in a neighborhood of {t = 0}. Then we see by Theorem 2.1 that this classical solution blows up at time t0 > 0 (Dx/Dy)(t,y°) = 0}. (Notice :

thatO
0

Next we consider the Cauchy problem (2. 13)-(2. 14) for general partial differential equations of first order. We assume the following hypotheses:

(A.I)' For any y E the Cauchy problem (2.15)-(2.16) has always a unique global C1-solution x = x(t,y), v = v(t,y), p = p(t,y) on the halfspace {t 0}.

(A.II)' On the solution curves of(2.15)-(2.16), it holds that x(t, y), v(t, y), p(t, y))

for any t e [0,TI, y E on T. For

.tvf( 1 + Ix(t,

andi = 1,2,... ,n where the constant M depends only

any t 0, we define a C -mapping H, from K 1

to K by x = x(t,y) def =

H,(y). Then Conditions (A.!)' and (A.!!)' guarantee that the mapping H, is proper. Therefore, Lemma 2.5 is also true for this H,.

Theorem 2.7. Under the assumptions (A.!)' and (A.!!)', the Cauchy problem (2. 13)-(2. 14) has uniquely a global C2-solutzon on the domain D {t 0, x E

§2.3. GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS

21

if and only if the Jacobian of the mapping x = Hg(y) does not vanish for any t 0 and y E

we can uniquely solve Proof. When (Dx/Dy)(t,y) 0 for any t 0 and y E the equation x = with respect toy for any (t,x) E D. Let y = y(t,x) be the def inverse function. A global C 2-solution of (2.13)-(2.14) is given by u = u(t,x) = v(t,y(t,x)). The uniqueness of this C2-solution follows from Theorem 1.6. (It can .

.

also be deduced from the method of proof used in Theorem 1.4.) The necessity of the above condition comes from Theorem 2.2. 0

Chapter 3 of Characterjstic Lurves and Prolongation of Classical Solutions §3.1.

Introduction

This chapter is continued from Theorem 2.2 in §2.2. Let us rewrite the equation which we will again consider here:

=0 in {t >0, 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°.

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

(B) The mapping H is bijective from a neighborhood of (t°, y°) to another one of (t°,z°) where By Condition (B), we can uniquely solve the equation x = z(t,y) with respect toy, and denote it by y = y(t,x) (for (t,x) in a neighborhood of (t°,x°), (t,y) in a neighborhood of (t°, y°)). The function y = y(t, x) is obviously continuous, though it may not be differentiable. Then we get the following.

Theorem 3.1. Under the hypothesis (A.I)', suppose (I)-(II) and (B). Then the solution u = u(t,x) remains a C'-solution of(3.1) in a neighborhood of(t°,x°), though it is not of class C2. Proof. Let V and U be open neighborhoods of (t°, y°) and (t°, x°), respectively, such that the mapping H is bijective from V to U. Consider S dcf = {(t, y) V

(Dz/Dy)(t,y) = 0) and H(S) = {H(t,y)

:

(t,y) E S}. By Sard's theorem,

the Lebesgue measure of H(S) is zero. Therefore U\H(S) is dense in U. We see that u = u(t,x) is of class C2 in the domain U\H(S). The reason is as follows. For any E V satisfying E U\H(S), there exists uniquely a point The inverse function y = y(t,x) is of class C1 0 and = in a neighborhood of which is contained in U\H(S). Here we recall Lemma 1.2 and Corollary 1.3 in §1.1, and we see that u = u(t,x) is a function of class C2 satisfying the equation (3.1) in the neighborhood of (t , Next we show that u = u(t,x) is continuously differentiable in the domain U. We pick up arbitrarily _o

a point (t , ) in H(S). Then we can choose a sequence {(tm,xm)}m of points in U\H(S) such that, when m tends to infinity, (tm,xm) is convergent to (t ). As the mapping H is bijective from V to U, there exists a unique point (t, ym) E V\S satisfying H(tm,ym) = (tm,xm) for each in. Since y = y(t,x) is continuous in U,

3. PROLONGATION OF CLASSICAL SOLUTIONS

26

0

o

it follows that ym = y(tm,xm) is convergent to y(t , ) when in tends to infinity. As u = u(t,x) is continuously differentiable at (tm,xm), we have by (1.13)

p(t,y) = Because

p = p(t,y) is continuous on the whole half space {t 0, y E R"}, we get lim

ôu

—(t,x) =

m-.oo Ox

lim

p(t,y) =p(i

).

The derivative On/Or of u = u(t, x) can thereby be continuously extended (in such a way that (Ou/Ox)(t, x) = p(t, y(t, x))) over U. That is to say, u = u(t, r) is of class C' in the domain U. Moreover, notice that the uniqueness of the above (extended) 0 C'-solution is assured by Theorem 1.6.

Remark. In Theorem 3.1, the assumption (A.I)' is a crucial point. It will be proved in Chapter 5 that, for quasi-linear equations of first-order, the assumption (A.!)' is not compatible with the property that the Jacobian vanishes somewhere.

§3.4. Sufficient conditions for collision of characteristic curves I For quasi-linear equations of first-order, Theorem 2.1 says that, if the Jacobian vanishes somewhere, the classical solutions blow up there. Therefore we cannot extend the classical solutions beyond the time t° when the Jacobian vanishes. This obliges us to treat weak solutions for t > t°. The typical singularity of weak solutions is "shock." As it will be shown in Chapter 5, the shock appears by the collision of characteristic curves. Therefore we try here to give sufficient conditions so that the characteristic curves meet after the Jacobian vanishes.

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

On

u(0,x) =

.

= ao(t,x,u) in on

{t =

0,

{t >0, XE R },

(3.6)

r E R'},

(3.7)

= a1(t,r,u) (i = 0,1) and are of class C1 in R3 and R1, = respectively. The characteristic curves for (3.6)-(3.7) are defined as solution curves where

§3.4. COLLISION OF

CURVES 1

27

of (2.7)-(2.8) for n = 1. Here, as in Chapter 2, we again assume Condition (Al) which assures the global existence of characteristic curves. Let x = x(t, y) and v = u(t,y) be the solutions of (2.7)-(2.8) for n = 1.

Theorem 3.2. Under Condition (A.!), ioe assume: (I)

=

and

0,

If (ôa1/Ou)(t°,x°,v°) $

0

0 fort < t°.

(II)

v(t°,y°), then the

x(t°,y°) and v0

where x0

characteristic curves meet in a neighborhood of (t°, x°).

Proof. This theorem follows immediately if the monotonicity of x = x(t, y) with respect to y is violated. First we prove that (Ox/8y)(t, y°) is negative for t > 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,

i.e.,

dt

<0.

Oy

with t — t° small. That is to say,

Therefore we get (Ox/Oy)(i, y°) < 0 for t >

(Ox/Oy)(t,y°) changes its sign at I = t°. We now define a function h = h(y) by setting h(y) inf{t : (OxfOy)(t, y) = 0) for each y. Then h = h(y) is of class C1 in a neighborhood of y = y0 In fact, as we have (Ox/Oy)(t°,y°) = 0 and (82x/OtOy)(t°,y°) <0, we can uniquely solve the equation (Ox/Oy)(i,y) = 0 with respect to y in a neighborhood of (t°, y°). Let us restrict our following discussions into a small neighborhood of (t°, x0, v°) only. We see by the same reasoning as in the above that, as a function oft, (Ox/Oy)(t, y) changes its sign at t = h(y) for each y.

Next, we will show that x = r(t,y) is not monotone with respect to y. Pick up a point
Ox-.. <0 and

Ox )

>0

3. PROLONGATION OP CLASSICAL SOLUTIONS

28

for t As this means that x = x(t,y) is not monotone with respect to y, the characteristic curves meet in a neighborhood of (t°, x°). When h = h(y) 0 in is constant in a neighborhood of y = y°, it follows that (Ox/ôy)(t°,y) a neighborhood of y = y°, i.e., x = x(t°,y) is constant there. This says that

the characteristic curves meet at the point (t°,x°). When there exists a point > t°, we can similarly see that x = z(t,y) is not monotone y = such that with respect to y in a neighborhood of y = y° for each t E (t°, Hence the

0

characteristic curves x = x(t, y) meet in a neighborhood of the point (t°, r°).

§3.5. Sufficient conditions for collision of characteristic curves H In this section we consider the same problem as in §3.4 for general partial differential

equations of first-order in one space dimension as follows: in

u(O,z) =

on

{t =

{t>0, rER ), 0, x

R'},

(3.8) (3.9)

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

are of class C2 in R4 and R', respectively. The = characteristic curves for (3.8)-(3.9) are obtained as solution curves of (2.15)-(2.16) for n = 1. Here we assume Condition (A.I)' (Chapter 2) which assures the global existence of characteristic curves. Let x = r(t,y), v = v(t,y), and p = p(i,y) be the solutions of (2.15)-(2.16) for n = 1.

Theorem 3.3. Under Condition (A.I)', we assume: (I)

and (II)

x(t°,y°), v° 0 where x0 v(t°,y°) and p° p(t°, y°), then the characteristic curves meet in a neighborhood of (t°, x°). If (02f/8p2)(t°,x°,v°,p°)

Proof. As in the proof of Theorem 3.2, we will show that x = r(t,y) is not monotone with respect to y. We repeat the same discussions in §1.1. When the Jacobian (ôx/Oy)(t,y) 0, we can solve the equation z = x(t,y) with respect to

§3.5. COLLISION OF CHARACTERISTIC CURVES II

29

y, and denote the solution by y = y(i,x). Then the solution of (3.8)-(3.9) is given by

u(i,x) def = v(t,y(t,x)) and x) =

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

x)

= —(t,y(t,x))

.

(_(t,y))

When a point (t,x) goes to (t°,x°) along the curve {(i,x)

:

x=

(Ox/Oy)(i,y(i,x)) tends to zero and p(i,y(t,x)) remains bounded. Therefore it must be true that (Ov/Oy)(t°,y°) = 0. Differentiating the system (2.15) with respect toy, we get a system of linear ordinary differential equations for Ox/Oy, Ov/Oy and Op/Oy just like (2.10). As the initial data (Ox/Oy, Ov/Oy, Op/Oy)(O, y) = (1,

are not zero, it follows that (Ox/Oy,Ov/Oy,Op/Oy)(t,y) (0,0,0) for any t 0. Hence (Op/Oy)(t°,y°) #0. We can see that (Ox/Oy)(t, y) satisfies the following differential equation: d (OX\ — 821 Ox + 82.f Ov + 82f Op di Oy) — OxOp Oy

OuOp Oy

Op2 a1i

Using the assumptions and the above results, we get —

di

Moreover, as (Ox/Oy)(t°,y°) = that

"

— Op2 0

,x 0 ,v0 ,p

,y°

and (Ox/Oy)(t,y°) >0 fort < to, it must be true

<0, —(—) dt \Oyi (t,p)=(t°,y°) —

i.e.,

d Ox

—(—) di Op

<0.

(i,ii)=(i°4i°)

(Ox/Oy)(t,y°) changes its sign at I = t°. Here we introduce a function h = def. h(y), just as in the proof of Theorem 3.2, by setting h(y) = inf{t (Ox/Oy)(t, y) = Hence

.

•

:

0) for each y. We consider the problem in a small neighborhood of (t°, x0, v°,p°) yO)•

0. We pick up a point By the same reasoning as in where 82f/Op2 we the above, we see that (Ox/Oy)(t, changes its sign at I = If > have and

Ox

for I E (t,h(j)). As this means that x = x(t,y) is not monotone with respect to y, the characteristic curves meet in a neighborhood of (10, x°). In the case where h = h(y) is constant, we can similarly prove that the characteristic 0 curves meet in a neighborhood of (t0, x°).

Chapter 4 Equations of Hamilton-Jacobi Type in One Space Dimension Nonexistence of classical solutions and historical remarks §4.1.

In this chapter we will treat general partial differential equations of first-order (3.1)-

(3.2) which satisfy Condition (A.!)' (Chapter 2); that is to say, the characteristic equations (2.15)-(2.16) have uniquely global solutions x = x(t,y), v = v(t,y) and The crucial point of Condition (A.!)' is the global p = p(t,y) for all y E existence of p = p(t, y). Let us sum up the principal results proved in Chapters 1, 2, and 3:

1) The Cauchy problem (3.1)-(3.2) has locally a unique C2-solution. 2) If the Jacobian (Dx/Dy)(t, y) of the mapping x = x(t, y) vanishes somewhere, it is impossible to extend the C2-solution beyond a point where the Jacobian vanishes.

3) Suppose that the characteristic curves do not meet in a neighborhood of the point where the Jacobian vanishes. Then the solution keeps being of class C1, but not of class C2, in the neighborhood of the point. Therefore, from now on, we consider the Cauchy problem (3.l)-(3.2) under the following conditions:

(I) (Dx/Dy)(t°,y°) = (I!) (Dx/Dy)(t,y)

0.

0 for t
E

I

where I is an open neighbourhood of

1$ = Y0.

(II!) The characteristic curves x = x(t, y) meet in a neighborhood of (t°, x°) where x0

§4.1. NONEXISTENCE OP CLASSICAL SOLUTIONS

31

In this section, the space dimension is always equal to one, though some subjects

are similarly discussed even in the case of several space variables. consider the case (III).

Now let us

Proposition 4.1. Suppose the above conditions (I), (II), and (III). Then the Cauchy problem (3.1)-(3.2) cannot admit a solution of class C1 in the whole space.

Proof. Let x = x(t,y), v = v(t,y), and p = p(t,y) be the solutions of (2. 15)-(2.16) for n = 1. By the assumption (III), there exist two points (a,a) and (a,/3) such that x(a, a) = x(a, /3) (a j9). As f = f(t, x, u, p) is of class C2, a Cauchy problem for (2.15) can not have more than one solution. Therefore we have

(x(a, a), v(a,a),p(a,a))

(x(a,j3),v(a,/3),p(a,/3)).

As x(a,a) = x(a,/9), it holds that (v(a, a),p(a, a))

(v(a,/9),p(a,/3)).

(4.1)

Let us assume that there exists a classical solution u = u(t,x) of (3.l)-(3.2) in the whole space. A. Ph proved in El 131 that a solution of class C' is generated by characteristic strips, that is to say

u(i,x(t,y)) = v(t,y), As

= p(t,y).

x(a,a) = r(a,/3) def = x , it follows that u(a, x°) = v(a, a) = v(a,

/3),

x°) = p(a, a) = p(a, /3).

This contradicts (4.1). When there do not exist classical solutions, we must introduce generalized solutions which contain singularities. As the solutions are not of class Ci, the solutions as regular as possible are required to be continuous. We recall by Condition (Al)' that p = p(t,y) remains bounded. Since p = p(t,y) is corresponding to the first

derivatives of solutions, the generalized solutions should be Lipschitz continuous. This suggests to us that, if the equations satisfy (A.I)', we would be able to construct generalized solutions which are Lipschitz continuous. This is the aim of the present

4. EQUATIONS OF HAMILTON-JACOBI TYPE

chapter. Before advancing further, we had better give briefly a historical survey on the global existence of generalized solutions for Hamilton-Jacobi equations.

Short historical survey. As an example of equations satisfying (A.!)', we are very familiar with "classical Hamilton-Jacobi equations." Here, when the function I = f(t, x,p) is independent of u, we call (3.1) a "classical Hamilton-Jacobi equation." On the other hand, notice that quasi-linear equations of first-order, for example

equations of the conservation law, do not satisfy (A.!)' (see Proposition 5.1 in Chapter 5). A global theory for classical Hamilton-Jacobi equations has been well studied and there are a lot of papers on this subject. At an early stage it has been considered under the convexity condition saying that the function I = f(t, x, p) is convex with respect to p. In this case generalized solutions are defined as "Lipschitz continuous and semi-concave functions" which satisfy the equations almost everywhere. A function u = u(t, x) is said to be semi-concave if it satisfies the inequality

u(t,x + y) + u(t,x — y)

—

2u(t,x)

(4.2)

for any x, y E RTh and t E R where K is a constant locally independent of x, y, and

The global existence and uniqueness of such generalized solutions for convex Hamilton-Jacobi equations have been well studied by several methods: for example, by variational method (E.D. Conway and E. Hopf [331), by methods of envelopes (E. Hopf 1641, S. Aizawa and N. Kikuch.i 141), by difference-differential method (A. Douglis [44]), especially by vanishing viscosity method (W.H. Fleming [51], A. Friedman [54], S.N. Kruzhkov [881, [92]). Non-linear semi-group approach has also been well developed by S. Aizawa [2], Y. Tomita [135]. The global theory for non-convex Hamilton-Jacobi equations has recently been considered by M.G. Crandall, L.C. Evans, P.-L. Lions ([35], [381, [100]). They have introduced the notion of "viscosity solutions" to define generalized solutions and characterized their properties. There are also many interesting works related to the viscosity solutions, for example, C. Barles [14], H. Ishii [67], P.E. Souganidis [123). By these contributions, the global existence and uniqueness of generalized solutions were established almost completely. Concerning more detailed references, refer to P.-L. Lions [100] and S.H. Benton [211. Historical remarks for equations of the conservation law will be given in Chapter 5.

§4.1. NONEXISTENCE OF CLASSICAL SOLUTIONS

33

As we have already written, generalized solutions contain singularities. Our principal aim in this chapter is to see the structure of singularities of generalized solutions or weak solutions. However, this problem could not be solved by the above many works. The reason comes from the methods. As the nice existence theorems for generalized solutions depend generally on vanishing viscosity method, the generalized solutions are obtained as the limit of solutions of non-linear heat equations. As the process of taking the limit is complicated, it seems to us that it would be difficult to get the informations on the singularities of generalized solutions by vanishing viscosity method. But this method is very effective to prove the existence of solutions. Let us review here many methods which are used to consider the propagation of singularities in linear wave propagation. To consider the singularities of solutions, one gives representation formulas for solutions, especially using Fourier-Laplace transform, pseudo-differential operators, or Fourier integral

operators. Next, one tries to analyze their expressions. Though this procedure is not always true, it is a typical approach to consider the propagation of singularities in linear wave propagation. Concerning this subject, refer to L. Hörmander 165], L. Gârding [55], and G.F.D. Duff [461. How can we construct concretely the solutions of non-linear partial differential equations of first-order? The method we know is the characteristic one. The weak point is that it is a local theory. The reason comes from the fact that the characteristic curves x = x(t, y) cannot be uniquely solved with respect to y in a neighborhood of a point where the Jacobian vanishes; that is to say, the inverse function y = y(t, z) takes many values in a neighborhood of a critical point of the mapping x = x(i, y). Therefore the "solutions" constructed by the characteristic method also takes many values there. As discussed in the above, we are looking

for one-valued and Lipschitz continuous solutions. However, a generalized solution

is not unique in the space of Lipschitz continuous functions (see Chapters 9 and 11). Therefore we impose a supplementary condition on generalized solutions which guarantees the uniqueness of solutions. It is the semi-concavity condition (4.2). What we would like to prove is that, if we assume (A.!)', we can choose only one value from many values of the "multi-valued solution" in order to get a solution that becomes one-valued and continuous. Then, if the equation is convex, we can show that the semi-concavity condition (4.2) is automatically satisfied. This is the program which we would like to develop in the following discussions.

4. EQUATIONS OP HAMILTON-JACOBI TYPE

34

From the results in this chapter, we will see that the equations with property (A.I)' have properties similar to that of classical Hamilton-Jacobi equations. Therefore we would like to call the equations with property (Al)' "equations of Hamilton-Jacobi type." Notice that S. Lzumiya [74J considers the differences between Hamilton-Jacobi equations and first-order quasi-linear partial differential equations from the geometrical point of view. M. Bony [23] has considered the propagation of weak singularities for non-linear partial differential equations. His method depends on some techniques elaborated to study linear problems. We think that the singularities which will be discussed here could not be treated by his method, because, for the definition of paradifferential operators which are the approximations of nonlinear equations, solutions must be at least classical solutions.

§4.2. Construction of generalized solutions To make clear our problem, we write it again. Consider the Cauchy problem in one space dimension as follows: Ou

.

u(O,x) = 4(x) on

in

{t>O, rER },

{t =

0, x

R1},

(4.3)

(4.4)

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

are of class C in R1 x R1 x R' x and = R1, respectively. For the construction of singularities of solutions, we need a little stronger assumption on the regularity of f = f(t,x,u,p) and 4 = Then the

characteristic curves for (4.3)-(4.4) are defined as the solution curves of (2. 15)-(2. 16)

for n = 1. We write them by x = x(t,y), v = v(t,y), and p =p(t,y). Suppose (A.!)'. This assures the global existence of x = x(t, y), v = v(t, y), and p = p(t, y). We consider the Cauchy problem (4.3)-(4.4) under three conditions (I), (II), and (III) introduced in §4.1. As (III) is not concrete, we assume the following sufficient condition for it to hold (see Theorem 3.3): 0

where

v=

v(t o ,yo) and

p0def = p(t o ,y ). o

(4.5)

§4.2. CONSTRUCTION OF GENERALIZED SOLUTIONS

35

From now on, we will construct the singularities of generalized solutions for 2> where t—t0 is small. First we solve the equation (ôx/ôy)(t,y) = 0 with respect to 2. As (Ox/Oy)(t, y) > 0 for t < t° and y E I (cf. (H)), it holds that (Ox/Oy)(t°, y) 0 for y E I. Moreover, as = 0, we get (02x/8y2)(t°,y°) = 0. We assume here (83x/0y3)(t°, y°) 0. This assumption is natural from the generic point of view. In this case, we see that

>0.

(4.6)

In fact, if (4.6) were false, we would have (82x/8y2)(i°,y) < 0 for y > y°, i.e., (Oz/Oy)(t°,y) < 0 for y > y°. As (ôs/Oy)(0,y) = 1, there exists 2 < 20 such that (Ox/Oy)(t,y) = 0. This contradicts (I). Let us draw a figure for the curve x = s(t,y) (2> t°) (see Figure 4.1). x

xl yl

ya

y

Figure 4.1

We explain the reason why the curve x = x(t,y) is drawn as in Figure 4.1. Taking the derivative of (2.15) with respect to y, we get a system of ordinary

4. EQUATIONS OP HAMILTON-JACOBI TYPE

differential equations concerning Ox/Oy, Ov/Oy, and Op/Oy just like (2.10). For example, the equation for Ox/Oy is written by

dOx

02f

=

Ox

621

+

02f

+

(4.7)

This system of equations is linear with respect to Ox/Oy, Ov/Oy and Op/Oy. Since

(0,0,0), we see (0,0,0) for any (t,y) R2. We

((Ox/Oy)(O, y), (Ov/Oy)(O, y), (Op/Oy)(O, y)) = (1, 4'(y), 4/'(y))

that ((Ox/Oy)(t,y),(Ov/Oy)(t,y),(Op/Oy)(t,y)) recall here Lemma 1.2:

(t,y)

= As (Ox/Oy)(t°,y°) =

0,

it follows that (Ov/Oy)(t°,y°) =

0,

R2.

hence that

0.

On the other hand, as (Ox/Oy)(t°,y°) = 0 and (Ox/Oy)(t,y°) > 0 for t < t°, the left-hand side of (4.7) must be non-positive at the point (t°, y°). Using (4.5) and the above results, we get =

<0.

(4.8)

Therefore we can uniquely solve the equation (Ox/ay)(t, y) = 0 with respect to tin a neighborhood of (t°,y°), and write the solution as a C°°-function t = p(y) of y. Then we have d

Ox

But (62x/0y2)(t°,y°) =

= 0

02x

,

(y) +

02x

= 0.

and (62x/OtOy)(t°,y°) < 0, it follows that p'(y°) =

0.

Similarly, since d2

Ox

=

02x

o

,y o )pu(yo ) +

03x

o

,yo) = 0,

using (4.6) we get p"(y°) > 0. As p'(y°) = 0, it follows that p'(y) > 0 for y > y° and that p'(y) <0 for y y° and strictly decreasing for y < y°, so the equation t = p(y) has two solutions

§4.2. CONSTRUCTION OP y

= yi(t) andy = y2(t)

(y2(t) < y° < y'(t)) fort

>

SOLUTIONS

37

= p(y°) where

i—ta is small.

Summing up these results, we get the following.

Lemma 4.2. In a nezghborhood of (t°,y°), the equation (Ox/Oy)(t,y) = 0 has two soluüons, y = y1(t) and y = y2(t); y1(t) > y3(t) for t > 2° with 2 — t° small. The two solutions y = y1(t) (i = 1,2) are of class C°° fort > and are continuous for t° (with yi(t° + 0) = y2(t° + 0) = y°). x(t, y1(t)) (i = = x(t°,y°) = x°).

The proof is obvious by the above discussions. Here we put

Then ri(t) <52(2) fort> t° (with +0) = we solve the equations = x(t,y) with respect toy for x E (x1(t),52(t)), 2 > 2° (t—t° small). Then we get three solutions y = g,(t,x) (i = 1,2,3) with g1(t,x) < g2(t,x) < gs(t,x). Define u1(t,x) 1! v(t,g,(t,x)) (i = 1,2,3). This means that the "solution" constructed by the characteristic method takes three values for x (xj(t),x2(t)) (2 > t° with 2— t° small). By the assumption (4.5), we treat here the 1,2).

Next

case where

> 0. Then we get the following.

Lemma

4.3. i) For

any 2 > 20 with 2—2° small and for any x E (xl(t),x2(t)),

we

have

ui(t,x) —U2(t,s) <0 and u2(t,x) ii) There exists uniquely x = ir(t)

—

>0.

(x1(t),52(t)) satisfying

ui(t,7(t)) =

Proof. i) By (4.8) and (4.9), we have (Op/Oy)(t°,y°) <0, i.e., (Op/Oy)(t,y) <0 a small neighborhood of (t°,y°). As gi(t,z)
(2, x)) > p(t, g2(t, x)) > p(t, gs(t, x)).

(4.10)

Using these inequalities, we get

-(uj(t,z) — u2(t,x)) = p(2,gj(t,x))

—

p(t,g2(t,x))

in

> 0,

38

4. EQUATIONS OF HAMILTON-JACOBI TYPE

with

=

(ui(t,z) —

0.

Hence it holds that ui(t,x) — u2(i,x) < 0 for x E Exi(t),x2(t)). In the same way, we obtain the inequality u2(t,x) — us(t,x) >0 for x E (xi(i),x2(t)j. ii) Using the results of i, we have (t, x) — us(t, x))

= p(i, gj (t, x)) — p(t, g3(t, x)) > 0,

with

(ui(t,x) —

z=za(t)

= (u3(i,x)

>0

—

and

(u1(i, x) — Therefore

(t)

= (uj(t, x) — u2(t,

(t)

< 0.

there exists uniquely x = 7(t) E (xi(t),x2(t)) satisfying ui(t,'y(t)) =

0

us(t,y(t)).

= 1,2,3) satisfy the We see by Lemma 1.2 that the three functionsu = xi(t). As equation (4.3) in their domain where i > 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
if if

deC10 u=ui(t,x)=ç I.

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

If f = f(t, x, u, p) is convex with respect to p, the supplementary condition would be the semi-concavity condition (4.2). When I = f(t,x,u,p) is concave, then it becomes the semi-convexity condition as follows:

u = u(t, x) is said to be semi-convex if it satisfies the inequality u(t, x + y) + u(t, x — y)

2u(i, x)

—

(4.12)

for any x, y and t where K is a constant locally independent of x, y and t. The supplementary condition (4.2) [or (4.12)] is very important, because it is related to the stability of solutions of the Cauchy problem (4.3)-(4.4). What we would like to claim here is that the solution constructed by (4.11) in §4.2 satisfies automatically the above supplementary condition; that is to say, we get the following:

Theorem 4.4. Assume

> 0 (resp.

< 0).

Then the solution u = u(t,x) extended by (4.11) is semi-concave (resp. convex) in a neighborhood of (t°,x°).

semi-

Proof. We consider the convexity case (4.9). It suffices to prove (4.2) in a small Assume that x = -y(t) and y > 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)

t >

>

(t) + 0) and

72(t) — 0) >

+ 0)

for t
=

(4.15)

As! = f(t,x,u,p) is convex with respect top, we have by (4.14) >

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
dt''

>

Therefore it follows that 71(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
Proposition 5.1. Assume (Oz/Oy)(t°,y°) =

Proof. Remember that Ox/Oy and Ov/Oy satisfy the system of ordinary differential equations (2.10) for n = 1. Since this system (2.10) is linear with respect to {Ox/Oy,av/ay}, we get (Ox/Oy,Ov/Oy) (0,0) for any (t,y) E R2. Therefore, if (Ox/Oy)(t°,y°) = 0, then (Ov/Oy)(t°,y°) 0. Moreover, we have by Lemma 1.2

=

(t,y) E R2.

Hence we can easily get this proposition.

This proposition means that, if the Jacobian vanishes somewhere, (A.I)' cannot be satisfied for quasi-linear equations of first-order.

5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER

48

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

guarantees the situation (N)

is

(5.7)

v(t°, y°). This is the result of Theorem 3.2. If (5.7) is violated, we

where v0

can construct an example in which characteristic curves do not meet though the Jacobian vanishes (see Example 1 in §3.2). Therefore, the solution of this case remains to be continuous, though it is not differentiable. First we solve the equation x = x(t, y) with respect toy for t > t0 in a neighborhood of (t°, y°). This procedure is almost the same as in §4.2. By the assumptions (I) and (II), we have (02x/0y2)(t°, y°) = 0. We assume (03x/8y3)(t°, y°) 0. This assumption is natural from the generic point of view. In this case also, we get, by the same way as the proof of (4.6), > 0.

(5.8)

Then the graph of the curve x = x(t,y) for t > t0 is drawn just as in Figure 4.1 (see §4.2). Therefore we use the same notations given in Figure 4.1. Here the functions y = y1(t) and y = y3(t) are the solutions of (Ox/Oy)(t,y) = 0 > def and z1(t) = x(t,y1(t)) (z = 1,2). Then Lemma 4.2 is also true for these y = y1(t) (i = 1,2). When we solve the equation x = x(t,y) with respect to y for z E (x1(t),xz(t)), we get three solutions y = g3(t,z) (g1(t,z) < g2(i,x) < gs(t,x)) and define x) x)) (i = 1,2,3). As we are looking for a single valued solution, we must choose only one value from {u1(t,x) i = 1,2,3} so that we can :

get a weak solution of (5.1). As we have written in §5.1, we can not get continuous

weak solutions under the condition (N). Therefore we look for a weak solution which is piecewise smooth. If u = u(t, x) is a weak solution of (5.1) which has jump discontinuity along a curve x = -y(t), it must satisfy the R.ankine-Hugoniot jump

condition (5.4). This suggests us that a nice weak solution jumps from a branch u = uj(t,x)} to the other {(t,x,u) u = u3(t,x)} along a curve { (t,x,u) :

:

§5.3. CONSTRUCTION OF SINGULARITIES OF WEAK SOLUTIONS

49

x = -y(t) on which condition (5.4) is satisfied. Therefore our problem is to solve the Cauchy problem for Rankine-Hugoniot's equation as follows: ( dx

—

f(t, x, uj(t, x)) — f(t, x, us(t, x))

di — I.

> 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°) <0. ii) For (i,x) E U, we get

<0,

ui(t,x) > u2(t,x) > u3(t,x) and

<0.

> 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 <

— VOIIL1.

(5.14)

Conversely, if (5.14) is true, then the condition (5.13) is satisfied. We would like to show that our solution (5.12) is reasonable in the above sense. That is to say:

Theorem 5.7. The solution defined by (5.12) satisfies the entropy condition (5.13) in a neighborhood of the point (t°,x°).

Proof. We consider the case where (5.10) is satisfied. By Lemma 5.2, we have u_(t) > 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
theorem, the canonical forms of fold and cusp points are expressed respectively as follows:

= =

—

x2 =

z2 =

Y2

in a neighborhood of a fold point,

(6.9k)

in a neighborhood of a cusp point.

(6.92)

§6.2. CONSTRUCTION OF SOLUTIONS

This means that the mapping

59

can be regarded as the mappings of (6.9k) and

(6.92) in a neighborhood of a fold point and a cusp point, respectively. Moreover he proved that any smooth mapping from R2 to R2 can be approximated by smooth mappings whose singularities are only fold and cusp points. We see by his result

(i = 1,2). has the cusps at X1 and X3 where the expressions When we solve the equation x = with respect toy for x E (6.9i) and (6.92) say that the solution y = y(t,x) takes three values. Write them for x E by y = (i = 1,2,3). Here we choose y = g2(t,z) as g2(t,x) E When we write u1(t,x) v(t,g1(t,z)) (1 = 1,2,3), the solution of (6.1)-(6.2) has three values {u1(t,x) i = 1,2,3)) for x E (see Figure 6.1 which shows these that the curve

:

situations).

V Ct. y)

H;

and

Figure 6.1. Curves

Lemma 6.2. i) ii)

-u;(t,x) = —

for x

(i = 1,2,3).

(g1(t,x) — g,(t,x))

< 0 for x

and

60

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

iii) ui(t,x) < u2(t,x) and us(t,x) < u2(t,x) for x E Proof. i) This is equivalent to Corollary 1.3 in §1.1. ii) Ftom the definition of gj(t, x), we have

xE

x = 91(t,x) +

(Ou,/c3x)(t,x) for i g1(t,x), it follows that (ôu;/Ox)(t,x) As g1(t,x) j. Using the convexity off = f(p), we get the inequality ii. into iii) We prove the first inequality. Divide the simple closed curve and write them by C1 and two parts joining two cusp points Xi and X2 of C3. Here we introduce the family of solution curves of the following differential equation: dx

T =gi(t,x)—gz(t,x). These curves start from C1 (or from C2) and end at C2 (or at C1, respectively), On each curve it holds that and the family of these curves covers the domain d —

u3(t,x)}

0u2

=

(j— —

(91 —92)

< 0.

Since ui(t,x) = u2(t,x) on C2 (or on C1), we get u1(t,x)
:

:

Lemma 6.3. Put I(t, x)

ui(t, x) — us(t, x). Then

{x

I(t, x) =

O}

and it joins two cusp points X1 and X2.

is a smooth curve contained in

Proof. In this case we introduce the family of curves defined by dx

T = gi(t,x)

—

gs(t,x).

These curves also start from C1 and end at Cz, and the family of the curves covers if we change the index "i" of g1(i,x) (i = 1,2,3), the above solution the domain curves may start from C2 and end at C1. On each solution curve, it holds that d

=

Ous\ —

(91

—ge) <0.

(6.10)

§6.3. SEMI-CONCAVITY OP THE SOLUTION u = u(t,z)

We see by Lemma 6.2 that the sign of

61

is different from that of

I(t,x) = 0 has a unique solution on each solution curve of (6.10). bbviously, I(t,x) = 0 at x = X1 and x = X2. As it follows from ii of Lemma 6.2 that for

we see that

= {x

I(t,x) = O} is a smooth curve joining the points Xj

a single valued and continuous solution, we define a solution

u = u(t,x) in {(t,x) u1(t,x) > 0} and

dl{x U (t

As

def

E

as follows: Writing = {x E u,(i,x) — u1(t,x) <0), we define

)

I

xE

:

:

us(t,x) —

:

—

us(t,x)

if x E if x

= {x = x(s)}. Then

is smooth, it can be parameterized as d

=

1ôu1

dx

OU3\

= 0.

.

This means that, though the first derivative of the solution u = u(t, x) has jump discontinuity along the curve it is continuous with respect to the tangential direction of rg.

§6.3. Semi-concavity of the solution u Let n(t, x) be a unit normal of

= u(t, x) and define at x E

advancing from ag,... to

—(t,x±0) def. = hm —(t,x±en). Ou

OU

e-4+O8X

Any C2-function satisfies the semi-convexity condition (6.4). Therefore, for the proof of (6.4), it suffices to treat the case where x and y = en (e > 0). Then

we have

u(i,x +y)+u(t,x — y) —0)—

—

2u(t,x) =

—

.sy))

+sy) yds +

—

+ 0)—

+0)). yds —0)) . y.

62

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

The first and second terms are easily estimated by K1y12. To get the inequality (6.4), it must be + 0) —

x—

0))

n(t,x) <0.

(6.11)

Contrarily, if (6.11) is true, then we can get (6.4). Hence (6.11) is equivalent to the semi-concavity property. On the other hand, we have, by the definition of us(t,x) — ui(t,x) >

if

xE

u3(t,x) — uj(t,x) <

if

xE

and

which show immediately that + sn) — ui(t,x +

0.

That is to say, —

.

n

0,

x E 1's.

(6.12)

From the definition of u = u(t, z), for x E flj, we have Ou

OiL

and

0U3

Substituting these relations into (6.12), we get (6.11). Thus u = u(t,x) is semiconcave. Summing up the above results, we get the following.

Theorem 6.4. After the time t0 where the Jacobian of the mapping

vanishes

for the first time, the solution takes many values. But we can uniquely pick up only one value from them so that the solution becomes single valued and continuous. Then the condition of semi-concavity is automatically satisfied.

Remark. Putting v(t,x) def = (Ou/Ox)(t,x) in (6.11), we have the condition on the jump discontinuity of v = v(t, x): .

(v(t,x+O)—v(t,x—O)).n<0, XE This is the entropy condition for the system of conservation law (6.5) given in the remark of §6.1.

§6.4. COLLISION OF SINGULARITIES

63

§6.4. Collision of singularities In this section we consider the collision of two singularities

and

construct-

ed in §6.2, assuming the hypotheses (S.1) and (S.2). Here we use the notations E,,g, for (i = 1,2) which correspond to for introduced in §6.2. We see that there exist three kinds of collision as described in Figure 6.2. Case (i): Consider the case where and collide as in (i) of Figure 6.2. Then By almost the solution becomes two-valued on a domain encircled by and the same discussions as in §6.2, we can uniquely pick up one value from two so that the solution is single valued and continuous. Then we can prove that the solution is semi-concave. In this case the new singularity appears as a smooth curve joining two points where and intersect each other. It is described as a dotted curve in Figure 6.2.

r. t

(iii)

(1)

Figure 6.2. Collision of singularities

6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS

64

Case (ii): Consider the collision of the type (ii) drawn in Figure 6.2. We put deC = and {y

(i = 1,2).

E ri,, and p E

at y = is a smooth simple closed curve which is tangent to A2,g is tangent to l',2 (i = 1,2). When the end point of r2,1 is on A where A = Y2,1 or A = 1'2,2 (see Figure 6.3).

Then

Y1,1

and

Yl,'

1A2

Figure 6.3. Relation between

and

does not take an extremurn at y = A, we get v(t,.) restricted on (Ov/äy)(i,y) Oat p = A; i.e., the curve CA v(t,y) = v(t,.)I.i} is (p E R2 As

:

smooth in a neighbourhood of p = A, and it intersects

A transversally.

takes an extremuni at p = A, the On the other hand, as v(t,.) restricted on curve CA is tangent to at p = A. This is in contradiction with the above. Hence the case (ii) does not happen. and meet at a time t = as shown in (iii) of Figure 6.2, drawn as in (i) of Figure 6.4. But, as the interior domain of the is U 1 + tAi(y) = O} is monotonously increasing, = {y E R2

Case (iii): When U E3,to is

curve

§6.4. COLLISION OF SINGULARITIES

65

described as in (ii) of Figure 6.4 for t > 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 x<0

(7.3)

(b, x>O,

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
x=x(t v

(7.4)

I y+tf'(b), y>O,

=v(t,y)={

y0.

a, b,

(7.5)

Assume that f'(b) — f'(a) > 0. Then the domain D {(t,x) tf'(a) <x < tf'(b)} is not covered by the family of characteristic curves (7.4). In the domain D, we look for a solution u = u(i,x) of the form u(t,x) = r(x/t). Then the function r = r(p) (p = x/t) satisfies :

Ou

+

8

x

=

,

1,x

+ I (r)]—r

= 0.

Therefore, if r = r(p) satisfies f'(r(p)) = p, then u = u(t, r) r(x/t) is a solution of (7.1). As this means that r = r(p) is the inverse function of u i—* f'(u), we must assume

f"(u)

0

for all u between a and b.

Now we define the solution u = u(i,x) by

x < tf'(a), r(x/t), (t,x) E D, b, x
u(t,x) 1

(7.7)

The solution defined by u = r(x/i) in the domain D is called "rarefaction wave." The solution (7.7) is continuous in {i > 0}, but it has jump discontinuity

at (t,x) = (0,0). Next we consider the case where f'(a) — f'(b) > 0. Then the domain R tf'(b) < x < tf'(a)} is covered two times by the family of characteristic

{(t, x)

§7.2. RAREFACTION WAVES AND CONTACT DISCONTINUITY

69

curves (7.4). As we look for a single-valued solution, we repeat the same discussions

as in §5.3. Let x = -y(t) be the solution of the following differential equation: —

f(b)—f(a) b—a

dt —

{

=

(7.8)

'

0.

Then we get easily -y(t) = {f(b) — f(a)}t/(b u = u(t,x) of (7.1)-(7.2) in the domain R by

—

a).

Here we define a weak solution

Ia, x = -y(t) satisfies the entropy condition (5.13), then this singularity is just "shock." But, if f = 1(u) is not convex, the inequality (5.13) is sometimes vio-

lated. In this case we use "contact discontinuity" which we will explain from now. F

b

0

C

Li

Figure 7.1

Assume that a > 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 and h'(y1) = 0. We

Then we have at A1 > 0

and

<0,

7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION

74

and at A2

<0 and f"(4'(y2)) >

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)
§7.4. BEHAVIOR OF THE SHOCK S1

75

Here we must pay attention to the order of this inequality which is contrary to the is negative, one given in Lemma 5.2. This is caused by the property that while it is positive in Lemma 5.2. As in §5.2, we consider the Rankine-Hugoniot jump condition as follows: ( dx dt I.

—

f(ui(t,x))

—

f(u3(t,x))

—

u1(t,z)

—

u3(t,x)

'

(7.12)

x(ti)=Xj.

Proposition 5.5 says that the Cauchy problem (7.12) has a unique solution x = which is the shock curve of S1. To extend the shock Sj, we must see the behavior of = (i = 1,3). We put uj(t) u(t,-y1(t) ± 0). Then = us(t, yj(t)) and u_(t) = uj(t,71(t)).

Lemma 7.1. As long as the entropy condition for S1 is satisfied, the function t gj(t,-y1(t)) is decreasing and t gs(t,-y1(t)) is increasing.

Proof. By the definition of y =

x=

x), we have x)))

x) +

(i = 1,2,3).

Taking the derivatives with respect to t and x, we have + f'(çi(g1)) ÷

0= and

1 = 0g1{1 +

Hence it holds that

x) =

x)

(i = 1,2,3)

which leads us to

(t)) =

{

—

(t)))}

(t)) (i = 1,2,3).

Here we recall

=

> 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)

=

<0,

=

ii)

0.

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

Proposition 7.6. For t > T where t

—T

is small, it holds that

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

so that Theorem 1.6 can simultaneously work therein. The reason is that the relat ions between L1,. . , and are bilateral and somehow awkward. In fact, .

are to some extent overdetermined by , Specifically, for each application of the theorem in such a procedure, must be ready-given. Thus, by (8.4), a

L1,. .

.

§8.2. A DIFFERENTIAL INEQUALITY OF HAAR TYPE

85

tuple (L1,. .. , is predetermined. However, the further condition (8.3) is needed for (i, x, u,p), (t, x, v, q) E x K, with K being a compact set in R1 x R" such that

(u,(t,x),(ôu,/Ox)(t,x)) E K for (t,x) E

j = 1,

2.

are necessarily Lipschitz constants with respect to ps,... , Pvi of the function f = f(i, x, u,p) restricted to x K. But these constants might unfortunately become large, say, substantially greater than the above-predetermined values L1 So we would reduce and hence possibly fail to touch the byperplane {t = 0, x E in such a procedure emanating from a point (t°, x°) E All the preceding remarks suggest that we should make an attempt to develop the theory of first-order partial differential equations. The aim of this section is to prove the following, which is in fact a generalization of Theorems 1.5 and 1.6. (We put off discussing its applications to the uniqueness problem to §8.3 and Chapter Therefore, L1,. .

.

,

11.)

Theorem 8.1. Let u = u(t,x) be a function in If there exist a nonnegative function p = p(x) locally bounded on and a nonnegative function £ = £(t) in L' (0, T) such that

8u(t, x)/OtI e(t) . [(1 + Ixl)IOu(t, x)/OxI + p(x)Iu(t, x)IJ

for almost every t E (0, T) and for all x E

u(t,x)I exp

[c(x)f £(r)dr]

.

(8.5)

then sup

u(O,y)I,

(8.6)

£(r)dr—i

where

T

C(x)

(Ip(y)I

:

(1 + xI)exPf £(r)dr — i}.

(8.7)

Remark 1. Inequalities (1.14) and (1.16) are usually referred to as Haar's differential ones (see [128, Corollary 37.1]). Therefore, we would like to call (8.5) "a differential inequality of Haar type."

Remark 2. We show by the following example that the Lipschitz continuity of u = u(t,x) is essential in Theorem 8.1.

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

86

Let J C [0, 1J be the Cantor set, i.e., the set of all numbers of the form (8.8)

where each ek is either 0 or 2. The set J is complete, nowhere dense on R1, and is

of Lebesgue measure 0 (see [42J).

We define a function w = w(t), which is called the Cantor ladder, or the Cantor function [53, p. 381, in the following way. For t E J given by (8.8), we take

w(t) def =

+00 k

where

def

=

If (a,f3) is an open maximal interval in (0, 1)\J, then J and = wCO). We set for t E w = w(i) w(t) def = is continuous on [0, 1] and that dw/dt = 0 almost everywhere in (0, 1). In fact, dw(t)/dt = 0 fort E (0,1)\J. we easily see that u = u(t,x) belongs to Setting for (t,x)

C1(((o,1)\J)

x

with u(O,x)

flC([0,11 x 0

V(t,x)

((0,1)\J)

x

The function u = u(t, x) thereby satisfies all the conditions of Theorem 8.1 except for the Lipschitz continuity. This explains why (8.6) does not hold: u(t,x) 0.

Proof of Theorem 8.1. For an arbitrary point (t°,x°) E Iu(t°,x°)I

exp

[C(x0)ft o

£(t)dt]

we must prove that u(0,y)[.

sup Ivt(1+Ie°I)exp

(8.9)

xI
the constraint x(t°) =

[0,

into

which satisfy subject to

x0.

From [29, Theorem VI-13], it follows that Ej(t°, x°) is a nonempty compact set in The sets Z(t,t°,x°) {x(t) x(.) E Ej(t°,x°)} and I'(t°,x°) respective{ (r, x) r E I, x E Z(r, t°, x°) } are therefore compact in R" and ly, for all t E I. Moreover, by the converse of Ascoli's theorem, the correspondence I t '-4 Z(t, t°, r°) C is a continuous multifunction. :

:

§8.2. A DIFFERENTIAL INEQUALITY OF HAAR TYPE

87

We now define a real-valued function g = g(t) on I by setting

g(t)

{Iu(t,x)I

:

xE

Z(t,t°,x°)}.

Then according to the maximum theorem (see [8, Theorem 1.4.16]), the fact that u = u(t,x) is continuous on r(t°,x°) implies that g = g(t) is continuous on I. In addition, we have: Lemma 8.2. For an arbitrarij number 9 E (0, t°), the function g = g(t) is absolutely Contznuou3 on [9,t0].

The following assertion will also be needed.

Lemma 8.3. We have for everij t E I the inclusion

Z(t,t°,s°) C Proof of Lemma 8.3. For each

0,

let

£(r)dr —1.

+ Ix°I

m,7(t)

(8.10)

.

The function rn,7 = rn,7(t) is absolutely continuous, positive on I with the derivative dm,7(t)/dt = —i(t). (i + rn,,(t)). To prove (8.10) we have only to show that

Ix(t)I <m,,(t) Vt E I for all x = x(t) in Ei(t°,x°) and for all Since > = 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 >

Vt

E (t', i°],

.

(i + Ix(t)I)

8. IMFPERENTIAL INEQUALITIES OP HAAR TYPE

88

if and only if m,,(t°) — m,,(t') = m,(t°) — Ix(t1)I > Iz(t°)I — Ix(t1)I. Hence

0

we obtain a contradiction. This proves Lemma 8.3.

Proof of Lemma 8.2. Since u = u(t,x) is locally Lipschitz continuous in there exists a number L 0 such that Iu(t1,x1) — u(t2,x2)I

—

t21 -+

Ir' — x21)

V (i',z1), (t2,r2) E (fO,t°] x By the absolute continuity of the Lebesgue integral, Lemma 8.2 will be proved if we can show that g(t1)

L [its —

— g(t2 )I

I+

(1 + 200.

f

£(t)dt . exp

(*1*21

j

£(i)dt] 8 12

Vt',t2 E [8,t°]. Now let

g(t')

g(t2)

and

g(t') = Iu(i', x(t1))I

for some x = x(t) in Ei(t°,z°). Since x(t2) E Z(t2,t°,x°), we have

0 g(t') — g(i2) = Iu(t', x(i' ))I — g(t2) u(t', x(t1 ))I — Iu(t2, x(t2))I

u(t1, x(t')) — u(t2, x(t2))I

L[1t1 — gal + lx(t1) — x(t2)l] =

L [hi

—

+

£(t)

(i + x(t)l)dt]

[*1 ,t2J

Therefore, (8.12) follows from Lemma 8.3. The proof is then complete.

0

Going back to the proof of Theorem 8.1, we set now for

t€(0,Tj.

By Lemma 8.3 and the definition of g = g(t), the inequality (8.9) will be obtained if we show that

g(t)

g(0) .exp[C(x°) . h(i)]

Vt E [0,t°].

(8.13)

§8.2. A DIPPERENTIAL INEQUALITY OP HAAR TYPE

For every

89

> 0, let

g,(t)

[g(0) +

exp { [C(x°) +

.

[h(t) + itt]

}.

To get (8.13), it suffices to prove that

g(t)
(8.14)

—g(t), where is temporarily fixed. Then (8.14) is equivalent Let w(t) to w(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')} <

and

A

A

w[0,

fl [0, t°]).

Let :

It is obvious that w(t.) = A, t. E (0,t')\G, and that w(t) > A Suppose that

g(t.) = Iu(t.,z.)I =

.

u(t.,x,),

e

Vt

[0,t.).

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

90

E Z(t.,t°,x°). Then one may find a function x = x(t) in Ej(t°,x°)

for some

so that x(t.) =

Choose a unit vector e E

(e,e

=

with

—

(8.16)

.

The system (of n ordinary differential equations)

has a C'-solution y = y(s) on R1 satisfying the condition y(h(t.)) = x•. Let x(t) def = y(h(t)) fortE [0,1']. Of course, x = x(t) is absolutely continuous on [0,1'], x(t.) = and = £(t) (1 + x(t)])

=

e

Vt E (0,T)\G1.

Therefore, the function j = .,x(t) defined by .x(t)

def

=

x(t)

E

I x(t)

if 0 < t

1.

Z(t,t°,x°) Vt E

This implies

u(t,z(t)) < Iu(t,x(t))I < g(t) = for all I E I0,t.).

Besides

— w(t) < g,7(t) — A

(8.17)

that,

u(t., x(t.)) = Iu(t.,x.)I = g(t.) = g,,(t.) — w(t.) = Furthermore, since I. E (0, T)\G, we see that: (i) u = ts(t,x) is differentiable at (ii) x = x(t) is differentiable at with

(iii) g,, = g,1(t) is differentiable at I. with

[C(x°)+q] .

— A.

(8.18)

§8.3. UNIQUENESS OF GLOBAL CLASSICAL SOLUTIONS

So it follows from (8.17)-(8.18) that [e

u(t, x(t))]

Consequently, e

/dx

Ou

i9u

[C(x°)+v7]

.g,1(i.).

Hence, Ou •

x.) + £(t.)(1 + Ix.I)

/

Ou •

[C(x°) +

.

x.)

[€(t,) +

. EIu(t.,x.)I + A].

0 and A > 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) <+00 (j = 1,2), OX

u2(t,x) in 11T•

Proof. Let u = u(t, x) be as in the proof of Theorem 8.4, and let r

max

<+co, K2

sup

(t,r)EflT Ox

Xk

x

... x (—k,k)c

(8.21)

L

C

(k = 1,2,...).

(8.22)

n timel

For an arbitrarily fixed T'

(0, T), we consider the sequence

of the

following parallelepipeds:

: 0
s'

c ... and

...

Next, take

=

[_3k,3k] C R1.

max max

(8.23)

(t,z)EP

We now define a function p = p(x) by setting

p(x)

def

I

if xE X', if xE Xk+1\Xk (fork = 1,2...).

=

It follows that p = p(x) is locally bounded on with the hypothesis of the theorem imply

= f(t, x, uj(t, x),

—

x)

= L(1 + lxi)

(8.24)

Moreover, (8.21)-(8.24) together

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

x) — u2(t, x)l

+p(x)lu(t,x)l in

(We may check this inequality first for (t,x) in P1, and then for (t,x) in each Theorem 8.1 therefore shows that u(t,x) is arbitrarily chosen, the conclusion follows.

0 in IZT.. Since T'

(0,T)

0

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

94

Corollary 8.6. Let I = f(i,x,u,p) belong to C'(riT x

such that the

x

funchon def

ii = &i(t,p) =

sup (z,u)Ek" xk1

+ lxi) (/11

finite and continuous on (0, TJ x I/ui = ui(t, x) and u2 = u2(t, x) are global C'-solutzons to the Cauchy problem (8.1)-(8.2) with zs

sup (t,z)EUT

then ui(t,x)

°uj

—(t,x) < +oo (j =

1,2),

ui(t,x) in 12T•

Proof. For any

convex

we see, by assumption,

compact sets K1 C R', K2 C

that zi(t,p) < +oo,

max

and that the function =PKt.K2(X)

Of

def

= (t,u,p)E(O,TlxKixK3

is continuous, and hence locally bounded on RTh. It is easy to check that (8.20) with LK3 and K2(x) in place of L and M, respectively, holds for any (t,x,u,p), (t, x, v, q) E 11r x K1 x 1C2. The corollary thereby follows from Theorem 8.5. 0

We conclude this section with the following result of continuous dependence on initial data for global C1-solutions. (Here the continuity is with respect to the topology of uniform convergence on compact sets.)

Theorem 8.7. Suppose I = f(t,x,u,p) satisfies the condition (8.20) in Theorem 8.4. Let u1 = u,(t,x) (j = 1,2) be globalC1-solutions to the equation (8.1) with the initial conditions

u,(0,x)=cb1(x) where

=

on

{t=0,

= 1,2) are given functions of class C° on

lui(t,x) — u2(t,x)(

exp(Mt) .

sup IyI(1+IzI) exp(L*)—1

for all(t,r) E uT. The proof of this theorem will be left to the reader.

Then —

§8.4. GENERALIZATIONS TO WEAKLY-COUPLED SYSTEMS

§8.4. Generalizations to the case of weakly-coupled systems We now examine how the case of systems of first-order partial differential inequal-

ities or equations can be treated by the preceding method. Let m be a positive integer. Consider the class x V(I1T).

x

,n time,

element of Vm(Ilr) is therefore a vector function, namely

Each

u = u(t, x) = (ui(t, x),. .

.

x))

,

into Rm such that u5 = u'(t, x) belongs to V(flT) for every

from 11r C

jE{1,...,m}. First, the following result may be proved in much the same way as Theorem 8.1.

Theorem 8.8. Let u = u(t,x) be a vector function in Vm(flr). If there nonnegative function p = p(x) locally bounded on £ = 1(t) in L'(O,T) such that

a

and a nonnegative function

max Iuk(t, x)I] (j = 1,... , m) Iôu,(t, z)/OtI < 1(t). [(1+IzI)lôu,(i, z)/OzI+p(z)k=1 (8.25)

for almost every t E (0, T) and for all x

then

£(r)dr]

max

3=1

max

sup

Iu,(O,y)I,

0

tohere

C(x) is given by the formula (8.7).

Proof. For an arbitrary point (t°,x°)

we must prove that

max

<exp [c(x0)JtOi(t)dt].

max

sup

0

m

uj(0,y)I. (8.26)

Ei(t°,x°), Z(.,t°,x°), h(.), G1

Let us continue using the notations

introduced in the proof of Theorem 8.1 and then define g(t)

max gk(t)

k=1

(8.27)

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

86

for t E I, where glI(j)

{Iuk(t,x)I

(k = 1,... ,m).

z E Z(t,t°,x°)}

(8.28)

It follows from Lemma 8.2 that for any number 6 E (0, t°) each function g" = is absolutely continuous on [6, t°] and so is the function g = g(t). Moreover, they

are all continuous on the whole I. Still as in the proof of Theorem 8.1, we see that (8.26) will be obtained if we can show that (8.13) holds. To this end, setting def w(t) = g,,(t) — g(t), with q > 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,T)\G 3

w(t) = A}

{t E (0,t']

and 1 <j
u1(t.,x,),

e

def

= signu,(t.,x.)

E Z(t.,t°,x°). Next, choose a unit vector e =

(8.30)

with

—

(8.31)

Finally, let y = y(s) be an function continuously differentiable on R1 such that y(h(t.)) = x. and dy/ds = (1 + $vI) e, and let x(t) y(h(t)) for E [0, T]. Analysis similar to that in the proof of Theorem 8.1 shows that

u,(t, x(t))

Iu,(t, z(t))I < g(t) = g,1(t) — w(t) < g,,(t) — A

S8.4. GENERALIZATIONS TO WEAKLY-COUPLED SYSTEMS

for all t

E

97

[0,t.), and that

e u,(t.,x(t4) =

—w(t.) Iu,(t.,z.)I = g(t.) = g,,(t.)

= g,7(t.) —

Consequently, [e

u,(t,x(t))}

This would give

> £(t.) [(1 + Ix. I)

+ C(x°)

[(1 + Ix.I) a

+

k—lm uk(t.,

)I]

k—lm Iuk(t.,

contradiction with (8.25). The proof is therefore complete.

Remark. Theorem 8.8 can be used to investigate the stability of global solutions to the Cauchy problem for weakly-coupled systems, i.e., systems of first-order partial

differential equations of the form

(j = l,...,rn)

thz,/Ot+f5(t,x,u,Ou,/ox) =0

in aT. These systems are of special hyperbolic type because each equation contains first-order derivatives of only one unknown function. Since (classical) solutions of elliptic equations do not depend continuously (with respect to the topology of uniform convergence on compact sets) on initial data, theorems of the non-stationary

type that we have studied in this chapter cannot be expected to apply to partial differential equations or inequalities of effiptic type. [First results on second-order partial differential inequalities of parabolic and hyperbolic types were obtained by Naguino and Simoda [108] and Westphal 1158].] For our next discussions, we need to extend the notion of comparison equations given in Szarski 11281 to the Carathéodory case. Consider an ordinary differential equation

= p(t,w),

(8.32)

where the function p = p(t, w) is defined on (0, +oc) x [0, +oo) = {(t, w) 2 > 0, u' 0}. The following Carathhodory conditions are always assumed. (1) For (2)

For

almost every

each w E

t E (0, +oo) the function

[0, +oo) the

function (0,

[0,

+cc)

-I-cc) 2

u'

p(t, w)

p(t, w) is

is continuous.

measurable.

98

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

(3) For any r E (0, +oo) there exists a function m,. = mr(t) in m,.(t)

+oo) with

Vw E I0,r]

for almost every t E (0,+co). A soluIn this situation we call (8.32) a Carathéodory differential equatson on 0, means a function w = w(t) tson of it on an interval I C (0, +m), with intl 0 absolutely continuous on each compact interval J C I (absolutely continuous on I for short) such that w'(t) = p(t,w(t))

almost everywhere in I. We refer to Coddington and Levinson [311 for what concerns

the local existence of a solution of (8.32) through any given point (t°,w°) E Moreover, every such solution can be extended (as a solution) over a [left, right] maximal interval of existence.

Definition 2. A Carathéodory differential equation (8.32), with p(t, w) 0 on 0 for almost all 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)
(8.36)

8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE

100

Now take

p(t,w) def =

I p(t,max{0,g(t)})

if t2 < t < t°, w max{0,g(t)}, if t2
(8.37)

The above-mentioned Carathéodory conditions (1)-(3) are clearly satisfied for /3 = on (t2,t°) x [O,+oo). Let w = w(t) be a solution through (t1,w1) of (8.32) with /3 in place of p, and let (t3, t'] c (t2, t'] be its left maximal interval of existence. We next claim that (8.38) (0 ) w(t) g(t) Vt E (t3,t1J.

Assume (8.38) is false. Then one would find a nonempty interval (t4, t5) C (t3, t1) such that w(t) > g(t) Vt E (t4,t5), (8.39) with

w(t5) = g(t5).

(8.40)

It follows from (8.36)-(8.37) and (8.39) that g'(t) < p(t,g(t)) = j5(t,w(t)) = w'(t) almost everywhere in (t4,t5). Thus (8.40) implies that g(t) w(t) for all t E (t4,t5), which contradicts (8.39). So (8.38) must hold. We proceed to show that t3 = i2. In fact, if (0 <)t2 < t3, then (8.37) together with Carathéodory's condition (3), where r max{g(t) t E It3, t']}, proves that the limit lim w(t) exists and is finite. Hence, w = w(t) could be extended (as a :

solution of (8.32) with /3 in place of p) over an interval (t6,t'J J [t5,t'], which is impossible.

Finally, (8.37)-(8.38) shows that w = w(t) is indeed a solution through (t1,w1) of (8.32) on (t2,t1] with lim w(t) = g(t2) = 0. Setting w(t) fort E [0,i2}, we

obtain a nonzero solution of (8.32) on (0,t') which tends to 0 as t goes to 0. Thus we arrive at a contradiction. This completes the proof. 0 We can now combine the method of §8.2 with the technique of Carathéodory comparison equations and prove the following.

Theorem 8.11. Let u = u(t,z) be a vector function :n Vm(1Zr) vnth u'(O,x)

0

(j = , m), and (8.32) a comparison equation. If there ezists a nonnegative function £ = £(t) in L1(0,T) suth that .

.

Ou,(t, x)/OtI

£(t)(1 + IxI) . Ou,(t, x)/OzI +p(t,

Iuk(t, x)I) (j = 1,..., m) (8.41)

§8.4. GENERALIZATIONS TO WEAKLY-COUPLED SYSTEMS

(0,T) and for all x E

for almost every i 1

then u,(1,x)

0 in

101

=

rn).

Proof. For an arbitrary point (t°, x°) E 11T, it suffices to prove that max

u,(t°,z°)l = 0.

(8.42)

[0,t°J, E1(i°,r°), Z(.,t°,x°), h(.), shall continue using the notations I G1 introduced in the proof of Theorem 8.1 (and, also, of Theorem 8.8) and letting g = g(t), g" = gk(t) be as in (8.27)-(8.28). Obviously, (8.42) will be obtained if one can verify that g(t°) = 0. Since g = g(t) is a nonnegative function absolutely continuous on (0,t°1, with limg(t) = g(0) = 0 (by assumption), Proposition 8.10 shows that we need only claim that We

g'(t)

p(t,g(t))

almost everywhere in (0,t°).

(8.43)

By the hypothesis of the theorem, one finds a set G2 C (0, T) of Lebesgue measure 0 such that (8.29) and (8.41) are fulfilled for any t E (0,T)\G2, x E R". Assume without loss of generality that g = g(t) is differentiable at any point of (0,t°)\G, where C C1 U C2. Now fix an arbitrary point t. E (0,t°)\G and take 1 <j < m such that (8.30) holds for some E Z(t,,i°,x°). Next, choose a unit vector e E R" satisfying (8.31). Let y = y(s) be an RTh-valued function continuously differentiable on R1 such that y(h(t.)) = x. and dy/ds = (1 + and let x(t) y(h(t)) for t E [0,1']. Of course (see the proof of Theorem 8.1), x = x(t) is absolutely continuous on [0,1'], x(t.) = x., and

= £(t) (1 + Ix(t)l)

.e

Vt

E

(0,T)\G.

(8.44)

Moreover,

x(t)

Z(t,t°,x°) Vt E [0,t.].

This together with (8.27)-(8.28) implies e

u'(t, x(t))

Iuj(t, x(i))I < g1(t) < g(i) for all t

(0, t.).

(8.45)

Besides that, by (8.30), e

= Iu,(t,,x.)I = g'(t.) = g(t.).

(8.46)

8. DIFFERENTIAL INEQUALITIES OF RAAR TYPE

102

Therefore, since t. E (O,t°)\G, it may be deduced from (8.45)-(8.46) that

g'(t.) <

[e

u,(t,x(t))]

.

Consequently, by (8.30)-(8.31), (8.41), and (8.44), we conclude that

g'(t.)

(Ou,(t.,x,)/Ot) + — £(t,)(1

+ ix.l) lOu,(t.,x.)/OxI

Iuk(ts,x.)I) = p(t., u,(t.,x.)I) = p(t,,g(t.)). Finally, because C has measure 0 and t. E (O,t°)\C must hold. This completes the proof.

is

arbitrarily chosen, (8.43)

0

Theorem 8.12. Let u = u(t,x) be a vector function in with u(O,x) 0 (j = 1,... ,m), and (8.34) be a comparison equation. If there ezut a nonnegative function = p(x) locally bounded on RTh and a nonnegative function £ = 1(i) in L' (0, T) suth that

iOu,(t,x)/Oti 1(t)

[(1

+ lxi) . iOu,(t,x)/ôxl +

(j = 1,...,m) for almost every t E (O,T) and for alix E

(8.47)

then u,(t,x)

0 in

QT (j =

Proof. For an arbitrary point (t°, x°) E it suffices to prove (8.42). Let us continue using the method (and notations) introduced in the proof of Theorem 8.11. We may extend the function £ = 1(t) over the whole (0, +oo) and assume essinf 1(t) > 0. Then by (8.47) [instead of (8.41)1 we get g'(t) <

almost everywhere in (0,i°)

[instead of (8.43)1 for some positive constant C. By Proposition 8.9(ü), the Carathéodory differential equation WI = C1(t)c(w)

is also a comparison equation. Thus (8.42) is straightforward as before.

Chapter 9 Hopf's Formulas for GloL?al

of Hamilton-Jacobi Equations §9.1.

Introduction

The aim of this chapter is to present some formulas for explicit global solutions of the Cauchy problem for Hamilton-Jacobi equations of the form

ôu/Ot + f(t,Ou/Ox) =

u(0,x) =

0

on

in

{t > 0, x

(9.1)

{i =

0, x

(9.2)

According to Theorem 1.4, the Cauchy problem (9.l)-(9.2) has locally a unique are of class C2-solution if the Hamiltonian f = f(t,p) and initial function = C2. However, there is generally no hope to find a global classical solution; therefore, one needs to introduce a notion of generalized solutions and to develop theory and methods for constructing these solutions. (For this, cf. Chapters 4-7.) During the past five decades, many mathematicians have obtained various global results by relaxing the smoothness conditions on the solutions. In particular, the global existence and uniqueness of (generalized) solutions for convex Hamilton-Jacobi equations were well-studied by several approaches (see §4.3 for the uniqueness of such a solution).

if the Hanultonian I = 1(p) is continuous and if the initial function = is globally Lipschitz continuous and convex with the Fenchel conjugate = Hopf [64] proved in 1965 that the formula E. u(t, x) = max{(p, x) — pER"

—

tf(p)}

(9.3)

determines a (generalized) solution of the Cauchy problem (9.1)-(9.2) in the sense

that this solution satisfies (9.1) at every point where it is differentiable. Since the solution is locally Lipschitz continuous, the well-known Rademacher theorem [116, Theorem 1.18] shows that (9.1) is then satisfied almost everywhere.

104

9.

HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

If the Haniiltonian f = f(p) is strictly convex with

lim f(p)/)p) = +oo and

if the initial function = çb(x) is globally Lipschitz continuous, E. Hopf [64] also established the following formula for a (generalized) solution of (9.1)-(9.2):

u(t,x) =

f((x — y)/t)}.

(9.4)

The above formulas are often associated with the name of Hopf, although (9.4) was actually first discovered for n = 1 by P.D. Lax [971 in 1957. Step by step, certain more general cases of Hopf's formula (9.3) will thoroughly be dealt with in this chapter under a standing hypothesis like (but somewhat more strict than) Carathéodory's condition on the Hamiltonian I = f(t,p). Section §9.2 concerns the case of convex (but not necessarily globally Lipschitz continuous) initial data. In Section §9.3 we consider the Cauchy problem with nonconvex initial

can be represented as the minimum of a = 4(x) is a d.c. function (i.e., it can be represented as the difference of two convex functions). Finally, Section §9.4 discusses Hopf's formula (9.4) in case = is just continuous. Most of the results presented here were originally published in [1421-[145] and data: first for the case where qS =

family of convex functions, and second for the case where

[1521. (For other results in the field, see for example [l1]-[12], [21], [33], [52], [100), and [124).) Our method is based on some techniques of multifunctions and convex

functions. The relevant material on these subjects from [8] and [117] (without proofs) can be found in Appendix II given at the end of the book. This makes our exposition self-contained.

We shall continue using the notation of Chapter 8. Moreover, in accordance with

to denote the set {0 < t < Chapter 2, when T = +00, we use both V and ((0,+oo)\G) x +00, x Further, for any G C R, put Va = {(t,x) V t G}. Consequently, V = V0. Let us recall that Lip([0, T) x fl CoO, T) x R"); accordingly, = Lip(D) fl C(D), where Lip(V) is :

the set of all locally Lipschitz continuous functions u = u(t, x) defined on V =

Definition. A function u = u(t, x) in Lip(10, T) x is called a global solution of the Cauchy problem (8.1)-(8.2) if it satisfies (8.1) almost everywhere in QT and if u(0,x) = çb(x) for all zE R't.

§9.2. THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA

105

§9.2. The Cauchy problem with convex initial data In this section we consider the Cauchy problem (9.1)-(9.2), with = ti(x) a finite the Fenchel conjugate function of convex function on Denote by = for pER", and

by E the effective domain of

= :

We assume the following two hypotheses:

(E.I) The

Hamiltonian

I = f(t,p)

is

continuous in {(t,p)

t E (O,+oo)\G, p E

:

R"} for some closed set G C R of Lebesgue measure 0. Moreover, to each N E such that (0, +oo) there corresponds a function 9N = gN(t) in

sup f(t,p)I gN(t)

pIN

for almost all t E (O,+oo).

(E.II) For every bounded subset V of that (p,x) —

whenever (t,z)

—

j I (r,p)dr

V,

> N(V).

there

exists a positive number N(V) so

—

<

—

f

f(r,q)dr}

(9.5)

Hypothesis (E.I) implies the i-measurability and p-continuity off = f(t,p) on is finite on R", this hypothesis allows {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) <

there

exist positive numbers r(t°,x°) and N(t°,x°)

(t, x)

max

IqIN(t°,z°)

V(t°, x°),

> N(t°, x°) (9.5')

where V(t°,x°) We

{(t,x) E

:

t — 201 + Ix

— x°I <

r(t°,x°)}.

proceed now to consider, for example, the Cauchy problem in

u(O,x)=x2/2

on

{t>O, xER},

{t=O, xER}.

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

= u(t, s)

in {O
— 2(t— 1)

(9.8)

R}. This solution can not be extended continuously beyond the time t = 1. We should note that in this case the set of all (j0, x°) E V such that (9.5') holds for some r(t°,x°) > 0 and N(t°,x°) > 0 is precisely the domain (0 2 < 1, x E R}. Moreover, if we try to apply Hopf's formula (9.3), ignoring the fact that the initial function here, = x2/2, is not globally Lipschitz continuous, then we also obtain the same solution as (9.8) in (0 2 < 1, x E R}.

§9.2. THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA

Remark 2. Let

= qS(x) be a finite convex function on Assume (E.I). Then (E.II) is satisfied if ço(t,x,p) tends to (—oo) locally uniformly in (t,x) E V as tends to (+oo), i.e., if the following holds:

E R and any bounded subset V of

(E.II)" For any

number N(A, V) so that

there

(t, x)

V,

exists a positive

> N(A, V).

Indeed, fix an arbitrary q° E 0 [117, Thm. 12.2]. (t,.r) '-+ p(t,x,q°) is continuous, it follows that

Since the finite

function V

deC

=

o

. rnf

)> —00

for any bounded subset V of V. Under Hypothesis (E.II)", we certainly find a

(for each such V) so that

number Nv

p(t,x,p)
(t, x)

> Nv, thus getting the validity of

Remark 3. Condition "gj.j = gN(t) be replaced by "9N = gN(i) E To

(E.fl).

in the hypothesis (E.I) could not see this,

consider the following Cauchy

problem au/Ox

Ou/Ot+

u(0,x)=x Here, n

def

=

=0 in {t>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

O<ö(t

e>O O<6<e

}.

Ii—.I<.

and

The quantities

are

called, respectively, the upper and lower

Dliii semiderivatives of (' = at the point in the direction e. if +oo > is said to be directionally > —oo for all e E Rm, then (' = = will be called its derivative differentiable at and is Lipschitz continuous at in the direction e. It can be shown that, when = near its upper and lower Dim semiderivatives may also be defined equivalently by — 5,1.0

and

lii

+ 5e) —

},

respectively.

Lemma 9.2. Let 0 be on open subset of

and w = w(e,p) an upper semscon-

to [—oo, +oo) with the following two properties: tsnuous function from 0 x (i) There exists a nonempty set E C RTh such that = w(e,p) is finste on 0 x E Moreover, to each bounded —oo where EC and that subset V of 0 there corresponds a positive number N(V) so that <

max

whenever

V, IpI

> N(V);

is differentiable in 0. (ii) For every fixed p E, the function w = is continuous on 0 x E. Besides that, the gradient Ow/Of = Then one has: is a locally Lipschitz continuous function in pE a) sI' = the domain 0;

§9.2. THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA

b)

is

=

109

directionally differentiable in 0 with

0,

= max

pEL(t)

e

E R)

where {p E

=

:

C E.

(9.9)

We shall also need the following.

Lemma 9.3. Let Condition (i) in Lemma 9.2 hold for a given function w = 0 (whenever p E E) and upper which we assume to be continuous in semicont,nuotzs with respect to the whole on 0 x R". Then (9.9) determines a nonempty-valued, closed and locally bounded multifunction L =

of

E 0.

is a compact-valued and upper semiRemark. Lemma 9.3 implies that L = continuous multifunction. In this lemma, the set 0 C R is not necessarily open.

Proof of Lemma 9.3. For any bounded subset V of 0, denote Nv

N(V)

(see (i)). Then as E V. This means that L = is locally C bounded. Moreover, = being upper semicontinuous, we can deduce (also from (i)) that the supremum = should always be finite pE and attained. Consequently, L = C E is a nonempty-valued multifunction of :

€0. To complete the proof, one needs only check the closedness of L = For this ph) purpose, let be a sequence convergent to a point p°) in Ox R" such ask = 1,2,3 By the definitions of 1' = and L = that p" we have

w

=

is upper semicontinuous in

and continuous in

(9.10) shows

that lim

L=

Thus p°

for all p. is therefore closed.

Proof of Lemma 9.2. a) Let V be an arbitrary compact convex subset of 0 and let N nonempty set

V, pick up an element p' in the N(V). For any two points CEn N) (cf. Lemma 9.3). Then —

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

hence the mean-value theorem gives —

where

A.

—

—

C V and A is a (finite) upper bound of

E

—

Vx Analogously, is thus locally Lipschitz continuous in 0. = b) For any E 0 and e E Rm, we find two sequences numbers convergent to zero such that

A

over The function

—

of positive

+ ake) —

= lim and

lim

C E and apply the mean-value theorem to

Let us take an arbitrary p obtain

+

+ a'e) —

—

A passage to the limit as k

where

for any p E

= +00 shows

+

e)

that

Hence

sup for each k = 1,2 Since the multifunction L = is closed and locally bounded (Lemma 9.3), by taking a subsequence Therefore, a passage to the if necessary, we can assume that p° Now choose an element

E

limit (similar to the above) in the inequality + like) —

<w(6° + I3he,p") —

13:,

(where

=

+

19k

(0,/3:,)) gives

< sup

(9.12)

§9.2. THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA

111

Finally, combining (9.11)-(9.12) yields

= max

=

pEL(C )

for any E 0, e E Rm. This implies the directional differentiability of and completes the proof.

=

0

We are now in a position to prove Theorem 9.1.

Proof of Theorem 9.1. It can be verified that the function w = x, p) (see (9.6)) satisfies all the assumptions of Lemma 9.3 where E dom def— def def 0[117, Theorem 12.2] andm = n+1, = (t,x). Here, weput0 = Dand conclude that the definition

L(i,x) d.f = {p E E :

= u(t,z)}

determines a nonempty-valued, locally bounded (and closed) multifunction L = L(t, x) of (t, x) E V. To this end, Our proof starts with the claim that u = u(i, x) is in take arbitrarily an r E (O,+oo) and denote V,. {(t,z) i + lxi < r}, N,. N(V,.) (cf. (E.II)). Let gp,r, = be as in Hypothesis (E.I). Then for any two points (t1, x'), (t2, x2) V,. we may choose an element p' of the nonempty set and get L(t1,x1) C u(t1,x1) — u(t2,z2)

N,... 1x1 where s,.

ess sup

= (p',x'

—

— x21

+ s,.

—

x2)

+f f(r,p1)dr

—

(t). Analogously,

tE(O,r)

u(t2, x2) — tz(i1,x') < N,.. 1x1 — x2i + s,.

t1 —

Thus, u = u(i,x) belongs to (0,0,... ,0,1) E (1,0,0,...,0), e1 Next, let e0 (0,1,0,. ..,0),..., def +1 It follows from Lemma 9.2 (we now replace 0Do, = the set G being as in R't (E.I)) that u = u(t,x) is directionally differentiable in V0 with .

= max{—f(t,p) = max{f(t,p)

p p

L(t,x)}, L(t,x)}

(9.13)

9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS

and (for 1
L(t,x)},

p

:

(9.14)

: pE L(t,x)}.

O_e•U(t,X) = max{—p;

On the other hand, according to Rademacher's theorem [116, Theorem 1.18], there

existsaset Q C Dof((n+1)-dimensional) LebesguemeasureOsuchthat u = u(t,x) is (totally) differentiable with Ou(t, x)/Ot

x),

ôeou(t, x)

(9.15)

Ou(t,x)/e9x1 = (3eeU(t,X) =

at any point (t,x)

V\Q. Hence, (9.14)-(9.15) show that the equalities (for 1

ôu(t,z)/0z1 = max{p1

p E L(t,x)} = min{p,

outside the null set 2def = (G x R )

:

p E L(t,x)}

Q; i.e., L(t, x) is precisely the singleton {Ou(t,x)/Ox} except on P. Consequently, (9.13) together with (9.15) implies that (9.1) must be satisfied almost everywhere in V. Further, by a well-known property of Fenchel conjugate functions [117, Theorem

hold

U

.

12.2), (9.7) gives

u(0,z) = max{(p,x) pER"

=

=

for all XE RTh.

From what has already been proved, we conclude that u solution of (9.1)-(9.2).

u(t, x) is a global

0

Remark. By the proof of Theorem 9.1, all partial derivatives of the global solution u = u(t, x) exist [(9.1) is then satisfied though the solution may fail to be differ-

entiable] at a given point (t°, x°) E D0 if L(t°, x°) is just a singleton. We shall investigate the smoothness of u = u(i, x) in greater detail in Under Hypothesis Corollary 9.4. Let
§9.3. THE CASE OP NONCONVEX INITIAL DATA

Proof. Only (E.II) needs verifying. Given any r + lxi
113

del

(O,+oo), denote V,. = {(t,x)

inf f(t,p)/(1 + p1). Then

f(r,p)dr — 4*(p)

—

for all (t,x) E V,., p

Lemma 9.13 later) that

that

(9.16)

ts,.(1 + lVI) < r(l + 21s,.l) max{1,

—

On the other hand, it being known (cf. Remark 2 of = +00, there exists a number N,. 1 such

urn

r(1 + 2ls,.I) + 1 whenever

N,.. Thus, (9.16) implies

provided

Finally, since r

E

(t,x)EV,.,

lpINr.

(0, +co) is arbitrarily chosen, Condition (E.II)" holds; hence so

0

does (E.II).

Corollary 9.5. Let f = f(t,p) be continuous on V and let =

be convex and

Then (9.7) determines a global solution of the

globally Lipschitz continuous on Cauchy problem (9.1)-(9.2).

Proof. Since = is convex and globally Lipschitz continuous, E should be bounded [117, §13.3] (and nonempty). Independently of (t,x)

D, it

follows that

= (p,x) whenever

is

—

—

I f(r,p)dr = —oo

Jo

large enough. Hypothesis (E.II) therefore holds while (E.I) is triv-

ially satisfied.

0

Remark. As we have mentioned in the introduction, Corollary 9.5 was proved by E. Hopf [64] in a different way for the case where I = f(p) depends only on p.

§9.3. The case of nonconvex initial data In this section we consider the Cauchy problem (9.1)-(9.2) under the more general assumptions that I = f(t, p) is still t-measurable and p-continuous as in §9.2, while = 4(x) is now a d.c. function, i.e., the difference of two finite convex functions on R". The class of d.c. functions plays an important role in the theory of global optimization. This class is rather large: It contains all the semiconvex

9. HOPP'S FORMULAS FOR GLOBAL SOLUTIONS

or semiconcave functions. We emphasize that it also contains all functions of class C2 (of the whole with second derivatives bounded either from below or from above. The reader is referred to [6] and [661 for a sufficiently complete study of d.c. functions. We first prove the following result.

Theorem 9.6. Let I be an arbitrary nonempty set and let Ua = ua(t, x) be a global solution of the Cauchy problem for the same Hamilton-Jacobi equation

u/Ot+f(t,3u/Oz)=O

{t>O, x€R'},

in

(9.1)

with the initial condition

u(O,x) =

on

{t =

(9.2a)

0, x E

for each a E I. Suppose that to each bounded subset V of 1) there correspond a set W(V) C V of Lebesgue measure 0, a nonnegative number M(V), and a subset J(V) of I such that all the functions Ua = ua(t, x) for a E J(V) are Lipschitz continuous in V unth a common Lzpschitz constant M(V) and satisfy (9.1) at every point of

V\W(V) and that infua(t,z) = aEI

u = u(t,x) where

for (t,x) E V. Then the function

mm

aEJ(V)

infua(t,x) is a global solution of the Cauchy problem (9.1)-(9.2)

aEJ

aEI

Proof. By assumption, u(t, x) =

mm

aEJ( V)

ua(t, r) on each bounded subset V of

Moreover, Iua(t',Z1) — Ua(t2,X2)I < M(V) .(It'

and any fixed (t',x'), (t2,x2) some a° E J(V). Then

—

+ Ix' — x2[) for any a E J(V)

V. Assume u(t',x1) 2 u(t2,z2) = u0o(t2,x2) for

0 < u(t1,x') — u(t2,x2) <

— Uao(t2,Z2) < M(V) . ([ti

—

+

—

This means that u = u(t,x) is in

Now denote V,, {(t,x) E V J(Vk), W,, t + xl < k}, W(Vk) for each k = 1,2,... and let Wo C D be a set of Lebesgue measure 0 such that u = u(t, x) is differentiable at every point of D\W0 (Rademacher's theorem). It will be shown that u = u(t,x) satisfies (9.1) except on the null set Q :

§9.3. THE CASE OF NONCONVEX INITIAL DATA

115

Indeed, given any (t°, x°) V\Q, we choose a positive integer k > 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
u(O,x) = exp(—IxI) = min{exp(x),exp(—x)}

{t = 0, x E R}.

on

By Corollary 9.8, a global solution of the problem is u

def

= u(t,x) = mm max{px — h,(p) s1,2 pER .

—

where def

h1(p)

Ipinp—p

if

I

if

=

0

—

1I}

p>O, p=O,

if p
L+oo

0

if p>O, if p=O,

—p ln( —p) + p

if

(+oo h2(p) =

tip 2

p < 0.

The solution can also be rewritten as — u(t,x) = min{max{pz — plnp +p — po = max{—pIxI—plnp+ p—tIp2 — Po

+ pln(—p) — p

—

tip2

—

lI}}

in which we adopt the convention that plnp = 0 if p = 0.

Example 2. Let h = h(o) be a finite-valued continuous function of o on a given compact set K C R't. We put for

min{h(ci) + Icil . ixi} aEK

x

R".

If f = f(t,p) belongs to C(V), depends only on t and and is decreasing with respect to ipi, then it follows from Corollary 9.7 that a global solution u = u(t, x) of the Cauchy problem (9.1)-(9.2) can be found by the formula u(t, x)

min{h(a) + max {(p, z) -

aEK

IpIIaI

= min{h(a) + Icii

f

f(r,p)dr}}

o

ri —

j f(r,s)dr}

for

(t,x) €

§9.3. THE CASE OP NONCON VEX INITIAL DATA

We now consider the Cauchy problem (9.1)-(9.2) in the main case of this section where 4, = 4i(x) belongs to the class DC(RTh); i.e., it has a representation of the form

4,(x)woi(x)—o2(x) on

(9.19)

= oi(x) and 02 = 02(X). We call (9.19) a d.c. representation of 4, = 4,(x). (Of course, there are always an infinite number of such representations for each d.c. function 4, = cb(x).) The notations = o(p) will signify the Fenchel conjugate functions of = 01(x) and 02 = a3(z). The effective domains of these conjugates are denoted by E1 and B2, for some finite convex functions

respectively.

Besides Hypothesis (E.I), we shall also assume the following ones. (E.III) To any bounded sets V C

and E C

there corresponds a positive number

N(V,E) so that (Pa(t,X,P) <

a E E,

max

IqIN(V,E)

> N(V,E).

Here,

'pa(t,X,p)

(p,x)

—

—

(E.IV) All the multifunctions

La(t,Z)

{p E

= La(t,X) of(t,x) E :

(9.20)

j f(r,p)dr. defined by

=

(9.21)

are actually single-valued in D\Q where Q is a certain closed set of ((n + 1)dimensional) Lebesgue measure 0 and is independent of a E

Remark. Let = o1(x) be a finite convex function on and I = f(t,p) be a p-convex function on {t > 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
u(0,x) = 4(x) on {i =

0,

x E R},

9. HOFF'S FORMULAS FOR GLOBAL SOLUTIONS

120

with f(p)

(for p E R) and

(1 +

f x3/3 1 x

—

if

(2/3)signz

x E [—1, 1],

if x

[—1,1].

We first note that neither the formula (9.3) of Hopf nor the formula (9.7) of Theorem

9.1 works in this case since the initial function here is not convex. Though the present Hasniltonian I = f(p) is in fact convex we should also mention that Hopf's formula (9.4) could not be applied directly to the problem, because =1

lim

+00.

In this case, however, it is easy to check the validity of d.c. representation (9.19)

where

10 x3/3

tx—2/3

if x<0,

if rE [0,1],

if x>1

and a2 = a2(x)

cr1(—x) are globally Lipschitz continuous on R. Further, we may invoke either (E.IV)" or (E.IV)' to deduce that (E.IV) holds. Therefore, by Corollary 9.10, a global solution u = u(t, x) of the problem can be found in the

form

u(i, x)

max{z(p —

mm

—

—

—

t(1 +

—

we have seen, Theorem 9.9 and its Corollary concern the Cauchy problem (9.1).(9.2) in case the initial function 4 = has a d.c. representation — o2(x) such that is bounded in R't. The following will be devoted to the case where dom o = R". As

Theorem 9.11. Let (9.19) such that

=

lim IrI-.+oo

be in the class with a d.c. representation = +oo. Under Hypotheses (E.I), (E.III), and

(E.IV), suppose that there exists a function g = g(t) in with the property that sup{f(t,p) g(i) for almost all t E (0,+oo). Then (9.22) p E R"} determines a global solution of the Cauchy problem (9.1)-(9.2). :

Proof. Since a2 = as(x)

is a finite

convex function on

with

lim a2(x)/IxI =

12H++OO

+m, so is its Fenchel conjugate function a = o(p); in particular, E2 R" (d. Remarks 1-2 of Lemma 9.13 later).

domo =

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

121

We shall continue using the notation Ua (t, z) introduced in the proof of Theorem 9.9.

{(t,x) E V

Let r E (0,+oo), v,.

and

.1.; g(r)Idr. Since

corresponds

:

t + ri

< r},

sup IzI
= +00, to any M E (0,+oo) there

lim

a number N,.,M 1 so that

o(a)/lal r+s,.+/4,.+M

as

al>

N,.,M.

(9.25)

(p,

By (9.20), it follows that, if(t,x) E V,., then Therefore, (9.24) and (9.25) imply

a(a)+max{(p,x) —o(p)} —rIal pElt

—8,.

—s,.

—s,. —rlal —8,.> 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)

<

rnin((t,x,w) < M < C(t,x,y) as (t,x) E V,

r we

> 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 <

locally uniformly in p'

r2

As a finite concave-convex function, I = f(p',p") is continuous on Theorem 35.1]; hence, def

= sup f(p',p")l < Ip'Iri

x

+00.

<+00. 1117,

(10.8)

IP"I'2 def So, with p = r2z ,, Liz"]

p, =def—r1z'/]z']], (10.5) [reap. (10.6)] together with

(reap.

(10.8) implies

f2(p', z")

inf

f'(z',p")

sup

(resp.

]z"]

z"i

Ip'Iri

+

z'

Ip"Ir2

4

as

—r1

z"] —*

as

+co +oo].

z']

Since r1, r2 are arbitrary, (10.7) must hold locally uniformly in p' p"

as

[reap.

required.

Remark. If (10.4) Ireap. (10.3)] is satisfied, then (10.5) (reap (10.6)] gives

IZ(p',zh') li/i

(resp.

fhl(z',p")

1,011)

>

-

I,)

<

li"]

—

uniformly in z"

-+

li/i

—+

—p

—oo

as

+00

(10.9)

(p"] —+ +001

(10.10)

[reap. z'

x R'%z. Besides Now let f = f(p',p") be a concave-convex function on = fdl(z',p"), we shall consider the and "partial conjugates" f2 = following two "total conjugates" of I = f(p', p"). The first one, which we denote by 7 = 7(z', z"), is defined as the Fenchel conjugate of the concave function

R"

p'

'—*

z"); more precisely,

f"(z' z") def = inf {(z',p') + p'

I

z")}

(10.11)

§10.2. CONJUGATE CONCAVE-CONVEX FUNCTIONS

for each (z', z") E x The second, conjugate of the convex function

f(z', z") —

for (z',z") E

131

= f(z', z"), is defined as the Fenchel _f*1(zI,pSl); i.e.,

sup {(z",p") + f1(z',pU)}

p"ER"2

(10.12)

x R"2. By (10.5)-(10.6) and (10.11)-(10.12), we have

f*(zI z")

= pmi sup {(z',p') + (z",p") — f(p',p")}, ER I p"ER"2

f(z', z") = sup

(10.13)

mi {(z',p') + (z",p") — f(p',p")}.

p"ER"Z p'ER'l

(10.14)

Therefore, in accordance with [117, p. 3891,f = T(z',z") and f = f(z',z") will be called the upper and lower conjugates, respectively, of f = f(p', p"). [Of course, (10.13)-(10.14) imply j'(zl,zhl) L*(zs,zh?).] For any z' E Rnh, the function x R'12

(p', z")

'-+

(z',p') + f3Q/, z")

h(p', z")

is convex. Thus (10.11) shows that T = T(z',z") as a function of z" is the image i—* (Ah)(z")

p', z")

:

A(p', z") = z"}

z". 3 (p', z") A(p', z") [117, Theorem 5.7]. On the other hand, by definition, T = is necessarily concave in z' E x Rn2. The same This upper conjugate is hence a concave-convex function on conclusion may dually be drawn for the lower conjugate f = I (z', z").

of h = h(p', z") under the (linear) projection

x

It follows that f = f(z',z") is convex in z"

We have previously seen that if the concave-convex function f = f(p',p") is finite on the whole x and satisfies (10.3)-(10.4), its partial conjugates f*2 = f*2(p',zI') and f*i = f'(z',p") must both be finite with (10.9)-(10.10)

holding. Therefore, Remarks 1-2 of Lemma 9.13 show that f = f(z, z") and f = f(z',z") are then also finite, and hence coincide by [117, Corollary 37.1.2]. In this situation, the conjugate

1* = f(z',

z") =

(10.15)

10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS

132

off

= f(p', p") will simultaneously has the properties:

= +00 for each z' E Rn',

lz"I-9+00

f(z',z")

urn

Iz'I

Iz'I—*+oo

For

(10.3)

Z

= —oo

(10.4)

for each z" E

the next discussions, the following technical preparations will be needed.

Lemma 10.1. Let f = f(pS,pfl) be a finite concave-convex functzon on with the property (10.3) [resp. (10.4)] holdsng. Then = +co locally uniformly in p' E

,lim IP' I—I+co

[resp.

x R'%a

Ip'

(10.16)

I

= —00 locally uniformly

lim Ip'I

p" E (10.17)

Proof. First, assume (10.3). According to the above discussions, (10.5) determines = f*2(pI,zhl). Further, [117, Corollaries 10.1.1 and a finite convex function 12.2.11 shows that

f(p',p") = sup {(z",p") — f3(pI, z")}

(10.18)

z"ER"2

x Let 0 < r,M < +co be arbitrarily fixed. As a finite convex function, f2 = f2(p',z") is continuous [117, Corollary 10.1.1], and hence

for any (p',p") E

locally bounded. It follows that sup If*2(pl,ZU)I < +oo. Ip'Ir Iz"IM So,

with z"

Mp"/Ip"I, (10.18)-(10.19) imply that >M—

Ip'Ir as

Ip"I

—

Ip"I

Ip"I -4 +oc. Since M > 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
f2(Ap' +

(1 —

A)q',z") = maxw(Ap' + (1

—

< max

z",p") + (1 — A) max

p EW'2

+ (1— A)f2(q',zh').

=

if all the equalities simultaneously occur, then there must exist a point p" E L(Ap' + (1

—

A)q', z") fl L(p', z") fl L(q', z")

with w(Ap' + (1

—

A)q', z",p") = Aw(p', z",p") + (1

—

A)w(q', z",p");

hence (10.21) implies

f(Ap' + (1 This would give p' =

—

A)q',p") = Af(p',p") +

(1

q?,

—

A)f(q',p").

and the convexity in p' of thereby strict. By duality, one easily proves the remainder of the lemma.

= f3(p',z") is

0

We are now in a position to extend Lemma 9.13 to the case of conjugate concaveconvex functions.

Proposition 10.3. Let f = f(p',p") be a strictly concave-convex function on

I

with both (10.3) and (10.4) holding. Then its conjugate f = (z', :") defined by (10.11)-(10.15) is also a concave-convex function satisfying (10.3)-(10.4). Moreover, f = f(z', z") is everywhere continuously differentiable with Zn)) + (au',

zhl))

—.

z"),

zht))

(10.22)

§10.2. CONJUGATE CONCAVE-CONVEX FUNCTIONS

Proof. For reasons explained just prior to Lemma 10.1, we see that 7(z',z") (z',z"), hence that (10.11)-(10.15) compatibly determine the conjugate f = I (z', z"), which is a (finite) concave-convex function on x with (10.3)(10.4) holding. We now claim that f = I (z',z") = I (z',z") is continuously differentiable everywhere. For this, let us again go back to Lemmas 9.2-9.3 and introduce the def def def def def temporary notations: n = n2, E = R m = ni+n2, 0 = Rn' +n 2, = (z ,z ), g

pt-f p". Since (10.10) has previously been deduced from (10.3), we can verify and that the function (z", p") + 1'' (z', p")

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

w=

(10.23)

meets Condition (i) of Lemma 9.2, while the other conditions are almost ready. In fact, as a finite concave function, f*a = f*1(z',p") is continuous (cf. [117, Corollary 10.1.1]) and so is w = w(z',z",p") (cf. (10.23)); moreover, Condition (ii) follows from (10.23) and Lemma 10.2. Therefore, by (10.2) and (10.15), this Lemma shows

that f = f'(z', z") = f(z', z") should be directionally differentiable in

x

with

8(e',e")f'(Z', z")

for (z',z") L=

e")

= p"EL(z'.z")

x Rn2, (e',e") E = L(z', z")

{p" E

x :

+ K.T(z',p"), es) }

(10.24)

where

w(z', z",p") = f(z1, z") = f(z', z")} (10.25)

is an upper semicontinuous multifunction (see Lemma 9.3 and its Remark). However, because f' = f1(z',ph') is strictly concave in p" E R'12 (Lemma 10.2), it may be concluded from (10.23) and (10.25) that L = L(z', z") is single-valued, and thus continuous in R'4 x Consequently, according to (10.24) and the continuity of the gradient mapping x 3 (z',p") '—i (Lemma 10.2),

the maximum theorem [8, Theorem 1.4.16] implies that all the first-order partial x derivatives of f = I z") exist and are continuous in (cf. also [128, Corollary 2.2]). The conjugate f = f(z', z") is hence everywhere continuously

10. HOPF.TYPE FORMULAS FOR GLOBAL SOLUTIONS

136

differentiable. In particular, since L = L(z', z") is single-valued, it follows from

(10.24) that

z")}

L(z', z") {

and therefore that z

0'

— )

ôf' (

,

' ôz"

= ôz'

(z , , z ,, )

Thus, (10.23) and (10.25) combined give

f*(zl

z")) +

z"))

Finally, we can invoke (10.20) to deduce that

z")

zu)) + (z',

(zi'

'

Oz'

zfl)) +

(cii

zIP))) +

-

( 0f

' Oz"' zIP))

Of

'\ '

z"),

—

The identity (10.22) has thereby been proved. This completes the proof.

zIP))

0

§10.3. Hopf-type formulas We now consider the Cauchy problem

On/Of + f(On/Ox) =

0

u(0,x) = d(x)

in V on

{t > 0, x = (x',x") E IR" x R't2},

{t =0, x = (x',r") ER"'

x

(10.26) (10.27)

An explicit global solution u = u(t,x) = u(t,x',x") of the problem will be found under the following three standing hypotheses.

r") is of class C° and the Hamiltonian (G.I) The initial function = j(x) = with (10.3)-(10.4) f(p',p") is strictly concave-convex on IR"' x f(p) = f= holding.

(G.II) The equality inf

sup i,

=

ml

i, ER 2

sup ((t,x,y)

§10.3. HOPF-TYPE FORMULAS

137

is satisfied in D, where

((t,x,y)

+t.

for(t,x) = (t,x',z") ED,

y)/t)

—

(10.28)

Here, f =

E

f(z) = f(z',z")

denote, the conjugate defined by (10.11)-(10.15) of I = f(p',p").

(G.III) To each bounded subset V of V there corresponds a positive number N(V) so that %D

sup ((t,x,w',w") < sup ç(txw'y")

mm ER

w'ER''

w'ER°l

inf C(t, X, w', w") >

max

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

*D"ER"3

w'I
whenever(t,x)

E R" x

V,

> N(V).

with

The main result of this section reads as follows.

Theorem 10.4. Assume (G.I)-(G.HI). Then the formula

u(t,x) def =

inf

sup

y'ER"a V"ER"2

((t,x,y) = inf

for (t,x) E D

sup p'ER"a

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

For the proof of Theorem 10.4, we need further generalizations of Lemmas 9.29.3 to the mixed sup-inf cases.

Lemma 10.5. Let 0 be an open subset of Rm and x = a = continuous function of y) = x with the following three y', y") E 0 x properties:

(i) The identity sup

= inf

inf

holds in 0.

sup

(ii) For every bounded subset V of 0, there exists a positive number N(V) such that mm

sup

< sup

mi

>

*0

Iw"IN(V) and

max

w'ER"l w"ER"2 Iw'IN(V)

V, y = (y',y") (iii) The function x = x mapping 0 x 3 whenever

E

mi

w"ER"2

x

is differentiable in

> N(V). E 0 with continuous gradient

10. HOFF-TYPE FORMULAS FOR GLOBAL SOLUTIONS

138

Then one has: (a)

sup

=

inf

=

mi

is a locally Lipschitz

sup

cont2nuous function in 0. is directionally differentiable in 0 with (b) = = =

y',

miii

e)

0, e

max v'EL'tt)

Rm)

where

{y' 1,

{y" E

:

unf ER 2

=

sup

=

(10.30)

Lemma 10.6. Let Conditions (i)-(ii) in Lemma 10.5 hold for a continuous function = x x Then (10.30) gives a nonempty-valued, closed x

and locally bounded multifunction L =

x

of

E 0.

Proof of Lemma 10.6. For any bounded subset V of 0, denote Nv

N(V)

(see (ii)). The function

0x

?

inf

'-÷ y

ER'2

is upper semicontinuous as the unfimum of a family of continuous ones. It follows from (ii) that for e V the supremum

=

sup

y'ER'

ml

should essentially be taken over all y' of just the compact set and hence be attained. This means 0 L'(e) C V. Analogously, (0', Nv) when L= is nonempty-valued and locally 0 bounded.

To complete the proof, we need only check the closedness of L = For this, let be a sequence convergent to a point in x x Rn2 such that as k = 1,2,3,... 0x

§10.3. HOPF-TYPE FORMULAS

Then one has and therefore 0

inf

139

= sup

=

ER 2

gets

,0

,y ,y

)

. = hm

x(e

k

,k

,y ,y

,,k

.

0

sup y'ER"I

By duality, one also gets

k

. liminf sup k-4+oo y'ER"

)

,y'y I

inf

,y, ,y

)

1,0

).

It follows from (i)

y

and the

definition

=

of

that

=

sup inf y'ER"l II ER"2

inf V

sup V'ER"l

0

inf V

hence thus

,

ItO

sup

ER 3y'ER"'

)

=

L=

that (y'°,y"°) E

closed.

is

U

,0 1,0 0 0 Remark. The above proof also shows that ,y ,y ) = E 0 ). Since is arbitrary, we have whenever (y', y") E L(e). By the way, the y") lI's =

function

takes finite values only.

=

Proof of Lemma 10.5. (a) Let V be an arbitrary compact convex subset of 0 and N N(V). For any two points e',e2 E V, pick up elements y'1 E L'(e') c N) [cf. the proof of Lemma 10.6]. Then (0', N), y"2 E L"(e2) C —

=

inf v

—

ER 2

,y

,,2

2

,y

= where

r

C

=

gi —

V and A is a finite upper

N) x r2(O"

Vx The function

sup y'ER"l

N).

—

over

bound of

Analogously,

is therefore locally Lipschitz continuous in 0.

A. e2

10. HOFF-TYPE FORMULAS FOR GLOBAL SOLUTIONS

140

E 0 and e E Rm, we find two sequences

(b) For any

of positive numbers convergent to zero such that

and

+ a'e) — k-4+øo

and + /3be) —

= urn

k-4+Oo

us arbitrarily take an element y'° E

and choose a sequence convergent to some with + abe) for k = 1,2,... (Such a E E choice is always possible in view of Lemma 10.6.] Then Let

p11k

+ abe)

inf

sup pER"

—

p

ER"2

ak

+ = where

for any y

E

+

y'°,

e)

(O,ak). A passage to the limit as k —, +00

E L,

o

), hence

shows

that

that sup

for some

E

Consequently,

inf

p"EL"(C°)

sup

(10.31)

It may dually be concluded that sup

inf

(10.32) )

Finally, combining (10.31)-(10.32) yields inf

sup

= =

= sup

inf

§10.3. HOFF-TYPE FORMULAS

0, e

for any

Rm. AU "inf" and "sup" are actually attained because the

nonempty set L(e°) = x

141

is compact (Lemma 10.6) and the mapping e) is continuous. The proof is thereby

x

(y', y")

y',

complete.

Proof of Theorem 10.4. By (G.I)-(G.III) and Lemma 10.3, (10.28) determines a continuous function ( = ((t, x, y) whose derivatives x, y),

x, y), .

. .

.x,

,

y),

x, y), .

exist and are continuous on the whole V x IRS' x

. .

x,

,

moreover, one may apply Lem-

def

def where m def = +fl2, e = (t,x), x(e,y) = V. Thus u = u(t, x) defined by (10.29) is locally Lipschitz continuous

ma 10.5 to the function x =

and 0

and directionally differentiable in V, with two quantities:

((i

—

x) equal to either of the following

(ii,

—

L'(t,x)

Here, e 0,

L"(t,x)

Rn2

{y"

:

{y' E Rfli

:

inf

y ER 2

((t,x,y',y") = u(i,x)}

sup C(t,x,y',y") = u(t,x)}

0 (Lemma 10.6).

But, according to Radamacher's theorem, U = u(t, x) is totally differentiable at any point outside a null set Q C V. It follows that = Ou(t,x)/Ot = hence, by (10.33), that Ou

—(t,x)= at

=

.

nun

max

p"€L"(t,z)

nun

max i,"EL"(t,z)

I I

cf

(s—) t

t

t

•fx—y\

i—i t / ".

\

Or \

t

t

L"(t,z) so

for (t,z) E V\Q. This would imply the existence of an element that t

/S'—Y'

—

t

'

t

t t

t

oak.

t

'

t

10. HOFF-TYPE FORMULAS FOR GLOBAL SOLUTIONS

142

for all y' E L'(i,x). Analogously, it can be deduced from (10.34) that Ott

—(t x)= Ot

.

max mm p'EL'(t,z) p'EL"(t,x)

=

for (t,x)

I •ix—y\

/x—y Ofix—y

I

/z—y Of*,x_y


max <1 I—J-(———,——I—--— \ t Oz \ t p"EL"(t,z) I \ t /

ruin

L'(t,x) with

D\Q, and that there exists

Ou

•

—

— y"\

/ x'

Of

—

— y"

—

'

/z"—y"

\ for all y"

' Oz"

t

"

t

t

"(10.36)

V'(t,x). Finally, by (10.35)-( 10.36), we get

(t,x)

V\Q, y

(y',y")

L'(t,x) x L"(t,x). In the same

L(t,x)

manner one may see that

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

V\Q, y = (y',y")

L(t,r).

Now, given any (t,x) 'P\Q, we pick up some yE L(t,x), then use (10.22) and (10.37)-(10.38) to obtain Ou(t,x)/Ot = —f(Ou(t,x)/Ox). The equation (10.26) is thus satisfied almost everywhere in V. As the next step, we claim that

limsup for each fixed x0 C

{

Indeed, let us first take a sequence (x'°,x"°) Rn1 x limsup u(t,x) = V converging to (0,x'°,x"°) such that and second choose arbitrary points

lim yk

L(t", z?k, ru1b)

Then it will be shown that y" —4 the sequence

(10.39)

for x

k = 1,2,...

On the contrary, since

is bounded (Lemma 10.6), we can suppose without loss of

§10.3. HOPF-TYPE FORMULAS

generality that y"

y0

d

143

x Rn2 with y0

z°. It is clear

from (10.28)-(10.29) and the above definition of L = L(i,x) that u(ik,rik,xutk) =

inf

y" ER"3 .

risk) + tk

,

u(t,x1k,xh11) = sup ((tk,xsk,xhIk,y1,yutk)

o")

(10.40)

c(jkx?kxiskxskyisk)

y'ER"l

+ jk .f* (o',

(10.41)

We need only consider the following two cases.

r'0. Then (10.4) and (10.40) show that

Case 1: y'°

*

Jim

u(tk,xuhl,xIilh)

Jim

k-4+oo

So,

implies

(10.41)

lirn tnhf*(Ot,(xhlk

—

((x ,k

I(r' =

—oo,

,k

k

,0 ii )

I.

—

y' )/t"I

=

a contradiction with

(10.3*). Case

y"°

x"0. Analogously, (10.3)

and

(10.41) show that

,k risk)

lim u(tk,

= +00,

hence that Jim i'f((z" — ysk)/tk 0") = +00, which contradicts (10.4). l,-++oo

The contradictions, which we have got in both Cases 1 and 2, prove that urn = x0. Therefore, a passage to the limit as k -+ +00 in (10.40)-(10.41) would give

(10.39). It may dually

be concluded that

liminf for

each

fixed r° = (z'°,x"°)

extension

over the whole

R" x

satisfying

Thus u = u(t,x) has a continuous

(10.27). The proof is thereby complete.

0

10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS

144

Remark 1. Hypothesis (G.II) says that a "saddle-point" (y',y") E L(t,s) of the function R" x ((t, x, y) [with respect to maximizing over p = (y', y") and minimizing over exists. If f = f(p', p") has a special d.c. representation

f(p',p")

92(P) — 91(p')

on

x

then we can use the "index of non-convexity" and a classical minimax theorem [7] to give sufficient conditions for (G.II) to hold. (See [12] concerning this question.)

Remark 2. If nj = 0, (10.29) gives the Hopf formula (9.4) and the formula (9.29). If n2 = 0, the Hopf formula for concave Hamiltonians [64] will also be obtained. Corollary 1O.T. Under Hypotheses (G.I)-(G.II), suppose that lan inf

Ix"I—*+oo

IX

lim sup

(10.42)

I

çb(s'

<+00 locally unsformly in x E Rn2.

,

Ix

IZ'I-4+00

. > —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 <

whenever

Ix°I

r, Iy'I

N.

§10.3. HOPF-TYPE FORMULAS

Finally, by (10.4), we certainly find a positive number c*(z'

'

/ 2(q

Putting

if(t,x)

—

A)

145

with the property that

as

ii.

+ v))1 we therefore conclude from (10.28) that,

V and Iy'I

N(V), then

inf C(t,x,y',w") <((t,x,y',x") = w€R2 + f*((xlyS)/t,0h1) [A

[A +

- A).

.

—

t

lx'—y'l]

y'I

(10.45)

Iy'I

because I(x' — y')/tI [r(1 + ii) — r]/r = ii, — y'I/Iy'I (Iy'I — r)/ly'I 1/2. Now, (10.44)-(10.45) show that the second condition in (G.III) holds. Analogously for the first one. Thus the corollary follows from Theorem 10.4. 0

Chapter 11 Global Semiclassical Solutions

of First-Order Partial Differential Equations §11.1.

Introduction

The present chapter is in principle a continuation of the previous three. However, it was actually originated in the following problem posed by S.N. Kruzhkov [931. Let a smooth [i.e., of class C'J function w = w(t, x) satisfy in the strip [0, T} x the inequality < NI&,(t,x)/Oxj,

N=

const.

0,

and the initial condition on

{i=0, rER1}.

Then it is easy to show (cf. Haar's Theorem 1.5) that w(t, x) the Cauchy problem for the first-order nonlinear equation

0 in Hr. Therefore,

+

f = f(p) is of class C1(R1), cannot have more than one solution in 11T, say, in the class of smooth functions with bounded derivatives. As Kruzhkov already remarked, the same conclusion may be drawn without appeal to the differentiability of w = w(t,x) (reap. of the solution u = u(t,x)) or the validity of (11.1) (resp. of the equation) at the points in any given finite union of straight lines {t = const., x E R} C Hi'. The following question arises naturally: to what extent can the condition on the smoothness of w = w(t, x) and on the validity of inequality (11.1) in the entire strip 11T be weakened? For example, the Cauchy problem for the equation Ou/Ot + (Ou/Ox)2 = 0 with the zero initial condition u(0, z) 0 has a continuum of piecewise smooth global solutions, such as = const. 0. Note that each function u = ua(i,x) min{O,alsI — a2t},

§11.1. INTRODUCTION

147

satisfies the corresponding inequality almost ev= erywhere in HT• Therefore, it is interesting to find intermediate classes (as wide as possible) between C'(flr) and Lip((0,T) x R'), in which only the zero function can simultaneously satisfy (11.1) and (11.2). These questions can be generalized to the multi-dimensional case.

The study of this problem suggests that we should single out the widest class between the class of continuously differentiable functions and the class of Lipschitz continuous functions in which the Cauchy problem for a first-order nonlinear partial differential equation has a unique global solution.

We shall assume 0 < T < +m (though most results here still hold in case and continue using the notation of Chapters 8-9. Our discussions in this T= chapter make an appeal to Theorems 8.1,8.8,8.11, and 8.12. The condition on the validity of inequality (8.5) is clearly much weaker than that of (11.1) in the entire domains of the corresponding functions under consideration. Moreover, it should be noted (see §8.1) that

C Lip([0,T) x The smoothness requirement on functions in is really weak enough: roughly speaking, these functions need only be absolutely continuous in time variable. By Chapter 8, the class V(QT) would be nominated as best candidate for our discussion concerning the above questions for the Cauchy problem

ôu/ôt+f(t,x,u,Ou/Ox) =0 in on

{t=0,

(11.3) (11.4)

We therefore arrive at the following definition of generalized solutions:

Definition 1. A function u = u(t,x)

in is called a global semiclassical solution to (ll.3)-(11.4) if it satisfies (11.3) for all x E JR" and almost all 2 E (0,T)

and if u(0,x) =4(x) for allx E The above definition allows us to deal with the case of time-measurable Hamil-

tonians f = f(t, x, u,p). The next section is devoted to the uniqueness of global semiclassical solutions in this case. Further, an answer to Kruzhkov's problem will be given. Section §11.3 discusses the existence theorems, whose proofs are much based on the results of Chapter 9.

148

11. GLOBAL SEMICLASSICAL SOLUTIONS

Most of the material presented here was published in [1411-11431 and 11501-1151].

The reader is referred to §2.3 and the references therein for the global existence of classical solutions.

§11.2. Uniqueness of global semiclassical solutions to the Cauchy problem The present section deals with stability questions concerning global semiclassical solutions of the Cauchy problem (11.3)-(11.4). Here, the initial data = is a given continuous function on The Hamiltonian I = 1(1, x, u, p) is always assumed to be measurable in t E (0,T) and continuous in (x,u,p) E x x The proof of the following uniqueness criterion is immediate from Theorem 8.1.

Theorem 11.1. Suppose that f = f(t,x,u,p) satisfies the follotinng condition: there exist a nonnegative function p = p(x) locally bounded on function £(t) in L1(0,T) such that

f(t, x,u,p) — f(t, x,

< £(t) [(1 + xI)Ip — qi + p(x)]u

and a nonnegative

— vi]

(11.5)

for almost every t E (0,T) and for all (x,u,p),(x,v,q) E x x If ui = u1 (t, x) and u3 = u2(t, x) are global semiclassical solutions to the Cauchy problem (11.3)-(11.4), then uj(t,x) u2(t,x) in 11T• Remark. Condition (11.5) is satisfied if and only if for some positive function = £(t) in L1(0,T), the function x R1 x

3 (t,x,u,p)

f(t,x,u,p)/[t(t)(1 + x])]

is Lipschitz continuous with respect to p uniformly in (t, x, u) E x R', and is Lipschitz continuous with respect to u uniformly in (t,x,p) E (0,T) xX x for every compact set X C R't [i.e., uniformly globally in (t,p) and locally in x]. A useful uniqueness criterion for global semiclassical solutions with essentially bounded derivatives is given by the next sharpening.

Theorem 11.2. Suppose that I = f(t, z, u,p) satisfies the following condition: for any compact sets K1 C R1, K2 C there exist a nonnegative function 42 = 42(t) in L'(0,T) and a nonnegative function PK1K2 = PK1K2(X) locally bounded

§11.2. UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS

and

on R° such that (11.5) with

149

p, respectively, holds for

xK2. Ifui = ui(t,z)

almost everyt E (O,T) and for all (x,u,p),(x,v,q) E

and u2 = u2 (t, x) are global semwlassscal solutions to the problem (11 .3)-( 11.4) with

<+00 (j = 1,2),

(t,z)Eflr

then uj(t,x)

u2(t,x) in

Remark. 1ff = f(t,p) depends only on t,p and is of class C1 on [0, T] x R't, then the condition of Theorem 11.2 is satisfied. In this case, Theorem 11.2 solves the problem of Kruzhkov (see Corollaxy 11.4 later). To prove Theorem 11.2 we need the following:

Lemma 11.3. Let

be a locally Lipschitz continuous function of x on = If it is differentiable in the whole then Oil,

= sup

zER

zER

Oily

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

X3

Proof of Lemma 11.3. Fix any i E {1,...,n}. It suffices to treat the case when def

= esssup

Oil,

UXj

<+m.

Let us write x = (x',z1) instead of x = x

where

, def

=

Then for almost all [with respect to the (n — 1)-dimensional Lebesgue measurel x' we have: <s1. —(x,.) L°'(R') —

Ox1

Since

the function il'(x',.) is absolutely continuous on each bounded segment, it

follows that — I

for almost all [with respect to the (n — 1)-dimensional Lebesgue measurej x'

and for all conclude that

E R1. From the continuity of il' = il'(x) and from (11.6), we —

—

11. GLOBAL SEMICLASSICAL SOLUTIONS

150

E R'1. Therefore,

for all

OtI,

<sj

0

This proves the lemma.

for all x

Proof of Theorem 11.2. According to the definition of

Lemma 11.3

shows that sup

ouj

for almost all t that

=

esssup rER"

ôti,

(j =

1,2;

= 1,... ,n)

(0, T). Taking the essential supremum over t E (0, T), we find esssup sup

ouj

= esssup

auj

(t,z)Eflr

zER

Consequently, by assumption,

j r def = rnaxesssup sup —(t,x) <+00. X

(11.7)

t€(0,T)

LC

Let X" be as in (8.22); K3

T' E (0, T), we consider the sequence

pk

x

For an arbitrarily fixed

£(.)

of the following paraflelepipeds:

= {(t,x) : 0< t
Continue using (8.23). Then the function = p(x) given by (8.24) is locally bounded on We now consider the function u = u(t,x) ui(t,x) — u2(t,x). Of course, u(O,x) 0. Moreover, in view of (11.7), the hypothesis of the theorem implies =

f(t, x, ui(t, x),

< t(t) .

[(1

+

z)) — —

f(i, x, u2(t, x), + ,z(x)Iui(t,x) — u2(t,x)I]

= £(t) . [(1 + x

u(t, x)

0 in

for almost all t E (0,T'). Therefore, Theorem 8.1 shows that (0, T) is arbitrarily chosen, the proof is complete. 0

Since T'

§11.2. UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS

151

Corollary 11.4. Let I = f(t,x,u,p) be measurable zn i E (O,T). contmuous in .r E R", and differentiable in (u,p) E R1 x

such that, for any compact set

K C R", the function

£K =€K(t)1+

sup

xK '-'P

is Lebesgue integrable on (0, T), and the function deC

1'K = i/K(x,u) = esssupsup

1

*€(O,T) pEK

x JR'. If ui = ui(t,x) and u2 = u2(t,x) are global semiclassical solutions to the Cauchy problem (11.3)-(11.4) with

is locally bounded on

('X

then uj(t,x)

Proof. Let

<+00 (.7 =

1,2),

in 11T• us

introduce the notation

,,K3(x)_

sup vK2(x,u)

uEK1

for any convex compact sets K1 c R1, K2 C R'. Then it is easy to check that (11.5) with 42 and PK,,K2 in place of€ and p, respectively, holds for almost every

t E (O,T) and for all (x,u,p), (x,v,q) E JR" x K1 x K2. The corollary thereby

0

follows from Theorem 11.2.

We leave it to the reader to prove the following criterion of continuous dependence on initial data for global seiniclassica.l solutions.

Theorem 11.5. Suppose f = f(t,x,u,p) satisfies (11.5). Let u3 = u,(t,x) (j = 1,2) be global semiclassical solutions to (11.3) with

on where

=

(j =

{t=0,

1,2) are given functions continuous on JR". Then

Iui(t,z) - u2(t,x)l <exp[C(x)Je(r)dTJ. £(r)dr—1

-

11. GLOBAL SEMICLASSICAL SOLUTIONS

C(x) being defined in (8.7).

Remark 1. The example in Remark 2 following Theorem 8.1 shows that the Lipschitz continuity of functions in the class V(Qr) also plays an essential role in the definition of global semiclassical solutions. The zero solution aside, this example gave no other global semiclassical solution to the Cauchy problem

Ou/Ot=O in Q1,

u(O,x)=O on {t=O, Remark 2. Consider the Cauchy problem Ou/Oi + (au/Ox)2 = 0

in

{0
u(0,x)=0 on {t=0, x€R1}. x )

0

Figure 11.1

By definition, if u = u(t, x) is a global semiclassical solution to the problem, then for almost every t (0, T), the function u(t,.) is differentiable on R'. Obviously,

§11.2. UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS

153

(1 1.8)-( 11.9) has a continuum of piecewise smooth global solutions, such as (see Figure 11.1) def

=

UA,e =

if

10

t
if T— e

— A2(t — T+e)}

<e <

T. For A > 0 the differentiability of the function UA,g(i,.) e) and at x = 0] if and only if belongs to the interval (T — e, T), whose Lebesgue measure is precisely e (positive but as small = uo,1(t,x) [i.e., A = 0] is the unique as we please). Thus, the zero function where

0, 0

A

fails somewhere [at x = ±A(t — T +

t

global semiclassical solution to (11.8)-(11.9) in the class of functions with essentially bounded derivatives (cf. the remark following Theorem 11.2).

Remark 3. The requirement that £ = £(t) be Lebesgue integrable on (0, T) is also essential to the above uniqueness theorems. To see this, we consider the Cauchy problem

equation):

(even for a linear

lOu

Ou

u

1_

u(0,x)=0 on Here, for 4x)

1, the

'Tj'

in

rE R'

{t=0, XER1}.

24P)

Hazniltonian f = f(t,u,p)

—

clearly satisfies

But we can easily check that the present problem admits the

(11.5) with £(i)

following two global (semi)classica.l solutions:

u=uj(t,x) def = t and u=u2(t,x) def = i(l+e ). The

explanation for this event is that the function £ = £(i) in this case is not

Lebesgue integrable on any (0, T). (We £ = £(t) for (11.5) to hold.) By the use of Theorems generalized to the case

0u1/Ot

8.8, 8.11,

of the

continuous

on

and

=0

+

data

Each

on

= Hamiltonian

qS =

8.12,

integrable

the results in this section can

be

Cauchy problem for weakly-coupled systems:

u(0,x) = Here, the initial

could not choose a Lebesgue

in

fir

{t =

(i =

,m),

0, x

,4m(x)) is a given

(11.10)

(11.11)

vector function

= f1(t,x,u,p) is always assumed to be

11. GLOBAL SEMICLASSICAL SOLUTIONS

154

measurable in t E (0,T) and continuous in (x,u,p) E R7t x x the definition of global semiclassical solutions for the problem.

Definition 2. A vector function u = u(t,x)

First, we give

is called a global serniclas-

in

sical solution of (11.10)-(1l.11) if it satisfies (11.10) for all r E

t E (0,T) and if u(0,x) =

and almost all

for ails E

We can now formulate some stability results for global semiclassical solutions of

the problem (l1.l0)-(1l.11) and leave the proofs to the reader.

Theorem 11.6. Suppose f, =

(i = 1,... ,m) satisfy the conditions as follows: there exzst a nonnegative function p = p(x) locally bounded on and a nonnegative function e = e(t) in L1 (0, T) such that

x,v,

+p(s) k=i max

< £(t) . [(1 +

Iuk

(11.12)

x Rm < for almost every t E (0,T) and for all (x,u,p), (x,v,q) E (i = m). Let u = n(t, x), ii = x) be global semiclass*cal solutions to (11.10) with the following corresponding initial conditions:

u(0,x)=4(x), ii(0,x)=cf4x) where

=

x), q5 =

{t=0,

x) are given vector functwns continuous on R'. Then

<exp[C(x)f e(r)dr} x

—

x

C(s) be;ng defined

on

sup

max

.=1

m

(8.7).

Corollary 11.7. Suppose f, = (11.12).

(i = 1,... ,m) satisfy the conditions If ii = u(t,r) and v = v(t,x) are global semiclassical solutions to the

Cauchy problem (11.10)-(11.11), then u(t,x)

v(t,x) in ciT.

Theorem 11.8. Let (8.32) be a comparison equation. Suppose f1 = (i = 1 m) satisfy the following conditions: there exists a nonnegative function = £(t) in L1(0,T) such that s,u,p) —

< e(t)(1 + IsI)Ip — qi + p(t, max

Iuk — vkl)

§11.2. UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS

155

for almost every t E (O,T) and for all (x,u,p), (x,v,q) E x x (i = 1,. , m). If u = u(t, x) and v = v(t, x) are global semiclassical solutions to the .

.

Cauchy problem (11.1O)-(11.11), then u(i,x)

v(t,x) in fZT.

Theorem 11.9. Let (8.34) be a comparison equation. Suppose = f1(t,x,u,p) (i = 1,... m) satisfy the following condition,,: there exist a nonnegative functzon p = p(x) locally bounded on and a nonnegative function £ = e(t) in L1(O,T) ,

such that x, u,p)—f1(t, x, v,

£(t)

Iuk—vkl)] (11.13)

(O,T) and for all (x,u,p), (x,v,q) x x R" (z = 1, . , m). If u = u(t, x) and v = v(t, x) are global semiclassical solutions to the Cauchy problem (11.lO)-(11.11), then u(t,x) v(t,x) in

for almost every t . .

In the case of systems, a useful uniqueness criterion for global semiclassical solutions with essentially bounded derivatives is given by the next sharpening.

Theorem 11.10. Let (8.34) be a comparison equation. Suppose f6 = (i = 1, . ,m) satisfy the following conditions: for any compact sets K1 C R, K2 C there ex;st a nonnegative function £K2 = (t) in L' (0, T) and a nonnegative . .

such that (11.13) with and in place of€ and p, respectively, hold for almost every t (0,T) and for all (x,u,p), (x,v,q) x K1 x K2 (i = 1,... ,m). If u' = u1(t,z) and u2 = u2(t,x) are global semiclassical solutsons to the problem (11.10)-(11.11) with function PK1.K2 = PKI,K3(x) locally bounded on

max max

3=1,2 t=1

then u1(t,x) With

<+00,

m

tz2(t,x) in mw in (8.34) [cf. the remark preceding Proposition 8.91, Theorem

11.10 gives:

Corollary 11.11. Let in x

= f1(t,x,u,p) be measurable in t (O,T), continuous R't, and differentiable in (u,p) Rm x such that, for any compact set

K C R, the function = £K(t)

i + max

+

sup OP

11. GLOBAL SEMICLASSICAL SOLUTIONS

is Lebesgue ;ntcgrable on (O,T), and the function def

= vK(x,u) = k;max 1

esssup sup

Ii

m *E(O,T) pEK

x R". If u1 = u1(t,x) and u2 = u2(t,x) are global is locally bounded on semsclass;cal solutions to the Cauchy problem (11.1O)-(11.11) with <+00,

max• max

j1,2 t1 then u1(t,x)

u2(t,x) in cT.

§11.3. Existence theorems This section is devoted to the existence of global semiclassical solutions to the Cauchy problem for a single Hamilton-Jacobi equation:

ôu/Ot + f(t,Ou/Ox) = u(O,x) =

on

{t =

(11.14)

in

0

0,

x E R"},

(11.15)

with = a finite convex function on Explicit representation of such solutions will be obtained by the use of Chapter 9. For definiteness, let us rewrite

the two standing assumptions (E.I)-(E.II) of §9.2 on the Hamiltonian I = f(t,p) and initial function = in a specified alternative form (when T < +co] as follows. (The Fenchel conjugate function of = is here denoted by 4' = as

usual.)

(E.I) The Hamiltonian I = f(t,p) is continuous in {(i,p) t E (O,T)\G, p for some closed set C C [0, TI of Lebesgue measure 0. Moreover, to each N E T) such that (0, +co) there corresponds a function = gN(t) in :

sup f(t,p)(

for almost all t E (0,T).

IpIN

(E.H) For every compact subset V of [0,T) x

there exists a positive number

N(V) so that (p,x) — q4(p)

—

f f(r,p)dr < max{(q,x) —

—

jf(r,q)dr}

§11.3. EXISTENCE THEOREMS

157

> N(V).

whenever (t,z) E V,

As we have mentioned in Chapter 9, (E.I) implies the t-measurability and pIn addition, since 4' = 4'(x) is continuity of I = f(t,p) on {0
(p, x) — 4'(p) —

(11.16)

j f(r,p)dr.

0 and deal with the function u = u(t, z) determined by the formulas:

We shall use the notation E dom 4' and multifunction L = L(t,x) of (t,x)

[O,T) x

pEE

L(t, x) def

Lemma 9.3, wzthm = n+1,

{p E E :

z,p) = u(t, x)}.

def

= (t,x), 0 def = [0,T)xR andw

def

= shows that L = L(t,x) is a nonempty-valued, closed and locally bounded multifunction of (t, x) [0, T) x R". This multifunction is therefore upper semicontinuous. An analogue of Theorem 9.1 says that (11.17) gives a global solution to the Cauchy problem (11.14)-(11.15). Next, set

((0,T)\G) x

= {(t,x)

QT

:

t

G}

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

Theorem 11.12. Let 4' = 4'(x) be a finite convez function on R". Assume (E.I)(E.II). Suppose that the multifunction L = L(t,x) given by (11.18) is continuous in G being as in (E.I). Then (11.17) determines a global semiclassical solution to the problem (11.14)-(11.15).

Proof. According to the maximum theorem [8, Theorem 1.4.16), the formulas (9.13)-(9.14) show that the derivatives ôu/Ot, Ou/Oxi,..., exist and are continuous in (see also [128, Corollary 2.2)). Hence, the global solution u = u(t, x) is continuously differentiable in

conclude that u = u(t,x) belongs to

Since C is of Lebesgue measure 0, we

An argument similar to that in the

proof of Theorem 9.1 now gives us the validity of (11.14) in

C]

II. GLOBAL SEMICLASSICAL SOLUTIONS

158

all partial derivatives of the global solution Remark. It can he proved that in u = u(t, x) exist [(11.14) is then satisfied though the solution may fail to he differentiable] if and only if the multifunction L = L(t,x) is single-valued. See §13.2 later for the smoothness of v = u(t, x).

Corollary 11.13. Let 4 =

be a finite convex function on R?Z. Assume (E.I)-(E.II). Then (11.17) determines a global semiclassical solution to the problem (11.14)-(11.15) if one of the following two condztions holds: = (i) is strictly convex on its effective domain E (ii) f = f(t.p) is strictly p-convex; more precisely, it is strictly convex with respect to p on E for almost every fixed t E (0, T).

Proof. Each of (i)-(ii) implies the strict concavity of p = p(t,x,p) with respect to p on E for every (t, x) E [0, T) x It follows that the upper semicontinuous

0

multifunction L = L(t, x) is indeed single-valued, and hence continuous.

In case n =

1.

an existence result is established as follows: def

Theorem 11.14. Let = be a finite convex function on IR, E = domç5 and let = be of class C2 in mtE. Assume:

(i) The Hamiltonian f = f(t,p) belongs to C°(((0,T)\G) x

*

with

esssuplf(t,p°)[ <+00 iE(O,T)

for some p° E E and some closed set C C IR of Lebesgue measure 0; it is, moreover, twice continuously differentiable in p E mt E with

max{esssup sup tE(O,T) pEint E

Of (t,p)/Opj, esssup sup 102f(t,p)/0p21} <+m tE(O,T)

E

(0, +oo). (ii) To each compact subset V of [0, T) x IR there corresponds a positive number N(V) E

so that

x, p) <

max

x, q) whenever (t, x) E V, p E

pEE

p1

> N(V). Here

qIN(V)

is defined by (11.16) withn=1, (p,x)=px. (iii) For every (t,p) E ((0,T)\G) x intE, it holds that d24

(11.19)

§11.3. EXISTENCE THEOREMS

Then (11.17) determines a global semiclassical soltttson to the problem (11.14)(11.15) with n = 1.

Proof. We first note that E is a nonempty convex set in R', and is therefore an interval with end-points, say, a,b E [—oo, +oo], a < b. Consequently, [117, = (p) to Corollary 7.5.1] implies the continuity of the restriction of Condition (i) shows that I = f(t, p) is t-measurable and p-continuous on {O < Moreover, we conclude from this condition that (11.16) gives a t < T, p E continuous function = on [0,T) x R x with values in [—oo,+oo); it being actually finite when p E E and twice continuously differentiable in p E mtE with the derivatives

=

z

('ôf

d4 —

7—(p) —

= —

f

(r,p)dr.

(11.20)

Under Hypothesis (ii), the method of Lemma 9.3 proves that the multifunction

L = L(t,x) defined by (11.18) is nonempty-valued and upper semicontinuous on [0, T) x R. It may be shown that

ess sup sup If(t,p)I <+00 tE(O,T)

IpIN

for

all N E (0,+oo), hence that u = tt(t,x) is in Lip([0,T) x R) (cf. Proof of

Theorem 9.1).

Finally, according to (11.19)-(11.20), for each (t,x) E the maximum set = cannot contain more than one point. Therefore, L = L(t,x) is really continuous in Analysis similar to that in the proof of Theorems 9.1 and 11.12 shows that u = u(t,x) belongs to and that (9.13)-(9.15) [with hold in Thus (11.17) determines a global semiclassical solution to the n = 11

L(t,z)

0

problem (11.14)-(11.15).

• Corollary 11.15. Let = be a finite convex function on R, E def = and let be of class C2 in intE. Under Conditions (i)-(ii) of Theorem =

11.14, assume that

>0

(11.21)

11. GLOBAL SEMICLASSICAL SOLUTIONS

160

intE and almost all t (0,T). Then (11.17) determines a global semiclasszcal solution to the problem (ll.14)-(11.15) with a = 1.

for all p

Proof. Since (11.21) implies (11.19), the conclusion follows. We conclude this chapter with the following example and remark.

Example. Let J c

[0,1) be the Cantor set and w = w(t) be the Cantor ladder (see Remark 2 after the formulation of Theorem 8.1). Define def

g(t) = min{It—sI

s€J}.

Clearly, g = g(t) is Lipschitz continuous on (0, 1) (with Lipschitz constant 1), and hence differentiable almost everywhere in (0,1); its derivative dg/dt = dg(t)/di being Lebesgue measurable. Next, consider the Cauchy problem On

+

dg

.

Ou

=

0

.

in

{0 < t <1, XE R }, 1

on {t=0,rER1}. Then we may invoke Theorem 11.1 to deduce that the function u = u(t,x) is the only global semiclassical solution to tius problem. Further, this —g(t) + problem has no classical solution even in the small [i.e., even in the local sense].

Remark. By the contributions on viscosity solutions, the global existence and uniqueness of generalized solutions to first-order partial differential equations have been established almost completely. (Look at the short historical survey in §4.1; see also the historical remarks for equations of conservation law in Chapter 5.) We shall show later in Chapter 12 that each global semiclassical solution is also a minimax, hence a viscosity solution since the last two are equivalent. The a priori estimates in Theorems 11.5-11.6 are certainly of much interest from various view-points (as we have partly seen in this book) with regard to numerical application inclusive. However, these estimates could be expected to apply only to sufficiently regular generalized solutions as they could in the case of global semiclassical solutions.

Chapter 12 Minima.x ot Par•tial Ditterential Equations with Time-measurable Hamiltonians §12.1.

Introduction

We have seen in the previous chapters that the main theorems in the classical theory of first-order nonlinear partial differential equations are valid only locally, in sufficiently restricted domains. The example in the preface shows that the Cauchy problem might fail to possess a global C'-solution even if the Hamiltonian and initial data are analytic. Therefore, the need for introducing generalized or weak solutions has arisen in the theory of first-order nonlinear equations and its applications. Such solutions have been investigated by many mathematicians in the past 50-70 years ([21-14], [21], [32]-[33j, 1441-145], [51]-(521, [54J, [63]-[64], [87]-

The notion of global solutions in Chapter 9, which was probably first used by Hopf [64], affords the existence but not uniqueness question. Some other supplementary condition like the semi-concavity or semi-convexity condition (see Chapter 4) must be introduced for the uniqueness of a global solution (in the case of convex or concave Hamilton-Jacobi equations). In a similar situation, entropy condition (see §5.4) is needed for the uniqueness question of weak solutions. In recent years the development of the theory of generalized solutions has been based to a significant degree on the concept of viscosity solutions, which was introduced by Crandall and Lions. Within this theory, [92], [94], [951, [97]-[98], 1112], [1181).

uniqueness and existence theorems have been developed for various types of equations and boundary-value problems, and also some applications to control problems and differential games have been studied ([51,1101-120], [28], [35]-[391, [47]-[50], 167]1721, [79], [991-11011, [122]-[i231, [131]). The concept of viscosity solutions is motivat-

ed by the classical maximum principle which distinguishes it from other definitions of generalized solutions.

In the present chapter, we develop another approach that can be considered as a nonclassical characteristic method, according to which a generalized solution

12. MINIMAX SOLUTIONS

162

is assumed to be flow invariant with respect to the so-called characteristic inclusjons. This direction has been suggested by Subbotin in [124]-[125J, and leads to the notion of minimaz solutions. The above term originates from the theory of differential games. It is justified by permanent presence of mm-max operations in investigations of these solutions, including the well-known Hopf formulas (see (9.3)-(9.4)), and in investigations based on idempotent analysis, which have been implemented in recent years by Maslov, Kolokol'tsov, and Samborskii [821, [105j. To facilitate access to the topics from motives of the characteristic method, let us go back to and consider the following "terminal" Cauchy problem (data being given at the end point t = T of time interval):

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

{t = T, x

on

(12.2)

is a given C1-function on After a change of variables (r, x) (T — t,x), from (12.1)-( 12.2) we indeed get the usual Cauchy problem (data being given at the initial point r = 0 of time interval). However, for the construction in this chapter, it will be more convenient to consider the terminal condition (12.2). We first assume that the Hanultonian I = f(t, x,p) is of the form Here,

=

f(t,x,p) with a = a(t,x) (a1(t,x),... Assume, further, that

(a(t, x),p), an

(12.3)

function of class C1 on

a(t,x)I
(12.4)

for some constant L E (0, -4-no). The characteristic system (1 .3)-( 1.4) of the Cauchy problem (12.1)-( 12.2) in this homogeneous linearity case is reduced to the system

( dx I

(12.5)

dv —

Ldt

0,

with the condition

x(T) =

y,

v(T) = o(y).

It is known that the problem (12.5)-( 12.6) has uniquely a solution

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

(12.6)

§12.1. INTRODUCTION

163

on some interval (T — e,T1 for each y E R". Under the hypothesis (12.4), the characteristic curves x = x(t, y) can be extended over the whole segment 0 t T (cf. Appendix I). Moreover, an easy argument (adapted from [62, Chapter V, Peano's Theorem 3.1, Corollary 3.1 and its Remark]) shows that the Jacobian

Dr/Dy of the C1-mapping RTh 3 y i-+ H1(y) does not vanish for any y RTh and 0 < t < T. Hence Lemmas 2.3-2.5 imply that x = is a diffeomorphism

[0,T). Therefore, the inverse function y = y(t,x) is and u = u(t,z) def = v(t,y(t,x)) = o(y(t,x)) is the only global

from R" to itself for any t —

of class C on C1-solution of (12.1)-(12.2) in the homogeneous linearity case (12.3). The above definition of the solution u = u(t, x) is based on the structure of 1

characteristics. In such a homogeneous linearity case, the family {(t, x(t, y)) 0 < of characteristic curves covers all of fZT and no two of them could intersect in c17; in other words, there exists one unique characteristic curve passing a given point (t, x) E For a nonhomogeneous linear Hamilton-Jacobi equation :

Ou/Ot + (a(t,x),âu/Ox) +g(t,x) =

0,

(12.7)

with g = g(t, z) a function in C1 (fir), we can continue using the above family,

{(t,x(t,y))

:

0

t

of characteristic curves and prove that def

u = u(t,x) = a(y(t,x)) +

I j g(r,x(r,y(t,x)))dr

is a unique global C1-solution of (12.7) and (12.2). Now, turn to the general case. Consider the nonlinear equation (12.1), with some twice continuously differentiable functions. In I = f(t, x, p) and = this case, the characteristic system (1.10) may be reduced to 81 I

I

Of

We shall assume that for any y

(z=1,...,n), (128

(z=1

n).

there has uniquely a solution

(x,p) = (x(t, y), p(i, y))

12. MINIMAX SOLUTIONS

164

of (12.8) on the whole segment 0

x(T) =

y,

t

p1(T) =

T satisfying the condition

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

of characteristic Moreover, suppose that the family {(t, x(t, y)) : 0 < t < curves covers and that no two of them could intersect in QT. Assume, further, that the inverse y = y(t, x) is of class C2 on Qr. Then it can be claimed that

u = u(t,x)

+ —

y(t,

(r, x(r, y(t, x)),p(r, y(t, x)))] dr

is the only global C1-solution of the Cauchy problem (12.1)-(12.2) [cf. Theorem 2.71.

must note here that for a general nonlinear equation (12.1) the above assumptions are not always automatically satisfied as they are in the linearity case (12.3). Sufficient conditions for them to hold could also hardly be given explicitly because the structure of characteristic curves is then via a projection (z, p) '—i x x from into R". Without these assumptions the preceding method of finding a global C1-solution breaks down. A relevant solution, which is expected to exist, We

should be understood in some generalized sense; say, one needs to relax the smoothness condition on it as usual. Subbotin's method of defining a minimax solution is based on a construction in the theory of positional differential games. According to this theory, the value function of a differential game is characterized by the properties of being u-stable and v-stable simultaneously (see (84]-[86]). It is also known that at the points of differentiability the value function satisfies a first-order partial

differential equation (called the Isaacs-Belhnan equation). The u- and v-stability properties can be expressed in various ways and used to introduce the notion of mmimax solutions. We shall see more concretely in the next section that one of these

ways is, to some extent, a relaxation of the classical characteristic method: The ordinary differential equations (12.8) to determine the characteristics x = x(t, y) can be slightly separated into two characteristic differential inclusions (see (12.14)

and (12.16) later). The first-integral condition, which means that u = u(t,x) is constant along each characteristic curve x = x(t, y), is accordingly separated into two inequalities (see (12.17) and (12.19) in the definition of a minimax solution).

§12.2. DEFINITION OF MINIMAX SOLUTIONS

165

Our aim here is to extend Subbotin's notion (given in [11 and (124]) of minimax solutions of first-order partial differential equations with continuous Hamiltonians to the case of time-measurable Hainiltonians and present the uniqueness and existence theorems for such solutions. The results in this chapter are new even when restricted to the case of continuous Hamiltonians. Almost of them were published in [132][1331 and 1146].

The outline of the chapter is as follows. In Section §12.2 we give the definition of minimax solutions to the (terminal) Cauchy problem for a general nonlinear evolution partial differential equation with time-measurable Hamiltonian. We investigate some properties of multivalued mappings which play a decisive role in the definition of minimax solutions. Section §12.3 is devoted to the relations between minimax solutions and global (semi)classical ones. We prove that a global semiclassical solution is also a minimax solution, and conversely, a minimax solution satisfies the

equation in the classical sense at the points of differentiability. Further, in Section § 12.4 we discuss the invariance of the definition with respect to the choice of concrete multivalued mappings. In Section §12.5 we establish the main theorem of this chapter on the uniqueness and existence of minimax solutions. Finally, Section § 12.6 concerns some generalizations to the case of monotone systems of first-order partial differential equations. Our method is based on the theory of multifunctions and differential inclusions, and on a sharpening of a well-known theorem on the Lebesgue sets for functions with parameters. We also use an implicit version of = 1}, Gronwall's inequality. Throughout, 0 < T < +00, S {p E :

def

B,. = {p E R

:

0), B def = B1.

§12.2. Definition of minimax solutions 10

Formulation of the Cauchy problem

Let us consider the Cauchy problem of the form

=0 in QTdI{0
on

{t=T,

Assume that the terminal data c = o(z) is of class C0 on xRx tonian f = f(t,x,u,p) depends on (t,x,u,p) E properties.

(12.9)

(12.10)

and that the Hamilwith the following

12. MINIMAX SOLUTIONS

166

Conditions:

a)

a!) For almost every (in the sense of Lebesgue measure) fixed t E (O,T), the Rx function 3 (x,u,p) i—+ f(t,x,u,p) is continuous. a2) For each (x,u,p) E R" x R x S, the function (O,T) 3 t '—* f(t,x,u,p) is measurable.

b) For any bounded sets D c R" and E C R, there exists a function A1,,8 = AD,E(i) in L1(O,T) with

A1,,e(t)ix—x'i

(12.11)

for almost alit E (O,T) and for allx,x' ED, u E E, p ES. c) There exists a function £ = 1(t) in L1(o,T) such that

sup{If(t,x,u,p)

—

—

1(t). (1 +

—

for almost all t E (0, T) and for all (x, u)

p,q E B} SO (12.12)

:

x R.

d) For almost all t E (0,T) and for all (x,p)

f f(t,x,u,p)

is positively homogeneous in p E R"; i.e.,

f(t,x,u,s for almost alit

R

=

s

.

f(t,x,u,p) Vs 0

(12.13)

(0,T) and for all (x,u,p) E R" x R x S.

Note that Conditions a2), c), and e) together imply:

a3) For all (x,u,p)

Rx

the function t '—*

f(t,x,u,p)

zs Lebesgue

integrable on (0, T).

2° Differential inclusions for supersolutions and subsolutions In the present section and the next one we shall often use the notations:

F(t,x)

for (t,x)

.

(1

+ lxi) . B,

F,j(t, x, u, a)

{z E F(t,x)

:

(z, a) f(t, x,

FL(t, x, u,

{z E F(t,x)

:

(z,

QT, u

R, o,f3 E S.

u,

< f(t, x, u,

a)},

(12.14)

§12.2. DEFINITION OF MINIMAX SOLUTIONS

167

Remark. For any multifunction and any G(t,x) C E denote by the set of all absolutely continuous functions x = x(t) from [0, T) into which satisfy almost everywhere in (0, T) the differential inclusion

E G(t,x(i)) subject to the constraint We shall always assume = that G = G(t, x) is nonempty convex compact valued, measurable in t, upper semicontinuous in x, and that IG(t, x)l on

sup {IzI

:

z E G(t, x)} < c(t) . (1

+ rI)

(12.15)

with c = c(t) a function in L'(O,T). From [29, Theorems 11.20, 111.15, and

VI.13j (see also [40, Theorems 5.2 and 7.1]), it follows that XG(i,,x.) is then a nonempty compact subset of C([0,T],lfr). Now let U XQ(t,x), 0 is compact, then by Lemma 8.3, C Xo({0} x Br) with r x B,.), which (r° + 1)ffc(f)di — 1. The compactness of follows from the upper semicontinuity of the compact valued multifunction R" r Xo(0,x) c (see [40, Theorem 7.1] or [29, Theorems 11.25 and VI.13]), therefore implies that of C 11T• If

C [0, TI x

Now it can be seen that the multifunctions (1,x) x IR x S R xS

F(t,x)

C

(t,x,u,a) '-+

C

(i,z,u,8) '-4 FL(t,x,u,(3) C

in (12.14) are nonempty convex compact valued, measurable in i, upper sernicontinuous in x. (It can be shown that they are indeed continuous in x.) As was mentioned in the above, the sets def

u,cs) = Xpu(.,.u,a)(i*, xe), XL(:t.,

u,

def

=

are always nonempty and compact for all (tm, z,) E

(12.16)

x,) So we may conclude the

following:

Definition 1. A supersolution of Problem (12.9)-(12.10) is a finite lower semicontinuous function u = u(t, x) on which satisfies the condition sup QES

mm

[u(r,x(r)) — u(i,x)J <0

(12.17)

12. MINIMAX SOLUTIONS

168

for all 0
and also the condition VxERTh.

(12.18)

Definition 2. A subsolution of Problem (12.9)-( 12.10) is a finite upper semiconwhich satisfies the condition tinuous function u = u(t, x) on inf

for

[u(r,x(r)) — u(t,x)]

max

BES

>0

(12.19)

—

all 0 < t < r
u(T,x)a(z) YxER".

(12.20)

The sets of all supersolutions and subsolutions of (12.9)-( 12.10) will be denoted by Solu and SOIL, respectively.

Definition 3. A function u = u(t,x)

Soli. is called a minimaz solution

in SOIU

of the Cauchy problem (12.9)-( 12.10). 30

Further properties of F0 = Fu(t, x, u, a) and FL = FL(t, x, u, /3)

It will be shown (see [124, p. 16]) that

Fu(t,x,u,a)flFL(t,x,u,fl)$O In fact, it follows from (12.13) that f(t,x,u,0) =

if(t,r,u,p)i Therefore, if (a,/3) = z

then a =

1,

f(t,x,u,a)

.

a =

for 0,

(1 + lxi)

all a,/3E

S.

(12.21)

hence from (12.12) that Vp E S.

(12.22)

and

f(t,x,u,/3)

Further, let 0 < (a,/3) < 1. Setting e

.

/3

e Fu(t,x,u,a)fl FL(t,x,u,/3). (1

—

(a,/3)2)_h/2 . ((a,[3) a — /3), we

get lel

= 1,

(e,cx) = 0,

(e,/3) = —(i

(a,/3)2)h/2 < 0.

(12.23)

f(t,x,u,a) a + £(t). (1 + lxi) e and conclude from (12.22)(12.23) that lzi2 = [f(t, x, u, a)]2 + [e(t) (1 + lxi)]2 < (1 + lxl)12, i.e., that z E F(t,x). On the other hand, (12.23) implies (z,a) = f(t,x,u,a), thus Next, choose z

.

DEFINITION OF MINIMAX SOLUTIONS Z

E

169

Fu(t,x,u,a). In addition, since (a,/3) 0, it follows from (12.13) that (z,/3) =

f(t,x,u, (a,/3) a) + £(t) (1 + si) . (e,f3). But, according to (12.12) and (12.23), (a,/9) a

—

2 1/2 /3( = (1— (a,/9) ) = —(e,/3),

f(t, x, u, (a, 13) a) Therefore,

f(i, x, u, /3) — £(t) (1 + si) (e, /3).

(z,fl) < f(t,r,u,/3); i.e., z E FL(t,x,u,/3). Hence, z E Fu(t,x,u,a) fl

FL(t, s, u,fi). we

Finally, let (a,/3) <0. Setting F(t,x) z (1 + Iii) . (a — /3)/(a —/31, obtain (z, a) = £(t) . (1 + (1— (a,/3))h/2 2 £(t) . (1 + si), because Ia — /3( = — (a,/3))1/2 (as al = $1 = 1). Thus, (12.22) gives (z,a) 2 f(t,s,u,a);

i.e., z E Fu(t,s,u,a). Analogously, z E FL(t,s,u,fl), and in consequence, z E Fu(i,s,u,a)flFL(i,x,u,/3). The equality (12.21) is thereby proved (for almost all t E (0, T) and for all (x, u) E x RI. It follows that

f(t,s,u,p) =

sup

mm

(z,p) = mi

max

$ES zEPL

a€S

for almost all t E (0,T) and for all (x,u,p)

(z,p)

(12.24)

R x R". Indeed, let p €5. By

(12.14), we easily see that sup

mm

aES

(z,p)

mm

Moreover, by (12.21) and again (12.14), we also have mm

zEF0(*,x,v,a)

(z,p) <

mm

ZEFU(i,z,u,a)flFL

(z,p)

f(t,s,u,p)

for any a E S. Thus, in view of (12.13), the first equality in (12.24) is satisfied, no matter whether p E S or not. Similarly for the second one. From the monotonicity and continuity in u off = f(i,x,u,p), it may also be deduced that 1)

U>,,

= Fu(i,s,v,a),

fl

= FL(t,s,v,/3)

(12.25)

Another property (0,T) and for all a,j3 E S, V E R, 5 E of the multifunctions = and FL = FL(t,s,u,/3) is given in the for almost all t following.

12. MINIMAX SOLUTIONS

Proposition 12.1. One supliminf aES

inflimsup ö\O

has 1 —

inf

1

sup

—

I

mm

ft

:EFu(r,z(r),u,a)

I

max

ZEPL(r,z(r),u,.8)

Jt

for almost all t E (0, T) and for all x E

(z,p)dr>f(t,x,u,p), —

(12.26)

(z,p)dr
U E R, p E

The proof of Proposition 12.1 will be based on the next two lemmas, which are sharpenings of a well-known theorem on the Lebesgue sets (see [119, p. 158]) for functions with parameters.

Lemma 12.2. For any r E (0, +oo), there exists a set A(r) C (0, T) of Lebesgue measure 0 such that 1

f(r,x,u,p)dr—f(t,x,u,p)=0

for alit E Ac(r)

x E B,. C R", u

(12.27)

C R, p

E,.

Proof of Lemma 12.2. According to b) and e), we find a function A = A(t) deC = AB,,E,(t) in L1(0,T) and a null set Aj(r) C (O,T) so that f(r,x,u,p) — f(r,x',u,p)] <

.

— x'I

A(r)

(12.28)

(0,T)\Aj(r). By al), there is a rE for 8.11 X,X' E Br, u E E,., p E null set A2 C (0,T) such that the function Rx ? (x,u,p) f(t,x,u,p) is everywhere continuous for each t A2 = (0,T)\A2. In virtue of Lebesgue's theorem (see, for instance, [119, p. 158]), we see that for any g = g(t) in T) the set Leb(g) of all t E (0, T) satisfying 1

is of measure 7'. Let Q A(r)

Aj(r) U A2 U

be

g(r)dr—g(t)=0

the countable set of all rational real numbers and take

(f(,,'p)) fl

fl Leb(A))

DEFINITION OF MINIMAX SOLUTIONS

a3) and c)]. Then mes(A(r)) =

[cf.

The only point remaining is to prove RTh. To this end, let e > 0 be

0.

(12.27) for any fixed t Ac(r), x Br, u p an arbitrary number. Since t Ac(r) C Leb(A) fl

i

1

sup

£(r)dr,

I

—

ft

sup

we have t+S

1

I

—

A(r)dT

)

+ 1 <+oo. (12.29)

J

> 0 such that

there must be

Because t

171

If(i,x',u',p') — f(t,x,u,p)I < e13 whenever max{Ix — r'I, Izt — u'I,

p' so

that

<). Now choose

p—

E

x'I,

—

—

— p'I}

(k = 1,2),

t



+ i)]}.

e/[6M(r + 1)1, Since

(12.30)

we could find 5(e) >

where

(12.31)

0 with the following

property:

1

f

t+6

f(r,

k = 1,2. (12.32)

> 0, we have only to show that

Finally, setting 1

For k

< e/3

—

f

= 1,2 1

t+o

f(r,x,u,p)dr — f(i,x,u,p) <e

(12.33)

one writes t+6

f

f(r,x,u,p)dr — f(t,x,u,p) t+a

1

=

+ +

[f(r,z,u,p)—f(r,x',u,p')]dr

j 1

(12.34)

[f(r,x',u,p') —

f

—

f(t,

u, p)].

12. MINIMAX SOLUTIONS

The first summand in the right-hand side of (12.34) is estimated

Let (5E as 1

f

(+6

[f(r,x,u,p) — f(r,x',

< 1

f

Ix —

g+6

(+6

e

e

e

by (12.12)-(12.13), (12.28)-(12.29), (12.31) and by the choice of

The

third and fourth summands are estimated as (12.30) and (12.32) (cf. (12.31)). Therefore, taking k = 1 in (12.34), we get 1

f

f(r,x,u,p)dr — f(i,x,u,p) <3

C

=

e,

(12.35)

because the second suminand in the right-hand side of (12.34) is then nonpositive by Condition d). Analogously, letting k = 2 implies 1

f(r,x,u,p)dr—f(t,x,u,p)>

—3.

C

=

—e,

(12.36)

since the second summand in the right-hand side of (12.34) is then nonnegative. Combining (12.35) and (12.36) gives (12.33), which completes the proof. 0

Lemma 12.3. There exists a set A C (0, T) of Lebesgue measure 0 such that 1

(+6

f(r,x(r),u,p)dr = f(t,x,u,p) uniformly in x(.) E Xp(t,x)

(12.37)

for alliE Ac,x where the sets A(k) are Proof of Lemma 12.3. It suffices to take A constructed in Lemma 12.2 and its proof. In fact, it follows easily from (12.27)because the (12.29) that (12.37) must hold for all i E Ac, x E u E R, p E function family Xp(t,z) is uniformly bounded and is equicontinuous at t. 0

Proof of Proposition 12.1. The integrals in (12.26) exist (for reasons explained just prior to (12.16); cf. also Remark 4 in Section §12.4 later). By the positive homogeneity in p off = f(t,x,u,p) (see (12.13)) and of(z,p), we need only prove

§12.3. RELATIONS WITH SEMICLASSICAL SOLUTIONS

(12.26) for almost all i (O,T) and for all x E to (12.14), one then obtains

U E

173

R, p E S. Indeed, according

fj supliminf aES J\O

ml

1 —

I

mm

zEFrj(r,g(r),u,a)

(z,p)dr> —

1

> liminf in! — I — i\O z(.)EX,(i,z)S ft

= liminf 6\O

in! lim sup

PES

sup

—

I

ft

1

in!

—

max

ZEFL(r,xfr),u.P)

limsup

ft

f(r,x(r),u,p)dr, —

—

1

=limsup

t4J

I

(z,p)dr

(z,p)dr < 1

sup

.\O

mm

zEF0(r,zfr),u,p)

/

Jt

sup — / ö\O z(.)EXp(t,z)'5 Jt

max

ZEFL(r,r(r),u,p)

(z,p)dr

f(r,x(r),u,p)dr.

0

Therefore, by Lemma 12.3, the proof is complete.

Corollary 12.4. We have supliminf

in!

in! limsup

sup

aES ö\Ü ,SES

1 —

d dt

ft/

for almost alit E (0,T) and for alix E

di

f(i,x,u,p),

f(t,x,u,p)

U ER, p E R'.

Proof. The proof is immediate from (12.26).

0

§12.3. Relations with semiclassical solutions The following result says that the notion of minimax solutions indeed generalizes "correctly" that of classical ones.

Theorem 12.5. Assume a)-e). Then i) every global semiclassical solution u = u(t, x) of (12.9)-( 12.10) 30 that the gradient mapping x i—* Ou(t, r)/Ox is continuous for almost all t (0, T) must also be

a minimax solution of the same problem,

12. MINIMAX SOLUTIONS

174

ii) there ezists a null set A C (0, T) such that, at each point (t, x) E ((0, T)\A) x where a certain minimax solution of (12.9)-(12.1O) is differentiable, the equation (12.9) must be satisfied.

Proof. i) Let u = u(i,z) be a global semiclassical solution of Problem (12.9)(12.10) such that z Ou(t,z)/Ox is a continuous function for almost every x S, we define t E (0,T). Based on (12.14), for (t,z,ci) E

F°(t,x,a)

{z° E Fu(t,x,u(t,x),ci)

(z°,Ou(t,x)/Ox) =

mm

xEFrj

(z,Ou(t,x)/Ox)}.

(12.38)

(t,x) '-+ Ou(t,x)/Ox is measurable in t, continuous in x and the Since x, u, ci) is indeed measurable in t (0, T), continuous in multifunction Fu = (x,u) E RTh x R, it follows from [8, Theorems 8.2.8, 8.2.11) and Berge's maximum theorem [22, p. 123] that the nonempty convex compact valued multifunction

F° = F°(t,x,s) is measurable in t E (0,T) and upper semicontinuous in x E deC

Given any ci E S, a certain estimate of the form (12.15) holds for G(.,.) =

F (., .,o)

because F°(.,.,a) C F(.,.). Hence, as was mentioned in the Remark 2°, for arbitrarily fixed 0 t < r T, x we could find at least one element

By (12.24) and (12.38), it is then easy to see that the continuous function [0, TI

i—*

is Lipschitz on every segment [e, T—eJ

u

(0 < e < T/2) with the derivative =

+ +

=

0

almost everywhere in (0,T). It is therefore decreasing on [0,T]. Thus, according to (12.25), if a function x(.) on [0,T] is chosen so that and = for some z(2)() E z,u(t,z), ci), then = z(.) E

Moreover, u(r,x(r)) — u(t,z) = u(r,z(')(r)) — u(t,x(1)(t)) 0. Hence, by definition, u = u(t,x) is a supersolution of (12.9)-(12.10). Analogously, it is also a subsolution, and is therefore a minimax solution of the same problem.

ii) Take A C (0,T) to be a null set satisfying (12.37) of Lemma 12.3. By [119, p. 158], we may assume that N C with £ = 1(t) the function

§12.3. RELATIONS WITH SEMICLASSICAL SOLUTIONS

175

mentioned in Condition c). Let u = u(t, x) be a minimax solution, differentiable at some

AC x R't, of (12.9)-(12.10).

Firstly, choose a S and 0 < s R so that ôu(t(°),x(°))/Ox = s a. Since u = u(t,x) is in Solij, for any 5€ (0,T—t(°)), there exists an XS(.) such that

+

(12.39)

0.

+ 5)) —

Let

def

=

=

(0

+ 5) —

dxö

AC is a Lebesgue point oU = £(i) and the function family is uniformly bounded, one has

Because

+oo>C def =

£(r)(1+Ixo(r)I)dr

sup .SE(o,T_g(°))

sup

1

—IyoI. (12.40)

1(°)

of u = u(t, x),

In accordance with (12.40) and with the differentiability at it follows from (12.39) that

+

C

(t(°),x(°))) < (1 + C) .5.

limes = 0.

(12.41)

By the definitions (12.14) of the inultifunction Fri = Fu(t,x,u(t(°),x(°)),a) and (12.16) of the set Xu(t(°),x(°),u(t(°),x(°)),a), we now see that /

1

1

1 —

= hence,

by

f(r,xo(r),u(t(°),x(°)),a)dr,

(12.13),

(1 S

I

Ox

I

Ox

Jt(O)

This together with (12.37) and (12.41) implies

+ f(t(°),x'°),u(t(°),x(°)),

Ox

< a.

12. MINIMAX SOLUTIONS

176

Analogously, we have

0,

+ f(t(°),x(°),u(t(°),x(°)),

provided that u = u(t,x) is in SOiL. The equation (12.9) must therefore be satisfied 0 at (t(°),x(°)). Theorem 12.5 is completely proved.

§12.4. Invariance of definitions There is a certain indefiniteness in the definitions (12.14) of Fu = Fu(t,x,u,a) = £(t). However, and FL = FL(t,x,u,/3). For example, we have a large choice this indefiniteness does not influence the basic Definitions in §12.2. As will be shown in the remainder of the chapter, instead of (12.14) one can, in fact, use any = Fu(t,x,u,a) Iresp. FL = FL(t,x,u,13)] in the following common families of multifunctions.

Let P and Q be arbitrary nonempty sets. We consider any multifunctions

(t,x,u,a) '-* Ftj(t,x,u,a) C R",

xRxP

fly. x R x Q 3 (t,x,u,f3)

FL(t,x,u,13) C

satisfying the general conditions below.

(i) The multifunctions Fu = Fo(t,x,u,a) and FL = FL(t,x,u,/3) are nonempty convex compact valued, measurable in t and upper semicontinuous in x. Moreover, x, u, cx) U FL(t, x, u, (3) C F(t, x)

for almost all t E (0, T) and for all x E

where c = c(t) in L' (0, T) is independent

of a, (3, u.

(ii) For almost alit

(0,T) and for dlx

fl Fu(t,x,u,a) = Fu(t,x,v,a), (iii) For almost all t

a

P,

Q, V ER one has

11 F,(t,x,u,f3) =FL(t,x,v,/3).

(0,T) and for all (x,u,p) E

R x R" the following

inequalitzes hold:

sup cxEP

mm

(z,p)
max

— PEQ ZEFL

(z,p),

(12.42)

§12.4. INVARIANCE OF DEFINITIONS

sup liminf

inf

inf limsup 5\o

sup

aEP

f(t,x,u,p),

dt

rt+o

/

177

f(t,x,u,p),

at

where, still as in (12.16), we let def

= XPu(.,.,u,a)(t,X),

def

XL(t,x,u,fJ) =

The family of all multifunctions = (resp. FL = FL(t,x,u,f3)) satisfying Conditions (i)-(iii) will be denoted by .Iu(f) (resp. FL(f)). If the Hainiltonian f = f(t,x,u,p) satisfies a)-e), then .Fu(f) 0, 0. In fact, under such hypotheses, we can choose any P, Q C with the property that

cEP,sO}={s./3

{s.a

and then use (12.14) to define a concrete pair of multifunctions = This pair would satisfy (i)-(iii). (See Part 30 of §12.2 for and FL = FL(t, x, u, the

case where Pdot = 5, Q def = S.)

Remark 1. By the Hahn-Banach theorem, it follows from (i) and (12.42) that

Fu(t,x,u,Q)fl Fj(t,x,u,/3) for almost all t E (0, T) and for all

E F,

E

(12.44)

0

Q, (x, u) E

x R.

Remark 2. Condition (12.43) holds if Fu = Fu(t,r,u,a) and FL = FL(t,x,u,/3) satisfy the inequalities 1

supliminf

in!

—

in! limsup flEQ 6\o

sup

—

aEP

1

p4+6

I

mm

(z,p)dr>f(t,x,u,p),

/

max

(z,p)dT
fg

for almost all t (0, T) and for all x E

(12.45)

U E R, p E

Remark 3. In the case where I = f(t,x,u,p) is a continuous function of its arguments, Subbotin [124] and Adiatullina and Subbotin [1] do not assume (12.43)

for Fu = Fu(t,x,u,a), FL = FL(t,x,u,P); they assume the upper semicontinuity in (t, x) of these niultifunctions (instead of the measurability in t and upper

12. MINIMAX SOLUTIONS

178

semicontinuity in x as we do) and assume the equalities in (12.42). However, it can be proved that (12.43) holds then. In fact, under such assumptions, for all there exists a E P so that e > 0, (t, x) E QT, U E R, p E mm

zEFu (*,z,u,a)

(z,p) f(t,x,u,p) —

e/2.

Given any x(.) Xp(i,x), the multifunction (O,T) 3 r Fu(r,x(r),u,a) is upper semicontinuous. From this and the maximum theorem, it follows that the (z,p) is lower semicontinuous. Hence, when function (0,T) r '—* miii 5 > 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


1

inf

—

inf

.j

ö\O z( .)EXp (t,)


(z,p)dr<

1

I

ft

j

(JEXF(t,z)o Jt

sup

mm

aEPZE Ptj (r,z(r),u,Q)

(z,p)dr

f(r,x(r),u,p)dr=f(t,x,u,p),

§12.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS

inflimsup

sup

LNo

1 —

,t+ö

I

(z,p)dr max zEFt(r,x(r),u,$) sup

—

limsup

sup

—

S\O

for almost all t E (0,T) and for all x E

For any

1

lim sup

=

in

1

p*+6

(z,p)dr

I

inf

/

f(r,x(r),u,p)dr=f(t,x,u,p)

Jt

u E R, p

max

R".

FL = FL(t,x,u,13) in

we have:

Definition 4. The set Solu(Fu) of supersolutions (relative to Fo) of the Cauchy problem (12.9)-(12.10) is defined to be the set of all finite lower semicontinuous functions u = u(t, r) on 11T which satisfy (12.18) and the condition sup

mm

aEP

[u(r,x(r)) — u(t,x)1

0

(12.46)

for any0
flEQ

max

[u(r,x(r)) —

0

(12.47)

and In Section 5 we shall prove that, for any Fu = Fu(t,x,u,a) in consists of a uFL = FL(t,x,U,/3) in IL(f), the intersection Solu(Fu) fl nique function, which is independent of Fij = Fu(t, x, u, a) and FL = FL(t, x, ti, f3). Therefore, in complete concord with the previous definitions, we may also give the following.

Definition 6. A function u = u(t,x) in Solu(Ftj) fl Soli.(Fz.) is called a minimax solution (relative to

and FL) of the Cauchy problem (12.9)-(12.10).

§12.5. Uniqueness and existence of minimax solutions The aim of this section is to establish the following main theorem.

12. MINIMAX SOLUTIONS

180

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

Theorem 12.6. Let a = a(.r) be continuous on

all Gonditzons a)-e). Then there exists one unique minimax solution of the Gauchy problem (12.9)-(12.1O).

the proof of Theorem 12.6 we need to make some preparations. Given any locally bounded function n = u(t,x) on clT, we define: For

ii

—

(t,x) def = — limiuf .

u(r,y) and

.

def u + (t,x) =

limsup

u(r,y)

Obviously, u = u(t.x) [resp. u+ = u+(t,x)] is then the largest lower seinicontinuous [resp. the smallest upper semicontinuous] function on ciT

for (t,x) E

which is at most [resp. at least] u = u(t,x). From now on, fix any =

in .Tu(f), FL = FL(t,x,u,Ø) in

FL(f) and let c = c(t), F = F(t,x) be as in Condition (i) relative to these Fu r E [O.T], u E R, o E F, /i E For (t.,x.) e Fu(t,x,u,o), FL = Q we set

D(t., x., r) DL(t,,

x.)},

{x(r)

:

x(.) E

r,ce) = {x(r)

:

r(.) E .Atj(t.,x,,u, o)},

r,

:

def

{.r(r)

x(.) E AL(t.,

u,

and shall be concerned in the sequel with the following two functions:

(t,x)

:

(t,x) i-+

:

yE D(t,x,T)},

124

yE D(t,x,T)}.

Lemma 12.7.

is a supersolution (relatwe to Pb) of the C'auchy = = = i,1'(1)(t,x), problem (12.9)-(12.10). Moreover, both the functions o(x). are continuous in 11T with one can easily prove Proof of Lemma 12.7. By the Remark in Part 2° of D(t, x, T) C (t, x) the upper sernicontinuity of the multifunction ciT On the other hand, the lower semicontinuity of this nonempty compact valued

multifunction follows from Filippov's theorem (see [8, Theorem 10.4.1] and [40, = = Lemma 8.3]). Thus, by the maximum theorem we see that satisfies = are in C(ciT). All that is left to show is that s E P and To this end, let 0 < t < r < T, x E (12.46) [with u replaced by x(.) E

C

Since D(r,x(r),T) C D(t,x,T), (12.48)

§12.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS

<

implies

181

in place of u holds. Lemma

i.e., (12.46) with

0

12.7 is proved.

Lemma 12.8. If u = u(t,x) is a locally bounded fimction on

satisfying (12.18)

and the condition inf

sup

aEP x(.)EXu (t,z,u(t,x)+e,a)

[u(r,x(r)) — u(t,x)] < e

(12.49)

—

r E [t,T], then u = u(t,x) is a supersolution (relative

for alle > 0, (t,x) E

to Fu) of the Cauchy problem (12.9)-(12.1O).

> ... > e(h) 0. Then there converging to (t,r) so that u(T,x) =

Proof of Lemma 12.8. Let x e

>

exists a sequence C urn u(t(k),x(k)). By (12.49), for any fixed a E P one can find functions k-+-foo

+ e(k),a)

E

with

u(t Since

def

= {(T, x), (t

,

x

),... , (t

Ic = ,x

), . . . } is

1,2,...

(12.50)

compact and since

C

we may (by the remark in Part 20 of §12.2) assume that x(k)(.) x(.) Xp(T, x) in Letting Ic -+ +00 in (12.50), we get a(x) < u(T,x). Thus, = u(t,x) satisfies (12.18) [with u replaced by u]. Fix now 0 < t < r ( T, x E R", a P. Then there exists a sequence

C [0,r) x

converging to (t,x) so that tC(t,x) = lim

We choose

> e(k) and find, by (12.49), functions x(k)(.) E 6(k),a) such that

u(r,

z(k)) +

For every e > 0, since lim [u(t(k), x(k))+e(k)] = k-4+oo

(ii) that x(k)(.)

Ic =

>

+

1,2,...

(12.51)

u(t, x), it follows from Condition

Xu(t(k),z(k),u(t,x) + e,a) when k is large enough. Therefore,

12. MINIMAX SOLUTIONS

182

again by (ii) and the Remark in Part 20 of §12.2, we may assume that x(")(.) —* x(.) E flXu(t,x,u(t,x) +e,a) = Xu(t,x,u(t,x),&). Letting k +00 in

(12.51), we get u(r,x(r))
in place of uJ; i.e., by definition, it is a supersolution (relative to F0) of the

0

Cauchy problem (12.9)-(12.10). The proof is then complete. Now let us define a function A = A(t, r) on 11T by the following formula: A(t,x) Since Du(t, x, u, T,

:

u(.,.)

D(t, x, T) for all (t, x) E 11r, u E R,

C

that u(t,x)

(12.52)

Solu(Frj)}.

P, it is easily seen

for all supersolutions (relative to F0) u = u(t,x) of the Cauchy problem (12.9)-(12.10). Therefore, A(t,x) on IT [by Lemma 12.7, = is a supersolution (relative to F0) of = the Cauchy problem (12.9)-(12.10)1. This together with the continuity of = implies that on

A(t, x) A(t, x)

x)

x)

(12.53)

Moreover, we have:

on

Lemma 12.9. A = A(t,x) is a lower semicontinuous supersolution (relative to F0) of the Cauchy problem (12.9)-(12.10).

Proof of Lemma 12.9. By (12.52)-(12.53) we need only show that A =

A(t,x)

is a supersolution (relative to Pb) of the Cauchy problem (12.9)-( 12.10). Moreover, since A = A(t,x) is a lower semicontinuous function equal to = i(x) on

{t =

T, x

(cf.

(12.53) and Lemma 12.7), all that is left to prove is that

(12.46) with u replaced by A 0.

holds.

Choose a sequence C [0,r) x converging to (t,x) so that = lim A(t(k),x(k)). Then by (12.52), there exists a sequence of functions

A(t,z)

= u(k)(t,x) in Solu(Fu) with s(")) <

+

(12.54)

hence, by (12.46) and (12.52), there exist x(k)(.) such that

A(r,

< A(r, x(k)(r)) <

< u(k)(t(k),

k

= 1,2,. (12.55)

§12.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS

Since end

183

urn uUI)(tUo),z(")) = A(t,x), we may assume, in the same way as at the

b-÷+oo

of the proof of Lemma 12.8, that the sequence

converges to some

s(.) E flXü(t,x,)C(i,r) + e,Q) = Xu(t,x,A(t,x),Q). Letting k -9 +00 in (12.54)-( 12.55), one gets

A(t,x);

k-++oo

i.e.,

= X(t,x) satisfies (12.46) [with

in place of ul. Lemma 12.9 is proved.

0 =

For any GE [O,T] and E Q, let us now consider a function which is given on x R by:

: yE

(12.56)

—

It is then easy from (12.53) that :

y E D(t,x,O)} —

<

t,b

: yE D(i,x,9)} (12.57)

and from Condition (ii) that pe,p = Beside x, is strictly decreasing in the function = we shall also consider another function x) defined on

by:

E R : p9,fl(i,z,1j) In virtue of (12.57)-( 12.58) and of the monotonicity in must have:

: yE D(t,x,O)} <

(12.58)

O}.

=

of

<

:

x, u'), one

y E D(t,x,6)}. (12.59)

Finally, let us investigate an auxiliary function aer I

=1

A(t,x)

=

z) by the formula:

if (t,x) E [0,9] x if (t,x) E [O,T] x

(Note that, by (12.56) and (12.58), ER x).) The following lemma gives us an important property of

:

(12.60) 0)

A(6,x) —

=

x).

12. M1NIMAX SOLUTIONS

184

Lemma 12.10.

is a supersolution (relative to

=

of the Cauchy

problem (12.9)-(12.10).

Proof of Lemma 12.10. The nonempty compact valued multifunction 3 (t,x) D(t,x,9) C RTh is locally bounded. Therefore by (12.59), taking account = (Lemma 12.7), we see that = are also locally bounded. Moreover, = and hence, = o(x). Thus, it follows from (12.53), (12.60), and Lemma 12.7 that according to Lemma 12.8, we have only to prove (12.49) with in place of u. of the continuity of

only to consider the following three cases: Case 1: 9
C Xu(i,x,p,p(t,x) + e,a)

x(.) E

= (12.49) [with u replaced by so that A(t,x) = is thus satisfied. 9. By (12.60), we have pe,p(t,x) = hence by (12.58) and Case 2: r we get pe,p(i, x, by the monotonicity in of = p,,p(t, x, x) + e) <0. Because of (12.44), there exists at least one —

+e,cx)flXL(t,x,/Ja,p(t,x)

z(.) E Of course, DL(T, x(r),

x)+E, 0, j3) C DL(i, x, pep(t, x)+e, 0, f3) and therefore,

by (12.56), it may be seen that

+ e)

+ e) < 0,

This together with (12.58) and (12.60) implies x) + e; (12.49) [with —

Case 3:

= e9,s(r,z(r)) in place of uJ is thus satisfied, too.

t < 9 < r. By Case 2, there exists x(')(.)

such that (12.61)

+ e.

such that

By Case 1, there exists x(2)(.)

pe,p(r,

x(1)(O))

(12.62)

S12.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS

Choose now x(.) E C([0,T],R") with (0,81

By

(12.61)-(12.62) and

— —

(0,8)

Condition

— Yl (8,T( —

(ii), we then see

that

Xu(i,x,,.te,,9(t,x)

x(.)

and

and

that

i.e., (12.49) is

also

satisfied with u def =

0

This completes the proof.

Lemma 12.11.

=

is a subsolutson (relative to FL) of the Cauchy

problem (12.9)-(12.10).

Proof of Lemma 12.11. By (12.53) and Lemma 12.7, A = A(t,x) is locally bounded with A(T,x) a(s). Let 0 < t < B < T, x /3 E Q, e > 0. We consider the auxiliary function = given in (12.60). Accordis a supersolution (relative to of the ing to Lemma 12.10, = Cauchy problem (12.9)-(12.10); hence by (12.52) and (12.60), one has A(t,x) = Thus, by (12.58) and the monotonicity in of = one also has p(t,x,A(t,x) — e) > 0. So, it follows from (12.56) that there exists x(.) E XL(t, s, A(t, x) — e, /3) such that A(O, x(O)) > A(t, x) — e. In so doing, we are led to another property of A = A(t, x), that is, inf

sup

fiEQ Z(.)EXL

for

all e > 0, (t,s) E

8

[A(9,x(O)) — A(t,x)] —e

[t,T}. Therefore, by arguments dual to the ones in

the proof of Lemma 12.8, we conclude that A+ = A+(i, x) is a subsolution (relative 0 to FL) of the Cauchy problem (12.9)-(12.10). The next lemma will play a crucial role in proving our main theorem. Adiatullina and Subbotin's method of proof ([1] and 1124]), based rather directly on Gronwall's inequality, seems to break down in the case of time-measurable Hamiltonians. Our

road to this result here is devious (by some "perturbation technique" on sets of Lebesgue measure 0), and proceeds via an implicit version of Cronwall's inequality. x) be super- and sub-solutions (relative Lemma 12.12. Let 17 = ii(t, x) and = to Fü and FL) of the Cauchy problem (12.9)-(12.10), respectively. Then ii(t, x) !L(t,x) on T.

12. MINIMAX SOLUTIONS

186

Proof of Lemma 12.12. Assume the contrary, that (12.63)

<

for some

E

x(.) E

t

Let D

be such that x(1) E D for all

B,. C

[0,7.'); and A = A(t)

where AD,E = AD,g(i) is

the function (in L'(o,T)) existing in Condition b) corresponding to the bounded sets

D and E def = Im (1) , m'2' ) with def < m(i) =

U

nun

(i,x)E(O,TIXD

— u(t,x)

def


=

< +oo.

max (,z)E(O,TIXD

(12.64)

There exists a set A of measure 0 such that (12.13) and (12.43) hold for all R", t E AC. We may assume that AC C Leb(c) fl Leb(A), that E R, p E R't, x

the function R u f(t, x, u, p) is decreasing for all p E t E AC, and xE that (12.11) is true whenever u E, p S, x, y D, i AC. For every e > 0, the set A can be contained in an open set U1 C (0, T) with

J8r(1 + r)c(t)dt <e. Let A(1)(t)

2A(t) + e and let

(12.65)

be the function given by

=

I18r(1+r)c(i)+e

if t U1, if tEll1.

(12.66)

For any t E [t(°), T) we consider the following (compact) set: x

E

—

J

.

A(1)(r)dr),

g(O)

<

Of course, N(')(t(°))

0. Setting

sup{t

< [t(°),T)

(12.67)

:

see that N(')(t(')) 0 because Xp(t(°),x(°)) is compact and II = ii(i,x)) is upper (resp. lower) semicontinuous.

0}, we

= i&(t,x) (resp.

and assume Let us prove that = T. To this end, find again the contrary, that t(1)
§12.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS

-

Case

1: i(')

Take

E Ue.

fixed o

tion, for any

S > 0 small enough so that

187

+ 5) C 14. By defini-

Q there exist

P, /3

E

with

+ 5))

+

+

<

+ 5)) (12.68)

where

<

(12.69)

Let x

2

if 0

I

def

t t(e),

ift(d) < t

deC I

2

if 0< t < ift(e)

T,

< t < T. (12.70)

Then E Xp(t(°),x(°)) xXp(t(°),x(°)). In addition, since N(e)(t(e)), it follows from (12.67) and (12.70) that + 5) —

=

+ 5)12

<

.

J

—

+

—

exp(J

+J

f

(

\

/

j

/

\Ji(O)

—

.exp(/

J .exp(J

J

g(O)

(12.71)

(Here, the first inequality follows also from the Bunhiakovskii-Schwartz inequality and the fact that Idx(t)/dtl < c(t)(1 + Ix(t)I) < (1 + r)c(t) a.e. in (0,T) for all we see, by (12.67)Thus, since again x(.) E E a contradiction. (12.71), that 0 N(e)(t(e) + 5) — Since j(c) Leb(c), Case 2: j(o) Eu:. Let us use (12.69) and —

one

def

has in =

sup

c(t)dt < +oo, and hence

sup

1

—/

d dt

dt

1

sup öE(O,T—t())

—/

c(t)(1+Ix(t)I)dt

z(.)EXF(t(°) ,z(°))

<(1+r)m. (12.72)

12. MINIMAX SOLUTIONS

188

Moreover, because we have

E

Leb(A), there exists

1

exp(_ 6

exp(f

6(1)

> 0 so that, for all 6 E (0,6(')],

2A(t)dt) >

J

A1(t)dt)

—

> exp(26A(t(1))) 1 + 26

(12.73)

On the other hand, the compactness of Xp(t(°),z(°)) implies the existence of 6(2)

with the property that Ix(t) — x(t(1))I <

! I 6

sup

—

(12.74) r)(1 + in)

< 6(2). Finally, since (12.43) holds at t

whenever z(.) E Xp(t(°),x(°)), It — one can find a E P, /3 E Q, and inf

16(1 +

> 0 so that

>

mm

zEFu(t,z(t),ii(°),a)

—e/8,

—

max

/

Jt()

+ e/8

(12.75)

<6 If we choose now 6 there exist E as 0

(12.68) is fulfilled.

(12.70).

min{6(1),6(l),6(S),T — i(t)} > 0, then by definition,

Let

x Xp(i(°),x(°)) be as in

E

It follows from (12.75) that

6 —

J

—

dt

f(t

By (12.76), setting p(t) dt

E/8,

(12.76)

f 6

such that

E

+ t E [O,T], we have

< f(t(e),x(O),u(O),p(O)) —

f(t (12.77)

Combining (12.72), (12.74), and

6

f

di


(12.77)

gives —

+ e/2,

§12.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS

189

and therefore,

+

8)12

—

—

+ e/2

A(t(e))

+e/2 =

+e/2

.

by (12.11), (12.13), (12.64), (12.68), and by the monotonicity in u of the function

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

Thus, taking account of (12.66), (12.67),(12.73), and of the fact E N(t)(i(1)), we see that

that

+ 8)12 [1 +

+ e6

.

.

exp(J

(j

A1(i)dt) +

.exp(f

A(M(t)dt)) •exp(f

A'(t)dt)+

t(O)

+1 A(o)(t)dt).

<J

(12.78)

Again, by (12.67)-(12.70) and (12.78), we get 0 N(s)(t(e) + 6) a contradiction. The contradiction8 we have got in both Cases 1 and 2 show that

that for any e > 0 there exists at least one

E

= T, i.e., It then

follows from (12.67) that .

—

Since

exp(JT A(*)(t)dt).

<Jt(o)

Xp(t(°), x(°)) is compact, one can find a sequence

such that

[notice that

= ii(t, x) (resp.

Letting k —+ +00 in = 0 (cf. (12.65)-(12.66)). the inequality —

in

<

=

x)) is lower (resp. upper) semicontinuous]

gives

<

<

—* 0

in C([0, T),

—+

(12.79) with e(k) in place of e, we obtain Hence, a passage to the limit as lv -+ +00

<

> ... >

>

(12.79)

<

<

12. MINIMAX SOLUTIONS

190

This is a contradiction, which says that (12.63) must be where x = 0 wrong. Lemma 12.12 is so completely proved. We are now in a position to prove Theorem 12.6.

Proof of Theorem 12.6. Since the Hamiltonian I = f(i,x,u,p) satisfies a)-e), Part 30 of Section §12.2 shows that Fu(f) 0. Let F0 = Fu(t,x,u,o) and FL = FL(t,x,u,[3) be in F0(f) and .FL(f), respectively, and A = A(t,x) be the function defined on IZT by (12.52). In virtue of Lemmas 12.9 and 12.11, = is in SolL(FL). Therefore, A(t,x) is in A = A(t,z) Lemma 12.12 together with (12.53) implies

A(t,x)

A(t,x).

=

Thus A = A(t,x) belongs to SolU(FU)flSolL(FL). Moreover, if

= = are also in .F0(f) and FL(f), respectively, and and then Lemma 12.12, applied to F0 = fl A(°)(t,x) belongs to gives .X(t,x) .\(°)(i,x) on 11T• Another F,j(t,x,u,cm) and = application of Lemma 12.12 to = and FL = FL(t,x,u,3) gives 0 A(°)(t,x) A(t,x), hence A(°)(t,z) A(t,z). The proof is complete.

Remark. Condition e) in Part 10 of Section §12.2 (positive homogeneity of the Hainiltonian I = f(t,x,u,p) with respect to p) can be omitted slightly by the method proposed in [1], (124] as follows. Let

f(t,x,y,u,p,q) deC = —

for

1

lim(q. f(i,x,u — y,p/q)) q\0

for

q=

0.

R and q R are two supplementary variables, and the limit is assumed to exist. It should be noted that the auxiliary Hamiltonian I = f(t,x,y,u,p,q) is positively homogeneous with respect to the variable (p, q) We also assume that the Haxniltonian f = f(t, x, u, p) satisfies conditions under which the positively homogeneous auxiliary Hamiltonian 7 = 7(t, x, y, u,p, q) has the properties indicated in Part 10 of Section §12.2. Then it can be shown that the minimax solution of the auxiliary Cauchy problem Here, y

011-

in on

{O
yER}

§12.6. TEE CASE OP MONOTONE SYSTEMS

191

the identity u(t, x, y) u(t, x, 0) + y. The function u = u(t, x) def_ = u(t, r, 0) can be considered as the minimax solution of(12.9)-(12.10) with non-homogeneous Hainiltonian f = f(t,x,u,p). For this case, an existence and uniqueness theorem (like Theorem 12.6) can also be established. satisfies

§12.6. The case of monotone systems In this section, we extend the notion of minimax solutions to the case of systems of first-order nonlinear partial differential equations with time-measurable Hamiltonians. We will only formulate the main results on the existence and uniqueness of such solutions for those systems satisfying a certain monotonicity condition (under which our systems are called monotone) and their relations with the (semi)classicai

solutions. The detailed proofs can be found in Thai Son, N.D., Liem, N.D., and Van, T.D. [133]. Our method here is to combine that of the previous four sections with some other techniques of monotonicity.

Inthesequel,misapositiveinteger. Givenu = (Ui,... ,um), v = (Vj,...,Vm)

E

1,...,m. Moreover,ifu u3 = v3 for some index 1 <j
10 Formulation of the Cauchy problem Let us consider the Cauchy problem for a weakly-coupled system of the form

=0 in u(T,x) = o(x) on

Qa,

(i = 1,...,m),

{t = T, x E

(12.80) (12.81)

Assume that the terminal data = cr(x) = .. ,Cm(x)) is a given vector function continuous on R", and that the Hamiltonians f1 = x, u, p) depend on (t, x, u, p) E (a)

x

x

with the following properties.

's Conditions: (al) For almost all (in the sense of Lebesgue measure) t E (0, T), the functions x x (x,u,p) i.-* f.(t,x,u,p) are continuous (1 i m).

12. MINIMAX SOLUTIONS

192

(a2) For any (x,u,p) E are measurable (1
x S, the functsons (O,T)

x

f1(t,x,u,p)

i

(b) For any lounded sets D C RTh and E C Rm, there exists a function AD,E = AD,E(t) in L'(O,T) with x, u, p) — f1(t, x', is,

AD,E(t)

for almost all t E (0, T) and for all x, x'

D, u

Ix — x'I

E, p E 5, i E { 1

m}.

(c) There exists a function £ = £(t) in L'(O,T) such that

sup{jf;(t,x,u,p) —f1(t,x,u,q)i —e(t) .(1 +

Ip—qI

:

p,q E B) <0

x Rm, tE {1,.. . ,m}.

for almost alit E (0,T) and for all(x,u)

(d) Monotonicity Condition (cf [72, §4, (A.1)&(A.3)]): For almost alit (0,T) u,v E Rm, if v,—u, = max(v,—u1) andforallj E {1 rn}, (x,p) E

0, then f,(t,x,u,p) f,(t,x,v,p). (e) The functions

p

—

—

f,(t,x,u,p) are positively homogeneous; i.e.,

f1(t,x,u,s .p) = s f1(t,x,u,p) Vs 0 for almost allt

(O,T) and for all (x,u,p) E

x S (i = 1,.. .,m).

x

Note that the monotonicity condition here implies:

(dl) Quasi-monotonicity Condition (ci. [72, Definition 2.2, Lemma 4.81): For if x S, u,v E almost all t (O,T) and for allj E (1,... ,m}, (z,p) u

v, then f1(t,x,u,p) f1(t,z,v,p).

(d2) The functions R 3 u1 '.4 creasing for almost all t E (0, T) and for all (u1,. . . .

.

. ,u,,.,,p) are de-

urn)

.

20 Differential inclusions for supersolutions and subsolutions First, for (t,x)

uE

a,/3 ES, 1

F(t, x)

.

(1

m, let

+ lxi) . B,

{z E F(t,x) {z

i

F(t,x)

(z,a) f5(t,x,u,ci)},

:

:

f1(t,x,u,$)).

(12.82)

§12.6. THE CASE OF MONOTONE SYSTEMS

Further, for any Rm-valued function on Qp, we set

where

=

=

193

z, u1, a)

(Fü)s(t, x,

x), a),

x,

(FL)1(t, x,

x), /3),

/3)

defined

. .

(12 83)

E R1 and x)

(t, x), .

. .

,

(t, x),

(t, x),

, Prn(t, r)) E Rm.

.

(12.84)

Then it can be seen that the multifunctions (1
x S 9 (t,r,u,c,) x Rm x S (t,x,u,fJ)

F(i,x)

C

x

C

(FL)1(t,x,u,/3) C

are nonempty convex compact valued, measurable in t and continuous in (x, u). On the other hand, it follows from (dl)-(d2) that

(Fu),(t,x,u,a) C

x, u, a) C

x, u1, .

,

+ e,

X, IL, i3) C

Z, U1,...,

— 6,

. .

if

u > v,

if

u .

v,

.

.

.

.

a) if

0,

if

0.

/3)

Therefore, by [133, Proposition 4.1], if

= 92(t,x) is lower (resp. upper) semicontinuous, one can check the measurability in t and the upper semicontinuity in x of the multifunctions (Fu)T(.,.,u1,a) (resp. (FL)T(.,.,u$,/3)) for 1 < m, u1 E R1, E S. And hence, according to the Remark 2°, the sets def

(Xe), (t.,x.,u1,a) = (resp.

(12 85)

def

(XL)1 (t.,z.,u,,/3) =

are all nonempty and compact for (t., x.)

So, we can make the following:

Definition 7. A supersolution of Problem (12.80)-( 12.81) is a lower semicontinuous function u = u(t,x) = (u1(t,x),. . .,um(t,X)) on which satisfies the condition

max sup

1
mm

[u1(r,r(r)) — u1(t,x)J < 0

12. MINIMAX SOLUTIONS

194

forallO
Vs E

(12.86)

Definition 8. A subsolution of Problem (12.80)-(12.81) is an upper semicontinuous Rm-valued function u = u(t,x) = (uj(t,x),. ..,Um(t,x)) on which satisfies the condition mm

mi

max

i<s<m$ES

0

Ius(r,x(r)) —

forall0
(12.87)

The sets of all supersolutions and subsolutions of ( 12.80)-( 12.81) will be denoted by Solu and SoiL, respectively.

Definition 9. An Rm-vaiued function u = u(t,x) in SOIIJflSO1L is called a mintmax solution of the Cauchy problem (12.80)-( 12.81).

3° Invariance of definitions Just as the invariance of definitions that has already been discussed in §12.4 for the case of single equations, instead of (12.82), we can now use any tuple Fu = [resp. FL =((FL)1,...,(FL)m)] in the following common families of tuples of multifunctions. Let P and Q be arbitrary nonempty sets. We consider any muitifunctions

xP

(t,x,u,a)

x Rm x Q

(t,x,u,8)

x

satisfying the general conditions below (1

C

'—

(FL),(t,z,u,$) C i

m).

i) The multifiinctions and = = (FL),(t,x,u,3) are nonempty convex compact valued, measurable in t and upper semicontinuous in (x,u). Moreover, (Fu)8(t, s,u, a) U (FL ),(t, x, u, 0) C F(t, x)

§ 12.6. THE CASE OF MONOTONE SYSTEMS

195

for almost alit E (O,T) and for aU(x,u) E R' x Rm, where c = c(t) in L'(O,T) is independent of i,a,

ii) For almost alit €(O,T) andforollx ER', a E P, /3 E Q, one has (vi,...,vm) E

J x, u, a) C

if u > v,

x, Ui,.. .

(FL)1(t, x, u,/9) C (FL)1(t,x, u1, .

=(ui,...,um),v =

> v,

if

(Fu)1(t,x,tL,a) C

u

.

U1 +

,

. ,

.

—

a)

zf

e > 0,

, u,,,/3)

if

> 0.

. ,Urn,

.

iii) For almost all t E (O,T) and for all (z,u,p) e R.' x Rm x

the following

inequalities hold: sup

mm

(z,p)
sup liminf

aEP J\O

mf limsup

$EQ

mi sup

max

(z,p),

—

—

1 —

1 —

I

d

\ di

p'dr > f,(t,x,u p) I

—

\ ç—(r),p1dr <

di

where

x, u, a)def = X(F,J ),

x),

clef

(XL)$(i, x, u, /9) = X(pL ).(.,.,',fl)(t, x).

The family of all tuples (rasp. FL =((FL)1,...,(FL)m)) satisfying Conditions i)-ill) will be denoted by .Tu(f) (resp. FL(f)). If the Harniltonians = (1
and then use (12.82) to define a concrete pair of tuples = ((Fu)j,. . and FL = ((FL)j,... ,(FL)m). This pair would satisfy i)-iii).

.

From now on, let us fix any (Fu, FL) E Fu(f) x .FL(f) and continue using the notations (12.83)-(12.85). It is [133, Proposition 4.1] (see aiso the Remark of §12.2-Part 2°) that makes the following definitions allowable.

12. MINIMAX SOLUTIONS

196

Definition 10. The set Solu(Fu) of supersolutions (relative to Fu) of the Cauchy problem (12.80)-(12.81) is defined to be the set of all lower semicontinuous Rmvalued functions u = u(t,x) = (ui(t,x),.. on which satisfy (12.86) and the condition .

max sup

1
mm

—

u(t,x)] <0 —

for

Definition 11. The set SOIL (FL) of subsolut:ons (relatzve to FL) of the Cauchy problem (12.80)-(12.81) is defined to be the set of all upper semicontinuous Rmvalued functions u = u(t,x) = (ui(t,x),. on 11T which satisfy (12.87) and the condition . .

mm I

for

inf

PEQ

max

[u1(r,x(r)) — u1(t,x)] > 0 —

any0
FL) E .Tu(f) x

the intersection

Solu(FrJ)flSolL(FL) consists of a unique (Rm-valued) function, which is independent of FL). Therefore, in complete concord with the previous definitions, we may also give the following.

Definition 12. An Rm-valued function u = u(t, x) in SolU(FU)flSolL(FL) is called a mlnimaz solution (relative toFu and FL) of the Cauchy problem (12.80)-(12.81).

4° Main results In this part, we formulate the main results on the existence and uniqueness of minimax solutions for monotone systems of first-order nonlinear partial differential equations with time-measurable Hamiltonians and their relations with the (semi)classical solutions (see Definition 2 in Chapter 11).

Theorem 12.13. Assume (a)-(e). Then (i) every global semiclassical solution u = u(t,x) of (12.80)-(12.81) so that the x i-+ 0u1(t,x)/Ox (1
(0, T) must also be a minimaz solution of the same problem,

§12.6. THE CASE OF MONOTONE SYSTEMS

197

(ii) there exists a null set A C (0, T) such that, at each point (t, x) E ((0, T)\A) x R' where a certain minimax solution of(12.80)-(12.81) is differentiable, the equations (12.80) must be satisfied.

and the Theorem 12.14. Let the initial data a = a(x) be continuous on Hamiltonians = f,(t,x,u,p), 1 < j < m, satisfy all Conditions (a)-(e). Then there exists one unique minimax solution of the Cauchy problem (12.80)-( 12.81).

Example. Now we present an example of monotone systems (12.80) of first-order nonlinear partial differential equations with time-measurable Hamiltonians. (The case of single equations is obtained when rn = 1.) It is left to the reader to check all the hypotheses (a)-(e) for those systems. x R such that R? .\ = G1(t,x,A) be of class C1 on Let (1 < i < m). Further, let h1 = he(s) be a is increasing for all t E (0,T), x E bounded, decreasing, and globally Lipschitz continuous function of s E R for any arctan(s) with a certain i {1,. . . ,m} (we can take, for instance, he(s) for each index i). negative constant Define (1 + 3x1)

.

andi€{1,...,rn}.

for

=

x) are given C'-functions satisfying on

0,

+

is in L' (0,T) and the conditions

and, c13(t,x) <0 if i

j.

Chapter 13 Mishmash In this final chapter we present various things, more or less "for completeness."

§13.1. Hopf's formulas and construction of global solutions via characteristics First, consider the Cauchy problem

Ou/*3t+f(Ou/Ox)=O in u(O,x) =

on

(13.1)

{t = 0, x E

(13.2)

under the following two hypotheses.

(P.1) The initial function = strictly convex on with lim

is of Class C° and the Hamiltonian I = 1(p) is = +00.

IpI—'+oo

(F.II) For every bounded subset V of D, there exists a positive number N(V) so that + t f((x — w)/t)} + t . f((x — y)/t) whenever (t,x)

V,

>

N(V). Here, f = f(z) denotes the Fenchel conjugate

function of f = 1(p). By Theorem 9.12, a global solution of (13.1)-(13.2) exists (and is found by Hopf's formula (9.29)) while the classical theory furnishes only its restriction to a neighborhood of {t = O} in which it is smooth if the Hainiltonian and initial data are supposed to be sufficiently smooth. In fact, as we have seen in Chapter 1, if are of class C2 on Cauchy's classical theory furnishes f = f(p) and = the solution by fitting characteristics of (13.1) to the initial data = These characteristics form a hypersurface x = x(t, y), v = v(t, y) in txv-space above lxspace such that x(0, y) = y, v(0, y) = 4'(y). For sufficiently small t > 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


= ü(i,x)+ e.

§13.1. CONSTRUCTION OF GLOBAL SOLUTIONS VIA CHARACTERISTICS

203

Because 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], p
ISO,

and cb(r)

Then I

x<0,

def I eZ,

=

Lx+1,

plnp—p, +co,

x>0.

pE[O,11,

and

ü(t,x) = sup{x It

—

4)(e')

is easily seen that the value ü(1, —2) = sup

—

tf(e")}. =

0

can not be attained in

(13.13) at any point yE R1. In the remainder of this section, we make the following assumption:

(E.V) For each (t,z)

1.),

the supremum in (13.13) is attained at some point

yE

Remark. Let 4) = 4)(x) be convex and of class C' on R". Assume (E.I)-(E.II). Then (E.V) is fulfilled, for example, if is an open set (cf. (13.15) and the Remark above). In particular, this appears when 4) = 4)(x) is a co-finite strictly convex function, i.e. when it is strictly convex with lim [4)(.Xx)/AJ = +co for all A-+-4-oo

See [117, Theorems 26.5-26.6].

Theorem 13.3. Let I = f(t,p) and 4) = 4)(x) be of class C2 on {t 0, p R respectsvely, such that 4) = s always posstwedefinite. Assume (E.II) and (E.V). Then the value u(t(°),x(°)) at each (t(°),x(°)) € ,

13. MISHMASH

V of the global solution u = u(t,x) given by Hopf's for,nula (9.7) of the Cauchy problem (13.8)-(13.9) can be determined as the largest values of i,.) =

y), 4,'(y));

the maximum being taken over ally such that the characteristic curves {(t, x(t, y)) 0} starting from (O,y) meet each other at (t(°), x(°)).

Proof. Since 4" is positive-definite, 4' = 4'(x) is convex on RTh. Notice that (El) is satisfied trivially. From Proposition 13.2 and its Remark, we also see that u(t,x) ü(t,x) on Moreover, it has been known (cf. (13.11)-(13.12)) that

= 4'(y) ÷ (4,'(y),x — y) —

f

f(r,4"(y))dr;

hence,

—cp(t,x,4"(y)) = (x

—

y —

f

(13.16)

In view of (E.V), the supremum

=

=

4,'(y))

sup pER'

is attained at some y(°) E R't, which must be a stationary point of We then have P.. (t(O) —0 4,'( Y )) Because

is

(13.17)

4,'(.)).

positive-definite, it follows from (13.16) that — —

j

therefore, (13.10) implies x(t(°),yt0)) =

=

0;

Notice that the maximum in (13.17)

is not affected if it is taken not for all y but only for those stationary y. This completes the proof.

0

Remark. To our knowledge, Hopf [64] was one of the first to ask whether the global solution (9.29) of (13.1)-(13.2) and the solution (9.7) of (13.8)-(13.9) could be determined by means of characteristics, i.e., whether they could be determined directly from the several values of v in (13.4) and (13.10), respectively. The answer

§13.2. SMOOTHNESS OF GLOBAL SOLUTIONS

20$

(essentially Theorem 13.1) was in the former case considered by Hopf [64] himself

while he had no answer in the latter.

§13.2. Smoothness of global solutions As we have seen (for example, by Theorems 2.1-2.2, Proposition 4.1 ... ), in general, classical solutions to the Cauchy problem for nonlinear partial differential equations

exist only locally in time, and global (generalized) solutions contain singularities. In the definition of most kinds of generalized solutions, such as weak, minimax, or viscosity solutions, the regularity condition is relaxed significantly. (A mimmax or viscosity solution needs only be continuous.) To get further information about the solution, it is therefore very important to investigate its smoothness. Theorems 2.6-2.7 and 3.1 can be considered, to some degree, as investigations in this direction. In particular, the theme of constructing the singularities of generalized solutions, or of weak solutions, is thoroughly studied in Chapters 4-7. In this section, we investigate the smoothness of the global solutions that are = given by Hopf's formulas. Consider the Cauchy problem (13.8)-(13.9), being a convex function on Let (E.I)-(E.II) hold. It has been shown in Chapter

9 that (9.7), or equivalently (13.14), yields a global solution u = u(t, x) of this problem. Recall that the definition

L(i,x) def = {pE R

:

= u(t,x)} C

determines a nonempty-valued upper semicontinuous multifunction L = L(t, x) of (t,x) E V. Here, = is defined in (13.12). We now prove:

Theorem 13.4. Let 4 =

be a convez function on Assume (E.I)-(E.II). Then the global solution u = u(t, x) given by Hopf's formula (9.7) of the Cauchy

problem (13.8)-(13.9) is continuously differentiable in an open set U C D0

{(t,x) E V

:

G} if and only if L(t,x) is a singleton for all (t,x) E U;

t

the set C being as in (E.I).

Proof. By the proof of Theorem 9.1, if u = u(t,x) is differentiable at (t(°),x(°)) E D0, then (9.14)-(9.15) give (for 1 i n)

=

:pE

= min{p1

:

pE

13. MISHMASH

206

is precisely the singleton {ôu(t(°),x(°))/Ox}. (Consequently, (9.13) together with (9.15) implies that (13.8) must be satisfied at (t(°),x(°)).) Conversely, suppose now that L(t, x) is a singleton for all (i, x) in an open set U C V0. Being single-valued, the mapping U (t, x) —+ L(t, x) is continuous. Hence, according to the maximum theorem [8, Theorem 1.4.161, (9.13)-(9.14) show that the derivatives Ou/Ot, Ou/Oxi,..., Ou/Ox,, exist and are continuous in U (see also [128, Corollary 2.21); i.e., the global solution u = u(t, x) is continuously differ0 entiable in U. So,

Remark. A special case of Theorem 13.4, when n equals 1, 4, = 4,(x) is globally

Lipschitz continuous, was considered by Kruzhkov, S.N. and Petrosyan, N.S. [94, Lemmas 1 and 3]. There, instead of the multi-valued function L = L(t, x), they used the single-valuedp =p(t,x)diinf sup L(t,x), and established some interesting properties of these single-valued functions. In particular, we have:

Proposition 13.5. [94, Lemma 51 For n =

suppose 4, = 4,(x) is convex and globally Lipschitz continuous, I = f(p) ss of class C1 on R. Let (t(°), x(°)) E {(t,x) x = (O,oo) x R, E L(t(°),x(°)), C + (t — t(°))f'(p(°)), 0 i t(°)}. Then L(t,x) for all (t,z) C. Further, if L(t(°),x(°)) is a singleton, 1,

:

so is L(t,x) for (t,x) E C.

Corollary

Under the hypotheses of Proposition 13.5, suppose the global * u(t,x) def = max{px —4, (p) — tf(p)} of(13.8)-(13.9) ss at

13.6.

soluhon u =

pER

(t(°),x(°)). Then it is also differentiable at (t,x) E C for all t E (0,t(°)).

Proof. Since u = u(t,x) is differentiable at (t(°),x(°)),the proof of Theorem 13.4 shows that L(t(°),x(°)) is a singleton. By Proposition 13.5, L(t,x) is also a singleton

its Remark, E (O,t(°)). Notice u = u(t,x) = max{px — qS(p) — tf(p)} is convex, because it is the maximum for all

the partial

that

(t, x)

C. Therefore,

derivatives

by the proof of Theorem 9.1 and

of the solution exist at (t,x)

C for

all t

pER

of a family of affine functions. Hence, according to [117, Theorem 25.21, it is 0 differentiable at the above-mentioned points (t, x).

ExampLe 1. Investigate the smoothness of the global solutio the problem On

On

in

{t>0,xER},

S13.2. SMOOTHNESS OF GLOBAL SOLUTIONS

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

(0, def I p, s 1,

207

xER},

p0, O1.

Note that 4) = x2/2 is convex, but not globally Lipschitz continuous; and we can not use Hopf's formula (9.3) for n = 1. In this case, however, (E.I) and (E.II)" hold trivially. (See Remark 2 following the formulation of Theorem 9.1.) Hence, by (9.7), a global solution of this problem is u =

u(i,x)

max{px —p2 /2—tI pER

More clearly, we have

u(t,x) = max {max{px — p2/2}, max{px — p2/2 — tp}, or

rnax{px

—

p2/2 —

t}},

2

if (t,x)ED1, if (t,x)EV2,

0

u(t, x) =

(x

—

t)2

(t,x)ED3,

2

if

(t,x)ED4.

Here,

x

zER}\(D1UV2UV3). This

solution is continuously differentiable in {t > 0, x E R}\(C), where the

"singularity" curve (C)

is

given by

(C)

.

—

5 x2/2,

x

2,

12(x—1)

This fact can be foreseen by noticing that L(t,x) = {0,x} if t = x2/2, x 2, and L(t,x) = {x — t,x} if t = 2(x — 1), 1 < x < 2, while L(t,x) is a singleton

13. MISHMASH

208

if (t,x)

(C). (Direct computation shows that L(t,x) = {x} for (t,x)

L(t,x) = {0} for (t,x)

V2\(C), L(t,x) = {z

V1,

i} for (t,x) E D3\(C), and

—

L(i, x) = {x} for (t, x) E V4\(C).)

Example 2. We consider the Cauchy problem

=0 in {t>0,x€R}, on

{t=0, x€R}.

A global solution of this problem is:

u = u(i,x)

—

pER

2

+t(1 +p2)112}.

By computing and relying on Theorem 13.4 we recognize that u = u(t,.x) is continuously differentiable in {t> 0, z R)\{(t,0) t 1). Using the method of characteristics, we see that when i > 1, characteristic curves intersect. Concretely, the two curves {(t,x(t, 1)) t 0} and {(t,x(t,2)) t O}, where :

:

:

x(i,y)=y—

ty + y2

starting from (0,1) and (0,2), respectively, meet each other at the point ç

Nevertheless, the differentiability of the solution is not broken down in some neighborhood of this point.

§13.3. Relationship between minimax and viscosity solutions As we have mentioned, 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 partial differential equations as well as in other types of partial differential equations. The primary virtues of this theory are that it allows merely nonsmooth functions to be solutions of nonlinear partial differential equations, that it provides very general existence and uniqueness theorems, and that it yields precise formulations of general boundary conditions. (See 151,

§13.3. RELATIONSHIP BETWEEN MINIMAX ANT) VISCOSITY SOLUTIONS I10]-[20], [281, 135]-[39], [47]-[50], [67]-[72], [79], [991-1101], [122J-[123), [131], and

209

the

references therein.) These contributions make great progress in nonlinear partial differential equations, where the global existence, uniqueness, and well-posedness of generalized 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, and it is based on replacing the equations by pairs of differential inequalities. In the research of both minimax and viscosity solutions, much attention is given to the construction of subgradients, subdifferentials, generalized derivatives, and so on, all of which are used in work with nonsmooth functions. For equations with continuous Hamiltonians, Subbotin and his coworkers ([1], [124]) have shown that minimax solutions can be defined by the use of inequalities for directional derivatives that are equivalent

in essence to inequalities for subgradients and supergradients used in the definition of viscosity solutions. We now examine the relationship between minimax and viscosity solutions in the case of equations with time-measurable Hamiltonians. In our approach, the definition of minimax solutions needs not be via inequalities for directional derivatives. It is enough to use only those given in Chapter 12. Let be a finite-valued function of near a point E Rm and let = e 6 Rm, -y 6 Rm. Recall that the upper and lower Dini semiderivatives of = at the point e are defined as: inf sup

e>O

+

O<J(e

—

(13.18)

inf

e>O

+

O<6(e

—

We introduce the following notation:

inf

sup

—

inf

—

Dt/'(e(°)) The mapping Rm A

—

—

—

Rm :

0},

{-y E Rm :

0}.

{i'

=

—

(13.19)

tf

i.e.,

—

(13 20)

6 [—co, oc] is of course positive-homogeneous,

e '—' (A

>0). Fore =

0,

the relation O(°))

{0,oo}

210

13. MISHMASH

holds. It also follows directly from (13.18) that this mapping is upper semicon-

tinuous. Analogous properties are enjoyed by the mapping Rm it is lower semicontinuous and satisfies the relations

=

A

e '—p

E {O, —oo} and

(A > 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>OO<6<e

=

—

0,

= oc;

hence,

inf sup

+

e>O

=

—

It follows that

0

inf sup

+

—

inf e>O

+

—

e>O O<ö(e

= 00

sup

—

il = 00,

—

a contradiction. In the converse direction, suppose we can choose a sequence C Rm, 0 urn

—

We

(y,e) Ye E Rm. Using (13.19), -4 0 as k

—

—

—

k-+oo

— h1

=

—

(e(k) — may assume without loss of generality that e(k) = e Rm (lel = 1) as k —+ oo. Since -4 0, (13.22) together with (13.18) implies —

= lim

+

= lim

+

k-+oo k—+oo —

(i',e)

—

co,

such that (13.22)

+

k)e(k), with

—

e)

0,

by (13.20). So the first equality in (13.21) has been completely proved. The proof of the second is similar. i.e., -y E

We are now in a position to give the definition of viscosity solutions to the Cauchy problem of the form

=0 in

(13.23)

13. MISHMASH

212

u(T,z)=c(x)

on

{t=T, xER'1}.

(13.24)

Here, 0 < T < oc. Assume that the terminal data = u(x) is of class C° on and that the Hamiltonian f = f(t,x,u,p) is measurable in t E (0,T) and continuous in (x,u,p) E

Rx

In accordance with [35] and [39], we propose

the following:

Definition. A viscosity solution of (13.23)-(13.24) is defined to be a function u = u(t,x) continuous on QT satisfying the terminal condition (13.24) and the pair of inequalities

a + f(t,x,u(t,x),b)

0 V(a,b) a +f(t,x,u(t,x),b) <0 V(a,b) Du(t,x)

(13.25) (13.26)

for almost all t E (0,T) and for all x

If the Hamiltonian f = f(t,x,u,p) satisfies the conditions a)-e) indicated in Part 10 of § 12.2, then it is known (Theorems 12.5-12.6) that a solution of (13.23)-(13.24) exists and is unique. For almost all t (O,T), the niinimax solution satisfies (13.23) at each point (t, x) where it is differentiable. Further, if a classical solution of (13.23)-(13.24) exists, then it coincides with the minimax solution. In this section, the main result on the relationship between minimax and viscosity solutions (in the case of equations with time-measurable Hamiltonians) is as follows.

Theorem 13.7. Under the hypotheses of Theorem 12.6, the m;nimaz solution of (13.23)-(13.24) is also a viscosity solution.

Proof. Take A C (0, T) to be a null set satisfying (12.37) of Lemma 12.3. By [119, (0, T)\A C Leb(t), with £ = £(t) the function p. 158], we may assume that AC mentioned in Condition c) (Part 10 of §12.2), and Leb(i) the set of all t E (0,T) satisfying lim

Let u =

1

J

£(r)dr

—

£(t) = 0.

be the minimax solution of (13.23)-(13.24) that exists and is unique by Theorem 12.6. Actually, we have shown more, namely that u = u(t, x) does not depend on the choice of the multivalued mappings x, u, cs) = and FL = FL(t,x,u,/3) (in Fu(f) and FL(f), respectively). From now on, we shall particularly use (12.14) for a concrete pair of such multivalued mappings and, accordingly, use Definitions 1-3 in Chapter 12.

§13.3. RELATIONSHIP BETWEEN MINIMAX AND VISCOSITY SOLUTIONS

Let us first prove (13.26) for t Ac,

Ac, x

R", (a,b) E Du(t(°),x(°)).

213

this end, suppose R so that S and 0 s

To

a

Choose

E

b=s•cx. x8(.) with

+

+ 5)) —

Because =f and the function family {xö(.)}ö C

+ 5) —

Let yo

(13.27)

0.

j(°)

E AC is a

1(0)

point of £ = £(t)

x(°)) is uniformly

bounded, (12.14) yields 1

sup

sup

—IyoI

5

£(r)(1 + x6(r)I)dr < oo.

I

Consequently, 1

=y

urn

5

for some y E R" and now see that

> 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<1

[u(t(°) +

+

—

u(t(°),x(°))]/5}

1•a + (y,b) a +

The inequality (13.26) has been proved for almost all t E Similarly

(0,

T) and for all x E

for the inequality (13.25). The proof is thus complete.

0

Appendix I Giobal Existence of Lharacteristic Curves In this appendix, we will give sufficient conditions which guarantee the global solv-

ability of the Cauchy problem (2.7)-(2.8) and (215)-(2.16). In [43], B. Doubnov gave sufficient conditions for the global solvability of Hamilton's equation. First, we report his problem and results. = {(i,x,p) Let f = f(t,x,p) be a C2-function defined on t E R, x E :

pE

Consider the Cauchy problem:

=

(1

= 1,2,... ,n),

(All)

1

(i=1,2

I x(0) =

(AI.2)

p(O) = p°.

Assume the following conditions: I p = 0(11), I

Op'

= 0(1 + IpI')

'

C'! I'

'Of Op —

for

Ii'I

Of 'Op

(AI.3)

with C = C(s) a certain continuous function defined on [0,oo), and rk 1. In (AI.3), all 0-estimations are uniform with respect to t and x. That is to say, p = 0(11) for -÷ 00 if and only if there exist positive constants R and K independent of (t, x) such that [p1

for

p[ R.

B. Doubnov [43] proved that, if (AI.3) is satisfied, the Cauchy problem (AI.1)x (AI.2) has uniquely a global solution on {t 0} for all (x°,p°) E We give here a brief explanation for this result. As it is well-known that there exists a local solution of (AL1)-(AI.2), we show that the solution does not blow up in a

GLOBAL EXISTENCE OF CHARACTERISTIC CURVES

215

finite time. Let x = x(t) and p = p(t) be solutions of (AI.1)-(AI.2). Then it holds that =

Since rk 1 in (AI.3), we get the estimate

If(t,x(t),p(t))I < where L and M are constants. As p = o(fk), p = p(t) does not blow up in a finite time. Using the third inequality in (AI.3), we see that x = x(t) does not blow up in a finite time. Therefore, we can say that the seminal idea of B. Doubnov 143) comes

from the following example: The Cauchy problem

(d/dt)x(t) = x(t)k,

with

x(O) = y,

is globally solvable for any y E R if and only if k 1. Now we consider the Cauchy problem (2.15)-(2.16). By almost the same reasoning as in [43), we assume the following conditions for (2.15).

(C.1) There exists a constant N > 0 such that Of

I

01

NIfI.

For any constant L > 0, if we put DL def = {(t,x,u,p) then there exists a constant M > 0 such that (C.2)

•

:

If(t,x,u,p)I < L},

and

+ on DL.

Proposition AI.1. Suppose (C.1) and (C.2). Then the Cauchy problem (2.15)(2.16) has a global solution x = x(t,y), v = v(t,y), p = p(t,y) for any initial data.

We leave the proof to the readers.

216

APPENDIX I

We conclude this appendix with another remark on [43]. B. Doubnov wrote that, if I = f(t, x, p) was an algebraic function of p such that lim f(t, x, p) = 00, 1,1—Poe

Condition (AI.3) would be satisfied. However, when S. Ouchi, A. Kaneko, and M Murata translated [43] into Japanese, they already pointed out that this is not true. Their counter example is as follows: for

n=2.

Though this example does not satisfy (AI.3), the Cauchy problem (AI.1)-(AI.2) has a global solution for any initial data. Therefore, we would like to give an example

where the Hamiltonian I = f(x,p) is a polynomial of p with

lim f(x,p) = 00,

but the corresponding Cauchy problem (AI.1)-(AI.2) cannot be solved globally with

respect to t.

Example. Let n = 1, and let a = a(x) be of class C°° on R such that a(x) = on l}, and that a(x) C = constant > 0 on {IxI 1}. Put f(x,p) a(x)p2. {(xI Then the Cauchy problem (AI.1)-(AI.2) cannot have a global solution for some initial data. For example, if x° > 1 and p° > 0, then it does not admit a global solution.

Appendix II Convex Functions, Mu Itifu nctions,

and Differential Inclusions In this appendix, for the convenience of the reader, we summarize without proofs

the relevant material on convex functions, multifunctions, and differential inclusions that we have used in an essential way since Chapter 8. For the proofs we refer the reader to [8], [22], [29], [40], [64], and [117].

§AII.1. Convex functions The Throughout, is the usual vector space of real n-tuples x = (x1,. . , Eucidean norm and inner product in it are denoted by .[ and (., .). A subset D of is called affine (respectively, convex) if (1 — A)x + Ay E D for any x E D, y E D and A E R (respectively, A (0,1)). Obviously, the intersection of an arbitrary collection of affine sets is again aifine. there exists a unique smallest affine set containing Therefore, given any D C .

D, namely, the intersection of the collection of alfine sets M such that M J D. This set is called the affine hull of D and is denoted by affD. The relative interior of a convex set D in which we denote by riD, is defined as the interior which results when D is regarded as a subset of its affine hull affD. In other words,

(x+eB)fl(affD)CDforsomee>0}, where B stands for the unit ball (centered at the origin) in

Theorem AII.1. [117, Theorem 6.11 Let D be a convex set in

andyED. Then (1—A)x+AyEriDforoA<1.

Let x

riD

APPENDIX II

218

Let 4 = 4(x) be a function whose values are real or ±00 and whose domain is a subset D of IR". The set deC epi4,={(x,p):xED,pER,p4,(x)}

is called the epigraph of 4, = 4,(x). We define 4, = 4,(x) to be a convex function (on A concave function is a function whose D) if epi 4, is convex as a subset of

negative is convex. An affine function is a function which is finite, convex, and concave.

The effective domain of a convex function 4, = 4,(x) on D, which we denote by dom 4,, is the projection on R" of the epigraph of 4, = dom4,

def

= {x

:

(x,p) E epi4, for some p E IR} = {x

4,(x) < +m}.

:

This is a convex set in IR", since it is the image of the convex set epi 4, under a and dom4, 0, linear transformation. if im4, {4,(x) x E D} C (—co, then 4, = 4,(z) is called proper. Trivially, the convexity of 4, = 4,(x) is equivalent to that of the restriction of 4, = 4,(x) to dom 4,. All the interest really centers on this restriction, and D itself has little role of its own. Moreover, one could limit attention to functions given on all of R", since a convex function 4, = 4,(x) on D can always be extended to a convex function on all of R" by setting 4,(x) +00 for x D. Therefore, by a "convex function," we shall henceforth always mean a "convex function with possibly infinite values which is defined throughout the space IR"," unless otherwise specified. The convexity condition can be expressed in :

several different ways. For example, we have:

Theorem AII.2. [117, Theorem 4.1] Let 4, = where

4,(x) be

D is a convex set (for example D =

a function from D to

JRtt).

Then

4, = 4,(x) is

convex on D if and only if —

A)x + Ày) < (1 — A)4,(x) +

whenever

A4,(y)

0

< A <

1

for any x and y in D.

Theorem AII.3. [117, Theorem 4.2] Let 4i = [—co, +oo]. Then 4, = 4,(x) —

is

çb(x)

be a function from R" to

convex if and only if

A)x + Ày) < (1 — A)a + Afi

for any A E (0,1)

§AII.1. CONVEX FUNCTIONS

219

whenever 4,(x)
Theorem AII.4. [117, Theorem 5.7] Let 3 x i—+ Ax E Rm be a linear transformation. Then, for each convex function 4, = 4,(x) on the function a = a(y) defined by

a(y) def. = inf{4,(x)

:

Ax = y}

is convex on

The function a = a(y) in Theorem AII.4 is called the image of 4, = 4,(x) under A, in symbols, a = A4,. The inequality in Theorem AH.2 is often taken as the definition of the convexity of a function 4, = 4,(x) from a convex set D to (—oo, +00]. (This approach causes difficulties, however, when 4, = 4,(z) can have both +00 and

—oo among its values, since the expression oo — oo could arise.) In this approach, furthermore, if 4, = 4,(x) is finite and if the inequality is strict for any two different

points x and y in D, then the function 4, = 4,(x) is called strictly convex on D. Of course, the condition in Theorem AII.3 could be used as the definition of convexity in the general case, but the definition given via epigraph seems preferable because it emphasizes the geometry which is fundamental to the theory of convex functions. Here are some elementary topological properties of convex functions. Theorem AII.5. [117, Corollary 7.5.1] For a lower semi continuous proper convex function 4, = 4,(x), one has cb(y)

= lirn4,((1

—

\)x + .\y)

for every x E dom4, and every y.

Theorem AII.6. [117, Corollary 10.1.1] A convex function finite on all of

is

necessarily continuous.

which we denote The (Fenchel) conjugate of a convex function 4, = 4,(x) on defined by the formula by qS = 4,(p), is another convex function on

4,*

4, = 4,(x) from to [—00, +ool can be defined by the same formula as above. It is actually a lower semicontinuous

APPENDIX II

220

convex function.) The following main facts about conjugate convex functions are from [117, Theorem 12.2, Corollaries 12.2.1-12.2.2, Theorems 23.4-23.5], or partly from (64, Theorem 4.1 and the addition].

Theorem AII.T. (i) Let

= be a convex function. The conjugate function is then a lower sernicontinuous convex function, proper if and only if

= = q5(x) is proper. (ii) The conjugacy operation induces a symmetric one-to-one correspondence in the class of all lower semicontinuous proper convex functions on For any = in this class, the supremum in (AH.1) is attained (maximum) in domq' zfpEri(domqS*). (iii) For any convex function = on one actually has :

pER's.

x

Further, we have:

Theorem AII.8. [117, Corollary 13.3.3] Let = be a proper convex function. In order that dom be bounded, it is necessary and sufficient that = be finite everywhere and that there exist a real number j.z 0 such that —

—

yj

Vx,y.

= 4(x) be a proper convex function that is (finite and) differentiable at some y E Then [117, Theorems 23.5 and 25.1] implies Let

=

—

This equality suggests that the conjugacy operation

is closely related to the classical Legendre transformation in the case of differentiable convex functions. (See § 10.2 for the definition of the Legendre transformation.) In fact, this relationship is in detail as follows. be any lower seznicontinuoiz.s Theorem AII.9. [117, Theorem 26.4] Let = proper convex function such that D is mt (dom non-empty and = is differentiable on D. The Legendre conjugate (B, i) of (D, gS) is then well-defined. Moreover, B is a subset of domq5 (namely, the range of the gradient mapping x i—÷ and a = a(p) is the restriction to B. =

§AII.1. CONVEX FUNCTIONS

221

Corollary AII.1O. (cf. [117, Corollary 26.4.1]) Let 4, = 4,(x) be any differentiable convex function on R". Then the Legendre conjugate (B, a) of 4,) is welldefined. One has

: xER't}Cdom4,. Furthermore, a = a(p) is the restriction of 4, = 4,(p) to B, and a = a(p)

is

strictly convex on every convex subset of B.

In general, the Legendre conjugate of a differentiable convex function need not

be differentiable or convex, and we cannot speak of the Legendre conjugate of the Legendre conjugate. As will be shown in the theorem below, the Legendre transformation does, however, yield a symmetric one-to-one correspondence in the class of all pairs (D, 4,) such that D is a non-empty open convex set and 4, = 4,(x) is a strictly convex function on D satisfying: (1) 4, = 4,(x) is differentiable throughout D. x(2),... is a sequence in D converging to (ii) lim 4,t(x(k))I = +00 whenever

a boundary point of D. For convenience, a pair (D, 4,) in the class just described will be called a convex function of Legendre type. By [117, Corollary 26.3.1], a lower semicontinuous proper convex function 4, = 4,(x) has x '—+ 4/(x) one-to-one if and only if the restriction of 4, = 4,(x) to D

mt (dom4,) is a convex function of Legendre type.

Theorem AII.11. [117, Theorem 26.5) Let 4, =

4,(x) be a lower semicontinuous def. •def. convex functwn. Let D = int(dom4,) and D = int(dom4, ). Then (D,cb) a convex function of Legendre type if and only if (DC, 4,) is a convex function of Legendre type. When these conditions hold, (D,4,') is the Legendre conjugate of (D,4,), and (D,cti) is in turn the Legendre conjugate of (D,çb). The gradient 4,'(x) is then one-to-one from the open convex set D onto the open mapping x convex set D, continuous in both directions, and cb' = (4,S)_1.

We now describe the case where the Legendre transformation and the Fenchel conjugacy correspondence coincide completely. A finite convex function 4, 4,(x) on R" is said to be co-finite if epi 4, contains no non-vertical half-lines, and this is equivalent (by [117, Corollary 8.5.2)) to the condition that

lim [4,(\x)/A] = +oo for all

x E RTh\{O},

APPENDIX II

222

or (by [117, Corollary 13.3.1]), to the condition that 4, = q5(p) is finite everywhere.

Theorem AII.12. [117, Theorem 26.61 Let 4, =

be a (finite) differentiable In order that x '— 4,'(x) be a one-to-one mapping from convex function on onto itself, it is necessary and sufficient that 4, = çb(x) be strictly convex and co-finite. When these conditions hold, j = çh(p) is likewise a differentiable convex which is strictly convex and co-finite, and = 4,'(p) is the same function on as the Legendre conjugate of 4, = 4i(x), i.e., 4,(x)

= (p, (4,')1(p)) — 4, =

4,

the following fact about differentiability of convex functions.

Theorem AII.13. [117, Theorem 25.2] Let 4' =

4,(x)

be a convex function on

4'(x) is finite. A necessary and sufficient

and let y be a point at which 4' = condition for 4' = cb(x) to be differentiable at y is that the n two-sided partial derivatives 84i/0x1 exist at y and are finite.

§AII.2. Multifunctions and differential inclusions is the family of all subsets of X. Given another 0 be a set. Then i-+ will be called a multifunction 0, a correspondence 0 E (a set-valued map, or a multivalued function). Sometimes, we permit ourselves to C X are the values of the multifunction; write briefly L = The sets allowing = 0 is (very seldornly) convenient for purely formalistic reasons only. on 0. A function p = Talk of "the single-valued case" means = We refer to with on 0 will be called a selection of L = Let X

set 0

L(0)

and

graph(L)

:

0, p

as the range and the graph of L = Rm; both R" and Rm being Throughout the book, X R" and 0 0 C Y endowed with the corresponding Euclidean metrics. Let P be a property of a subset of a metric space (for instance, closed, measurable). We shall say as a general rule

SAIL2. MULTIPUNCTIONS AND DIFFERENTIAL INCLUSIONS

223

that a inultifunction satisfies P if and only if its graph satisfies P. For instance, a multifunction is said to be closed (respectively, measurable) if and only if its graph is closed (respectively, measurable) in the product metric space Y x X. (Whenever we deal with measurability, we consider on Y = Rm the cr-algebra of all Lebesgue measurable subsets and on X = R?i the cr-algebra of Borel subsets.) if the values of a multifunction are closed, bounded, compact, and so on (in X), we say that it is closed-valued, bounded-valued, compact-valued, and so on. Of course, L =

is

is bounded for any bounded subset fl of

called locally bounded if

0. We shall consider only nonempty-valued multifunctions. Ftirther, as a rule, multifunctions investigated in this book are always compact-valued. For such miiitifunctions, we can use:

Definition 1. A multifunction L =

is said to be upper semicontinuous if :

is (relatively) closed in 0 whenever A C W is closed.

Definition 2. A multifunction L =

is said to be lower semicontinuous if L1(V) is (relatively) open in 0 whenever V C R't is open.

Definition 3. A multifunction L =

is said to be continuous if it is simulta-

neously upper and lower semicontinuous.

Evidently, upper (respectively, lower) semicontinuity is nothing else than continuity if L = is single-valued. Here are useful tests for upper semicontinuity:

Proposition AII.14 L= L

(ci. [40, Propositions 1.1-1.2]) Let L =

is upper semicontinuous and 0 is closed, then L = is upper semicontinuis closed L=

otis.

have compact convex Proposition AII.15. (cf. [29, Theorem 11.20]) Let L = values. Then it is upper semicontinuous if and only if the function

sup(p,x)

,EL(t)

APPENDIX II

224

is upper semicontinuous for every x

Proposition AII.16. (cf. [29, Theorem 11.251) Let L = be an upper semi continuous multifunction with compact values. Then L(Q) is compact for any compact subset Q of 0.

Let L = be given. We associate with any real-valued function w = w(e,p) defined on 0 x R" the following marginal function: =

sup

0.

for

(AII.2)

The maximum sets will be denoted by {p

for

0.

(AII.3)

Berge's maximum theorem concerning the above marginal function can be formulated (see [22, p. 123], or [8, Theorem 1.4.161) as follows.

Theorem AII.1T. (Maximum Theorem) Let L = L = and u = are lower semicontinuous, so is the marginal function; (ii)

if L =

and w =

are upper semicontinuoua, so is the marginal

function; (iii) if L = and w = w(e,p) are continuous, so is the marginal function; moreover, defined by (AII.3) is then an upper semicontinuous (nonempty= valued) multifunction. We are now concerned with calculus of measurable multifunctions. As we have mentioned earlier in this section, a multifunction is measurable if and only if its graph is measurable. Notice that measurable multifunctions with closed values can also be defined in the following way:

on a non-empty Lebesgue measurable set 0 C Rm is said to be measurable if L—1(V) is (Lebesgue) measurable

Definition 4. A closed-valued multifunction L = whenever V C

is open.

Especially, the above definition agrees with the classical one for measurable functions if L =

is single-valued.

§AIL2. MTJLTIFUNCTIONS AND DIFFERENTIAL INCLUSIONS

225

Theorem AII.18. (cf. [8, Theorem 8.2.8]) Let 0 E Rk be a measurable single-valued map, and let G = z) be a multifunction measurable in E 0 and continuous in z E IRk (with compact values in IRA). Then the multifunction

0

is measurable.

Theorem AII.19.

[8, Theorem 8.2.11]) Let be a real-valued = E 0 and continuous in p E and let L = be a measurable multifunction on 0 with closed (non-empty) values in Then the margmal function defined by (AI1.2) is measurable. Furthermore, the = (cf.

functson measurable in

multifunction L° =

defined by (AII.3) is also measurable.

Theorem AII.20. (cf. [8, Theorem 8.2.14] and [29, Theorem 111.15]) Let 0 be a Lebesgue measurable subset of If L = is a measurable inultif unction on O with closed (non-empty) values in

then the function

sup(p,x) is measurable for every x E The converse statement holds true if the values of L= are convex and bounded.

In the remainder of this appendix, given J

Jx

[0, T] C R, a multifunction

(t, x) i-+ G(t, x) C R", we are looking for absolutely continuous solutions

of the differential inclusion

E G(t,x(i)) almost everywhere on J.

(AII.4)

For any E J x RTh, denote by Xo(t.,x.) the set of all absolutely continuous functions x = x(t) from J into which satisfy (AII.4) subject to the constraint

x(t.) =

Topological properties of the solution sets will be investigated in the Banach space C(J, of continuous functions given on J with values in R". (The norm in C(J, is the usual "max" one.) We have:

Theorem AII.21.

[40, Theorems 5.2 and 7.1]) Let G = G(t, x) have nonempty closed convex values and be measurable in t E J, upper semicontinuous in x 6 R" such that G(t,x)I

(cf.

:

z E G(t,x)} < c(t)• (1 + ri) for t 6 J, x E

APPENDIX II

226

with c = c(t) a function in L'(J). Then Xcg(t.,x.) is a non-empty compact subset

of C(J,R") for each Further, the multifunctzon EJx Xo(0,x) C C(J, is upper semtcontinuous.

3x

We conclude this appendix with Filippov's theorem for differential inclusions, which is as important as Gronwall's lemma for ordinary differential equations.

Theorem A1L22. (Filippov) [8, Theorem 10.4.1]

(cf. [40, Lemma 8.3]) Let G = G(t, x) have closed non-empty values and y = y(t) be an function absolutely continuous on J, and let 8 > 0. Assume: (i) G = G(t,x) is measurable in t E J; (ii) there exist an r > 0 and a nonnegative function £ = £(i) integrable on J such

that

G(t, r1) C G(t, x2) + £(t)Ix' — x2JB

Vx', x2 E y(t) + rB

for almost all t E J; (iii) the function t f—* 1(t)

p(t)

exp(j

—

£(r)dr)

p E G(t, y(t))} is integrable on J. (8

and

+f

If i7(T) < r, then for every E y(0) + SB, there exists a solution x = x(t) in of(AII.4) such that

VtEJ Ix'(t)

—

y'(t)I < £(i)zj(t) + -y(t)

almost everywhere on

.J.

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Index

equation of the conservation law, 12, 45 Cantor function (ladder), 86 Cantor set, 86 Filippov's theorem, 180, 226 Carathéodory differential equation, 98 Carathéodory's conditions, 97, 166, 191 fold point, 57 characteristic generalized solution, 32, 55 curves, 1 general PDE of first-order, 4 equations, 2, 4 global C'-solution, 92 strips, 4 global semiclassical solution, 147, 154 system of differential equations, 4 1 global solution, 104 C"-solution, 7 collision Haar's differential inequality, 85 of characteristic curves, 26, 28 Haar's theorem, 7 Hadamard's lemma, 19 of singularities, 41, 63 Hopf's formulas, 104 comparision equation, 98 concave-convex function, 128 conjugate (Fenchel) concave-convex, 131

Jacobian, 2 Jacobi matrix, 2

(Fenchel) convex, 105, 219-222

cusp point, 58

Lebesgue's theorem, 170 Legendre transformation, 128, 220-222 life span of classical solutions, 12

function, 113 d.c. representation, 117 differential inclusion, 225 differential inequality of Haar type, 85 Dini serniderivative, 108, 209

marginal function, 224 maximum theorem, 224 minimax solution, 168, 179, 194, 196 monotonicity condition, 192 rnultifunction, 222

Legendre, 128, 220-222 contact discontinuity, 70

d.c.

Dini semidlifferential, 210 directional derivative, 108

closed, 223 closed-valued, 223 continuous, 223

effective domain, 105, 218 entropy condition, 52 epigraph, 218 equation of Hamilton-Jacobi type, 34

locally bounded, 223 lower semicontinuous, 223 measurable, 223-224 upper semicontinuous, 223

INDEX

non-degenerate singularity, 58 proper, 123, 218

quasi-linear PDE of first-order, 1 quasi-monotonicity condition, 192 Rankine-Hugoniot's condition, 41 rarefaction wave, 68 relative interior, 217 semi-concavity, 32, 39, 56 shock wave, 53

strict convexity, 118, 122, 219 subdifferential, 210 subgradient, 210 subsolution, 168, 179, 194, 196 superdifferential, 210 supergradient, 210 supersolution, 167, 179, 193, 196 viscosity solution, 212

Wniewski's theorem, 9 weakly-coupled system, 97 weak solution, 45

237

classical solutions lhc ch.tracteristic meihod s tekls the local theory to lirst-ordcr nonlinear partial ditfereniial equations. The global theory has principally depended on the vanishing s iscosity method. ftc authors between the local and global theories bridge the ol this by using the characteristic method as a basis for settiliC a theoretical framework for the studs of global generaltied solutions. That is. they extend the smooth solutions obtained by the characteristic method. Within ork. they oIler material on the life span of classical such a solutions, the construction of simzularities of generali,ed solutions, new existence and uniqueness theorems on minimax solutions. inequalities of' Haar pe and their application to the uniqueness t global explicit iornnilas tor global semi—classical solutions, and iiopi solutions.

i'liis soltirne represents a comprehensixe exposition 01 the authors' works ox'er the last decade. concentrating on sonle basic liicts and ideas of the general lied characteristic methods br study tug global si lutions. Suitable as a text. the book is sell-contained and assumes as prerequisites only . and ordinary differential equations. lhe basic iileasurc I het . tt appendices pros ide necessary material, primarily on nonsmooth analysis and the t he ry of di fb'ere it ial inclusions.

Reiu/ers/iij,: \latheniaticians.

sicists. and engineers: in n in I near partial di I Icrent al equations, ill terent ,il inequalities. multix aiucd

analysis. diltcrential gatnes. and related topics in applied analysis: upper- les ci undergraduate and graduate students in these disciplines.

irait l)uc

is l)iicctor of the Ilaititi Institute of \latliciii:ities and

I 'rol'essor. I )epait ilicilt of Partial 1)1 fferential

I

ions

\Iikio l'suji is a Professor ol' \lathcitiatics at the laLtilty of Sciences ol ot crslty Ky oto Ihai Son is .in Associate i'rolcssor of \Ijthcnuttcs. College \gti%tiI ol Sciences, line tnixersity.

CHAPMAN & HALL/CRC

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