πCH… H山C Monographs a.nd Surveys in Pure a-ndApplied Mathematics
101
THE CHARACTERISTIC METHOD AND ITS GENERALIZATIONS FOR FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TRAN DUCVAN MIKIOTSUJI NGUYEN DUYTHAI SON
CHAP问AN
& HALUCRC
Monographs and Surveys in
Pure and Applied Mathematics
Main Editors H. Brezis , Université de Paris R. G. Douglas, Texas A&M University A. Jeffrey, UniversiηI of Newcastle upon 乃ne (Founding Editor)
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π CH 队4认附酬帕 州柳 AP 仰棚附 P刊仲 tv…L丛 肌 ω 山/尼 Lυ C
and Surveys 阳 in Pure and Applied Mathematics
f问w叮、》叶1onographs
I0I
THE CHARACTERISTIC METHOD AND ITS GENERALIZATIONS FOR FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS TRAN DUCVAN MIKIOTS叫 l
NGUYEN DUYTHAI SON
CHAPMAN & HALUCRC Boca Raton London New York Washington, D. C.
Library of Congr咽s Ca taI oging-in-Publication Data Tran, Duc Van. The characteristic method and its generalizations for 且rst -order nonlinear p缸tia1 differentia1呵uations / Tran Duc Van , Mikio Tsuji, and Nguyen Duy Thai Son. p. cm. -- (Chapman & Ha1 VCRC mongraphs and surveys in pure and applied mathematics ; 101) lncIudes bibliographica1 references and index. ISBN 1-58488-016-3 (a1k. paper) 1. Di fferenti a1 equations , Nonlinear--Numerica1 solutions. I.Tsuji , Mikio. 11. Nguyen , Duy Thai Son. 111. Title. IY. Series QA374.T65 1999 519.5'.353--dc21 99-27321 CIP 四 is book contains information obtained from authentic and high1 y regarded sources. Reprinted materi a1 is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable e仔urts have been made to publish reliable data and information, but the author and the publisher cannot assume responsib且ity for the va1idity of a11 materia1s or for the consequences of their use. Neither 由is book nor any part may be reproduωd or transmitted in any form or by any means, eIectronic or mechanica1, incIuding photocopying, microfilming, and recording , or by any information storage or retrieva1 system, without prior pem咀ssion in writing from 由e publisher. The consent of CRC Press LLC d∞s not extend to copying for genera1 distribution, for promotion, for creating new works , or for res a1e. Specific pen回ssion must be obtained in writing from CRC Press LLC for such copying. Direct a11 inquiries to CRC Press LLC, 2α)() N. W. Corporate Blvd. , Boca Raton , F1 0rida 3343 1.
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Contents
Contents Preface Chapter 1. Local Theory on Partial Differential Equations of First-Order 1.1. Characteristic method and existence of solutions 1. 2. A theorem of A. Haar 1. 3. A theorem of T. 巩Tazewski
1 1
7 8
Chapter 2. Life Spans of Classical Solutions of Partial Differential Equations of First-Order 2. 1. Introduction 2.2. Life spans of classical solutions 2.3. Global existence of classical solutions
12 12 13 18
Chapter 3. Behavior of Characteristic Curves and Prolongation of Classical Solutions 3. 1. Introduction 3.2. Examples 3.3. Prolongation of classical solutions 3 .4. Su伍cient conditions for collision of characteristic curves 1 3.5. Su伍cient conditions for collision of characteristic curves 11
22 22 23 24 26 28
Chapter 4. Equations of Hamilton-Jacobi Type in One Space Dimension 4. 1. Nonexistence of classical solutions and historical~arks 4.2. Construction of generalized solutions 4.3. Semi-concavity of generalized solutions 4 .4. Collision of singularities
30 30 34 38 41
Chapter 5. Quasi-linear Partial Differential Equations of First-Order 5. 1. Introduction and problems 5.2. Difference between equations of the conservation law and equations ofHamilton-Jacobi type 5.3. Construction of singularities of weak solutions 5 .4. Entropy condition Chapter 6. Construction of Singularities for HamiltonJacobi Equations in Two Space Dimensions 6. l. Introduction
44 44 47 48 52 55 55
CONTENTS 6.2. Construction of solutions 6.3. Semi-concavity ofthe solution u = u(t , x) 6 .4. Collision of singularities
Chapter 7. Equations of the Conservation Law without Convexity Condition in One Space Dimension 7. 1. Introduction 7.2. Rarefaction waves 创ld contact discontinuity 7.3. An example of an equation of the conservation law 7.4. Behavior ofthe shock 51 7.5. Behavior ofthe shock 52 Chapter 8. Differential Inequalities of Haar Type 8. 1. Introduction 8.2. A differential inequality ofHaar type 8.3. Uniqueness of global classical solutions to the Cauchy problem 8 .4. Generalizations to the case of weakly-coupled systems
56 61 63 67 67 68 70 74 77 82 82 84 91 95
Chapter 9. Hopf's Formulas for Global Solutions of Hamilton-Jacobi Equations 9. 1. Introduction 9.2. The Cauchy problem with convex initial data 9.3. The case of nonconvex initial data 9 .4. Equations with convex Hamiltonians f = f(p)
103 103 105 113 121
Chapter 10. Hopιτ丁ype Formulas for Global Solutions in the case of Concave-Convex Hamiltonians 10. 1. Introduction 10.2. Conjugate concave-convex functions 10.3. Hopf-type fo口丑ulas
127 127 128 136
Chapter 11. Global Semiclassical Solutions ofFir吼-Order Partial Differential Equations 146 11.1. Introduction 146 11. 2. U niqueness of global semiclassical solutions to the Cauchy problem 148 156 11. 3. Existence theorems Chapter 12. Minimax Solutions of Partial Differential Equations with Time-measurable Hamiltonians 12. 1. Introduction 12.2. Definition of minimax solutions 12.3. Relations with semiclassical solutions 12 .4. Invariance of definitions 12.5. U niqueness and existence of minimax solutions 12.6. The case of monotone systems Chapter 13. Mishmash 13. 1. Hopf's formulas and construction of global solutions via characteristics
161 161 165 173 176 179 191 198 198
CONTENTS 13.2. Smoothness of global solutions 13.3. Relationship between minimax and viscosity solutions
205 208
Appendix 1. Global Existence of Characteristic Curves
214
Appendix 11. Convex Functions , Multifunctions , and Diπ'erential Inclusions AI I.1. Convex functions AI I. 2. Mu1t ifunctions 缸ld differential inclusions
217 217 222
References
227
Index
236
Preface One of the main results of the classical theory of first-order partial differential equations (PDEs) is the characteristic method which asserts that under certain assumptions the Cauchy problem can be reduced to the corr臼ponding characteristic system of ordinary differential equations (ODEs). To illustrate tlús , let us consider the Cauchy problem for the nonviscid Burger equation: θuθu 一一 +u 一一= 御街
0,
t
> 0,
u(O ， x)=l巾)，
x
(1)
ξR ，
x
εR
We try to reduce the problem (1)-(2) to an ODE along some curve x precisely, let us find x = x(t) such that
(2)
= x(t).
More
去M By the chain n白， we may simply require dxjdt can be defined by
u , and so the characteristics
z 二 x(t)
生 =u(t， x). dt
(3)
Along each characteristic x = x( t) we have d叫 dt = 0 , i.e. , u = u(t , x(t)) takes a constant value and then the characteristic must be a straight line with slope given by (3). Thus , by the initial data (匀， the characteristic passing through any given point (0 , s) on the x-axis is (4) x = s + h(s 沛， on which u has the constant value:
u
= h(s).
(5)
Hence , if the Cl-norm of h h(s) is bounded , then , by means of the implicit function theorem and (份， we can get
s=s(t , x)
(6)
PREFACE
for small values of t. solution)
Substituting (6) into (5) gives the classical solution (C 1 -
u
=
h(s(t , x))
(7)
to our Cauchy problem (1)-(2). However , in general , this solution exists only locally in time. In fa叽 if h = h(s) is not a nondecreasing function of s , there exist two points (0 , Sl) and (0 , S2) on the x-axis such that
Sl < S2
and
h(Sl) > h(S2).
(8)
Then the characteristic curves beginning from (0 , Sl) and (0 , S2) will intersect at tíme
t-
s2- S ~>O.
h(s t} - h(S2)
Since the solution u u( t , x) is constant along each of the two curves but has different values h(s t} and h(S2) , respectively, at the intersection point , the value of the classical solution cannot be uniquely determined. Hence , in this case the Cauchy problem (1)-(2) never admits a global classical solution on {t 主 O}; in fact , the classical solution will blow up in a finite time no matter how smooth and small the initial data h = h( s) 缸e. On the other hand , if h h( s) is a nondecreasing function of s , then the 巳haracteristics emanating from distinct points (O , Sl) and (0 , S2) on the x-axis will not intersect , and thus the solution u = u( t , x) will exist globally for t 主 O. The previous example shows ，但仅ally speaking , 出 t ha 剖t for (陆如 firsωtιω创卧心 创 O r时 伽削臼叫) non d 血 1 partial dωif宦fe臼ren 时 1曲tia 叫1 equations or systems , classical solutions to the Cauchy problem exist only locally in time , while singularities may occur in a finite time , even if the initial data are sufficiently smooth and small. Therefore , the notions of generalized solutions or weak solutions have been introduced. In fact , the global existence and uniqueness of generalized solutions have been well studied from various kinds of viewpoints. In the 1950s-1970s , the theory and methods for constructing generalized solutions of first-order PDEs were disω叮叮ed by Aizawa , S. [2]-[4], Bakhvalo巾， Benton , S.H. [2 月， Conw町， E.D. [32]-[33], Douglis , A. [44]-[剑， Evan , C. , Fleming , W且 [51]-[52 ], Friedman , A. [54 ], Gelfru吨I. M. ， Godunov , S. K., Hopf, E. [63]-[64], Kuznetso叭 N.N. [95], Lax , P.D. [9η-[98]， Oleinik , O.A. [112], Rozdestvenskii , B. L. [118] , and other mathematicians. Among the investigations of this period we should mention the results of Kruzhko巾， S.N. ([8可 -[92] ， [94]) , which were obtained for Hrunilton-Jacobi equations with convex Hamiltonian. The global existence and uniqueness of generalized solutions for convex Hamilton-Jacobi equations were well studied by several methods: variational method , method of envelopes , vanishing viscosity method , nonlinear semi-group method , etc. Since the early 1980s , the concept of viscosity sol包tions introduced by Crandall and Lions has been used in a large portion of research in a nonclassical theory of first-order nonlinear PDEs as well as in other types of PDEs. The primary virtues of this theoηr are that it allo啊s merely nonsmooth functions to be solutions of nonlinear PDEs , it provides v芭ry general existence and uniquen
PREFACE
yields precise fonnulations of general boundary conditions. Let us mention here the Crandall , M.G. , Lions , Pι. ， Aizawa，丘， Barbu, V. , B町di ， M. , Barles , G. , Barron , E.N. , Cappuzzo-Dolcetta , 1., Dupuis , P. , Evans , L. C. , Ishii , H. , Jensen , R. , Lenhart ，丘， Osher，旦， Perthame , B. , Soravia，卫， Souganidis , P. E., Tataru , D. , Tomita , Y. , Yamada , N. , and many others (s白间， [10]-[20 ], [28 ], [35]-[39] , [47]-[50 ], [67]-[72 ], [79] , [99]-[10 月， [122]-[123 ], [131] , and the references therein) , whose ∞n tributions make great progress in nonlinear PDEs , and where the global existence and uniqueness of viscosity solutions have been established almost completely. The concept of viscosity solutions is motivated by the classical maximum principle which distinguishes it from other definitions of generalized solutions. Another direction in the theory of generalized solutions is motivated by differential game theory as suggested by A. 1. Subbotin. This leads to the notion of mzmmαx solutions of first-order nonlinear PDEs. As the tenninology "minimax solutions" indicates , the definition of such global solutions is closely connected with the minimax operations. This definition is b田ed ， to some extent , on the so-called "characteristic inclusions" (a generalization of the classical characteristic system in this 此uation). Subbotin 矶d his coworkers ([1 ], [124]-[127 ], [129]-[130]) developed an effective theory of minimax solutions to first-order single PDEs 缸ld gave nice applications to control problems and differential games. The research of minimax solutions employs methods of nonsmooth analysis , Lyapunov functions , dynamical optimization, and the theory of differential games. At the same time , the research contributes to the development of these branches of mathematics. A review of the results on minimax solutions and their applications to control problems and differential games is given in [124]-[125]. We also want to mention the investigations on PDEs based on the idempotent analysis and Cole-Hopftransformation , which have been discovered by V.P. Maslov and his coworkérs. Indeed , V.P. Maslov , V.N. Kolokol'tsov , S.N. Samborskii , and others developed a nonclassical approach to define the weak global solutions to first order nonlinear PDEs , in which by a suitable structure of new function serr让modules ， nonlinear operators become "linear" ones. In this direction , based on the methods and results of the well n创丑es:
PREFACE
As for the theory of partial differential inequalities , the first achievements were obtained by Haar [6 月， Nagumo [107] , and then by Wazewski [154]. Up to now the the。可 has attracted a great deal of attention. (The reader is referred to Deimling [40] , Lakshmikantham and Leela [96 ], Szarski [128 ], and Walter [153] , for the complete bibliography.) It must be pointed out that the characteristic method gives us the local existence and uniqueness of classical solutions to first-order nonlinear PDEs. We would like to use tms method as an important basis for setting the global generalized solutions. This book is devoted to some developments of the characteristic method and mainly represents our results on first-order nonlinear PDEs. Our aim in the first seven chapters is to fill a gap between the local theory obtained by the characteristic method and the global theory which principally depends on vanishing viscosity method. This is to say, we try to extend the smooth solutions obtained by the characteristic method. Our first problem then is to deter mine the life spα ns of the smooth solutions. Next , we want to obtain the generalized solutions or weak solutions by explicitly constructing their singularities. In Chapter 1, we present the classical results wmch are necessary for our follow ing discussions: the characteristic method , existence of local solutions , and Theorems of Haar and Wazewski on the uniqueness of solutions to the Cauchy problem in C 1 -space. Chapter 2 is devoted to the life spans of classical solutions of the noncharacteristic Cauchy problem. Our method depends on the analysis of the smooth mapping obtained by the family of characteristic curves. Even if the Jacobian of the mapping may vanish at some point , we can sometimes extend the classical solution beyond the point where the Jacobian vanishes. Therefore , we are obliged to consider very often some properties of the inverse of the mapping in a neighborhood of a singular point. This is the subject of Chapter 3. In Chapters 4 and 5, we consider the extension of solutions beyond the singularities of solutions in the case where the dimension of space is equal to one. Then our principal problem is to construct the singularities of generalized solutions or of weak solutions. The theme of constructing the singularities of solutions is picked up again in Chapter 6 for convex Hamilton-Jacobi equations in two space dimensions. The difference between Chap
PREFACE
"characteristic bundles" are invoked instead of characteristic differential equations and characteristic curves. Chapters 9-10 are devoted to the study of Hopf-type formulαs for global solutions to the Cauchy problem in the case of non-convex, non-concave Hamiltonians or initial data. In Chapter 9 , we first consider the case where the initial data can be represented as the minimum of a family of convex functions , and next the case where it is a d.c. function (i.e. , it can be represented as the difference oftwo convex functions). In Chapter 10 , the Hamiltonians are concα ve-convex functions. The method of Chapters 8-10 allows us to deal with global solutions , the condition on whose smoothness is relaxed significantly. In Chapter 11 , we propose the notion of globα 1 semiclassical solutions , which need only be absolutely continuous in the time variable , and investigate their uniqueness and existence. By the way, an answer to an open uniqueness problem of S.N. Kruzhkov [93] is given. In Chapter 12 , we extend the notion of Subbotin 's 口IÏnimax solutions to the case of first-order nonlinear PDEs with time-measurable Hamiltonian. The uniqueness and existence of such solutions are investigated by the theory of multifunctions and differential inclusions. Our road here is devious (by some 平 erturbation technique" on sets of Lebesgue measure 的， and proceeds via an implicit version of Gronwall's inequality and via a sharpening of a well-known theorem on the Lebesgue sets for functions with par臼neters. The results are new even when restricted to the case of continuous Hamiltonians. Generalizations for monotone systems w i11 also be considered. Finally, in Chapter 13 we examine Hopf's formulas in relations with the construction of global solutions via characteristics and the smoothness of the solutions. In this chapter , the relationship between minimax and viscosity solutions is also investigated. We have to say that this book is not designed as an introduction to , or a guidebook on , the general theory of 如st-order nonlinear PDEs. Our goal is not to try to cover as many subjects as possible , but rather to concentrate on some basic facts and ideas of the generalized characteristic methods for studying global solutions. Suitable as a text , the book is self-contained and assumes as prerequisites only calculus , linear algebra, topology, ODEs , and basic measure theory. In the appendices at the end of the book we collect nec
Chapter 1 Local Theory on Partial Differential Equations of First-Order ~1.1.
Characteristic method and existence of solutions
Partial differential equations of first-order have been studied from various points of view: for example , classical mechanics , variational method , geometrical optics , etc. In this chapter we will always suppose that the equations and solutions are realvalued. The classical method to solve the equations is the characteristic method. As this is the fundamental tool in our following discussions , we will give here a brief explanation of the method. For more detailed results and geometrical meanings , refer to , for example , R. Courant and D. Hilbert [34] and F. John [80]. First we consider a quasi-linear partial differential equation of first-order as follows:
内，事内
去+罢αi(川)23 斗。(川 u(O , X)
= 4> (x)
on
def
of {(O , x ,4> (x)) and
4>
=
x E Uo} in
4> (x) are of class
C1
U,
Uo~' {x ε IR n : (O ， x) ε U} ，
where U is an open neighborhood of (t , x)
IR n +2.
in
= (0 , 0).
(1.1) (1. 2)
Let V be an open neighborhood
Assume that aí = 咐，吼叫 (i=O ， l ， … ， n)
in V and Uo, respectively. A function is said to be
of class C k if it is k-times continuously di晶rentiable ， and Ck(U) is the fan均 of functions being of class C k in U. A Ck-function means that it is a function of class
Ck. Characteristic curves of (1. 1)-(1. 2) 缸e deßned by solution curves of the following system of ordinary differential equations:
t (仔 ιdtι= 坠 可… G
生 = ao(巾， υ) . dt
(1. 3)
1. LOCAL THEORY
2
In accordance with (1 功， the initial condition for (1. 3) is given by 引 (0) =的
(i=1 ， 2 ，...， n) ，
v(O)= φ(y).
The ordinary differential equations in (1. 3) are called the
(1. 4)
"chαracteristic
equations"
for (1. 1) , where we use v
= v(t , y) instead ofu = u(t , x) to avoid confusion. In the following discussions , u = u( t , x) is a solution of (1. 1) and v = υ (t ， y) is a solution of (1. 3)-( 1. 4) which is equal to the value of u = u(t , x) restricted on the corresponding solution curve x = x(t , y) of (1.3)-( 1.4). As ai = ai(t , x , V) (i = O ， l ，...， n) 皿d cþ= φ( x) are of class
C 1 in V and 吨， respectively, the Cauchy problem (1. 3)-( 1. 4)
has a system of solutions xi class
C1
=
Xi(t , y) (i
in a neighborhood of {(O , y)
= 1, 2,... , n)
and v = υ (t ， y) which are of
y ε Uo }.
Let us fix our notations on derivatives of functions. i.e. , x
=
t(XllX2'...'Xn). Therefore dxjdt
=
A vector x is vertical ,
t(dxddt , dxddt ,..., dxnjdt). On
the other hand , given any real-valued functionφ = cþ(叫， we write gradcþ( x ) 御用x
x
=
= cþ'(x) =
(御用xll âcþjθX2 ， …，御用x n ).
For an n-vector valued function
x(y) of an n-vector y , we defi.ne its Jacobi matrix and Jacobian , respectively,
by
θXl
θXl
θXl
θ'yl
θy2
θyn
(ZL=13.Y and
去(y) 哩叫去儿，j=1， 2 ，. .， n
We will sometimes write the Jacobi matrix simply by we see that (DxjDy)(t , y) U ε Uo } ，
=
1 for t
= 0, Y
t.\ I((予X :,-. J.
\θyj}
Since x(O , y)
= y,
E Uo . In a neighborhood of {(O ， ν)
:
as (DxjDy)(t , y) does not vanish, we can uniquely solve the equation
= x(t , y) with respect to y and write the solution by y = ν (t， x). Puttingu(t， x) 哇f v(t , y(t , x)) , we will prove that u = u(t , x) satisfies (1. 1)-( 1. 2) in a neighborhood of x
the origin. Theorem 1.1. The Cauchy problem (1. 1)- (1 功 hαs un叩tely α solution in a neighborhood
01 the
origin.
01 class Cl
~l.l.
CHARACTERISTIC METHOD AND EXISTENCE OF SOLUTIONS
3
Proof. We use the notations introduced in the above. The following discussions are true only in the definition domain of y
= ν (t ，
x). This domain is a neighborhood of
the origin , where the Jacobian (Dx/Dy)(t , y) does not vanish. As x
= x(t , y(t , x)) ,
we have
(~二儿，j=1， 2 ，0 0 ， n (缸，户口， JI(叫
( 1. 5)
0
。x . (θXi 、
θy
a\θyj ) i ，j=l ι.. ，n.
ðt
(1. 6)
As u(t , x) = υ (t ， ν (t ， x)) , we have θuθuθv
一=一 ， y) ðt ðt (t '0'07/
åu
+ åu ;，~ . ~ ðt (t , x).
(1. 7)
•
By (1. 7) , using (1. 5) and (1. 6) , we get 百 (t ，
x)
= αo(巾 ， u(t ， x)) 一
θv
(åUi \
. (一三二 l
τ
θν\θXj ) 1 三'，}三 n å
=α o(t， x , u(t , x)) 生 (t， x) . 生 ðt θx'-'-/
=句川一去川去 As u(O , x) = v(O ， ν (O ， x))
rþ(x) , we see that u
u(t , x) satisfies the Cauchy
problem (1. 1)-( 1. 2) in the above neighborhood of the origin. We will show the uniqueness of solutions. Let u =
u 衍，
C10f(11)-(12) , and put Z(tJ)qzf u(t， zO， ν)) where x the solutions of (1. 3)-( 1. 4). Then the
x) be any solution of class
= x(t， ν) and v = v(t， ν) are def
~
differenceω (t ， ν) ~. u(t ， ν)
- v(t , y) satisfies
the following Cauchy problem:
(d一 ω(川) = L: (αj(t ， x ， v) 一 α (t， x 叫)一+(咐， x ， u) n/~\θu } = 1 \ } \ ", W'~!)θ Xj
ω (O ， y)
I
飞
咐 ， x ， v)) ,
= O.
As the right-hand side of this differential equation can be estimated by
Mlu -
vl =
MI叫， we get ω( t ，的三 0 ， i.e. u(t , y) ==咐， ν) for any (t ， 的 in a neighborhood of
the origin. This rneans that the solution of C 1-class is unique a10ng the curves
x
= x(t ， 的.
That is to
say，出 10吨 as
the Jacobian
the solution of (1. 1)-( 1. 2) is unique in the
(Dx/Dy)(t ， ν)
C 1 -space.
does not vanish , 口
1. LOCAL THEORY
4
Next we consider the Cauchy problem for general partial differential equations of first-order as follows:
去 +f(t， x， u， 去) = 0 u(O , x) = <þ (x)
(1. 8)
U
in
Uo ~f {x ε IRn : (O， x) ε U} ，
on
(1. 9)
where U is an open neighborhood of the origin. Let V be an open neighborhood of
x E Uo} in IR x IR n x IR x IRn. Assume that f =
{(O ，川(x) ，<þ'(x))
and
<þ = <þ (x) are of class
C2
Characteristic strips for
f(t ， 川 p)
in V and Uo, respectively.
(1. 8)-( 1. 9) 町ede直ned
as solution curves of the following
system of ordinary differential equations: dXi
δf
百=再;(t，川 p)
(i = 1, 2, . . . , n) ,
!; åf 一=艺 Pi τ (t ， x ， v ， p)
dt
i~".åp
-
f(t , x , v , p) ,
dPi δfδf -二一( t , x , v , p) - Pi -;" (t , x , v , p)
dt
åXi
(1. 10)
(i = 1, 2, . .. , n) ,
with
Xi(O) = 钩，
v(O) = <þ( ν) ，
Pi(O) = 矿 (ν )
We remark that system (1. 10) is called the equαtions ，"
or simply
"chαracteristic
(i=1 , 2,..., n).
"chαracteristic
(1. 11)
system of differential
equations ," for equation (1. 8). As f and <þ
are of class C 2 , the Cauchy problem (1. 10)-( 1. 11) has uniquely the solutions x
x(t , y) , v = of class
C1
v(t ， ν)
and p
with respect
= p( t , y) to (t , y).
in a neighborhood of t As x(O , y)
= y , we
=
=
O. Moreover , they are
have (Dx/Dy)(O ， y) 二 1 for
any y ε Uo . Therefore , there exists an open neighborhood W of {(O , y) where the Jacobian (Dx/ Dy)(t , y) does not vanish and the equation x
y ε Uo }
= x(t , y) can
be uniquely solved with respect to y. Denote the solution by y = ν (t ， x) ， and put
u(t , x)
have
~l.l.
CHARACTERISTIC METHOD AND EXISTENCE OF SOLUTIONS θXl
p
ι'bu
)
Mud
一一
‘飞
..,,-Ttw
，，a‘飞
noxo U-v
δXl
δXl
θν1θν2
δνn
)
Hg
(1.12) δX n
θX n
δν1δy2
θ yn
θX n
仙 ere
5
p(t , y) = (Pl (t , y) , P2(t , y) , . .. , Pn( t , y)).
Proof. We put θXl
δXl
θXl
δyl
。y2
θyn
。马二
0年二
。土二
θyl
θ'y2
θνπ
defθu
z(t , y) ~θu (t ， y) - p( t , y)
Using (1. 10) , we have
(dθf dtz(t , y) = 一再 (t ， X(t , y) , V(t , y) , p(t , y)) z(式的， z(O ， y) 二 o.
As this is a linear ordinary di fIerential equation concerning z z(t ， y) 三 o.
口
Remark. Fix any y E UO, and let J solutions X
Z(t , y) , we get
X(t , y) ,V
c
V(t , y) and P
IR be an interval around 0 on which the =
p(t , y) of the characteristic equations
(1.1 0)-( 1. 11) exist. Then (1.12) is true for each t vanish at (t , y). Recall that in W we have (Dx / Dν )(t ， y)
手 0，
ε J even if the Jacobian m町
and we can uniquely solve the
= x(t ， ν) wi th respect to y. The sol u tion has been denoted by y = y (t , x) and used to de鱼ne u(t , x) 乍fu(tJ(tJ)).
equation x
Corollary 1. 3. In the definition domain of y = ν (t ， x) ， we have
主 (t， x) = pi(t， y(t， x)) 问 2，...
, n)
Using Lerruna 1. 2 and its corollary, we get the following:
(1.13)
1. LOCAL THEORY
6
Theorem 1. 4. Suppose f εC 2α nd <þ ε C 2 • Then the Cαuchy problem (1. 8)-( 1. 9)
has uniquely a solution of class C 2 in α neighborhood of the origin. Proof. We 岳rst prove the existence of solutions. Let u( t , x) 哲 υ (t ， y(t ， x)) as in the above notations. By Lemma 1. 2 and Corollary 1. 3 , using (1. 10) , we have
坐(t， x) = 生(t， y(t，冽冲 Z叫x叫)川)+' 去川， 冲冲 Z叫x叫)川) 鱼0 &t,-,i>,-'-,, å 街 θ t'-'-'
1J
ιδf i=l "'1"'&.
rl.
1.
11
\ \
f
= >.如 (t ， y(t ， x)) τ - f(t , x , v , p) - p(t ， 仰， x)) 忙二 飞~ ~3
飞 /
τ
l:::;i ,j :::;n
二 θf = L，Pi(t ， y(t ， x)) 再一月 ， x ， v ， p) - p(t ， y). 否J ~Y'
= -f(t , x , v(t , y(t , x)) , p(t , y(t , x)))
=肌叫树，去。 Moreover , by (1. 1 月， u(O , x)
=
v(O ， ν (O ， x))
= <þ (x). It
follows that u
= u(t , x)
is a
solution of (1. 8)-( 1. 9). As Pi = Pi(t , y) (i = 1, 2,... , n) are of class C 1 in W and y= ν (t ， x) is of class
C1 in its definition domain , we see by (1. 13) that u = u(t , x)
is of class C 2 in the definition domain of y = 仰， 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(t , x) be a solution of class C 2 of (1. 8)-( 1.时， and z = z(t , y) = (Zl (t , y) , . . . , zn( t , y)) be a solution of
(dzθf 一 = :J 'V (t ,_, ， z ， u(t ， z) dt
θIpi
Zi(O)
一 (t ， z))
_'V ，_" θz
(i =
1 ， 2 ，...，叫，
= yi.
We putω (t ， y) ~f u(t , z(t , y)) and q(t , y) ~f (仇/δx )(t , z(t , y)) , then we get 一 = -f(t ， 川， q) + >~q队 ν) :~ (t , z , w , q) , |ωιθf dt 孚Jθ'Pi
i=l
dqi θfθf 一=一一 (t ， z , w , q) - qi(t , y) 一(毛巾 ， q) dt
with
ω(0 ， ν)=φ(ν)
θ的
and q(O , y)
= φ， (ν).
(z ， ω ， q)
=
(i=1 , 2 ,..., n) ,
Hence
(z(t ， 的， ω ( t ， 的， q(t ， ν))
~1. 2.
A THEOREM OF A. HAAR
7
satisfies the Cauchy prob1em (1. 10)-( 1. 11). By the uniqueness of solutions of ordinary differentia1 equations , we have v(t ， ν)=ω (t ，
x(t , y) = z(t , y) ,
y) ,
p(t ， ν)
=
q(t ， ν) ,
where (叭叭 p) = (x(t ， y) ， v(t ， ν ) ， p(t ， y)) is the solution of (1.1 0)-( 1.1 1) which has a1ready appeared in Lemma 1. 2. This says that the solution of class C 2 is unique a10ng the curves x = x (t , y). Therefore , as
10吨 as
the Jacobian
(Dx/Dy)(t ， ν)
not vanish , the 叫ution of class C 2 of (1. 8)-( 1. 9) is unique.
does 口
31. 2. A theorem of A. Haar We have seen by Theorem 1. 4 that the Cauchy prob1em (1. 8)-( 1. 9) has a solution of class C 2 in a neighborhood ofthe origin. But , as the equation (1. 8) is of first-order , we wou1d 1ike to discuss the existence and uniqueness of solutions in C 1 -space. In the following , we write a solution of class C k as a Ck-so1ution , C 1 -s01utions will be sometimes called
吐出sica1
solutions."
One of our aims is to consider what kinds of phenomena may appear when we extend the classica1 solutions of (1. 8)-( 1. 9). In this procedure , we need the uniqueness of solutions in C 1 -space. This subject had been well studied by A. Haar [61] and T. Wazewski [154]-[155]. Moreover , their results are indispensab1e to deve10p our discussions. But , as it seems to us that they are not well-known , we would like to introduce them. In this section , we report a theorem of A. Haar [61]. We consider the Cauchy prob1em (1.创刊1. 9) in one space dimension , that is x E R 1 . Let ß be a triang1e defiued by ß={(t ， x):O 三 t 三 α ，
where α> 0 , L ~三 0 ， c {(u ， p)
<
c 十 Lt 三 z 三 d - Lt} ,
d and 2Lα 三 d - c. Moreover ，反 is a cornpact set in
εR 2 }.
Theorem 1. 5. (A. Haar [61]) Suppose that the function f α Lipschitz
condition
= f(t , x , u , p) satisfies
α s folloωs:
If(t , x , u , p) -
f(t ， x ， v ， q)1 三 Llp
- ql
+ Mlu - vl
1. LOCAL THEORY
8
for all
(t ， x ， u ， p) α nd
(t , x ， 叫 q) in ß
x 允 Let
αnd (Uj(t ， 叫， (θUj/θx)(t ， x)) ε 反 for
the
eqωtion
domaiηß
(1. 8) in the
Ul(t ， X) 三 U2(t ， X)
Uj = Uj(t , x) (j = 1, 2) be in C 1 (ß)
all (t , x)
εß.
扩 Uj(t ， x)
(j
and Ul(O , X) = U2(O , X) for x
= 1, 2) satisfy ε [c ， 司 ，
then
inß. f(t , x , u , p) , the difference
Proof. According to the Lipschitz continuity of f
Z
<- L
AEKW
十
AZU
叫的h
内的a
z(t , x) ~f Ul(t , X) - U2(t , X) satis自es the inequality Z
MZ
AZLU
Z
(1.1 4)
Assume that z 二 z( t , x) takes some positive values in ß. We putω (t ， x) ~f
e-atz(t , x) for
α > M.
P E ß at which w
As ω=ω (t ， x) is continuous on ß , there exists a point
= ω (t ，
x) attains the maximum. The point P is not on the initial
ω(0 ，
x) == O. Let us consider the derivatives of ω=ω (t ， x) at
line {t = O} because
P with respect to the directions (-1 , - L) and
(一 1 ， L).
As
ω=ω (t ，
x) takes the
maximum at the point P , we have θωθωθωθω L -,,- (P) 一 τ (P) 三 O.
-L τ (P) 一 τ (P) 仆 ， ux cn
Hence it holds that
Rewriti吨 this
ux
στ
主 (P) 三 LI~:(p)1
for z = z(t ， 叶， we get
去但)主 ω(p)+LI::(p)1 As z(P)
> 0 and
α >
M , this contradicts (1. 14). Therefore z(t ， x) 三 o for
all (t ， x) εß. Since we can similarly prove that z(t ， 叫主 0 ， we obtain Ul(t ， X) 三
U2(t , X) in
ß
口
31. 3. A theorem of T. Wazewski In this section we report a theorem of T. Wazewski [154] which is a generalization of Haar's theorem to arbitrary space dimensions. Let us consider the Cauchy problem (1. 8)-( 1. 9) in a pyramid ß which is defined by
~1. 3.
where α> 0 , Li ?: 0 , ci
A THEOREM OF T. WAZEWSKI
< di
and (2L i )α 三(命 - Ci) for all i = 1, 2 ,.. . , n. A set 筑
is a comp配t set in {(u , p)
U 巳Iæ， p ε Iæ n }.
Theorem 1. 6. (T. W aZewski [154]) Suppose that f follo仇ng
9
f(t , x , u , p) satisβes the
Lipschitz condition: If(t , x , u , p) - f(t ， x ， v ， q)1 三汇 Lilpi - qil
for (t ， x ， u ， p) αnd (t , x , v , q) in ß x .fì. Let Uj
=
+ Mlu 一叫
Uj(t , x) (j
= 1, 2)
be in C 1 (ß)
αnd (Uj(t ， x) ，( θUjj θx)(t ， x)) ε .fì for all (t , x) 巳 ß.lfUj=Uj(t ， x) (j =1 ， 2) αre
solutions of (1. 8) in the domain ß and Ul(t , X)
=
U2(t , X) on ß
n {t = O}, then
Ul(t ， X) 三句 (t ， x)inß.
As preparation for the proof of this theorem , we give a fundamental lemma which plays an important role in his many works. Lemma 1. 7. Let
~
be the
pνra αm 旧iω d defin 附ed 切 in 伪 t he α abo ω 盹 v y 唱
def
(t , x) εß}. Let U = u(t , x) be in C 1 (ß) αnd put ω (t) ~. max{u(t , x)
x ξ ßt }
for each t 巳 [O ， a]. Then ω=ω (t) is differentiable from the right on [0 ， α) ， αnd
叫(t) = 尝(们)一全Li I~~. (t , x)1 …… for some x ε ß t ωith ω (t)
= u(t , x).
Proof. As the first step , consider the case where Li = 0 , ci for all i
-1 and d i
1
1, 2,… , n. By the definition of ω=ω (t ， x) ， we easily see that it is
continuous. Let U (t) ~f {x 巳 ß t
ω (t) = u(t , x)} for t ε[0 ， α). Then the sets
U(t) (0 三 t 三 α) are closed , bounded , and non-empty. As U = u(t , x) is in C 1 (ß) ,
for any fixed t O ε [O ， a) ， we.can pick up a point x O ξ U(t O ) C
生 (t O ， xO )
=
ßto
such that
max 些 (tO ， x).
"'EU(t O ) θt
We will prove thatω=ω (t) is differentiable from the right at t O withω+ (tO) = (θuj街 )(t O ， x O ). For any x ε U( t O ) , we have
ω (t O ) = U内) and 去川三言川)
LOCAL THEORY
1.
10
Define z(t) ~f [ω (t) 一 ω (tO)l/ (t - t勺， and put
ß~flimsupz(t)
α 乞f limi~fz(t).
and
t一斗ftO
t-+t t>t O U
t>tO
As ω (t) 一 ω (t O ) ::::: u(t , x O) - u(tO , x O), we see that α 主 (θu/θt)(tO ， X O ). By the definition of ß, we can pick up a sequence {tm}m
c (t O, a)
with lim t m = t O and ，π-+00
JlyiJ(tm)=F.For each tm , t k arbitrarily zmε U(t m ) ， so that u(tm , xm) ω (t m ).
8ince the set
{(t m ， 俨
=
= 1, 2 ,...} is bounded , we can assume that
m
the sequence{(tm , zm)}m is convergent to a point(to , ZO)in A.As the functions u = u(t , x) and ω=ω (t) are continuous , this implies that u(t口， 20)=ω (t O ) ， i.e. , ε U( t O), hence that (θu/θt)(tO ， ~。 x-) 三 (θu/ât)(tO ， X O ).
Z
ω=ω (t) ， we have ω (t O ) 主 u(t O ，
By the de负rútion of
xm) for all m = 1, 2,.... Using this inequality, we
get
z(t m )
=
ω (t m ) 一 ω (t O ) 。
tm
-
tU
ω ( t O)
u (tm ， xm)
tm
-
tO
u (tm ， xm)-u(俨 ， x m )θ U .......m 。 =τ (t , x m ) , t m - tU ät ......m
ith )i m (t
,..
......0θu
, xm) = (t O,X--)./" It_follows that ..._-
→∞，-
~
.~..~.. ~
ß::::: r---'
一(tO ， ;;U)I ---' :::::一(tO ât \ _ ,8t ， xO). There-
fore ,
去川)三 α 三比去川) 80 ω=ω (t) is differentiable from the right at t O, and ω't (t O) = (θu/δt)(t O ， x O). Next , we consider the same problem in a general pyramid ð. To do so , we take the transform of coordinates a.s follows:
{ :二 S! 向=三 [Ci + di + Yi(di -
Ci -
2Lρ) 1
Then the set ð is mapped to the set {怡 ， y)
(i
=
1, 2, . .. , n).
0 三 s 主 α ，一 1 三 ω 三 1
(i =
1, 2,... , n)}. Combining the result obtained in the first step and the above transform of coordinates , we complete the proof of the
lemma.
口
~1. 3.
A THEOREM OF T. WAZEWSKI def
Proof of Theorem 1. 6. We put u(t ， x)~' Ul(t ， X)
max{u(t , x)
x 巳 ßtl.
and
问 (t ， x) ，
11
and defineω (t) ~
For t ε [0 ， a) , let x(t) ε ß t be such thatω (t) = u(t , x(t))
叫 (t) = … 尝(t， x(t)) 一天Li I 芒。， x(叫 …
(cf. Lemma 1. 7). Then (户tw(t))~
_, IθuJ、
= e- at
|θu ，
,,,
1
,,,
1
~页(机 (t))- ~Lilð;i(t ， x(t))1 一 αu(t ， x(t)) ~.
(1.1 5)
On the other hand , it follows from the hypotheses of the theorem that
1: 川) 1 兰叫去川)1 + Mlu(t , x(t))I. If we choose
α >
(1.1 6)
M , then from (1.1 5)-( 1.1 6) we get
(
川ω (t))~ 俨 ργ? 三 俨 e 严〔一叮 叫 d 时t气 a 叫(MIμω 叫州(仅仲 t
0, from which it may be concluded that ω (t) 三 0 ， i.e. , Ul (t, x) 三句 (t ， x ), for all ω(仰 0) =
(t , x)
巳 ß.
句 (t ， x)
Since we can simi1a r1y prove U2(t , x) 三 Ul (t , x) , we finally get Ul (t , x) 三
on ß.
口
We will rewrite Theorem 1.6 in a general form. A function f
= f(t , x , u , p)
is
said to be locally Lipschitz continuous with respect to (u , p) if, for any comp肌t set .Iì in lR x lRn x lR x lR n , there exist constants Ll' L2' … , Ln and M such that
If(t , x , u , p) - f(t , x , v , q)1 三汇 Lilpi 一创 + Mlu -vl i=l for all (t , x ， 飞 p) and (t ， 叭叭 q) in .Iì. Theorem 1.8. Suppose 陀esψ T :pect 归 to 仙 ( u ， 时p) .扩 u
伪 t.hαωt
f
= f(t ， 凯 x， 叽 u， 叫 p)
u(t ， x)andv
neighborhood of the origin satisfying u(O , x)
is
loc ωα l闯 l仿 yL μzp 严schi必tz cω o时 n timω包旧s ω t伪 h
υ (t ， x) αre 三 v(O ， x) ，
neighborhood n of the origin, such that u(t , x)
三 v(t ， x)
C 1 -s01utions of (1. 8)
in α
then there exists an open in
n.
We willleave the proof to the readers. Remark. The uniqueness of C 1 -s01utions is assured by the Lipschitz continuity of
f = f( t , x , u , p) with respect to (u , p). But this condition is not sufficient to get the solvability of the Cauchy problem (1. 8)-( 1. 9). Wazewski [156] has given necessary and sufficient conditions which guarantee the local existence and uniqueness of classical solutions of (1. 8)-( 1. 9).
Chapter 2 Life Spans of Classical Solutions of Partial Differential Equations of First-Order 32. 1. Introduction As we have shown in Chapter 1, the Cauchy problems (1.1)-(1. 2) and (1. 8)-( 1. 9) have locally classical solutions. It is well-known that the solutions may generally have singularities in finite time even for smooth initial data. For some equations of the conservation law , the life spans of classical solutions have been exactly calculated , for example by E. Hopf [63] , P.D. Lax [97]-[98 ], E.D. Conway [32 ], etc. The principal aim of this chapter is to determine the life spans of classical solutions of general partial differential equations of first-order. Moreover , using the results on the life spans of classical solutions , we will give necessary and
su面cient
conditions
which guarantee the global existence of classical solutions for (1. 1)-( 1. 2) 皿d (1. 8)-
(1. 9). As an example , we consider a simple equation of the conservation law as follows:
生+于α巾)生 =0
{t>O， 叫町，
in
…一….
u(O , x)= 4> (x) where ai
αi(U)
on
{t=O , x
(2.1)
巳]R"}，
(2.2)
(i = 1, 2 ,... , n) and 4> = 4> (ν) are of class C 1 in ]R and ]R",
respectively. The equation treated in E. Hopf [63] is the case where n = 1 and E.D. Conway considered the above equation
(2.1)-(2功 in
α l(U)
= u.
[32]. 1n this case , the
characteristic curves which are the solutions of (1. 3)-( 1. 4) are written by x= ν +tα(4) (ν) )，
υ (t ， y) =4>(ν)
(2.3)
where α(u) 哲 (α1 (u) ， α2(U) ，... , a,, (u)). Then the Jacobian of the mapping x =
x(t , y) is given by
(Dx/D训， ν)=1+ 训ν)
with
Obviously, the Jacobian (Dx/ Dy)(t , y) does not
>.(y) 哩艺 αH4>(州θ4>/8Yi) (ν)
var山h
in a neighborhood of (t ， ν)=
~2.2.
,
13
LIFE SPANS OF CLASSICAL SOLUTIONS
,
,
(0 0). Therefore as we have shown in 31. 1 the Cauchy problem (2.1)-(2.2) has a unique C 1 -so1ution u
u(t , x) in a neighborhood of the origin. Since u(t , x)
4> (y( t x)) where y
x) is the solution of the equation x =
,
= ν (t ，
x(t ， ν) ，
we have
(茬，却=品而(茬，... ，茬儿(t，.， ) Then the Jacobian (DxjDy)(t O ， 的= 0 for t O ~f 1 j ( →λ(俨)). We see by (2 .4) that when (t x) goes to (to x O) along the curve x = x(t ， 俨) where Assume λ(俨)
< O.
,
,
,
,
zotf z(tO ， ν勺， at least one of the first derivatives of u = u( t x) tends to infinity. Therefore , if the Jacobian vanishes somewhere , then the Cauchy problem (2.1)(2.2) can not admit a global solution of class C 1 • In 32.2 and 32.3 , we will give similar results on the life spans of classical solutions for general partial differential equations of first-orde r. What we would like to remark here is to point out that there exist some differences between quasi-linear equations and general partial differential equations. See Theorem 2.1 and Theorem 2.2. In 32 .4, we will consider the global existence of classical solutions. These existence results are corollaries of theorems given in 32.2 and 32.3.
32.2. Life spans of classical solutions Let us consider a quasi-linear partial differential equation of first-order as follows:
2 十也川)去= aO川 ln 书> 0, x E ~n}，
,
u(O x)= 4> (x)
on
,
{t=o x
, ,... ,n) and 4> =
where 向=句 (t ， x ， u) (i = 0 1
ξ ~n} ，
4>(ν) are of class
(2.6) C 1 in
~ x ~n X ~
and ~n ， respectively. The characteristic equations for (2.5)-(2.6) are written by (L dt A t u ) ( = 1 2 n )
去 =αO(t， x ,v) ,
(2.7)
with the initial conditions
, ,...
Xi(O) = 衍。= 1 2
，叫，叫 0)
= φ (y).
(2.8)
2. LIFE SPANS OF CLASSICAL SOLUTIONS
14
We write the solutions of (2.7)-(2.8) by x = 仰 ， y) and v = 咐， ν). Then v = 咐， ν) means the value of a solution of (2.5) restricted on the curve
x 二 x(t ， ν).
Here we
assume the following condition: (A .I) The Cauchy problem (2.7)-(2.8) has uniquely a global
v = v(t , y) on {t
~
O} lor
sol1巾 on
x(t ， 的 F
x
a叼 yε Jæ.n.
It is not easy to write down sufficient conditions which guarantee the assumption (A .I). Co配erru吨 this subject , see for example B. Doubnov [43] and M.S Krasnosel'skii [83]. We wi11 consider this in Appendix 1. When we assume (A .I), we
= x( t ， 的
get a smooth mapping x
from
Rn to Jæ.n for each t
~
O. The 1ife span of
classical solutions of (2.5)-(2.6) is determined by the followi吨: Theorem 2. 1.
Un 叫 de旷r
Condition (A .I), s包叩 ，ppose
山阳/川 均νω D 州州川)(附队 圳札阳 (tt式， 川阳 y 0川10川 T川t 01 the x
solution u
<俨 t oO
= 0
αn 叫 d
仙 T阳川川 het轩m … Tη川叫 b川α时呻t leaα… E叫 01 叫川…tiω 去0
= u( t , x) tends to
in.βmty 叫 en
t goes to t O - 0 along the
c包rve
= x(t , yO) def
r
,.
Proof. Let us put L ~. {(t ， 俨) : 0 三 t
t
伪 t hαωt (Dx/Dy 剖)(t俨O ， 俨 y 0)
<
t O}.
,. < tO} and• COdef ~. {(t , x) : ~
r
x 口 x(t ， y勺， 0 三
By the assumption and by the theorem of inverse functions , we can get an
open neighborhood V of L so that the Jacobian (Dx/Dy)(t ， 的 does not vanish on
V ~fV 门 {O 三 t
< t O}
= x( t， 的 can be uniquely solved ((t , x) x = x(t , y) , (t , y) E V}. Let
and that the equation x def
with respect to y for any (t , x) in U ~.
def
us write the solution by y = 州 ， x) and define u(t , x) ~叫 t ， y(t ， x)). We already proved in 31.1 that u = u(t , x) is a C 1-s01ution of (2.5)-(2.6) in the domain U. The uniqueness of C 1-s01ution of (2.5)-(2.6) is assured by Theorem 1.1. Here we have
θuιθuθνk åXi 仨:百;瓦 θυ
二、 θU åXk
句J
主:θxk θ的
Tr
ln
(2.9)
u
(2.9')
On the other hand , from (2.7) ， θxdθ的 and θυ/θ的 (i ， j
= 1, 2 ,... , n)
satisfy
the following system of 1i near equations:
(ddt 巳 åa; åx 川 \ 句JJ 仨1θXk θ屿 'θυθyj
一.一一一一一-
d ( δ。 \2飞 θα。 θXk . θα。 θu 一.一一一一一
dt 飞 θ的/仨1θXk θ屿
'θυθ的
(2.10)
15
32.2. LIFE SPANS OF CLASSICAL SOLUTIONS
with the initial data
结(0)=(;::; 二;
and
( tt ) UUJly
、 /''1
= rank
h
…
,-,
ôY
仇一句
「BEEtEEE
町的θ 一θ
rank
EEEEBEEt
aTb
叮
n一 m句
By the linearity of (2.10) , we have
θuθ￠ ~- (0) 口二 (y).
ÔYi
nu
专 (0)
= n
t 2: 0 ,
for all
(2.11)
一 θ
生 (0) σν
itθXi ( θXi e 一一一一.
‘ θXi) and θ~=(θv θ~) -:.~ = I 一一，... .一一 l θν 飞 δYl ,... ， θνnJ Assume that all the components of 仇 /θX remain bounded along the curve Co, ‘一~ I θu\θνδYnJ
then we can pick up a sequence {tm}m C [O , t O ) and one
C
=
ε IR n
, , … , Cn )
(Cl C2
such that
1) lim t m = 沪， and m--->OO
2)J记去川川。))口 C From
(2. 叭，
we get
ZW)= 立守川) As (Dx/D ν)( 户，俨)
= 0, (2.12)
(2.12)
contradicts (2.11). This means that at least one
component of (仇 /δz 帅 ， x(t ， 俨)) tends to infinity when t
•
tO
-
0 along Co .
口
Next we consider general partial differential equations of first~order as follows:
去 +fhuz) 二 o in 书> 0, u(O , x) = <þ (x)
on
{t = 0, x E
x E
IRn }，
IRπ} ，
口 13) (2.14)
where f = f(t , x , u , p) and φ = <þ( ν) are of class C 2 in IR x IR n x IR x IRn and IR n , respectively. The characteristic equations for (2.13)-(2.14) are written by
2. LIFE SPANS OF CLASSICAL SOLUTIONS
16
dXi
ôf
dt
ÔPi
一 =70 ， z ， υ ， p)
~
ôf
(i=1 , 2,..., n) ,
r 三Pj 百 (t ， x ， v ， p) dPi
一= dt
-
f(山，~，
~臼)
θfθf
一 (t ， x ， v ， p) 一一 (t ， x ， v ， p)
(i = 1, 2,… , n) ,
θ￠ υ(0) =的)，以 0) =一 θ的
(i=1 ， 2 ，...， n).
θXi
θu
with 的 (0) =钩，
(2.16)
We assume here the following condition: (A .I)' The Cauchy problem (2.15)-(2.16) has uniquely a global solution on for any y
{t 主 O}
ε Rn.
We denote the solution of (2.15)-(2.16) by x = x(t , y) , v = v(t , y) , and p
=
p(t , y). As (Dx/Dy)(O , y) = 1 for all y ε Rn ， the Jacobian does not vanish in a
neighborhood of {t = O}. Therefore we can
iquely solve the equation x = x(t , y)
Ull
with respect to y , and denote it by y = ν(t ， x). Define u(t , x) ~f v(t , y(t , x)). Then u = u(t , x) is a C 2 -s01ution of (2.13)-(2.14) in a neighborhood of t = O. Moreover ,
we can see by Theorem 1. 4 that there does not exist another C 1 -so1ution of (2.13)(2.14) in a neighborhood of {t = O}. When we extend this solution for large t , we get the following. Theorem 2.2. Under Condition (A .I)', s叩pose that (Dx/Dy)(t O ， 俨) = 0 and (Dx/ Dy)(t , yO) 手 o for t
< t O•
Then
πlθ2U
1
L: 1 一一一 (t ， x) 1 tends to infinity 叫 en t z户11θXiθzj'| ,
,
goes to t O - 0 along th e curv
Proof. As the proof is similar to that of Theorem 2.1 , we use the same notations def
introduced there. Put L ~I {(t ， yO) neighborhood V
n {O
~ t
。三 t
<
t O}. Next we choose an open
V of L so that the Jacobian (Dx/Dy)(t , y) does not vanish on V
< t O} , and
~f
that the equation x = x(t , y) can be uniquely solved with
resp田t to y for any (t , x) in U 乞f{(t， z):Z =z(tJ) ， (tJ)ε V} (theorem of inverse functions). We write the solution by y = y(t , x). Here we define u(t , x) ~f 叫 t ， y(t ， x)). Then u = u(t , x) is a C 2 -so1ution of (2.13)-(2.14) in the domain U.
32.2. LIFE SPANS OF CLASSICAL SOLUTIONS
17
Our aim is to show that , when t goes to t O - 0 along the curve x = x 衍，俨)， at least θt叫』旷 /â θx one of {仰 Differentiatir吨 (2.1臼5 时叫)
line町 ordinary θ'pdθ的
i， j
with
respe 时ct
= 1, 2,... , 叫 ,
(j
we get a system of
= 1, 2,... , n} like (2.10). As this system ofequations is linear , we 用…用 mu 用…用 bh 阶句部
Udy
-n
=n
t 二::
for any
o.
(2.17)
1/
(δIPn/θν )(t)
叭川 j川 nunu
,
呻飞、们/、
r an 'k
川们
(θυ/δ的 (t) (δ'pdθν )(t)
、l//
-
(θx n /θy)( t)
nυnU
UUU
、 lJ
( ( (: ( ( ) ) ) ) (( i( ( -nh
(θxdθy)( t)
I
的
differential equations of first-order concerning {θXi/ θ屿， θυ/θ屿，
have
rank
to
Since (2.17) is true at (t ， ν) = (t O, yO) , we can choose n vectors in {âx d句， θx n /句，如/句， θ'pd句，... ， âpn/θy}
such that they are linearly independent at
(t , y) = (沪，的. We denote them by 叫t ， y)~f(δ/δy)bi(t ， y) (i = 1, 2,… , n) , where bi(t , y) is one of {Xk(t , y) ， υ (t ， y) ， pk(t ， y): k=1 , 2,..., n}. As(Dx/Dy)(tO , yO) 0, we can find v(t , y) or some pk(t , y) in {b 1 (t ， y) ，...， ι( t , y)}.
=
Here we recall
Corollary 1.2:
去 (t， x) = 白 (t， y(t， x))
1 ，丸
U (j =
in
， n).
(2.18)
On the other hand , we get in U 「
，
+ι
,( < -1 < -d
飞
n<
,,.,.
一
一
n
咱自晶
寸
一 叫 σ θ
)
、-BEI/
•••
-.,,
1l ·LH<
Z 一 Uν
‘‘‘
、i/
J?bj
Z4h
J
EEEEEEE'EEEEEEEBBBBBE.
?tbU，?ι.·
，
」
盯ν
ι ，，
IIll--IIll--L
,,,, .EEEE
}
3
1|lJUU
FEEEEEEEEEEL
」『
?ι
，
δ 忏的盯 f 旦d 归 旦一 川 叮 nd文0!h
.•
EEEEEEEEEEEEEEEEEaEEEE
，
?ι
』「
yuνUυνUυνUUνυ
Gααααα
n
一一一
<一
」
1 <-o--J,,
「 Il--Ill--Il--
b
、‘自目 r，，
，， a··、、
，。
... ''b UU(,, z )
叮 ''''EEEEEE
θ-h
rzaa-aEEEEL
(( )( )( )( )( ))
12n12n
Since {α1 (t ， 的， α2(t ， y) ，...， α衍。，的} are linearly independent in a neighborhood of (tO ，俨) and
(Dx/Dy)(tO , yO) = 0 , it follows that at least one component of the
2. LIFE SPANS OF CLASSICAL SOLUTIONS
18
matrix [(θ/θXj) 句 (t ， y(t , X))ll :'S i，江 n tends to infinity when t goes to t O - 0 along the curve x = X(t , yO). But we see by (A.I)' and (2.18) that
ð~j v(t , y(t , x)) }}I..一…=瓦川， yO)) = Pj(t， yO) 阳， 2，... , n) (θjv(t， 1..=.dt."O\θu remain bounded (though the Jacobian vanishes)j therefore , some Pk (t , y) must be contained in {b 1 (t ， y) ， b2 (t ， 时，... , bn(t , y)}. Hence we get Theorem 2.2.
口
Theorem 2.2 says that , if the Jacobian vanishes at a point (to , y勺， then the second derivatives of classical solutions blow up at time t
t O • But this does
not prevent the existence of C 1 -s01utions 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.
32.3. Global existence of classical solutions As we have shown in 31.1, the Cauchy problem for a partial
d泪汪rential
equation
of fÌr st-order has locally a classical solution. We have also seen in 32.2 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 ,
扣 f悦讪 fo11 削110wi 呐m 吨 g Ts叫 吐 u j
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 C 1 -so1utions on the half space {t 主 O} ， using Hadamard's lemma on the diffeomorphism ofEuclidean spaces. Another proof was given by Li Ta-tsien and Chen Shu-xing [102]. Qin Tie-hu [114] treated the case where the functions ai (i
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)叩ace and the linear dependence of the functions
al = α1( u) ,. • . ， αn
αn(U).
Now we review the results on the diffeomorphism of the Euclidean space ]R n. Concerning this subject , we must go back to J. Hadamard , but here we fo11ow W.B. Go 创，rdon巾 note [58]. Let
H be a C 1 -mappi吨 from
]R n
to ]R n defined on the whole
~2.3.
GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS
sp缸e. Then the mapping
H is said to be proper if H- 1 (K) is
19
comp配t whenever
K is compact. We have: Lemma 2.3. (J. Hadamard) The
mα:pping
H
is α diffeomorphism
from ]Rn to ]Rn
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, Ixl ε ]Rn.)
denotes the Euclidean norm of x
(A .I) For any y
ε ]Rn ，
the Cauchy problem (2.7)-(2.8) has always a unique global
C 1 -s01ution x
= x(t , y) ,
(A.II) On the
sol1巾on c旷ves
v 二 v(t ，
y) on {t ~ O}.
of (2.7)-(2.8) , it holds that
问 (t ， x(t , y) ， 咐， ν)) 1 主 M(l
for any t ε [0 ， T ], y ε ]Rn and i
+ 1吨， ν) 1)
1 , 2 , . . . , n ωhere the constant M depends only
on T. When Conditions (A .I) and (A.II) are satisfied , we can define , for any t 主 0 ， a C 1 -mapping
H t from ]Rn to ]Rn by x
Lemma 2.4. Under the for
= x(t ， y) 暂且 (ν). Then we get:
αssumptions (A .I)α叫 (A.II) ，
the mapping H t is
pr叩er
αny t 主 O.
Proof. As we have
兰 Ix(t， y)1 =一」一〉月 (t， y)αi(t， x(t , y) , v(t , y)) Ix(t ， y)1 匀 主 -nM(l
we get Ix(t , y)1 三 (1
+ ly l) e- nMt -
+ Ix(t , y )l),
1. Therefore Ix(t ， ν) 1 →∞ as Iyl →∞. This
means that H t is proper.
口
U sing Lemma 2.3 , we have: Lemma 2.5. For ]Rn 可 αnd
αny
fixed
t 主 0，
the mapping H t is a diffeomorphism from IRn to
only íf íts Jacobían does not vanísh at any poínt y
ε ]Rn.
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.II) , the (2.6) has a global onlν 矿 the α nd
y
C 1 -s01ution
Cαuchy
problem (2.5)-
uniquelν on the domain D ~f {t ~ 0 , x 巳Jæn} 可 αnd
Jacobian (Dx / Dν) (t ， 的
of the
a叼 t 主 O
mapping H t does not vanish for
ε Jæn.
Proof. When
(Dx/Dy)(t ， ν) 并 o
for any y
ε Jæn
and t > 0, we can see from
Lemma 2.5 that H t is a diffeomorphism from Jæn to Jæn for any t inverse function y = ν (t ， x) of x
= Ht (ν) is a function of class
C1
主 O.
Hence the
defined on D and
u = u(t , x) ~f v(t ， ν (t ， x)) is the unique global C 1 -so1ution of (2.5)-(2.6). (For the
uniqueness , follow the proof of Theorem 1. 1 or 1.6.) Next , for the necessity, suppose that (Dx / Dy)( t ， 俨)
= 0 for some t
ε(0 ，∞)
= 1 for all νε Jæn ， we can get a unique C 1 -so1ution of (2.5)-(2.6) in a neighborho6d of {t = O}. Then we see by Theorem 2.1 that this classical solution blows up at time t O 乞f inf{t > 0 (Dx/ Dy)(t ， 俨) = O}. (Notice O that 0 < t 手 t.) 口 and yOε Jæn. As (Dx/Dy)(O , y)
Next we consider the Cauchy problem (2.13)-(2.14) for general partial differential equations of first order. We assume the following hypotheses: (A .I)' For
α叼 νε Jæ飞 the Cαuchν problem
(2.15)-(2.16) has
alwaνsα unique
global
C 1 -s01ution x = x(t ， ν) ， v = v(t ， ν) ， p=p(t ， ν) on the half space {t 主 O}. (A.II)' On the solution curves of (2.15)-(2.16) , it holds that
|去 (t， x(t , y) , v(t , y) , for
a叼 t
E [0 , T ], y
ε Jæn
and i
1, 2 , ..., n
叫例 如 constant
M depends only
on T.
For any t H巾).
~
0, we define a C 1 -mapping H t from Jæn to Jæn by x
= x(t , y)
def
~
Then Conditions (A .I)' and (A.II)' guarantee that the mapping H t is proper.
Therefore , Lemma 2.5 is also true for this H t . Theorem 2.7.
Under the
αssumptions
(2.13)-(2.14) has uniq包ely a global
(A.I)' and (A.II)' , the
C 2 -solution
C.αuchy
problem
on the domain D 哲 {t ~ 0 , x 巳Jæn}
~2.3.
GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS
21
if and only if the Jα cobian of the mapping x = Ht (ν) does not vanish for αny t 主 O and y
ζ ]Rn.
Proof. When
(Dx/Dy)(t ， y) 笋 o
the equation x
= Ht (ν) with respect to y for any (t , x) ι D.
for any t
~
0 and y
ζ ]Rn ，
we can uniquely solve Let y
= y(t , x) be the
ínverse functíon. A global C与olutíon of (2.13)-(2.14) ís gíven by u 咐 ， y(t ， x)). The uníqueness ofthís
C 2 -so1utíon
= u(t , x) 彗
follows from Theorem 1.6. (It can
a1so be deduced from the method of proof used ín Theorem 1.4.) The necessíty of the above condítíon comes from Theorem
2.2.
口
Chapter 3 Behavior of Characteristic Curyes and Prolongation of Classical Solutio-ns 33. 1. Introduction This chapter is continued from Theorem 2.2 in 32.2. Let us rewrite the equation which we will again consider here:
去 + f(t , X， u， 去) = 0 in 书> 0, u(O , x)= 4> (x) where f
= f(t , x , u , p)
and 4>
=
on
4> (ν) are of class
respectively. The characteristic equations (2.15)-(2.16). Let x
{t=O , x
(1) (DxjDy)(t O ， 俨 )=0 ，
and
(3.1)
ξ ]R.n} ,
C 2 in
]R.
x ]R. n
(3.2) X ]R.
x ]R. n and ]R. n ,
correspondi吨 to (3.1)-(3.2) 町e
= x(t , y) , v = v(t , y) , and p =
(2.16). We consider the Cauchy problem
x E ]R. n} ,
p(t ， ν)
(3.1)-(3 勾 in
given by
be solutions of (2.15)-
the following situation:
(II)(DxjDy)(t ， yO) 笋 o for
t < t O.
We put x O 乞:f x(tO , yO). Theorem 2.2 says that , when (t , x) goes to (t o, X O) along the curve x = x(t , y勺， one of the second derivatives of the solution u = u(t , x) of (3.1 )-(3.2) tends to infinity. But this does not prevent the existence of C 1 -so1ution in a neighborhood of the point (t O, x O ). Our problem is to see whether or not we can extend the classical solution u
u(t , x) beyond the time t O. On the other
hand , we will show later that , if the characteristic curves meet in a neighborhood of (t O ， 的， then the Cauchy problem (3.1)-(3.2) cannot admit a classical solution there. Therefore , it is necessary for us to consider whether or not the characteristic curves meet in a neighborhood of (t O , X O ) , i.e. , whether or not there exist two points
yl and y2
(yl 乒 y2)
satisfying x(t , yI) =
x(t ， 的)
for some t. In 33.2 , we will give two
examples in which characteristic curves do not meet though the Jacobian vanishes. In 33.3 , we wi1l consider the case where we can extend classical solutions of (3.1) (3.2) beyond a point where the Jacobian vanishes. In 33 .4 and 33.5 , we wiU give
~3.2.
EXAMPLES
23
sufficient conditions so that the characteristic curves meet in a neighborhood of the point (t口， X O ).
~3.2.
Examples
Example 1. We consider the Cauchy problem for a quasi-linear partial differential equation as follows:
( 2仨卜卜←忖叫叫J( α叫伊(呻巾 川 t， u ux
u(O , x)=x where a(t , u) ~rα'(t)e-tu
on
(3.3)
{t=O , x
+ ß'(t) ε -3t u 3
ε ]Rl },
and the two functions α=α(叶， β = ß(t)
satisfy the following conditions:
ß = ß(t) are in C1 ( 1R1 ). 三 o for each t ， α(0) = 1, andα( t) = 0 for all 主 o for each t , ß(O) = O. + β (t) 并 o for all t ε 1R 1
1)α=α (t) and 2)α (t)
3)
ß(t)
4)α (t)
t 三 K
= constant
> O.
Then the characteristic curves for (3.3) are written by
x = x(t , y) = α ( t) ν + ß(t)y3
and
v = v(t ， ν) = ety ,
(3 叫
from which we easily see that 吗
瓦 (t ， y) =α (t)+3ß(t)y~ ，
and
Dx
一 (t ， O)=O forall
Dy
t?K.
But we can also see from (3 .4) that the characteristic curves x = x(t , y) do not meet for all t 三 O. In this case the solution u = u(t , x) is represented as
u(t , x) = ß(t)-1/3 et x 1/ 3 for
t 主 K.
This representation says that the solution contains algebraic singularity at x = 0 , and that the singularity of shock type does not appe缸 though the Jacobian vanishes.
3. PROLONGATION OF CLASSICAL SOLUTIONS
24
Example 2. We consider the following Cauchy problem:
(去忏叶= 0 in 书> 0, x E R 咐叫 =jz2on
1
}
(3.5)
{t=0, zd1} ,
where
f贝(阳川川 ， 叫p) 彗 ;←护扣扒 α旷叽印，气勺(t)扩 ♂叶 f γ泸p2+叶咐呐 t与 肉州，气勺 βr 例 (tt叶) 产 and the functions
α=α(吟， β=β (t)
are the same functions that we introduced
in Example 1. This example is not of quasi-linear type , and it satisfies Conditions (A .I)' and (A.II)' given in
~2.3.
The characteristic curves for (3.5) are written as
x= 巾) =州 +4β(旷 and u=ty(M)=jα(t)e t y2 + 切(呐 Therefore the J acobian (Dx / Dy)( t , y) = α(t) + 12ß(t)y2 vanishes on L 哇f {(t ， ν)
t ;:: K and y = O}. But x = of y = 0 for each
t 主 K.
x(t ， ν)
is a bijective mapping defi.ned in a neighborhood
In a neighborhood of L , the solution u = u(t , x) is written
by
u(t , x) = const.β (t)-1/3 e t X 4 / 3
for
t 主 K.
This says that the solution u 二 u(t ， x) is of class C I, but not of class C 2, in a neighborhood of L.
~3.3.
Prolongation of classical solutions
As we have shown in
~3.2 ，
there exists the case where the characteristic curves do
not meet in a neighborhood of points where the Jacobian vanishes. In this case we can uniquely extend classical solutions even if the Jacobian may vanish. This is the problem which we would like to prove in this section. Now let us make clear the situation under which we consider the Cauchy problem (3.1)-(3.2). We always assume Condition (A .I)' (Chapter 2) which assures the global existence of characteristic curves. Let x
x (t , y) ,
v
υ (t ， y)
and p = p(t , y) be
the solutions of (2.15)-(2.16 ), and define a mapping H from Rn+l to Rn+1 by
H(t ， 的乞f (t , x(t ， ν)). Suppose: (1)
(DxjDy)(t O ， 俨) = 0 ,
33.3. PROLONGATION OF CLASSICAL SOLUTIONS
25
皿d
(II) (Dx/ Dy)(t ， 俨)手 o for t
< t O.
In trus section we consider the case where the characteristic curves do not meet. Therefore , furthermore , we assume the following condition:
(B) The mapping H is bijective (tO ， X O ) ωhere x O~f x(tO , yO).
from α neighborhood
of
(t O ， 俨)
to
αnother
one of
By Condition (B) , we can uniquely solve the equation x = x(t , y) with respect to y , and denote it by y = ν (t ， x) (for (t , x) in a neighborhood of (沪 ， x勺， (t , y) in a neighborhood of (t O ， 俨)). The function y = 仰， x) is obviou句 continuous ， though it may not be differentiable. Then we get the following. Theorem 3. 1. Under the hypothesis (A .I)飞 suppose (I)-(II)α nd (B). Then the
solution u = u( t , x) 乞f v(t , y(t , x)) remains a C 1 -s01ution of(3.1) in α neighborhood of (t O, x勺 ， though it is not of class C 2 • Proof. Let V and U be open neighborhoods of (to, yO) a叫(tO ， xO) , respectively, such that the mapping H is bijective from V to U. Consider S ~f {(t ， ν)ε V
(Dx/ Dy)(t ， ν) = O} and H(S) = {H(t , y) (t ， ν)ε S}. By Sard's theorem , the Lebesgue measure of H(S) is zero. Therefore U\ H(S) is dense in U. We see that u = u( t , x) is of class C 2 in the domain U \ H(S). The reason is as follows. For any (t ， x) ε U \ H( 町， there exists uniquely a point (t , y) ε V satisfying
(Dx/ Dy) (i, y) 并 o and
x= x(i , y). The i盯erse functionν = y(t , x) is of class C
1
in a neighborhood of (t , x) wruch is contained in U \ H(S). Here we recall Lemma 1. 2 and Corollary 1. 3 in ~1. 1 ， and we see that u 二 u(t ， x) is a function of class
C 2 satisfyi吨 the equation (3.1) in the neighborhood of (t , x). Next we show that
u(t , x) is continuously differentiable in the domain U. We pick up arbitrarily ~O _n a point (t , i) in H(S). Then we can choose a sequence {( t m , xm)}m of points in u
=
~O
n
U\ H(S) such that , when m tends toinfinity, (tm , xm) is co盯ergent to (t ， 王). As the mapping H is bijective from V to U , there exists a unique point (t m satisfying H (t m ， νm)
= (tm , xm) for each m. Since y
= ν (t ，
, y Tn)
εV\s
x) is continuous in U ,
3. PROLONGATION OF CLASSICAL SOLUTIONS
26
= y(tm , Xm)
it follows that ym
is convergent to
~O def
y
~.
infinity. As u = u( t , x) is continuously differentiable at
~O ~O
y(t , x-) when m tends to (t飞 xm) ，
we have by (1. 13)
p( Because p
= p( t ， ν)
is continuous on the whole half space {t 主 0 ， y ε IRn} ， we get
h 生 (t m ， xm) = ......c白 ux
The derivative
θu/θx
of u
a way that (θu/θx) (t , x) C1
= u(t , x)
liIIl p(川m)
m......o。
~O ~。
= p( t
， ν)
can thereby be continuously extended (in such
= p( t ， ν (t ， x))) over U.
That is to say, u
= u (t , x) is of class
in the domain U. Moreover , notice that the uniqueness of the above (extended)
C 1 -s01ution
is assured by Theorem 1. 6.
口
Remark. In Theorem 3.1 , the assumption (A .I)' is a crucial point. It will be proved in Chapter 5 that , for quasi-linear equations of first-order , the assumption (A .I)'
i日
not compatible with the property that the Jacobian vanishes somewhere.
33 .4. Suflìcient conditions for collision of characteristic curves 1 For quasi-linear equations
of 岳rst-order，
Theorem 2.1 says that , if the Jacobian
vanishes somewhere , the classical solutions blow up there. Therefore we cannot extend the classical solutions beyond the time t O when the Jacobian vanishes. This obliges us to treat weak solutions for t
>
t O • The typical singularity of weak
solutions is "shock." As it wi11 be shown in Chapter 5, the shock appears by the collision of characteristic curves. Therefore we try here to give sufficient conditions so that the characteristic
cu凹es
meet after the Jacobian vanishes.
In this section we consider quasi-linear equations of first-order in one space dimension as follows:
主 +αl(t， x， U)去 =α巾 ， u) in 川， u(O , x) = ~(x)
0 , 1) and ~ respectively. The characteristic curves for where ai
αi(t ， x , u) (i
on
x E IR 1 } ,
{t = 0, x εIR 1 } ，
(~.6) (3.7)
~(y) are of class C 1 in IR3 and IR\ (3.6)-(3.7) 缸e defined 田 solution
curves
~3 .4.
of
(2.7)-(2.8)
for n
COLLISION OF CHARACTERISTIC CURVES 1
=
2, we
1. Here , as in Chapter
which assures the global existence of characteristic v= υ (t ， y)
be the solutions of
(2.7)-(2.8) for
Theorem 3.2. Under Condition
=
n
27
again assume Condition (A .I) cu凹es.
x(t , y) and
Let x
1.
(A .I)， ωEαssume:
吟川=斗0， α7叫 (11咋 I l厅f 仰 ( θα叫dθu叫)(t俨O ， 泸 xO ， 俨 v 0) 手 0 ω 叫her何e ♂ x o
= x(t , y) with
respect to y is violated. First we prove that (θxjθy)( t , yO) is negative for t
> tO
(2.7) with respect to y , we get the system
with t-t O small. Actually, differentiating
of ordinary differential equations (2.10) for n
=
1. Then (2.10) is linear with respect
to 8xj句 andθ旷句. As the initial data are not zero , we get θzθu
(一 (t ， y) , :" (t , y)) 并 (0 ， 0) θY \ """ ， θy\"'"'' T \~，
for any
t 川， yE Jæ1 .
Hence we have (θυ/θy)( t O ， 俨)并 O. By the assumption , we get d / θz\l 月a， 一(一)|=」 (t勺 O ， V O ) 一(t0 ， yO) 笋 O dt \δν J I(t ，y)=(沪 ， yO)
Moreover , as (δxj句 )(t O ，俨) d
= 0 and
/ θz 飞
(
~..
)1
dt 飞 θν J
θu
l(t ,y)=(tO ,yO)
Therefore we get (θxjθν)( t ， ν。)
(θxj句) (t , yO)
剑
.....:~，
<
i.e. ，
.._.,
0 for t
> 0 for t < t O, we get
d / θz\|
~(~..
11
dt 飞 θy J I(t ， y)=(沪，苗。)
< O.
> t O with t - t O small.
That is to say,
(θxj句) (t ， 俨) changes its sign at t = t O. We now define a function h = h(y) by
setting h(ν) 哲 inf{t
θxj句 )(t ， y) =
Cl in a neighborhood of y
(δ2xj街句)(t0 ，俨)
< 0, we
O}
for each y. Then h
= h(y)
俨. In f:缸t ， as we have (θxj句 )(t O ， 俨)
can uniquely solve the equation (θxj句 )(t ， y)
is of class
= 0 and = 0 with
respect to y in a neighborhood of (t o, yO). Let us restrict our following discussions into a small neighborhood of (t O ， 沪，俨) only. We see by the same reasoning as in the above that , as a function oft , (θxj句 )(t ， y) changes its sign at t
= h(y)
for each
y.
Next , we will show that x a point y
并 y
o
. If h(y)
= x(t , y)
is not monotone with respect to y. Pick up
< t U , then we get
字 (t， y) < 0 and 生 (t， yO) > 0 8y θy
3. PROLONGATION OF CLASSICAL SOLUTIONS
28
for t ε (h( 剖 ， t O ). As trus means that x
=
x(t ， νis not monotone with respect
to y , the characteristic curves meet in a neighborhood of (t O , X O ). When h = 州的 沪， it follows that (θx/句 )(t O ， y) 三 o in
is constant in a neighborhood of y a neighborhood of y
俨
i.e. ，
x(tO , y) is constant there. This says that
x
the characteristic curves meet at the point (沪 ， X O ). When there exists a point
y=
Y such that h(y)
> 沪， we can similarly see that x
= x(t , y)
is not monotone
with respect to y in a neighborhood of y = 俨 for each t ε (t O ， h( y) ). Hence the characteristic curves x = x(t , y) meet in a neighborhood ofthe point (tO ， x O ).
口
33.5. Sufficient conditions for collision of characteristic curves 11 In this section we consider the same problem as in equations of first-order in one space
dimension 出 follows:
些主 +f(t， x， u 旦旦 )=0 θr
J \ '"
....."
'"""θz
u(O ， x) =φ(x)
where f
= f(t , x , u , p)
33 .4 for general partial differentia1
on
in
{t>O， 川剧，
(3.8)
{t = 0 , x ε Iæn ，
(3.9)
and <þ = 功(y) 町e of class C 2 in JR.4 and JR. l , 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 = x(t , y) ， the solutions of (2.15)-(2.16) for n Theorem 3.3. Under Condition
υ= 咐， ν) ，
and p =
p(t ， 的 be
= 1. (A .I)'，切Eαssume:
咔内0) =才0， αmηd (11咔 I 扩(阳 θ2f/θp 泸2)(t俨O ， 泸 x oυ沪O ， pO) 笋 o 1ψIJher陀e 泸 x o
O ~fυ (tO ， yO) and pO p(t O ， υ勺 ， then the characteristic c包rves meet in α neighborhood 0/ (t O, XO). Proof.
V
As in the proof of Theorem 3.2 , we will show that x
monotone with respect to y. We repeat the same discussions in
x(t ， νis
31.1.
def
not
When the
Jacobian (θx/句 )(t ， y) 笋 0 ， we can solve the equation x = x(t , y) with respect to
53.5. COLLISION OF CHARACTERISTIC CURVES 11
y , and denote the solution by y
= ν (t ，
29
x). Then the solution of (3.8)-(3.9) is given
by u(t ， x) 乞f u(tJ(t , z))aad
叫树)=去 (tJ)=2川峰0 =
芋(t， y(阳呐)). (~宇己(t川 t ， ω) y)川)-1 dy
飞 dy'
-, I
百 =y(t ，:z:)
When a point (t , x) goes to (t o, x O) along the curve ((t , x) x = x(t , yO)} , 但'x/句 )(t ， y(t ， x)) tends to zero and p(t， ν (t ， x)) remains bounded. Therefore it must be true that (θυ/θy)(tO ， yO) = O. Differentiating the system (2.15) with respect to y , we get a system of linear ordinary differential equations for ðx /δy ， θυ/θν andθ'p/句 just
like (2.10). As the initial data (θx/θy， θ。/θ'y ， ðp/δy)(O，
y) = (1 ，4>'(ν) ，4>"(ν))
are not zero , it follows that (δ'x/θν ， θ旷θy ， θIp/θν)( t , y) 并 (0 ， 0 ， 0) for any t 2:
o.
Hence (θ'p/θν )(tO ， yO) 并 O. (θx/句) (t , y)
We can see that
satisfies the following differential equation:
d ( θz 飞伊 1
ðx . δ21 θυIθ21 θp
一.一.一一一一二
dt 飞 θyJ
δzθpθν 『 δuδpθ'y
，
θ'p2 θν-
U sing the assumptions and the above results , we get d ( θz 飞 |δ21 1 , 0U
I :,-
J 1.
dt 飞 θy J l(t ，y)=(tO ，yO)
Moreover , as (θx/句 )(tO ， yO)
0
0
0\
ðp I , 0
0
一τ (t ， XU, VU , pU) 一 (tU ， yU) 并 O. δpθν
= 0 and
(θx/句 )(t ， 俨)
> 0 for t < t O, it
must be true
that d f δz 、 一{一)
1. , ,. ., ~ 0 , i.e
dt 飞 θν/I(t ，y)百 (tO ，y勺..~.，
d
θx ， 1
一(一)1.， dt δ'y
'1 (t ，y)=(沪，宙。)
< O.
= t O. Here we introduce a function h = h(y) , just as in the proof of Theorem 3.2 , by setting h(的管 inf{t : (θ'x/句)(t ， y) = Hence (θx/句 )(t ， 俨) changes its sign at t
O} for each y. We consider the problem in a small neighborhood of 仰，沪，沪 ， pO)
δ2//θ'p2 乒 O. We pick up a point y (y 并 J)·By the same reasoning as in the above , we see that (δx/θy)( t , y) changes its sign at t = h(y). If h(y) > t O, we where
have
生 (t， y)>O and 生 (t， yO)
for t 巳 (t ， h(y)). As this means that x = x(t , y) is not monotone with respect to
y , the characteristic curves meet in a neighborhood of (tO , X O). In the case where h(y) < t O or h = h(y) is constant , we can similarly prove that the characteristic curves meet in a neighborhood of 川， x O ).
口
Chapter 4 Equations of Hamilton-Jacobi Type in One Space Dimension 34. 1. Nonexistence of classical solutions and historical remarks In this chapter we will treat general partial differential equations of first-order (3.1)(3.2) which satisfy Condition (A .I)' (Chapter 2); that is to say, the characteristic equations (2.15)-(2.16) have uniquely global solutions x = x(t , y) , v p
=
= υ ( t ， ν)
and
p( t , y) for all y ε ]Rn. The crucial point of Condition (A .I)' is the global
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 C 2-so1ution. 2) Ifthe Jacobian (Dx/Dy)(t , y) ofthe mapping x
= x(t , y)
vanishes somewhere , it
is impossible to extend the C2-so1ution 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 C 1 , but not of class C 2 , in the neighborhood of the point. Therefore , from now on , we consider the Cauchy problem (3.1)-(3.2) under the following conditions:
(1) (Dx/Dy)(tO , yO) (11)
=
O.
(Dx/Dy)(t ， ν) 并 o
for t
<
t O and y εIωhere 1 is an 叩en neighbo旷hood of
y= νo
(111) The characteristic c町ves x = x(t ， ν ) meet in a neighborhood of (tO , x O) where zotf z(tO ，俨) .
~4. 1.
NONEXISTENCE OF 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.
Now let us
consider the case (111). Proposition 4.1.
Suppose the α bo时 conditions (1) , (11) ， αnd (III). Then the
Cauchy problem (3.1)-(3.2) cannot αdmit a solution of class C 1 in the ωhole spαce.
Proof. Let x = x(t ， y) ， υ=υ (t ， y) , and p = p( t , y) be the solutions of (2.15)-(2.16) for n = 1. By the assumption (111) , there exist two points (α， α) and (α ， ß) such that x( α， α) = x( α ， ß) ( α 并。). As f = f(t , x , u , p) is of class C 2 , a Cauchy problem for (2.15) can not have more than one solution. Therefore we have (x(α， α) ， v(α ， α ) ， p( α， α)) 并 (x(α ， ß) ， v( α， ß) ， p( α ，
As
x(α， α)
=
x(矶的，
ß)).
it holds that (υ(α ， α ) ， p(α， α)) 手 (v( α ， ß) ， p( α ，
ß)).
( 4.1)
Let us assume that there exists a classical solution u = u( t , x) of (3.1 )-(3.2) in the whole space. A. Plis proved in [113] that a solution of class C 1 is generated by characteristic strips , that is to say
叫t， x(t， y)) = υ(式的， ZMJMM As x(α， α) = x(α， ß) ~f x O, it follows that θu
u(α ， zo)= 巾， α)= 巾，的，一 (t ， X O ) = 帅， α) = p( α，的. δz
This contradicts (4.1).
口
When there do not exist classical solutions , we must introduce generalized solutions which contain singularities. As the solutions are not of class C 1 , the solutions as regular as possible are required to be continuous. We recall by Condition (A .I)' that p = p( t , y) remains bounded. Since p = p( t ， ν) is corresponding to the fust derivatives of solutions , the generalized solutions should be Lipschitz continuous. This suggests to us that , ifthe 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
32
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 .I)', we are very familiar with "classical Hamilton-Jacobi equations." Here , when the function
f =
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 .I)' (s臼 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 f
= 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+ ν )+u(t ， x-y)-2u(t ， x) 三 KI ν1 2
for any x , y
ε Jæn
and t
ε Jæ
(4.2)
where K is a constant locally independent of x , y , and
t. 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 [33]) , by methods of envelopes
(E. Hopf [64] , S. Aizawa and N. Kikuchi [4]) , by difference-differential method (A. Douglis [44]) , especially 均 b y va 叩 rus副 hin 吨 g viscosity method (仰 W.H. Fleming [归阿 盯l叫叫]， 5 A. Frie 由 ed 缸 man [恻 叫 54 4] ， S.N. K 且 ruz l址抛 皿 zlt also been well developed by S. Aizawa [阴 句]， Y. Tomi让ta [口135 2 叫]. The global theory for non-convex Hamilton-Jacobi equations has recently been considered by M.G. Crandall , L. C. Evans , P •. Lions ([35] , [38 ], [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 , G. 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 [21]. Historical remarks for equations of the conservation law wil1 be given in Chapter 5.
94. 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 仕om
the methods. As the nice existence
theorems for generalized solutions depend general1y 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 con sider 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 [65 ], L. Gårdi吨 [55] ， and G 卫 D. Duff [46]. 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
= y( t , x) takes many values in a neighborhood of a critical point of the mapping x = x(t , 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 缸ld Lipschitz continuous solutions. However , a generalized solution function y
is not unique in the spaεe of Lipschitz continuous functions (see Chapters 9 and 11). Therefore we impose a supplement 町 condition on generalized solutions which guarantees the uniqueness of solutions. It is the semi-concavity condition (4.2). What we wo吐d like to prove is that , if we assume (A .I)飞 we can choose only one value from many values of the "multi-valued solution" in 0
4. EQUATIONS OF HAMILTON-JACOBI TYPE
34
From the results in this chapter , we will see that the equations with proper ty (A .I)' have properties similar to that of classical Hamilton-Jacobi equations. Therefore we would like to call the equations with property (A .I)' "equations of Hamilton-Jacobi type." Notice that S. Izumiya [74] considers the differences between Hamilton-Jacobi equations and
fir协 order
quasi-linear partial differential e-
quations 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
singul缸ities
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.
34.2. Construction of generalized solutions To make clear our problem, we write it again. Consider the Cauchy problem in one space dimension as follows:
生 +f(t， x ， u， 生) = 0 θt u(O , x)
= <þ (x)
in
{t > 0，刊的，
{t = 0 , x ε ]Rl },
on
where f = f(t , x , u , p) and <þ = <þ (y) are of class C∞ in ]R l ]R l
(4.3) ( 4 .4)
X
]R l X ]R l X ]R l
and
, respectively. For the construction of singularities of solutions , we need a little
stronger assumption on the regularity of f characteristic curves for
1. We write them by x
(11) , and (111) introduced in
= x(t , y) ，
and <þ
=
<þ(ν).
Then the
as the solution curves of (2.15)-(2.16)
v(t , y) , and p = p(t , y). Suppose (A .I )\This assures the global existence of x = x(t , y) , v = υ (t ， y) ， and p = p(t , y). We consider the Cauchy problem (4.3)-(4 .4) under three conditions (1) ,
for n
=
= f(t , x , u , p)
(4.3)-(4 .4)町'e de缸ed
34. 1.
υ =
As (III) is not concrete , we assume the following
sufficient condition for it to hold (see Theorem 3.3): δ2 f f , 0 0 0 0 哥王(tU ， X U , VU, pU) 并 0
where v O 乞f 叫 tO ，俨) and pO 营 p(tO ， yO).
( 4.5)
~4.2.
CONSTRUCTION OF GENERALIZED SOLUTIONS
35
From now on , we will construct the singulaxities of generalized solutions for t > t Owhere t _t Ois small. First we solve the equation (θx/θy)(t ， y) = 0 with respect to t. As (θx/ 句 )(t ， ν)
o for
> 0 for t < t Oand y ξ 1 (cf.
y E 1. Moreover ，出
(θx/句 )(t O ，
(II)) , it holds that (θx/ 句 )(t O ， y) 主
yO) = 0 , we get (θ2X/句 2) (tO ，俨) = O. We
assume here (θ3 X /θy3)( tO , yO) 并 O. This assumption is natural from the generic point of view. In this case , we see that J::l 3_
二亏 (tO ， yO)>O.
(4.6)
ây
< 0 for y > yO , i. e. , < 0 for y > yO. As (δx/θy )(0 , y) = 1, there exists t < t O such that
In fact , if (4.6) were false , we would have (θ2X/θν2)(t O ， ν) (δx/θν)(t O ， ν) (θx/θν)( t
, y) = O. This contradicts (1)
Let us draw a figure for the curve x = x(t , y) (t
> t O)
(see Figure 4.1).
×
』
。
y,
〉♀
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 OF HAMILTON-JACOBI TYPE
36
differential equations concerning
θxjθy ， θυ/θν ， andθ'pj θνjust
example , the equation for
is written by
d
f θz 飞
θ2f
一{一)=一一 (t ，
θxjθy
θzθ2f
like (2.10). For
θuθ2f
x , v ， p) 一十一一 (t ， x , v ， p) 一 +τ (t ， x υ ， p) θν 8p
θp
( 4.7)
dt 飞 θν/δzθpθνθuθ'p \V ，_，
θxj句， θ旷θyand δ'pj θy.
This system of equations is linear with respect to ((θxjθν )(O ， y) ，( θυ/θν)(0 ， ν) ， (δ'pj θy)(O ， y))
Since
= (l ， rþ'( ν) ， rþ叮 y)) 并 (0 ， 0 ，时， we see
that ((θxj句 )(t ， y) ， 但υ/θy)(t ， y) ， ( θpj 句 )(t ， ν)) 并 (0 ， 0 ， 0) for any (t ， y) ε ]R2. We recall here Lemma 1. 2:
p( t , y) 生 (t， y)
= 生( t , y)
θνδu
(t , y)
for any
ε ]R2.
As(θxjθν)( t O ， 俨) = 0 , it follows that (θ旷θν )(t O ， 俨) = 0 , hence that
θ'P 1.0 uy
0
τ (tU ， yU) 并 O.
0 for t
< t O, the
left-hand side of (4.7) must be non-positive at the point (t o, yO). Using
(4.5) 阻d
On the other hand , as (θxjθν )(tO ， yO) = 0 and (θxjθν )(t ， yO)
>
the above results , we get θ/δX\ Iθ2 f
1.0
0
0
0\
8p 1.0
0
( ~- ) 1 ，一τ (tU ， XU ， VU ， pu)~rw ， yU) 街\句 ) 1 (t，y)=(t O ，的 UjJ
Therefore we can uniquely solve the equation
(θxjθν )(t ， y)
< O.
(4.8)
= 0 with respect to t in
a neighborhood of (t O, yO) , and write the solution as a C∞ -function t = ρ (y) of y Then we have -J
窍，鸭、
~2_
~2 币
二{二二 (ρ(ν) ，的)=二二二 (ρ( 抖， ν)ρ， (的+云云 (ρ( 的，的= O. dy 飞 θν/θtδy
But (θ2xjθ的 (tO ， yO) = 0 and (θ2xjθtθ的 (t O ， yO) Similar衫，
< 0, it follows that
ρ， (ν。) =
O.
since
d2 f θz 飞 |θ2X 1"0 0\_11 。 θ3 X 一一 (tU ， yU) ρ(ν) 十一丁 (ρ (tO ， yO) =0 , (ρ (y) ， y))1 一(一 dy2 飞句 川自 =yO at句'" 8y /r-'
\"
/
'
> O. As ρ'(yO) = 0, it follows that ρ'(y) > 0 for ν > yO and that ρ， (ν) < 0 for y < yO. Thus the function t = p(ν) is strictly increasing for ν > yO and strictly decreasi吨 for y < yO , so the equation t 二 ρ(ν) has two solutions using (4.6) we get ρ"(yO)
~4.2.
y= 的 (t) and
CONSTRUCTION OF GENERALIZED SOL UTIONS
< 俨 <
y = Y2(t) (Y2(t)
37
yl(t)) for t > t O =ρ(yO) where t - t Ois small.
Summing up these results , we get the following. Lemma 4.2. In α neighborhood of (to ， ν。)， the equation (θx/句) (t , y) = 0 has two
y = 的 (t); νl(t) > Y2(t) for t > t O with t - t O sma ll. The two solutions y = Yi(t) (i = 1 ， 2)α re of dαss C∞ for t > t O ， αnd are continuous for t ~ t O (with Yl (t O+ 0) =的 (t O + 0) = y O). solt巾 ons， ν= 的 (t) αnd
The proof is obvious by the above discussions. Here we put Xi(t) ~f x(t , Yi(t)) (i
= > t O (with Xl(t O+ 0) = X2(t O+ 0) = x(t O ， ν。) = xO). Next we solve the equation x = x(t , y) with respect to y for x ε (Xl(t) ， X2(t)) ， t > t O (t - t O small). Then we get three solutions y = 9i(t , X) (i = 1, 2, 3) with gl(t , X) < def g2(t , X) < g3(t , X). Define Ui(t ， X)~' V(t , gi(t , X)) (i 二 1 ， 2 ， 3). This means that 1, 2). Then Xl(t) < X2(t) for t
the "solution" constructed by the characteristic method takes three values for x (Xl(t) ， 句 (t))
ε
(t > t O with t - t Osmall). By the assumption (4.5) , we treat here the
case where
θ2 f
f10
0
0
0
苟言 (tU ， XU ， VU ， pU)
> O.
(4.9)
Then we get the following. Lemma 4.3. i) For αny t
> t O with t - t O
smαII
and for
αny
x
ε (Xl(t) ， X2(t)) ， 切t
hαve
Ul(t , X) -U2(t , X) < 0αnd U2(t , X) - U3(t , X) > O. ii) There exists uniquely x
Ul (t ， γ (t)) =
= γ (t) ε (Xl(吟 ， X2(t)) sα tisfyin
U3(t ， γ (t)).
Proof. i) By (4.8) and (4.9) , we have (θIp/θν )(t O ， 的< 0, i.e. , (θ'p/句 )(t ， y)
P(t , gl(t , X)) > p(t , g2(t , X)) > p(t , g3(t , X)). Using these inequalities , we get
￡川 x) 一州
(4.10)
4. EQUATIONS OF HAMILTON-JACOBI TYPE
38
with
(咐，仆地(机))|Fh(t)=0 < 0 for 川队 (t) ， X2(t)). In the we obtain the inequality U2(t , X) - U3(t , X) > 0 for X 巳 (Xl(t) ， X2(t)]. Hence it holds that Ul(t , X) - U2(t , X)
same w町，
ii) Using the results of i , we have
￡MZ)- 州 with
(咐， x) 一句 (t， x))1叫。=(咐， x) 一句(机))1一例 >0 and
(Ul(t , x)
- U3(t , x))1
,.
I "'="'1 (t)
Therefore there exists uniquely
= (Ul(t , x) - U2(t , x))1
" I",=",t{ t)
γ (t) ε (Xl (t) , X2( t))
x
< O.
satisfying Ul (t ， γ (t))
U3(t ， γ(t)).
= 口
We see by Lemma 1.2 that the three functions
U
= Ui (t , x)
(i
= 1, 2, 3) satisfy the
> tO(t-t Osmall) and X2(t) 主 Z 主 Xl(t). As single-valued and continuous solution , we extend the solution
equation (4.3) in their domain where t we are looking for a U
=
u(t , x) into the above domain by defining if x 三 γ (t ), if x> γ (t).
(4.11)
This extended solution is obviously Lipschitz continuous. It satisfies the equation
(4.3) except on the curve {( t , x)
t 主 tO ，
x
= γ (t)}. The next problem is to prove
the uniqueness of generalized solutions. This is the subject of the following section.
34.3. Semi-concavity of generalized solutions First we give an example of non-uniqueness of Lipschitz continuous solutions which is due to Y. Tomita (see also Remark 2 of Theorem 11. 5 in Chapter 11). Example. Consider the Cauchy problem:
(生+(去)二。 ln … E ~1} U(O , x)
=
0
on
{t = 0 , x E
~1}.
~4.3.
SEMI-CONCAVITY OF GENERALIZED SOLUTIONS
Then it has a trivial solution u =
def
UO(t ， x)~'
39
0 , and also
u= 咐， x) 纠。
0:::: t 三
if
l -t + IXI
if t>
Ixl ，
IXI.
The above example shows the non-uniqueness of generalized solutions in the space of Lipschitz continuous functions. Therefore we must introduce a supplementary condition on generalized solutions which guarantees the uniqueness of solutions.
If f
f( t , x , u , p) is convex with respect to p , the supplementary condition would
=
be the semÏ-concavity condition (4.2). When f
=
f(t ， x ， 矶 p)
is concave , then it
becomes the semi-convexity condition as fo l1ows:
u = u( t , x) is said to be semÏ-convex if it satisfies the inequality u(t , x + ν) for 缸ly X
+ u(t , x -
y) - 2u(t , x) ~ KI ν1 2
(4.12)
, Y and t where K is a constant locally independent of x , y and t.
The supplementary condition (4.2) [or (4.12)] is veηimporlant ， because it is related to the stability of solutions of the Cauchy problem (4.3)-(4 .4). What we would like to claim here is that the solution
constn川ed
by (4.11) in 34.2 satis-
fies automatically the above supplementary condition; that is to say, we get the fo l1owing: θ2 f 1.0 _0 _.0 _0θ2f Theorem 4 .4. Assume 一一 (tU ， xU ， VU ， pU) > 0 (resp 一τ (tO ， XO ， VO ， pO) θIp2'"' ,..., ,..... 'r 1"" ..... '...........r. θp
Then the CωO η 俐 ve口z 功 叫)
in
solt出on
u
u(t , x) extended by (4.11) is
<
0).
sem 胁 Z
of 川 ( t俨O ， 川 X 0) .
α nezg 抖 hb 加 or 呻 hood
Proof. We consider the convexity case (4.9). It suffices to prove (4.2) in a small neighbor hood of {( t , x)
x
= γ (t)}. Assume that
x
= γ(t) and
y
> 0 (y
is small).
Then
u(t , x ={u(t ， x+ ν)
+ ν)
- u(t , x
+ u(t , x -
y) - 2u(t , x)
+ O)} 一 {u(t ， x -
0) - u(t , x - y)}
={去(机+句)一去(川的)叫卖(仙一 0) 一去(t， x-øy)}ν +{去 (t， x+O)- 去(机一 0) }y , where 0
< (}, (} < 1.
4. EQUATIONS OF HAMILTON-JACOBI TYPE
40
To estimate the first and second terms , we must calculate the second derivative of u = u(t , x):
jL(tJ)= ￡(叩斗(机)=￡POJ(tl))
=(去(t， y)/::(t， y))ly=y(t，，，) By (4.8)-(4. 时，但p/ 句 )(t ， ν) is negative in a neighborhood of (t O, yO). From one of the assumptions , it follows that 伽/句)(tO ， yO) = 0 and (θx/句)( t , yO) > 0 for t < t O. Hence , when (t , x) goes to (t O, x O) along the curve x = x( t , Y勺， (θ2U/θx 2 )(t ， x) tends to 一∞， i.e. ， θ2U/θX 2 )( t , x) is bounded from above in a neighborhood of (tO , X O). By (4.10) and the definition of u(t ， 叫， the third term is estimated as follows:
ZM)+0)ZM)-0)zpMM-PM 川<。但 Summing up the above results , we see that there exists a constant K satisfying u(t ， x+y)+u(t ， x-y)-2u(t ， x):::::KIν1 2 .
In the case where (θ2f/θ'p2)(t O ， x O ， V O ， pO) is negative , we can similar1 y prove that the solution is semi-convex in a neighborhood of (tO , x O ).
口
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 ofthe solution u = u(t , x) lies on the curve {(t , x) x= γ (t)}
where Ul(t ， γ (t)) = U3(t ， γ (t)). Taking the derivative ofthis equality with
respect to t, we get
{}z n 同 旦旦。， γ(小去。， γ(t))) dγθUl (t)=-OJ(小旦旦旦(们(t)) dt
at
n
By the definition of u = u( t ， 功， it holds that
旦 (t， ì (t)) = 一 (t ， ì(t)-O)
θU3
一 (t ， ì(t))
θz'θz
=
::'-(t, ì (t)
+ 0).
~4 .4.
= Ui(t , X)
As u get
,
d
(i
f..\
-:，~ (t) 一
= 1, 2)
COLLISION OF SINGULARITIES
41
satisfy the equation (4.3) in their definition domain , we
fhuZM)+0))-f(川去川) 字(们(t)+O) 一字(t， ,(• ux ax
dt
where x defγ(t) ， u def u(t ， γ(t) ).
0))
(4.14)
0)
The equali ty (4.14) is corresponding ωthe
Rankine-Hugoniot j 1llllP condition for equations of conservation law. Remark. We can extend the singularity of the solution for large t by the same method. But , if (θ2f/θ'p2)(t， 民 u ， p) changes its sign , the solution may generally 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.
g4 .4. Collision of singularities In this section , we assume that
f 二 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)
Let Wi
= {x
= 节(吟 ，
t>
ti}(i 二 1 ， 2) be si吨ula副es of the solution u 口 u(t ， x).
Assume that W 1 and W 2 meet at a time t
= T;
that is to say, let γ1 (t) < γ2(t) (t
T) andγl(T) = γ2(T). By (4.10) and (4.13) ，阴阳 that θu ZMt)-0)> 轨 γl(t) + 0) and 一川t) θz
for t
< T.
Taking the limit as t
•
0) >
~ - (t, , (川) 2
T - 0 in the above inequalities , we get
θuθu 2川(川 T盯)一0创) >坐川川阳+刊0)仨=一 (σ川 T， γ% 划甘响 '22纠J 川 川 (σ T盯)-0创) >一川阳0创)归 θz'θz
As f = f(t , x , u , p) is convex with respect to p, we have by (4.14) dγ
d"Y
云 (T)>Ef(T) ，
ie ， γ1 (t) > γ2(t) for t > T
<
4. EQUATIONS OF HAMILTON-JACOBI TYPE
42
Our problem is how to extend the solution
U
= u( t , x) for t > T
x l nD
E
tJ np 『J
马(t)~---
O
Y
Figure 4.2 First , see Figure 4.2 which expresses the behavior ofthe curve x =
x(t ， ν)
for t
>
T. We use the notations indicated in Figure 4.2. Each part AiBi (i = 1, 2 , 3) ofthe curve x = x(t , y) can be
u叫uely
solved with respect to y , and we write y = gi(t , X)
def
(i = 1, 2 , 3). Put Ui(t ， X)~' V(t , gi(t , X)) (i = 1, 2 , 3). As γ1 (t) > γ2(t) for t > T , the solution
U
= u(t , x)defined by (4.11) takes two values for x
εb2(t) ， γl(t)J.
define I(t , x) ~ Ul(t ， 叫一句 (t， x). By Lemma 4.3 , we get I(t ， γ2(t))
I(t ， γl(t))
= Ul(t , ì2(t)) -
U3(t门2 (t))
=Ul(t ， γ'2(t))
U2(t ， γ2(t))
-
< Ul(t , ìl(t)) - U2(t , ìl(t)) = 0 ,
= Ul(t ， γl(t)) - U3(t ， γ1 (t))
= U2(t ， γ1 (t)) -
U3(t门l(t)) > U2(t ， γ2(t))
-
U3(t ， γ2(t)) = O.
Moreover , it fol1ows from (4.10) that δI
否E(t ， z)=p(t， g10， z))-p(t ， gs(t ， z))>0.
We
~4 .4.
COLLISION OF SINGULARITIES
Hence there exists uniquely x
43
γ (t) ε[γ2(t) ， γ1 (t)] satisfyi吨 I(t ， γ (t))
= o. As
we are looking for a continuous and single-valued solution , we define the solution
u = u(t, x) for x
ε[γ2 (t) ， γl(t)] as follows: -O
(( )) ZZ uu if b ，，
唱A
，?心
rEE、 eEE 、
Z
、‘，，，，
ATtu
一一
，，.‘、
u
if
x 主 γ (t) ，
x < γ (t).
u(t , x) is Lipschitz continuous and semi concave. Taking the derivative of I(t ， γ (t)) = 0 with r臼pect to t , we also get the Then we can similarly show that u
same jump condition (4.14). The collision of a finite number of singularities can be treated by the above method. For example , let {x
= ìi(t) , t > td
(i
= 1, 2,... , k)
be singularities of
the solution u = u(t , x). Suppose (1) 悦。) <γ2(t) < ... < ìk(t) for t < T , and (II)γl(T) = γ2(T)
= ... =γk(T). Then we have
但 dt (T) \- > 但 dt (T) \ - > …>但 dt (T). l'
l'
...,
Therefore it follows that γ1 (t) > γ2(t)>...> γk(t)
for
t > T.
Next , we can prove that there exists uniquely x = γ (t) εbk(吟， γ1 (t) 1satisfying Ul(t ， γ(t))
=
Uk(t ， γ (t)).
A new generalized solution may be defined in the same
way as before. Except the case where an infinite number of singul町ities collide simultaneously, we can repeat the above discussions forever. 1f (δ211δIp2)(t ， x , u , p) changes its sign, the situation may generally become complicated. The phenomena are locally almost the same as the above , but the global behavior of singularities is generally quite different. We wi11 consider this subject in Chapter 7.
Chapter 5 Quasi-linear Partial Differential Equations of First-Order 95. 1. Introduction and problems In this chapter we consider the Cauchy problem for quasi-linear equations of firstorder in one space dimension as follows:
2+忖α向1 毗 叫(O ， x) u
= cþ(x)
on
{杠t
= 0, x
ε Jæl} ，
(5.2)
where 向 =αi(t ， x ， U) (i = 0, 1) and cþ = cþ(y) are of class C∞ in Jæ l
X
Jæ l
X
Jæl
and Jæl , respectively. Our principal interest is on the construction of singularities of weak solutions of (5.1). Therefore we put a little strong assumptions on the regulari ty of 向 =αi(t ， x ， U) (i = 0 , 1) and cþ = cþ(ν). On the other hand , there 缸e
many works on the global existence and uniqueness of weak solutions for quasi-
linear equations of first-order , especially for equations of the conservation law. For example , refer to P.D. Lax [98], J. Smoller [121] , A. Majda [104] , A. Jeffrey [77]. Though the difference method is very important from the point of view of numerical analysis , the best result on the global existence of weak solutions was obtained by the vanishing viscosity method (see O.A. Oleinik [112 ], S.N. Kruzhkov [88] , [92]). By the same reason stated in Chapter 4 , it seems to us that it would be
di缸cult
to get the informations on the singularities of solutions by the vanishing viscosity method. Therefore , in this chapter as we l1, we construct the singularities , especially of "shock" type , by the analysis of characteristic curves. The characteristic curves for (5.1)-(5.2) are the solution curves x
v = v(t , y) of (2.7)-(2.8) for
n 且1.
= x(t , y)
and
Here , we assume (A .I) (stated in 92.2) , which
assures the global existence of characteristic curves. We have seen by Theorem 2.1 that , if the Jacobian of the mappi吨 x = x ( t , y) vanishes at a point (t o , y勺， then the classical solutions blow up at a point (沪，的， where X O 哲 x(tO ， yO). Therefore we
~5. 1.
45
INTRODUCTION AND PROBLEMS
must introduce weak solutions for equation (5.1). To define it , we rewrite equation (5.1) 出 follows:
θuθ
一 + ~f(t ， x ， u) = g(巾 ， u)
(5.1')
θtθz
where (θf/θu)(t ， x ， u) = α1 (t ， 吼叫 and g(t ， x ， u) 一 (θf/θx)(t ， x ， u) = α。 (t ， x ， u). If g(t ， 吼叫三 0 ，
Equation (5.1') is of conservation law. Let
ω =
w(t , x) be locally
integrable in lR.2. The function ω=ω (t ， x) is called a weαk solution of (5.1 )-( 5功 if, for any k = k(t , x) in CO" (JR勺， it holds that
fω|θkθk 叫一 + f(川)一
__1 dtdx + 户(x )k(O , x )dx = 0, + g(川kl
|挠'加|
~
(5.3)
~
where JR~ 乞f{(t， z):t >0 ,
Z
εJRl}. Let ω =
w ( t , x) be a piecewise smooth
weak solution of (5.1) which has jump discontinuity along a curve x
= γ (t).
Then ,
by the definition of weak solutions , we get the following Rankine-Hugoniot jump
U :l:
问u h-
where
I'一 d
-a '归
condition:
I'd-u
(5 .4)
(t) = 吨， γ(t) 土 0) 誓 :EU(的(批 ε)
From now on , we will consider the Cauchy problem (5.1)-(5.2) in a neighborhood of (t o, x O) (x O = x(tO , yO)) under the following two assumptions:
(1) (Dx/Dy)(t O ， 俨)
=
O.
(11) (Dx/Dy)(t ， ν) 手 o for t
< t O and y
εIωhere 1
is an 叩en neighbourhood of
y= νo
Our problem is to see what kinds of phenomena may appe町 for t
>
t O• As
Example 1 in 33.2 shows , we have to consider two cases as follows:
(I1I) Though the Jαcobian of x
x(t ， ν) vanishes at α point (t O ， ν。)， the chα 旷r、-
创 α cter 付i臼 巾sti 时化 C CU1 旷 阿 r、
XO ~f x(tO , yO).
(IV) For t > t O, 叫 ere (Dx/Dy)(tO , yO) = 0, the chαracteristic CU1 旷附r. meet in α neighborhood of (沪 ， x O).
5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER
46
In the case (I II) , we can uniquely solve the equation x = x( t , y) with respect to
y and denote it by y = ν (t ， x) which becomes a continuous function. Put u(t , x) ~f v(t , y(t , x)). Then Theorem 2.1 means that u
=
u(t , x) is not a classical solution of
(5.1) , however , we can see that it is a continuous weak solution. The proof follows. The continuity of u = u( t , x) comes from the continuity of v
y = y(t , x). Put S ~f {(t , y)
= υ (t ，
is 出
y) and
(Dx/Dy)(t , y) = O} and H(S) 哇 {(t ， x)
x =
x (t , y) for (t ， υ)ε S}. Then , by Sard's theorem , the Lebesgue measure of H(S)
is zero. As a corollary of Theorem 2.1 , we equation (5.1) outside H(S). Hence u
ca且 see
= u(t , x)
that u = u(t , x)
satis丑es
the
is a continuous weak solution of
(5.1)-(5.2). The important problem remaining is to characteristic curves meet , that is to
s叮，
co日sider
x(t , yd
=
the case (IV). Suppose that the
x(t , y2) where
yl 并 ν2.
By the
uniqueness of solutions of (2.7)-(2.8) , we have (x( t , y仆， υ ( t ， 的))并 (x(t ， y2) ， υ (t ， 的)) . As x(t ， 的) = X(t , Y2) , it follows v(t ， yd 并咐， y2). This means that we will not be able to get continuous weak solutions. Therefore we willlook for piecewise smooth def
weak solutions. As we will see a little later , the solution u = u(t , x) ~υ( t , Y(t , x)) in the case (IV) takes several values after the Jacobian vanishes. As we are
looki吨 for
a single-valued solution , our problem is how to choose one appropriate value from these so that the solution becomes a may jump from a branch of
si吨le-valued
solu川川 tio ∞ I丑1 tωoa 皿 I且10 时the 盯r
weak solution of (5.1). If
0丑
one , the jump discontinuity must
satisfy Rankine-Hugoniot's equation (5 .4). Solving the Cauchy problem for (5 .4), we can get a curve of jump discontinuity. J. Guckenheimer [59] and G. Jennings [78] took this approach. But they forgot to pay attention on the uniqueness of solutions for the Cauchy problem to (5 .4). In fact , the right-hand side of (5 .4) is not Lipschitz continuous at an initial point (see Lemma 5.3). The aim of 35.3 is to prove the
of solutions for the Cauchy problem to (5 .4)
u叫ueness
The generic property of singularities was studied in D.G. Schaeffer [120] and applyi吨 Thorr山 "Catastrophe
T. Debeneix [41] by
theory." The construction of
singularities in two space dimensions for Hamilton-Jacobi equations was studied by M.
Ts毗 [137].
By a similar method , S.
Naka时 [109]-[110]
singularities of shock type in several space dimensions.
constructed the
But , as he treated the
singularities of fold and cusp types only, his results are essentially those in the case of two space di
~5.2.
EQUATIONS OF HAMILTON-JACOBI TYPE AND CONSERVATION LAW
47
example , [74]-[76]) give the generic classifications for the bifurcations of singularities of geometric solutions.
!ì 5.2.
Difference between equations of the conservation law and equations of Hamilton-Jacobi type In this section we wí11 give some property of equation (5.1). This will characterize a difference between Equation (5.1) and equations satisfying (A .I)\ A difference between characteristic Equations (2.7) and (2.15) is that (2.7) does not contain an equation concerning p = p(t , y). Let x = x(t , y) and v = v(t , y) be the solutions of (2.7)-(2.8) for n (δuj 8x )(t , x(t , y)) ,
1.
As p
p(t , y) is corresponding to
an ordinary differential equation 岛r p
= p(t , y) is written as
follows:
dp -r dt
8al 'l c 8ao 8α1 一」 (t ， z ， u)pz+{ 」 (t ， z ， υ) 一一位， υ )}p+ ~，，-U(t ， 川 1.
\
I •
θuθU
\-1 -
1 - /
8x
θz
p(O) = 矿 (y).
(5.6)
The equation (5.5) corresponds to the last one of (2.15). see by Corollary 1. 3 that , if u
(5.5)
More concretely, we
u(t , x) is of class C 2 , p
p(t , y) is equal to
(θujθx)(t ， y(t ， x)).
for t < t O. Then the solution p = p(t ， 俨 ) of (5.5)-(5.6) tends to infinity when t goes to t O- o.
Proposition 5. 1. Assume (δxjθy)( t O ， 俨)
= 0 and
( δxjδ的 (t ， 俨)并 o
Proof. Remember thatθxjθyand θ旷句 satisfy the system of ordinary differential 1.
equations (2.10) for n
Si且ce
this system (2.10) is linear with respect to
{δxjθy， θ旷θy} ， we get (θxjθy ， θ旷 θν) 并 (0 ， 0) for any (θxjδy)(tO ， yO)
= 0 , then
(t , y) εIR 2 . Therefore , if
(θυ/θν )(t O ， 俨)手 O. Moreover , we have by Lemma 1. 2
p(t , y) 生 (t， y) θu
= 些( t , y)
for all (t , y)
Hence we can easily get this proposition.
ε Il~.2 . 口
This proposition means that , if the J acobian vanishes somewhere , (A .I)' cannot be satisfied for quasi-linear equations
of 岛的 -order.
5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER
48
35.3. Construction of singularities of weak solutions In this section we will consider the case (IV) and construct the singularities of weak solutions which are called "shock" or "shock wave." A sufficient condition which guarante臼 the
situation (IV) is
37(t川 V O ) 并 0，
(5.7)
where v O 哇f v(t O ， 俨). This is the result of Theorem 3.2. If (5.7) is violated , we can construct an example in which characteristic curves do not meet though the Jacobian vanishes (see Example 1 in 33.2). Therefore , the solution of this case remains to be continuous , though it is not differentiable. First we solve the equation x = x( t , y) with respect to νfor t
> t Oin a
neighbor-
hood of (t O ， 的. This procedure is almost the same as in 34.2. By the assumptions
(1) and (11) , we have
(θ2xj句 2)(t O ，
yO) = O. We assume
(θ3 x j句 3)(户，俨)并 O. This
assumption is natural from the generic point of view. In this case also , we get , by the same way as the proof of (4.6) ,
去川 >0 Then the graph of the curve x = x(t , y) for t
(5.8)
> t O is drawn just as in Figure 4.1
(see 34.2). Therefore we use the same notations given in Figure 4. 1. H盯 the functions y = 的 (t) and y = Y2(t) ( ν1 >的) are the solutions of (θxj句 )(t ， y) = 0 and Xi(t) 乞f X(t , Yi(t)) (i
Yi(t) (i Z
1, 2). Then Lemma 4.2 is also true for these ν=
1, 2). When we solve the equation x
ε (Xl (功 ， X2(t)) ， we get three solutions y =
x(t , y) with respect to y for
9i(t , X) (91(t , X) < 92(t , X) < 93(t , X))
and define Ui( t , x) ~f v(t , 9i(t , x)) (i = 1, 2, 3). As we are looking for a single valued solution , we must choose only one
value 仕om
{Ui(t , X)
i = 1, 2 , 3} so that we can
get a weak solution of (5.1). As we have written in 35.1 , we can not get continuous weak solutions under the condition (IV). Therefore we look for a weak solution which is piecewise smooth. If U = u( t , x) is a weak solution of (5.1) which has jump discontinuity along a curve condition
{(t , x , u)
(5叫.
u
x
γ( 抖，
it must satisfy the Rankine-Hugoniot jump
This suggests us that a nice weak solution jumps from a branch
uI( t , x)} to the other ((t , x , u)
u = U3(t , X)} along a curve
35.3. CONSTRUCTION OF SINGULARITIES OF WEAK SOLUTIONS x= γ (t)
on which condition (5 .4) is
satis直ed.
49
Therefore our problem is to solve the
Cauchy problem for Rankine-Hugoniot's equation as follows: (生
f(t ， x 叫 (t ， x)) - f(t ， x 问 (t 川
dt x(tO) = XO.
> t O.
where Xl(t) 三 x(t) 三 X2(t) for t def
r
t > t O,
~' j (t , x) ,
Ul(t , X) - us(t , x)
(5.9)
j(t , x) is obviously O t > t and Xl(t) < x < X2(t)} and
The function j .n
differentiable in a domain U ~' {(t , x) 1.
is continuous on U (the closure of U). As we will show in Lemma 5.3 , it is not Lipschitz continuous at the end point (t O ， 俨). As j
= j(t , x)
is continuous in U , we
can see the existence of solutions by the classical theory. But we can not get the uniqueness of solutions. In J.
G旧ke出eimer
[59] and G. Jennings
[7时，
they did not
pay attention to this point. This is one of our problems which we will consider in this chapter. As we put the hypothesis (5.7) , we assume here more concretely
:?川 ， V O ) > 0 Though j is in
=
C 1 (U).
(5.10)
j(t , x) is not Lipschitz continuous at the point (t o, x勺， j
=
j(t , x)
Therefore , if the solution could enter into the interior of the domain
U , we can easily extend it further. Hence we restrict our discussions in a small
neighborhood of (t O , X O ). Lemma 5.2. i) (θvj句 )(t O ， 俨)
< o.
ii) For (t , x) E U , we get θUl
θU2
Ul(t , X) > U2(t , X) > us(t 'θX ， x) α nd -",-' (t , x) < 0 一 (t ， x) > 0 旦旦 (t ， x) < O. \....,....., ............， θz'θz iii) When (t , x) goes to (tO , XO) in U , then 阴阳/θx)(t ， x) (i
tend to
= 1, 2 , 3)
infiη ìty.
Proof. i) As (θxjθν) (t ， ν。)
d
I θz\
一(一川
> 0 for t < t O and
(θxjθν )(tO ， yO) = 0 , we have
Oa ，
二二土 (t勺。 ， V O ) τ (tO ，
dt 飞 θν) l(t ， y)=(tO ， yO)θx
uy
yO).
Since ((θxjθy)(t ， ν) ， (θvjθy)(t ， 的)并 (0 ， 0) for all (t ， ν) ， i t holds that (θvjθν )(tO ， yO) 笋 o.
5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER
50
Hence we get i) by (5.10).
9i(t , X) (i 1, 2, 3) , we have 91(t , X) < 92(t , X) < 93(t , X) and Ui(t , X) = V(t , 9i(t , X)) (i = 1, 2, 3). Using the property i , we get the first half of ii. As 91(t , X) < 归 (t) < 92(t , X) < 仇 (t) < 93(t , X) for z ε (Xl(t) ， X2(t)) ， we have ii) By the definition of Y
生 (t， 91(t， X)) > 0
些(t， 92(t， X)) < 0
θν'θν
As (δu;/âx)(t ， x)
，::J ~\U ，..v IJ
..............，
生 (t， 93(t， x)) > o. θν
= (âv/句 )(t ， 9队 x ))/(θx/θν )(t ， 9i(t ， X)) ，
we get the second part
of ii , and also iii.
口
= j(t , x) is continuoωly di.fferentiable in the domain __ def " . U and is continωω on U 叫 ere U~' {(t , x) t > t O , X2(t) > x > Xl(t)}. But it O is not Lipschitz continuous at the point (to , x ). ii) For t > t O, j = j (t , x) is decreasing with respect to x.
Lemma 5.3. i) The function j 一-;-
Proof. The first part of i is obvious. Taking the derivative of j (t , x) with respect to x , we have 1θfθf
:; j(t , x) = 一一一{一 (t ， x , ud 一 1 - U3 θ'x ，-，-，
θz
(5.11)
(t , x , U3)}
十二石[去川1)-t♂川d-f川 r 1δf I~ J âu +二-二|二一-{f(t ， 吼叫一 f(t ， x ， U3)} _ :'J (巾 ， U3) Iτ主 (t， x). Ul - U3
I Ul
- U3
au
I
ax
When (t , x) goes to (tO , x O) in U , the first term of (5.11) is convergent to (伊 f/δzδu) (t o, x O, UO) where U O 乞r u(tO , x O). Therefore it is bounded in a neigh borhood of 川， X O ). When (t , x) (t O, x O) in U , the coeffi. cient of (θUl 月x)(t ， x)
•
tends to (θ2f/θ2U)(t O ， 沪， u O )/2 (δU3/âx)(t ， X)
Lemma 5.2 to
=
(θα1 月u )(t O ，沪， u O )/2
has the same property as
(仇 dâx)(t ， x).
> O. The
coe伍cient of
Here we apply ii and iii of
(5.1 月， 出 t hen
tends to 一∞. This means that j = j(t , x) is not Lipschitz continuous at (tO , x O) , and that it is monotonically decreasing with respect to x in
JJemma 5.4. The functions Xi = Xi(t) properties:
i) 轩仲
def
~. x(t ， 以 t))
U.
(i = 1, 2)
sα tisfy
口
the following
~5.3.
CONSTRUCTION OF SINGULARITIES OF WEAK SOLUTIONS
dx? α叫一三 dt >j(t ， X2(t))
ii) 旦~ dt <' j(t , . ,, -Xl(t)) • , . 11 J
51
for t>t O.
Proof. i) Since the functions Y = 的 (t) and Y2(t) (Yl(t) > 的 (t)) are the solutions of (θx/θν )(t ， y)
= 0 for t > t O, we see that dYi ~~Xi(t)= 一 ( t "" ， 的 (t)) ， Yi(t))~:"(t) dt - ., . I ðt ,. \- + τ ôy (t \. "" , . dt /1
'
/1
=去。， Yi(t)) = 向川 ii) By the definition of j(t , X) , we have j(t , Xl(t))
= α 1(t ， Xl(t) ， U3(t ， Xl(t))
+ 8(Ul(t ， xd 一句 (t ， Xl)))'
0<8< 1.
As (δαd仇 )(tO ，沪 ， VO)>o ， αl(t ， x ， U) is strictly increasing with respect to u in a neighborhood of (tO ，沪， U O ). Moreoverwehaveul (t, x) >问 (t ， x) for (t , x) ε U and
V(t , Yl (t)) = U3(t , Xl(t)). Hence we get j(t , Xl(t))
> αl (t， Xl(t)， US(t， Xl(t))) = 兰 Xl (t). dt 口
We can similarly obtain the second inequality. Using the above lemma , we can prove the following. Proposition
U 乞f ((t , x)
5品 The
t
Cauchy problem (5.9) has a unique solution in the
domαm
> t O, X2(t) > x > Xl(t)}.
Proof. We extend the definition domain of j
= j (t , x)
to the whole space keeping
the following two properties: (1) j = j(t , x) is continuous on decreasing with respect to x for t
> t O•
JR2j
and (11) j(t , x) is
Then it is obvious that the Cauchy problem
(5.9) has a solution x = γ (t). From ii of Lemma 5 .4 and Lemma 5.3 , we get easily
Xl(t) < γ (t) < x 2 (t) for t > t O • Next we will prove the uniqueness of solutions. Let x= γl(t) and x = ì2(t) be two solutions of (5.9). As j =舟 ， x) is decreasing with respect to x , we have
二 [γl(t) γl (tO) 一
ì2(tW = 2 h'1 (t) - ì2(t)] [j (t , ìl (t)) γ2(tO )
j(川) )]三 0，
= o.
Hence we get γ1 (t) 三 γ2(t) for t ~ tO.
口
5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER
52
Now we define the weak solution of (5.1) in the interval (Xl (t) , X2(t)) for t
> tO
by
(t , x) =
(Ul(t , X)
if x 主 γ (t) ，
l U3(t , X)
if x> γ (t).
~
(5.12)
Remark. It is impossible to jump from the first branch {u second one {u
U3( t , x)} so
This is corresponding
Ul (t, X)} to the
刷 t hat 创t Ra 础 出山 nl 1垃ki 虫时 n 肘 e-Hu I吨 缸 u19 阴 创m O ∞o
tωo Lem卫皿la
6.2 for Hamilton-Jacobi equations.
35 .4. Entropy condition We first give an example which shows the non-uniqueness of weak solutions of (5.1)(5.2) in the space Lloc (I~.2). Lloc (IR2) is the space of measurable functions which are integrable on any compact set in IR2. Example. Consider the Cauchy problem
(θωθ -+-d=Oin 御街
ω (O ， x)=O
on
{t>0，叫则，
{t=O , x
Then this problem has a trivial solution ω (t ， x)
一-
Z
= 0 and a
01-
、
{(t , x) on {(t , x) on {(t , x) on
E4 唱
iEEtr 、 EEE
r，，‘、
ω
ATb
εIR}.
weak solution as follows:
叫主 t 主 O} ，
t>x>O} , t>-x>O}.
def
We get the above example by putting w(t ， x)~' (仇 jðx)(t ， x) where u = 吨 ， x) is the function given in "Example" of 34.3. As weak solutions of the above example are not unique , we must impose the entropy condition which guarantees the uniqueness ofweak 时utions in the space
Lloc( 1R2). Concerning the entropy condition, we follow
here O.A. Oleinik [112] and P.D. L缸 [9可Consider the equation (5.1') and let u
= u(t , x)
be a weak solution of (5.1')
which has jump discontinuity along the curve x = γ(t). Put U::l: (t) 哲 u(t ， γ(t) 士 0). Then the entropy condition is
expressed 田 follows:
For any value v between u+(t)
and u_(吟， it holds that S[v ， u-l;三 S[u+ ， u-l
where
ef f(υ)
- f(u)
S[v , u] ~υ -u
(5.13)
~5 .4.
ENTROPY CONDITION
53
A jump discontinuity of weak solutions of (5.1) satisfying (5.13) is called a "shock" or "shock
wα ve."
The condition (5.13) is also important from the viewpoint
of stability. For example , B. K. Quinn [115] has proved the following. Theorem 5.6. Assume that g(t , x , u) defined in (5.1') is u(t , x)
αnd
v
υ (t ，
(5. 川 .1'丁) foT' αllx αn 叫 d
x) t
αT'e
>
pzecewzse
identicallν ze T'o. 扩 u
cωOη 时ti饥 ηm ωO包 包 ωs1切引 d 副iffi 万 1e 阿'ff T' η 时且此tiωα b1e 叨tα k s01ωt 包L巾 tio旧
of
0ωωi必th initiωα1 dαtα uo(归 x) αηdυo(x) 1ψvhich αT'e pzece切 ωzse
Cω o时 n ti饥 n1ωωl仿 yd 码i伊 如 ffi 托e 陀 T T'tη tiωα b1扣tαn 叫 d Ll_斗tη 时tε叩 grab1扣e in x , and ifu = u(t ， x) αndv = υ (t ， x)
satisfy the condition (5.13) a10ng
discontin包ity CU T' ves ,
then it h01ds
thαt
Ilu(t) - v(t)llu 三 Iluo - vollu. ConveT'sely， 矿 (5.14)
(5.14)
is t T' ue , then the cond巾on (5.13) is satisfied.
We would like to show that our solution (5.12) is reasonable in the above sense. That is to say: Theorem
5 工 The
solution defined by (5.12) sαtisβes the entropy condition (5.13)
zn α neighboT'hood of the point (t O, x O)
Proof. We consider the case where (5.10) is satisfied. By Lemma 5.2 , we have u 一 (t)
> u+(t).
As the inequality (5.10) means the convexity of
f 口 f(t ， x ， u)
with
respect to u in a neighborhood of (t O, x O, v O ) , we get
f' (t , x , u_) > f' (t , x , u+) along a jump discontinuit
口
Next we extend the weak solutions 如r large t. If (δ2f/θu 2 )(t ， x , u) does not change the sign , we can extend the solutions with singularities by the above method and also treat the collision of shocks just as in
34 .4.
But , if (θ2 f/ 仇 2)(t ， x ， U)
changes its sign , the solutions may sometimes lose the entropy condition. Then we must introduce other types of singularities , for example "contact singularity." This Ís the subject which wi1l be discussed in Chapter 7. Remark 1. From the above discussions , we can say that the essential difference between equations of conservation law and Hamilton-Jacobi equations is the global solvability of ordinary differential equation (5.5)-(5.6) with respect to p
=
p(t , y).
5. QUASI-LINEAR EQUATIONS OF FIRST-ORDER
54
This property determines whether the singularities of generalized solutions , or of weak solutions , are continuous or not. Remark 2. We solved the ordinary differential equation (5.9) to for quasi-linear differential equations of first order (5.1).
constn时 shocks
But , for equations of
conservation law in one space dimension , we can reduce the construction of shocks to the singularities of solutions for Hamilton-Jacobi equations. Suppose that u
= u(t , x)
satisfies the following equation of conservation law: θuθ
一 +
;;_f(t , x , u) = 0 ðt ' 8x
(5.15)
Put u(t , x) = (θ叫θx )(t , x). Then ω=ω (t ， x) satis直es
苦 +f(t， x，
(5.16)
For Harnilton-Jacobi equation (5.16) , we can con由uct the singularities of generalized solutions , as done in g4.2. In this procedure , we do not need to solve ordinary differential equations. Then we see by (4.9) and (4.13) that u
=
(伽 /θx )(t , x)
is a
weak solution of (5.15) which has jump discontinuity satisfying locally the entropy condition (5.13). B.L. Rozdestvenskii had written this idea a little in [118]. But we cannot apply this idea to quasi-linear equations of first-order which are not of conservation law because the above transform u( t , x) well to
aηive
=
(如 /8x)(t ， x)
does not work
at Hamilton-Jacobi equation. Moreover the equations treated in [59]
and [78] do not depend on (t ， 吟， i.e. , f
= f(u).
By these re出ons ， the discussions
in Chapter 5 are necessary to construct the singularities of shock type for general quasi-linear partial differential equations
of 如st-order.
Chapter 6 Construction of Singularities for Hamilton-Jacobi Equations in T wo Space Dimensions 36. 1. Introduction Consider the Cauchy problem for a Hamilton-Jacobi equation in two space dimen S10口 s
as follows:
。uδu
一 ðt +' f( 一) òx J
\
u(O , x) where f
= f(p)
=0
= φ(x)
is of class C∞ and
4> =
{t
m on
> 0,
x ξ Jæ2} ，
(6.1)
{t 工 0 ， x ζR 2 } ，
4>(ν) is in
(6.2)
S( Jæ 2 ). Here , S( Jæ 2 ) is the space of
rapidly decreasing functions defined in R 2 . We assume that f = f(p) is uniformly convex , that is to say, there exists a constant C edθ2 f
f" (p)~'
1
IiθIPiÒpj "':十一 J1 "5， i ，j 三 2
> 0 such that ?C.I>O
(6.3)
This chapter is continued from Chapter 4. Our aim is to construct the singularities of generalized
soh山 ons
of (6.1)-(6.2) in two space dimensions. The
differe配e
between Chapter 4 and this one is the dimension of spaces. The crucial part in our analysis is to solve the equation x = x(t , y) in a neighborhood of a singular point. To do so , we must see the canonical forms of singularities of smooth mappings in a neighborhood of a singular point. Though the singularities of smooth mappings are simple in the case of one space dimension , this subject is di ffi.cult and complicated in the case of higher space dimensions. Here we apply the well-known results of H. Whitney [159] to get the canonical forms of si吨ular points of the smooth mappings. This is the reason why we restrict our discussions to the case of two space dimensions. First , let us repeat the definition of generalized solutions of (6.1). Deftnition. A Lipschitz continuous function u called a generalized solution of (6.1)-(6.2) if
u(t , x) defined on Jæ 1 x Jæ 2 is
a叫 only
if
56
6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS
= U(t , x) satisfies the equation (6.1) almost everywhere in JRl x JR2 and the initial condition (6.2) on {t = 0, x ξJR2} ii) u = u(t , x) is semi-concave, i.e. , there exists a constant K such that i) u
u(t , x + y)
+ u(t , x -
y) - 2u(t ， x) 三 Klyl2
Remark. Put Vi(t ， X) 哲 (θu/θ町)(t ， x) (i
x , y εJR2 ， t > O.
for all
= 1, 2).
(6 .4)
Then the equation (6.1) is
written down as a system of conservation law: θθ 一句+ ",,-
ât -.
, åXi
f(v)
=0
(i
= 1, 2).
(6.5)
The inequality (6 .4) turns into the entropy condition for (6.5). See a remark in
36.3. 36.2. Construction of solutions Characteristic
cu凹es
for
(6.1)-(6.2) 町e de负ned 臼 solution
curves of ordinary dif-
ferential equations as follows:
ldzθf dt 瓦 (p) ， I
句 (0)
dv ~;
\
= Yi , v(O)
= - f(p)
= 的)，
+ (p , f' (p)) ,
~. = 0
θ￠
Pi(O) = τ (y)
(i =口)，
(i = 1, 2) ,
~IH
where f' (p)
=
(θf/θ'Pl ， θf/θIp2) and 巾， q)
is the scalar product of two vectors P
and q. Solving these equations , we have
Z 二 ν +tf'(φ'(y)) 乞f Ht(y) , v = v(t , y) = cþ(y)
(6.6)
+ t[- f(矿 (y)) + (矿 (y) ，f' (cþ'(y)))].
(6.7)
Then H t is a smooth mapping from JR2 to JR2 and its Jacobian is given by
玄机 y) = det[I + tf"(内附y)] We write A(ν) ~f 1" (φ， (ν ) )cþ叮y) and denote the eigenvalues of A(ν) by Àl(Y) 主 λ 2(Y).
When the sp配e dimension is one，
λ l(Y)
=
1" (φ'(y))φ"(y).
Since f" 忡， (ν))
>
56.2. CONSTRUCTION OF SOLUTIONS
o and <þ =
<þ (y) is in
S( 则，
it follows that
À 1 = λ1 (y)
57
must take negative values at
some points. In this case also , we can prove that Itnh(ν)=λ1(俨) =
Put t O 乞f 11M. First assume t
-M < O.
< t O• As
去M 并 o for 町 Ud2， we can uniquely solve the equation (6.6) with respect to y and write y = y(t , x). Thenu = u(t， x) 乞f 叫t， y( t , x)) is a unique classical solution of (6.1 )-( 6.2) for t Our problem is to construct the solution for t
< t O•
> t O•
Suppose that t - t O is positive and sufficiently small , and consider the equation (6.6) in a neighborhood of (t O ，俨). The Jacobian of H t vanishes on I:t 乞r {y ε IR2
1 + tÀ 1 (y) = O}. Assume the condition:
(S.1)λ 1(y) 并 λ2(ν) ， gradλ1(ν) 并 o
~t; α nd ~t
on
is a simple closed
c也内
In this c出e， ~t is parameterized 出马 ={ν= t(的 (s )，的 (s)) : s ε I} ， where 1
is a closed interval and
的 =
C∞ (I)(i=1 ， 2).
Yi(S) is in
Put
,
习;卢乞彗纠 2 f气切 {yU川(
By the definition of H. Whitney [159 ], a point in ~t\ 习 is a fold point of the mapping Ht , i.e. ,
二x(t， y(s)) i= 0 for 作) E ~t\习， because it holds that
':;-v( t ， ν (s)) =
δv
:~
δ'u
du
d
. -，~ (ν (s)) .~x(t ， y(s)) ds = <þ' T ，~ , ds
并 O.
1/
Lemma 6. 1. Assume that the number
01 elements 01 ~t
去(t， y(s)) = (I + 仁的))艺= 0
is two. Then
ν(s) ε2:
lor
Proof. Put
)) 『
R
、
飞
••,,Eeae
-
9"
、、.，，，
ft
EE咱
wuo
。4
,
Ttw
、‘，，，
α
」
吨，"
...
，， at
NSU3 EEEEEEEE
'A
αα
E
， ?b ，， ι '''1、，，，飞、
....
,,
一一
A(
「BIll-L
ATb
凹3
+
、‘
rt
(6.8)
58
6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS
Then α dt ， y) and α2 (t ， ν) 町e li时axly dependent on ~t. As they are smooth in the interior of ~t ， they can not take every direction of JR2. That is to say, when (t , y) moves in a small neighborhood of (t O, yO) ， 向( t , y) (i = 1, 2) remain in a small neighborhood of 叫 tO ，俨) (i = 1, 2) where α 1(t O ， yO) andα 2(t O ， 俨) axe linearly dependent. Contrarily, when the point y = y(s) makes a round of
~t ，
(dyjds)(s) takes all
the directions. Therefore (dj ds )x( t ， ν (s )) vanishes at least at two points. But we see by (6.8) that the points where it vanishes are contained in
~r
Hence we get 口
this lemma. Assume here the following condition: (S.2) 习={巧，凡}，
cusp points
i. e. , the number
01 elements 01 ~~
is two;
and 巳( i 二 1 ， 2)
are
01 Ht , i.e. , -12
ds 2X (t ， y(s)) 并 o
at
忡忡巨
(i =口)
Remark. Assume:
01 >'1 =λ1(ν)αre Hessian 01 >-1 =λ1(ν )
(C.1) The singularities
non-degenerate; i. e. ，
a point y , then the
is regular there.
可 grad>-dν)
= 0 at
(C.2) (θvjθν) (tO , yO) 并 O. Then , for t
>
t O where t - t O is small , I: t becomes a simple closed curve and the
number of elements of
~~
is two.
We denote the restriction of v
υ( t ， ν)
on ~t by vE 工 VE(t ， ν). We see by def
(6.8) that 叩 = VE(t ， ν) takes its extremum on I:~， especially if we put v(t , Yi) ~ ci
(i = 1, 2) and suppose C1
its maximum at y
= 几.
< C2 , then VE(t , y) takes its minimum at y
Denote by Dt the interior of the curve I: t and by Slt the
interior of Ht(~t). Then the curve {νεJR2 ν= 巨 (i
= 旦回d
v(t ， ν) 口 cd is ta吨ent to
Dt at
= 1, 2). Here we apply the results ofH. Whitney [159]. Accordi吨 to his
theorem , the canonical forms of fold and cusp points are expressed respectively as follows: X1
讨
X1 = Y1Y2 - yf ,
X2 =
y2
X2 = y2
in a neighborhood of a fold point ,
(6.9 1)
in a neighborhood of a cusp point.
(6.9 2)
59
96.2. CONSTRUCTION OF SOLUTIONS
This means that the mapping (6 乌)
Ht can be regarded as the mappings of (6 乌) and
in' a neighborhood of a fo1d point and a cusp point , respective1y. Moreover he
proved that any smooth mapping from ]R2 to IR 2 can be approximated by smooth mappings whose singu1arities are on1y fo1d and cusp points. We see by his result that the curve Ht( í:, t) has the cusps at X 1 and X 2 where Xi 哩 Ht( Yi) (i When we solve the equation x = Ht(y) with respect to y for x E
= 1, 2).
nt , the expressions
= y(t , x) takes three values. Write them by Y =9i(t , X) (i = 1, 2 , 3). Here we choose y =92(t , X) as 92(t ， X) ε Dt for x εnt def When we write Ui(t ， X)~' V(t , 9i(t , X)) (i = 1, 2 , 3) , the solution of (6.1)-(6.2) has three va1ues {Ui( t , x) i = 1, 2 , 3)} for x εnt (see Figure 6.1 which shows these (6 乌) a时 (6.9 2 ) S可 that
the solution y
situations).
X
,
H
vCt ,
y)=C
,
H"'
‘
vCt ,
,
y)=C
Figure 6. 1. Curves 马 ， Ht( í:，t) and Ht- 1 (Ht (í:, t)) θ￠
Lemma 6.2. i) 瓦 Ui(t ， X) = 瓦 (9i(t ， X)) for 川岛 (i '…·….、
= 1, 2 , 3).
ii) (去 (t ， x) 一去。 ， x) ) . (ω (t ， x) - 9j(t , x)) t 并 J.
<
0 for x ξnt and
60
6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE Dl MENSIONS
< U2(t , X) and US(t , X) < U2(t , X) for x
iii) U1(t , X)
ε !lt.
Proof. i) This is equivalent to Corollary 1. 3 in !ì1.1. ii) From the definition of 9i(t , x) , 啊 we have 户 Z =9 纠蚓 i(川
As 9i(t ， X) 并 9j(t ， X) ， it follows that (θUdθx)(t ， x) 并 (θUj/θx)(t ， x) for i 并 J. Using the convexity of f
= f(p) , we
get the inequality ii.
iii) We prove the first inequality. Divide the simple closed curve θ!lt into θ岛，
two parts joining two cusp points X 1 and X 2 of
and write them by C 1 and
C2 • Here we introduce the family of solution curves of the following differential
z
equation:
=g1(tJ)-hM
These curves start from C 1 (or from C 2 ) and end at C 2 (or at C 1, respectively) , and the family of these curves covers the domain !l t. On each curve it holds that
;- {U1(t , X) - U2(t , X)} =
/刁H 唱
8U2 \
(-=;一土一一二)
飞 θzθxJ
. (91 - 92)
Since 问 (t ， x) = 问 (t ， x) on C 2 (or on Cd , we get U1(t , X)
< U2(t , X)
in !lt.
口
We are looking for a continuous solution. The iii of Lemma 6.2 means that we
U = U1 (t , x )}
cannot attain our aim by jumping from the first branch {( t , x , u) to the second one {(t ， 吼叫 third one {( t ， 吼叫
U
= U2 (t , x)}
U
问 (t ，
and also from the second branch to the
x)}. The last choice is to advance from the first
branch to the third one. dd
_
dd
一-
Lemma 6.3. Put I(t , x) ~'U1(t ， x) - U3(t , x). Then ft ~' {x ε !lt
is a smooth curve
contαined
in
!lt α nd
it joins two
cωp
Proof. In this case we introduce the family of curves 口3
， 丁G d
GU'A(''tu Z
、、，J ，
-
z-T
:
I(t , x) = O}
points X 1 and X 2.
de鱼ned
by
( z)
qdw
aTb
These curves also start from C 1 and end at C 2 , and the family of the curves covers the domain !l t. If we change the index "i" of 9i(t , x) (i = 1, 2 , 3) , the above solution curves may start from C 2 and end at C1 . On each solution curve , it holds that 18凹， βU~ 飞 ~I(t ， x(r)) 二( -<>-_' - -<>-_~ ) • 飞 θzθx J
(91 - 9s)
< o.
(6.10)
~6.3.
SEMI-CONCAVITY OF THE SOLUTION u
= u(t , z)
61
We see by Lemma 6.2 that the sign of l(t , x)l_ is different from that of 10.
l(t , x)I_. Therefore l(t , x) = 0 has a unique solution on each solution curve of ''"'2
(6.10). Obviously, l(t , x) = 0 at x = X 1 and x = X 2 • As it follows from ii of Lemma 6.2 that grad"， l(t ， x) 笋 o
we see that ft
=
and X 2 in the domain
ε f2 t ，
= O} is a smooth curve joining the points X 1
1 (t , x)
{x ε f2t
x
for
f2 t.
口
As we look for a single valued and continuous solution , we define a solution
u= 吨 ， x)in{(t ， x) x 巳 f2t} as follows: Writing f2 t ,+ ~f {x ξ f2t def U1(t , X) > O} and f2 t ,- ~'{x ε f2t U3(t , x) - U1(t , x) < O} , we define
f
U1(t , X) U3(t , X)
(t ， x)~' ~
l
As ft is smooth , it can be
if x ξ f2t汁， if x ξ f2t ，- .
parameterized 出口 = 18凹
{x = x(s)}. Then
8u，. 飞
d/(t , x(s)) = ~舌- ð~~) This means that , though
the 直rst
discontinuity along the curve direction of f
口，
U3(t ， X) 一
dx
石=
= u(t , x) has jump
derivative of the solution u
it is continuous with respect to the tangential
t.
!ì 6.3. Semi-concavity of the solution u = 仰 ， x) Let n(t , x) be a unit normal of ft advancing from f2 t ,- to f2 t ,+, and define at x εft
uu ,.
. _,
def
,.
UU
去 (t ， 让 O)~' li~_ :.= (t, x 土 cn). (J~丁~.....→-U (J~丁
Any C 2 -function satisfies the semi-convexity condition (6 .4). Therefore , for the proof of (6 .4), it we have 吨， x
s咄咄
to treat the
c甜 where x εft
(11 θu ，
+ y) 十 u(t ， x-y)-2u(t ， x)= JoI
( 一 (t ， x
飞 ax'
ax
/
= cn (é > 0). Then
8u
\
+ sy) - -;':(t, x + 0))
飞 θzθx\~'-'-)J
(11 θu ， 8u \/θu ，. 十 I ( τ (t ， x - 0) 一云一 (t ， x - sy) ) . y ds + (了。 ， x In
and y
飞 ax
au
+ 0) 一了。 ，x ax
. yds 飞
0)) . y.
/
62
6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS
The first and second terms are easily estimated by KI ν1 2 • To get the inequality (6 .4), it must be
(去(机 +0)-2(tJ 一州 n(t， x)::; 0 Contrarily, if (6.11) is true , then we can get
(6 .4).
(6.11)
Hence
(6.11) is equivalent to the
semi-concavity property. On the other hand , we have , by the
de丑 nition
of Dt ，土，
>
if x ξDt汁，
US(t , X)-Ul(t , X)<
if x εDt ，- ,
ua(t , x) -Ul(t , X) and
which show immediately that
二 [U3(t， x + sn) -
Ul(t , x
+ sn)ll.=o
~0
That is to say,
(320， z)-370， z)) 川 x E ft From the definition of U = u(t , x) , for x 页。 ， x
+ 0)
εDt ，
we have
.δu ，
=工土 (t ， x)
Substituting these relations into
and
(6.12)
8u~
一 (t ， x - 0) θx'-'-, =王三 8 (t ， x).
(6.12) , we
(6.11).
get
concave. Summing up the above results , we get
Thus u = u(t , x) is semi
the 岛llowing.
Theorem 6.4. Aβer the time t O ωhere the Jacobian of the mapping H t vanishes for the first time , the solution
tα kes
many values.
But ωe
can uniquely pick up only
one value from them so that the solution becomes single Then the condition of semi-concα vity is Remark.
def
= 仰，
(v(t , x
continuous.
automaticallν satisfied.
Putti吨 v(t ， x) U~. (θujδx)(t ， x)
jump discontinuity of v
valued αnd
in
(6.11) , we
have the condition on the
x):
+ 0) -
v(t , x - 0)) . n
三 0，
x
εft.
This is the entropy condition for the system of conservation law (6.5) given in the remark of 36. 1.
56 .4. COLLISION OF SINGULARITIES
~6 .4.
Collision of singularities
In this section we consider the collision of two singularities ed in ~i ，t ， ~6.2.
63
~6.2 ，
口 ，t 缸ld
f 2 ,t construct-
assuming the hypotheses (8.1) and (8.2). Here we use the notations
f2 i ,t , Di ,t for
fi ,t
(i
= 1, 2) which correspond to
~t ，
f2 t , Dt for
ft
introduced in
We see that there exist three kinds of collision as described in Figure 6.2.
Case (i): Consider the case where
口 ，t
and f 2 ,t collide as in (i) of Figure 6.2. Then
the solution becomes two-valued on a domain encircled by
口 ，t
and f 2 ,t. By almost
the same discussions as in 36.2 , 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 口 ，t and
口 ，t
intersect each other. It is described as a dotted curve
in Figure 6.2.
口， t 、.，，
、1''
，，，‘、
(·1·l-1
.•. ‘
Figure 6.2. Collision of singularities
64
6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS
Case (ii): Consider the collision of the type (ii) drawn in Figure 6.2. We put
ZLtf{丑，1 ， Yi, 2} and Ai，t 乞f{U:Hdu)ε 口，t and y 巳 Ht- 1 (ni，t) \Di，t} (i = 1, 2). Then Ai ,t is a smooth simple closed curve which is tangent to
~i ，t
at y
= 1, 2). When the end point of 口，t is on 口 ，t ， A 2 ,t is tangent end point y = A where A = 巧， 1 or A = 巧，2 (see Figure 6.3). 巧，2
(i
= 轧1
and
to A 1 ,t at the
VH R/ 』
‘
^2
,
t
Figure 6.3. Relation between A 1 ,t and A 2 ,t As v(t ,.) restricted on A 1 ,t does not take an extremum at y
(θv/θy)(t， ν) 并 o at y
A , we get
v( t， ν) = v(t , .)IA} is y = A transversally. On the other hand , as v(t ,.) restricted on A 2 ,t takes an extremum at y = A , the
= A;
i.e. , the curve CA ~f {y ε 1R.2
smooth in a neighbourhood of y
= A , and it intersects A1 ,t at
curve C A is tangent to A 2 ,t at y = A. This is in contradiction with the above. Hence the case (ii) does not happen. Case (iii): When r 1,t and 口 ，t meet at a time t I; l ,tO U I; 2 ,tO is drawn as in (i) of Figure 6.4. curve I; t
=
{νε ]R2
= tO
臼 shown
But ，由 the
in (iii) of Figure 6.2 ,
interior domain of the
1 +队 1(ν) = O} is monotonously increasing ,
I; l ,t U I; 2 ,t is
96 .4. COLLISION OF SINGULARITIES
described as in (ii) of Figure 6.4 for t
> tO.
65
When it satisfies the condition (8.1)
and (8.2) , we can construct the singularity of solution by just the same way as in
36.2. Remark on Figure 6 .4. Assume that I: 1 ,t and I: 2 ,t meet at y = yO = (a , b) and that the si吨ularities of λ= 人 1(ν) are non-degenerate. As À1 = À1 (y) does not take minimum and maximum at y = yO , we can suppose by Morse's lemma λ l(Y) = λ1 (y O) + (y1 一 α)2 一(的 -b户
1
+ t O λ 1(ν。) = Q.
= 1, 2) have singularities at y = yO. But , for t > t O, the curve 1 + t λ r( y) = O} is smooth in a neighborhood of y = yO.
Therefore I: i ,tO (i {y εR 2
芝 Lt O
(i)
(ii)
Figure 6.4. Change of I: 1 ,t U I: 2 ,t with respect to time 8umming up these results , we get the following. Theorem 6.5. Assume that the assumptions (8.1) αnd (8.2) are alωays satisfied. Then , even if two singularities collide with each other, we cαn uniquely pick up one value from two values of solution so that the solution becomes single-valued and continuous. In this case also , the condition of semi-concavity is satisfied.
66
6. CONSTRUCTION OF SINGULARITIES IN TWO SPACE DIMENSIONS
Remark. What we have done until this point is thelocal construction of singularities of generalized solutions or weak solutions. The next problem is to consider the global behavior of singularities. This subject has been considered in several cases. For single conservation laws , see D.G. Schaeffer [120J for n = 1 and B. Gaveau [56J for n
= 2 where
n is the space dimension.
Chapter 7 Equations of Conservation Law without Convexity Condition in One Space Dimension 37. 1. Introduction We consider the Cauchy problem for an equation of the conservation law in one space dimension as follows:
ôu
ô
一 +;;f(u)=O 御街
u(O , x) where f
=
f( u) is of class
=
<þ (x)
C∞.
in on
{t>O ， 川的，
(7.1)
{t=O , x εJRl },
We assume the initial data <þ
(7.2)
= <þ( x)
to be in
Cgo(JR1) , except in 37.2 where the Ri emann problem will be discussed. Even in the case where f
= f(u)
is not convex , the global existence of we a.k solutions of
(7.1)-(7.2) has been well-studied , for example by
0λ Oleir此 [112]，
S.N. Kruzhkov
[88 ], [92]. Our interests are in the singularities of wea.k 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 we a.k solutions. Our next problem is to extend the we a.k solution in the large. In this chapter , we consider the Cauchy problem (7.1)-(7.2) in one space dimension. If f
= f(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 34 .4 and 36 .4. But , if f叮 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.
68
7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION
97.2. Rarefaction waves and contact discontinuity In the Cauchy problem (7.1)-(7.2) , we assume in this section that the initial data nunU <> ZZ
(7.3)
飞
，。
) =
』〈 lt
where a and b are constants
Z
α
，， ‘ az、、
AMY
rt
<1> = <1>( x) is given by
withα 笋 b.
The Cauchy problem (7.1)-(7.2) with the
initial data (7.3) is called "Riemann problem." In this case , characteristic curves are written as
x
=
x( t , y)
( y 十 tf'(α) ， = ~ " . . rl
l y + tf'(b) ,
ν< 0 ,
y
> 0,
υ=UM=(:;;; Assume that f'(b) - 1'(α)
> O.
(7 .4) (7.5)
Then the domain D 乞f {(t , x)
tf'(α )<x<
t f' (b)} is not covered by the family of characteristic curves (7 .4). In the domain D , we look for a solution u = u(t , x) of the form u(t , x) = r(x/t). Then the function r = r(p) (p = x/t)
satis直es 。u
â
a
θx
一十一 f(u)
=
[一十 f' (r)].;r'( 一)
t
" " t 't
= O.
f' (r(p)) = p, then u = 仰， x) 彗 r( x /t) is a solution of (7.1). As this means that r = r(p) is the inverse function of u 叶 1'( u) , we must Therefore , if r
= r(p)
satisfies
assume for all
u
= u(t , x)
by
f" (u) 并 o
Now we define the solution u
{α
between αand b.
x
< tf'(α) ，
畔， z)qf{ 巾 /t) ，
l b,
x
(7.6)
(t , x) ε D ，
(7.7)
< t f' (b).
The solution defined by u
r(x/t) in the domain D is called "rarefaction wave." The solution (7.7) is continuous in {t > O} , but it has jump discontinuity at (t , x) = (0 , 0). Next we consider the case where 1'(α) -
((t , x)
t f' (b) < x <
tf'(α)}
f' (b) > o. Then the domain R ~f
is covered two times by the family of characteristic
~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 35.3. Let x = γ (t) be the solution of the fo l1owing differential equation:
( 生 UUU=JJJf 只卅(仙b)卜一→f贝( b一 α
dt γ(0)
(7.8)
= O.
Then we get easilyγ (t) = {f(的一 f(α )}tj(b 一 α). Here we define a weak solution
u = u(t , x) of (7.1)-(7.2) in the domain R by x< γ (t) ， x> γ (t).
If
x
γ (t)
satisfies the entropy condition (5.13) , then this singularity is just
"shock." But , if f = f(u) is not convex , the inequality (5.13) is sometimes violated. In this case we use "contact discontinuity" which we will explain from now. F
O
b
d
C
G
u
Figure 7.1 Assume that a
> b, and
that the entropy condition (5.13) is not satis负.ed. Sup-
= f( u) is drawn 出 in Figure 7. 1. We draw tangent lines L + and L - passing the points 忡， f(b)) and (α ，f (a)) ， respectively. Let L- and L+ be tangent to the curve f = f(u) at (c ,f(c)) a时间， f(d)) ， respectively. See Figure 7. 1. Then the slopes of L- and L+ are equal to f勺)皿d f'(d) , respectively. We denote C+ 乞f{(t， z):z zf'(d)t， t 三 O} and C- ~f {(t , x) x = f'(c)t , t 兰的· pose that the graph of f
10
1. CONSERVATION LAW WITHOUT CONVEXITY CONDITION
Put fl ~ {(t , X)
f' (c)t < x < f' (d)t , t > O} which is contained in R. The bound-
ary θfl is equal to C+ U C-. The shock curve x = γ (t) comes into the domain fl. We assume here that 1'( u) is monotonously decreasing on [d , c]. For any (t , x) E fl ,
f' (V) = x/t. If we use the function r = r(p) introduced in the definition of rarefaction wave , we have v = r(x/t). We define a weak solution u = U(t , X) of (7.1) in the domain R as follows: we can pick up a value v ε [d， c] satisfying
(b , f'(α)t > x > f' (d)t , u(t , x) = < r(x/t) , f' (d)t > x > f'(c)t , {α ， f' (c)t > x > f' (b)t.
(7.9)
Though the solution u 二 u(t ， x) has jump discontinuity along the curves C+ 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. "
37.3. An example of an equation of the conservation law Before giving an example , we explain the reason why we consider "contact discontinuity." The propagation of singularities for equations of conservation law without convexity condition has been studied , for example , by D 卫 Ballou [剖， J. Gucken heimer [59] , and G. Jennings [78]. Their method is to construct locally the ities ofweak solutions for the Cauchy problem
(7.1)-(7功 where
singul町
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 Ri emann problem for (7.1)-(7功.
Though the initial data are sufficiently smooth in their discussion , a
rarefaction wave
appe町 s
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 farnily 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
37.2. 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 prob lem
(7.1)-(7功 with
the smooth initial data. In answer to the above two questions ,
~7.3. AN EXAMPLE OF AN EQUATION OF THE CONSERVATION LAW
we will consider the following example: Assume that the graphs of f
=
71
f( u) and
cþ = cþ(x) are drawn as in Figure 7.2 (i) 皿d Figure 7.3 (i) , respectively. The graphs
of their derivatives appear in Figure 7.2 (ii)-(iii) and Figure (ii).
f= f'( υ 〉
cE
αl
υ
Figure 7.2 (i) f' f" =f'(υ 〉
Figure 7.2 (ii)
72
7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION
F"
Figure 7.2 (iii) l
且
hv
。
Figure 7.3 (i)
×
97.3. AN EXAMPLE OF AN EQUATION OF THE CONSERVATION LAW
73
, AV
×
Figure 7.3 (ii) In the above case , it holds that f( 向) 1, 2) and <þ"(c)
= <Þ "(d) = O.
=
f'(向)
=
0 (i
= 1, 2) , f" (bi) =
0 (i
=
Moreover , we assume max <þ (x) >屿，
because , if not , our problem is equivalent to a case where f = f(u) is concave on the whole IIt1 . For the Cauchy problem (7.1)-(7.2) , the characteristic curves are written by
x
=
x(t ， ν)
= y + tf'(<þ( ν)) ，
v=v(t , y)
= <þ (y).
(7.10)
Therefore it follows that AV UU ，，，‘、
HJ
、‘.，，，
ATb
E
牛
E
+ rJ,( AV, (
、‘，，， 、、 ，J
噜
aTb
一一
nO玄。
z-u ( wu)
(7.11)
We set h(y) ~r f" (仰))内)， and assume that the graph of h = 协) is drawn 出 ín Figure 7 .4. Next , take Ai Then we have at A 1
= (yi , h(ω))
矿 (Yd
>0
and
(i
= 1, 2)
where h(Yi)
f"(<Þ (Yl)) < 0,
< 0 and h'(Yi) = o.
74
7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION
and at A 2 的 Y2)
<0
and
f"(r:þ (Y2))
> o.
h
y
AI
Figure 7.4 def
We now put ti ~. -1/h(的) and Xi~' X(ti , Yi) (i
!ì 5.3 , we see that a shock appears at each point it by Si (i
= 1, 2).
condition for t
= 1, 2).
(钉 ， Xi)
(i
As we have shown in
= 1, 2) , and
we denote
We have proved in !ì 5 .4 that each shock Si satisfies the entropy
> ti
where t - ti is small. Our problem is to see what kinds of
phenomena may happen when we extend the shocks
马 (i=1 ， 2)
!ì 7 .4. Behavior of the shock Sl In this section we will extend the shock 51 for large t. To explain the situation , we repeat briefiy how we have constructed the shock Sl. As the graph of x for t
> ti
is drawn as in Figure 4.1 , we use the same notations appeared there.
Solving the equation x three solutions y 咐， ν)
= x(t , y)
= r:þ (y)
x(t ， ν)
= 9i(t , X)
(i 。
with respect to Y for x ε (Xl (t) , X2(t)) , we get
< 92(t , X) < 93(t , X). As r:þ (9i(t , X)) (i = 1, 2 , 3). Then
1, 2 , 3) with 91(t , X) def
for a l1 (t ， ν)ε 腔， we define Ui(t ， X)~'
we have by Lemma 5.2
Ul(t , X) < U2(t , X) < U3(t , X).
~7 .4.
BEHAVIOR OF THE SHOCK Sl
75
Here we must pay attention to the order of this inequality which is contrary to the one given in Lemma 5.2. This is caused by the property that
f" (<þ (y I))
is negative ,
while it is positive in Lemma 5.2. As in 95.2 , we consider the Rankine-Hugoniot jump condition as follows:
(生一川)一川) dt x(t I)
=
U1(t , X) - U3(t , X) X 1•
(7.12)
Proposition 5.5 says that the Cauchy problem (7.12) has a unique solution x = ì1(t) which is the shock curve of Sl. of ui(t , ì1(t))
U+(t)
= <Þ (9i (t, ì1(t)))
= 问 (t ， γl(t))
Lemma 7. 1. As t 叶 gl (t , ì1 (t))
τ'0
extend the shock Sl , we must see the behavior
= 1, 3). We and u_ (t) = U1(t , ì1(t)).
long αs
(i
the entropy condition for Sl is satisfied, the function
is decreasing and
Proof. By the definition of y
x
put U:l: (t) ~f U(t ， ì1(t) 土 0). Then
t 叶 g3 (t , ì1 (t))
= gi(t , x) , we
is
increαsmg.
have
= gi(t , X) +tf'(<þ (gi(t , X)))
(i
= 1, 2, 3).
Taking the derivatives with respect to t and x , we have
。=等+印刷 and
1= 尝{1 +矿"制)矿(gi)} Hence it holds that
号子(t， ← - 1'( Ui(t , x)) 学(t， x) ar ax
(i
=山 3)
which leads us to gi(t ， γl(t))
=
， θ'gi {一 - f' (Ui(t γl(t)))} (t , ì1(t)) dt
毡
::1.
'θz
Here we recall
U+(t) =
U3(t ， γl(t))
> U1(t ， γl(t)) = u_ (t).
Then the entropy condition (5.13) gives us
与去价三阳 (t， ì1(t))) = f'(u一附
(i = 1, 2, 3).
(7 叫
76
7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION
皿d
生!.(t) 三 f'(us(t， ')'1 (t))) = f'(u+(t)). dt Since (θ'9i/8x)(t ， x)
> 0 for
i = 1 and i = 3, we get
主g1(t， γ1 (t)) 主 O. dï''''-' γ1(t)):::;0 ,.,-" -' - and 兰gs(t， dt 口
The proof is complete. The initial function <þ 出 the
= <þ( x)
has the prop创 ies 田 drawn in Figure 7.3. As long
entropy condition (5.13) is satisfied , we see by Lemma 7.1 that <þ (g1 (t , ')'1 (t)))
advances to 0 , and that <þ(gs(t ， γ1(t))) goes to its maximum. We write here P+(t)
=
(u+(t) ,j(u+(t))) and P_ (t) = (u_ (t), f(u- (t))). The shock S1 enjoys the entropy condition for t > t1 where the graph of f = f(u) lies entirely in the upper side of the chord p+p_ joining two points P+(t) and P_ (t). If we extend the shock S1 further , then the chord p+p_ may be tangent to the curve {(包 ， f(u)) u 巳I1~.1} in finite time. We will assume so , i.e. , assume that
P+P一 is
tangent to the curve
f = f(u) at t = T. Then we see obviously that p+p_ is tangent to the curve f = f(u) at the point P+(T) , but not at the point P_(T). Therefore it holds that
守问=队 (T))
(7.14)
Le 曰 e… nE
Proof. i) Using the definition (7.12) of (dγl/ dt)(t) ， we have
1一一 毛主 (t) = 一 ( d21(t)) = rrLu+f' f' ((u+) 2 ,-, dt' dt ,-" - u_
f(u+) - f(u- )l 卫土 l d (t) (u+ - u_ )2 J
V(U+) 一只 u_)
(u+ - u_)2
一
As u+(t) = Us(t , ì(t)) , w可 get by (7.14)
去+凡T=[尝 +371flLT
f' (u 一)
1|一二 du (t).
u+ - u-J
~7.5.
Put yí
=
gl(T， γl(T)).
BEHAVIOR OF THE SHOCK S2
As u_ (t)
=
Ul(t ， γ1 (t))
77
4>(gdt ， γ1 (t))) , we obtain by
=
(7.13)
二乡仨← 叫一(仲 L t =斗{1守f 扣 非(t叶) 一→干1'(仙叫-斗(附旷州州 u_队 刨 (ωω g仇1川 圳 (川3尝?叫 "="n (t)t<) This leads us to
生(T) =
1 1 f' (u_(T))r u_,..(T) - u_(T) II 一(T) dt ,- 1 - ,,--,I
rdγ1
rT1\
I rn\
rll
I rn\ \
I~
rT1\
/J
4>'但)组(T，'Y1 (T))
Here we recall that the starting point ofthe shock Sl is the point (t 1 , X t). Therefore it follows that 扩(巧) > 0, u+(t) - u_ (t) > 0 and (句1 月x)(t ， x) >
o.
Hence we get
(d? ìddt2)(T) > O. 口
Part ii is easily obtained by (7.15). By Lemma 7.2 , we have
fp(们队 This implies that the entropy condition is satisfied. Summing up the above resu1ts , we have the following. Proposition 7.3. The shock Sl starting from the point
( 仇 ， xt) αlωα ys
satisfies
the entropy condition.
37.5. Behavior of the shock S2 In this section we extend the shock S2 which has appeared at the point (句 ， x2 ).
x(t , y) for t > t 2 can be drawn as in Figure 4. 1. Though x = x(t , y) in this section is different from x = x(t , y) in 34.2 and 37 .4, we use
The graph of x
the same notations introduced there. As in 37.4, we solve the Cauchy problem (7.12) with the initial condition X(t2) We write here also u:!:(t) and u_ (t)
=
=
4> (gl(t ， γ'2(t))).
= X 2, and denote
the solution by x = γ2(t).
U(t ， ì2(t) 土 0); that is to say,
Put
P，土 (t)
the entropy condition (5.13) for t
=
(u 土 (t) ， f(u土 (t))).
> t 2 where
t 一句 is
u+(t) =
4>(如何， γ2(t)))
The shock S2 satisfies
small.
In this case , as
78
7. CONSERVATION LAW WITHOUT CONVEXITY CONDITION
!"((Y2)) > 0 , (5.13) says that the graph of f = f(u) lies entirely in the lower side of the chord p+p_. Lemma 7.4. As long as the entropy
is decreasing and
t 叶剑 (t ， γ2 (t))
cond凶on
for S2 is satisfied,
t 叶 91(t ， γ2 (t))
is increasing.
We omit the proof, because it is similarly obtained as Lemma 7. 1. When t gets larger ,
(93(t ， 有 (t)))
tends to 0 and
(91(t ， γ2 (t) ))
tends to the maximum of =
(x). Therefore we assume that , though the entropy condition is satisfied for t < T， 瓦 p_ becomes tangent to the curve {(u , f(u))
follows that
豆豆 dt (T) \- = I
Lemma 7.5.
d2 γ2 ，fT'1\ i) 亏f- (T)
f'((93(t I \ r -,,ì'2( t)))) I JJI I
J
\.:1 õ)
< 0,
I ~ \ -
\
..\a
r1t
u ξ ]Rl} at t = T. Then it
_ = f'(u+(T)) 11\\1
ii) 扩 (u+(t)) I 可 =0
The proof is almost the same as that of Lemma 7.2. By this lemma , we get the following: Proposition 7.6. For t
> T where t - T is small, it holds that J牛
U~2 (t) < f' (u+ (t)). dt This proposition means that the entropy condition is violated for t > T. To overcome this point , we use "contact discontinuity" explained in 37.2.
。
V
Figure 7.5
U
u
57.5. BEHAVIOR OF THE SHOCK 52
79
Consider the equation υ
EJ 一
rJ-u u-)rt1、-
叫:
(-
E' ， 、、‘
ff.飞 1
rJ,
, u > V.
(7.16)
We solve the equation (7.16) with respect to v , and denote the solution by v
h(u). 5ee Figure 7.5. To construct a contact discontinuity starting at the point (t , x)
= (T， γ2(T)) ，
we solve the Cauchy problem as follows:
(22f(州的l(t， x)))) x(T)
(7.17)
= γ2(T) def
We denote a solution of (7.17) by x = γc(t) and put Sc~' ((t , x)
x
= γc(t) ，
t2
T}. Let W be a domain sUITounded by the curve Sc and a characteristic line x= 俨十厅I (4) (yO)) which passes through the point yO must satisfy the equation "1 2(T)
=
yO
(t , x) = (T， η (T)). 50, the point
+ T l' (4)(俨)).
Hence yO
=
g3(T， γ2(T) ).
Figure 7.6 explains this situation.
t x=γ/t)
T
× 。
Figure 7.6 Our problem is how to define a weak solution u
= u( t , x)
in the domain W. As
the domain W is not covered by the family of characteristic lines whose starting
80
7. CONSERVATION LAW WITHOUT CONVE Xl TY CONDITION
points are on the initial line {t
= 吗，
we introduce a family of characteristic lines
which start from the curve Sc , i.e. , we construct the "contact discontinuity." For any point (t , i) ε Sc ， we first define -.. ..
we determine u+ = u+ (t , i) by u+
,,-.... . . . . def u_ by .u_ def ~' 4>(i.J) where y ~. 91( t , i).
Next
def .. ,"""
h(u_) and draw a characteristic line which passes through the point (t ，;;) 臼 follows: x = ;;
~.
+ (t -
i) f'(';;+).
On the characteristic line , we define the value of the solution u
= u( t , x)
by u (t , x) =
u+. As the family of the above characteristic lines covers the domain W , the weak solution u = u( t , x) is completely defined on the domain W. It would be obvious that the entropy condition is satisfied (see Figure 7.7).
t
。
x=y+tF'(υ+(
T) )
O
Figure 7.7 Repeating the above discussions , we can extend further the weak solution of non-convex conservation law (7.1).
What we would like to insist in the above
construction is that we did not use "rarefaction waves." Example. For the Cauchy problem (7.1)-(7.2) , we assume that the initial function
4>
=
4>( x) is of a form as shown in Figure 7.8 , and that it
satisfies 飞皿 4> (x)
> b2
57.5. BEHAVIOR OF THE SHOCK 52
and min a
~(x)
<
81
b1 • Then , after the collision of two shocks , we will arrive at the
situation in Figure 7.1 (see
~7勾. AU·
Figure 7.8 Remark. Recently, S. Izumiya [73] and S. Izumiya & G.T. Kossioris [75]-[76] 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 丑rst-order partial differential equations , the results of [76] would be the best at today's point.
Chapter 8 D ifferentia I Ineq ua Iities of Haar Type s8. 1.
Introduction
The theory of ordinary differential inequalities was originated by Chaplygin [30] and Kamke [81] and then developed by Wazewski [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.
Rβsu1t s
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 8zarski [128] for a systematic study of such subjects. As for the theory of partial differential inequalities , first achievements were obtained by Haar [6 日， Nagumo [107 ], and then by Wazewski [154]. Up to now the theory has attracted a great deal of attention. (The reader is referred
tωo Deim m 江 11 叫 山 lin
[4 咀州 叫]， La 0 此 kshrnika 扭 z且1时 tham and Le 白ela 创 [阳 96 叫]， 缸 8 z缸 arsk 副 ki [口128 剖]， and Walt 此te 臼r [1臼 53 剖]. 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 Haar-Wazewski 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 mu1t ifunctions and differential inclusions , to investigate the uniqueness problem.
This method
allows us to deal with global solutions , the condition on whose srnoothness 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.
83
INTRODUCTION
The structure of the chapter is as follows. In Section !ì 8.2 we introduce a socalled differential inequality 01 Haar type. (See (8.5) 1ater. Note that (1. 14) and
(1. 16) were usually referred to as Haar 's differential inequalities.) An estimate via initia1 va1ues for functions satisfying this differenti81 inequ81ity will be estab1ished (cf. (8.6)-(8.7)). As an app1ication , Section !ì 8.3 gives some uniqueness criteria for globa1 classica1 solutions to the Cauchy prob1em for first-order nonlinear partia1 differentia1 equations. In this way, moreover , the continuous dependence on the Fin811y, Section !ì 8 .4 concerns some
initia1 data of solutions can be examined.
genera1izations to the case of weakly-coupled systems. Most ofthe resu1ts presented here were published in [141 ], [147]-[149 ], and [151]. The re1evant materia1 on
mt削functions
and differentia1 inclusions from [8] and [29]
may be found in Appendix 11 given at the end of the book. Throughout this chapter , 0
< 十∞，
and
。T~f(o ， T)x lRn={(t ， x)
O
The notation θ/θx will denote the gradient (θ/θXl ， .. . ， θ/θzπ). Let 1.1 and (.,.) be the Euclidean norm and sc81ar product in
lR飞 respective1y.
Denote by Lip(n T ) the set of all 10cally Lipschitz continuous functions u
州， x) defined on
Ft川her， set Lip([O , T) x lR n ) ~f Lip(n T ) 门 C([O ， T) x lR n ). For every function u = u (t , x) de缸ed on nT , we put
nT .
Dif(u) ~f {(s ， ν)εn T
u
u (t , x) is differentiab1e at 怡， ν)}.
=
We shall be concerned with the following class of Lipschitz continuous functions:
v(n T ) ~f {u(. ，.) εLip([O ， T)
x lRn )
:
::3
G C [0 , T]
mes( G)
= 0,
Dif(u) ~ nT\(G x lRn )}. (Here , "mes" signifies the Lebesgue measure on lR 1 .) In other words , a function u = u(t , x) in Lip([O , T) x lRn ) be10ngs to on1y iffor almost 811 t , it is differentiab1e at any point (t , x).
v(n T )
if and
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
84
s8.2. 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
Wazews挝、 Theorem
1.6 or Ha缸's
Theorem 1. 5. Our comments wi1l concem the Cauchy problem in the large for a general first-order partial differential equation as follows: θujθt
+ f(t ， x ， u ， âujθx) = 0
u(O , x) = rþ(x)
{t = 0, x
on
in
f! T ,
(8.1)
ξIR n } ，
(8.2)
where the Hamiltonian f = f(t , x ， 矶 p) is a function of (t ， x ， u ， p) εn T x IR1 andrþ=φ(x)
is a given function of x
X
IRn
ε ]Rn.
Assume the function f = f(t , x , u , p) to be locally Lipschitz continuous with respect
f! T
X
to 仙 ， p)
IR 1
X
in the sense specified in
S1. 3;
i.e. , for any bounded set K*
IRn there exist nonnegative numbers Ll' . . . ， Ln 缸ld M such that
If(t ， x ， u ， p) 一 f(t ， x ， v， q)1 三 LLilpi - qil for all two
c
(t ， 民矶时 and (t ， 吼叫 q)
C 1 -s01utions
in
K气
+ Mlu - vl
Let Ul 二 Ul(t ， X)
and U2
(8.3) U2(t , X) be
on the whole 百T of (8.1)-(8.2). Then Theorem 1. 8 assures the
equality Ul(t , X) = U2(t , X) in a neighborhood of {t = 0 , x εIR n } in f! T. The question arises as to whether this equality can be extended to the entire domain
f! T. 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 川 ， X O ) ε f!T to the hyperplane {t = 0,
D. ~f {(t ， x)
x 巳 IRn}
by consecutively gluing pyramids of the form
α1 < t 三句 ， q+L川三岛主出一 Lit
(i = 1,... , n)} ,
(8均
where 。三 α1 <α2 < T , Ci < d毡， 0 三 2Liα2 主仇 -Ci
(i=1 , 2 ,..., n) ,
so that Theorem 1. 6 can simultaneously work therein. The reason is that the relations between L 1 ,..., Ln and D. are bilateral and somehow awkward. In fact , Ll' . . . , Ln are to some extent overdetermined by D.. Specifically, for each applica-
tion of the theorem in such a procedure , D. must be ready-given. Thus , by (8 .4), a
~8.2.
tuple (L 1,.
••
A DIFFERENTIAL INEQUALITY OF HAAR TYPE
85
, Ln) is predetermined. However , the further condition (8.3) is needed
for (t , x , u , p) , (t ， 叭叭 q) εð x
K , with K
being a compact set in
(Uj(t ， x) ， 但Ujjδx)(t ， x)) ε K
for
]æ1
x ]æn such that
(t , x) εð ， j = 1, 2.
Therefore , L1' . . . , Ln are necessarily Lipschitz constants with respect to P1 , . . . , Pn ofthe function f = f(t , x , u , p) restricted to ð x K. But these constants might unfortunately become large , say, substantially greater than the above-predetermined values L 1,. • • , Ln. 80 we would reduce ð , and hence possibly fail to touch the hy pe叩lane{t=O ， x ξ ]æn} in such a procedure emanating from a point
(to, x O )
ξ DT .
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
11. )
Theorem 8. 1. Let u = u(t , x) be a function in V(D T function μ=μ (x)
locally bounded o7Ì
]æπαnd
). /f there exist a nonnegative a nonnegative function l = l(t) in
L 1(0 , T) such that |δu(t ， x)j街|三 l(t). [(1 + Ix l) lδu(t ， x)jδxl+μ (x)lu(t ， x)l] for αlmost
every t
ε (0 ， T) αnd
lu(t , x)1 :::; exp IG(x)
I
for
αllx ε ]æn ，
l(r)drl.
C(z)tfsup{|内)1
then sup
lu(O , y)l ,
(8.6)
1111 三(1+ 1 "，1) exp J~ l( -r )d -r -1
Jo
where
(8.5)
:
1们 (1+lxl) 叫飞(材一 1}.
(8.7)
Remark 1. Inequalities (1.1 4) and (1.1 6) 缸e usually referred to 出 Haa内 dif
.ferentialones
(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 , 1] be the Cantor 叫 i.e. ， the 则 of all numbers of the form +∞
t 营艺Ck/ 3 1c where each
CIC
(8.8)
is either 0 or 2. The set J is complete , nowhere dense on 1肘， and is
of Lebesgue measure 0 (see [42]). We define a
functionω=ω (t) ，
which is called the
w er e
ε J
C.αntor ladd叫 or
the
C，α ntor
given by (8.8) , we take M=
ε
,,,, 9"
LEW
h 噜
F』
ω
"=
词汇阳
function [53 , p. 38], in the following way. For t
,,,,, 9"
pbLEW
If (α ， ß)
is an open maximal interval in (0 , 1)\ J , then α ， ß ε J and ω(α) =ω (ß). def We set for t ξ(α ， β):ω (t) ~ω(α) =ω(的= const. It follows thatω=ω (t) is continuous on [0 , 1] and that dω (t)/dt
dω /dt
= 0 almost everywhere in (0 , 1). In fact ,
= 0 for t ε(0 ， 1)\ J.
Setting u(t , x) ~fω( t) for (t , x) εnl， we easily see that u = u( t , x) belongs to
C 1 ((仰，
1)\J)
x IRn)
n C(归， 1] x IR n) with u(阳)三 0 ，
θu(t ， x)/θt
=0
V(t ， x) ε((0 ， 1)\ J) x IR n .
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) 弄 o. Proof of Theorem 8. 1. For an arbitrary point (t o , X O ) ε nT ， we must prove that lu(tO ， xO)1 三 exp IC(x O)
I JO
Let 瓦=
H;. ~f {x
t' (t)dtl.
sup
lu(O , y)l.
(8.9)
Iyl 三(1+ 1 .， 0 1) exp f~o l(t)dt-l
εIR n
Ixl ~ r} , r ~ O. Denote by ~I(tO ， x O) the set of def ，~ .n all absolutely continuous functions x = x(t) from 1 ~. [0 , t O] into IR n which satisfy
dx.
almost everywhere in 1 the differential inclusion d~ (t) ε B l(t).(l+1 削1) subject to the constraint x(t O ) =
X O•
From [29 , Theorem VI-1坷， it follows that ~I(沪 ， X O ) is a noner叩ty compact set def( .n n, 1 .n n\def in C (I,lR n ). The sets Z(t , tO , x O) ~I {x(t) x(.) ε ~I(tO ， XO)} and r(tO , x O) ~ _.........I'
1
T"""\I'
{(r , x) : r ε 1， x ε Z (r , t O, XO)} are therefore ∞mpact in ]R n and ]R n+l , respectively, for all t ξ 1. Moreover , by the converse of Ascoli's theorem , the correspondence I 川叶 Z(t ， 沪 ， X O ) C ]Rn is a continuous mt山ifunction.
~8.2.
A DIFFERENTIAL INEQUALITY OF HAAR TYPE
We now define a real-valued function 9
87
= g(t) on 1 by setting
g(t) 乞fm皿 {lu(t， x)1 : x ε Z(t ， tO ， X O )}. Then according to the maximum theorem (see [8 , Theorem u
1ι16]) ，
the fact that
= u (t , x) is continuous on r (t o, X O ) implies that 9 = g( t) is continuous on 1. In
addition , we have: Lemma 8.2. For αηαrbitrary number ()ε (O ， t O ) ， thefunctiong
= g(t) is αbsol1山 ly
continuous on [(), t O]. The fo l1owing assertion will also be needed. Lemma 8.3. We have for
everνt ε 1
the inclusion
Z(t , tO , XO) C B (l+l Proof of Lemma 8.3. For each
η>
",
f
o l) exp ,'" l(r)dr 一
(8.10)
0 , let
时间 1+1巾 The function dm可 (t)/dt
叫=
m T/(t) is absolutely continuous , positive on 1 with the derivative
= -R(t) . (1 + m可 (t)).
To prove (8.10) we have only to show that
Ix(t)1 for all x
= x(t)
> Ix(t)1
'V t εI
(8.11)
in ~I(tO ， XO) and for all η> O.
Since 叫(tO) m可 (t)
< m可 (t)
>
Ix(tO)I , there exists a number 0 ε (0 ， t O ) such that
Ixol
whenever t ε (t O - 0, t O].
Assume that (8.11) were false , so that there exists t' ε[0 ， t O) such that mη ( t') 三 Ix(t')I. Setting t 1 ~f sup {t ε[O ， t O ) : mT/ (t) 三 Ix(t)l} < t O, we would have: Ix(t 1 )1
=
m可 W); Ix(t)1
< mη (t)
'V t ε (t 1 ， t O],
阳d
dm可 (t)/dt
= -R(t) . (1 + m可 (t))
三 -R(t) .
主一 Idx( t)/ dtl 三 dlx( t) 1/ dt almost everywhere in (t l, t O). On the other har吨
f 写ilia>I: 守卫 dt
(1 + Ix(t)l)
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
88 if and only if
mη(tO ) - mη (t 1 ) = mη (t O ) _IX(tl)1
>
Ix(tO)1 一 IxW )I. 口
Hence we obtain a contradiction. This proves Lemma 8.3. Proof of Lemma 8.2. Since u there exists a number L
lu(t 1 , x 1 )
-
u( t , x) is locally Lipschitz continuous in
nT ,
> 0 such that
u( t2, x 2 )1 三 L(ltl _ t 2 1 + Ix 1 \f
_
x 2 1)
(t 1 , X 巧， (t 2 , x 2 ) ε ([8， t O] x Rπ) 门 r(tO ， X O ).
By the absolute continuity of the Lebesgue integral , Lemma 8.2 will be proved if we can show that
|到t 1 )
_
g( t 2 ) 1 三 L[ltl_t21+(1+1叫 (8.12) \f t 1 , t 2 ε [8 ， t O].
Now let
g(t 1 ) 主 g(t 2 )
and
g(t 1 ) = lu(t 1 , x(t 1 ))1
for some x = x(t) in ~I (to ， XO). Since x( 户 )ε Z(沪，沪 ， x勺， we have O 三 g(t 1 )
_
g(t 2 )
=
lu(t\ x(t 1 ))1_ g(t 2 )
三 IU(t\X(tl))I-lu( t2， x(♂ ))1 三 lu(t 1 ， x(t 1 ))
υ[lt 1 -
t2 1+ Ix(t 1 )
-
u( t2, x(t 2 ))1
呐 1] = L[I川叫 J ~; (t)dtIJ [t 1 ,t 2 ]
三 L[I川 Therefore , (8.12) follows from Lemma 8.3. The proof is then
complete.
口
Going back to the proof of Theorem 8.1 , we set now
h(t) ~f
I
R(r)dr
for
By Lemma 8.3 and the definition of 9 =
g(吟，
t ε [O， T].
JO
the inequality (8.9) will be obtained
if we show that
g(t) ::; g(O) . exp [C(X O) . h(t)]
\f t ε[0 ， t O].
(8.13)
~8.2.
For every
η>
A DIFFERENTIAL INEQUALITY OF HAAR TYPE
89
0 , let
gTJ(t) 乞f [g(O) + η] 叫忡忡。 )+η] . [h(t) 十 ηtl} To get (8.13) , it suffices to prove that
g(t) < 9η (t)
Vtε[0 ， t O].
(8.14)
Let ω (t) 哩 gTJ(t) - g(t) , where ηis temporarily fixed. Then (8.14) is equivalent toω (t) > 0
ω(0)
Vtε[0 ， t O]. Obviously， ω(0) =η> O. We shall show thatω (t) 主
Vtε[0 ， t O]. Assume this is false , so there exists t' ε(0 ， t O] such thatω (t')
<
ω(0).
It is well-known that there exists a set G 1 C (0 , T) of Lebesgue measure 0 with the property that
dh(t)jdt = C(t)
Vt E (0 , T) \ G 1 •
By the hypothesis of Theorem 8.1 , we find a set G 2 C (0 , T) also of Lebesgue measure 0 such that
nT \ (G 2
X ]R
n) C
Dif( 叫 and
that (8.5) holds for all t
ξ
(0 , T) \ G 2 , X ε ]Rn.
Since the image of a null set under an absolutely continuous mapping is also a null set , Lemma 8.2 implies
me个 (G n [8 , t O ])) 口 o V8E(0， t勺， where G qzf G1U G2·So
mes(ω(Gn 归Jolo=jv作 (G 门 [8， t O]))
(8 叫
= O.
From (8.15) and the continuity of ω=ω (t) on 1 we conclude that there is a number λwith max{O ， ω (t')) < λ<ω(0)
and
λεω [0 ， t'] \ω(G 门 [0 ， t O]).
Let
九 4些f inf{ t
E
[0 , t']
ω(t) = λ}.
It is obvious that ω( 儿 )=λ ，九 ε (O ， t') \G， and that ω (t) > λVt Suppose that
g(t.) = lu(t ., x.)1 =ε . u(i. ， x.) ，
εtf signu(thh)
E
[O , t.).
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
90
for some x. E Z(儿 ， t O ， x O). Then one may find a function 与=与 (t) in ~I (tO ， XO) so
that 与 (t.) =
x.. Choose a unit vector e
\
y(s) on Jæ. l satisfying the =
(8.16)
ρlv
= x"
/，，‘、
EA
(
咱
+ UU s) )
一一
equations)
d-d U-s) s 飞 J''
differe出al
x(t) 生f y(h(t)) for t E [0 , T]. Of course , x x(九)
with
e， ε 去川)) =一|去川) I
The system (of n ordinary
has a C 1 -so1ution y
ε Jæ.n
cond山on y(h(儿)) =
x.. Let
x(t) is absolutely continuous on [0 ，町，
and
dx. dt (t) \. /
dh. du (t) ~，" (h(t)) \. / . ds
= -~; dt
Therefore , the function
.x
=
.x (t)
口 C(t)
. (1
+ Ix(t)l) . e
Yt E (0 , T) \ G 1 •
defined by if
0:::;
t 三儿，
if 儿三 t 三 t O belongs to ~I( t O, X O). Hence , x(t) 巳 Z(t ， tO ， X O )
Ytε[0 ，叫.
This implies ε . u(t ， x(t)) 三 lu(t ， x(t))1 三 g(t) = 9η (t) 一 ω (t)
for all t
ε[0 ，
入
(8.17)
t.). Besides that ,
ε .u(t. ， x(九))
Furthermore , since
= lu(t 扪 x.)1 = g(t.) = 9叮(九)一 ω( 九 )=gη( 九)一 λ(8.18) με (0 ，
T) \ G , we see that:
(i) u = u(t , x) is differentiable at (t. , x.) (ii) x = x(t) is differentiable at t. with
生 (t.) = C(儿) . (1十 Ix. 1) . e dt (iii)
< 9η (t)
9η =gη (t)
is differentiable at
守(牛
t.
with
~8.3.
UNIQUENESS OF GLOBAL CLASSICAL SOLUTIONS
91
80 it follows from (8.17)-(8.18) that
二 lε u(t， x(t))] It=to 2 等(九) Consequently,
ε 去(儿， z(tJ)+ 〈 Zw 去(儿， x(t*)) ) 主 [C(X O ) + η]
. [f(儿 )+η] . g (t*). .,,
Hence ,
2(0tω 川川. ，川山 叫 ， 叫.)忡 x. +f川 +什忡训 |μ队 叫* x.
ε
主 [C(x ♂0勺)+η 叶]
Because
η>
. [l u(t. , x*)1 + λl
EA 咱
)l
r，.‘、
Z
nδ
u(''tv
事
.. ,,,,
晶亭
‘‘、
+ C..,,,Z
、、‘
Z*
。
...• *
、、3，，，
u-z ,,, ‘、
*
''ι
+Z
θ-θ
[
、、EE，，，
> nz( * )
EA 唱
Z* )
+ η]
0 , the last inequality together with (8.16) implies /，，‘飞
*
λ> ''ι
''ι
仇-a
(
0 and
. [f(t.)
ny)
On the other hand , sinc坦 hε Z(儿 ， t O ， x O) , Lemma 8.3 yields Ix.1
~ (1 忡。1)呻 f 珩同T 一 l 三 (1 + I 巾
Therefore , the formula (8.7) gives C(X O ) 之 |μ(x.)I ， which shows that (8.19) contradicts (8.5). It follows that there exists no t' ε [0 ， t O] with w( t') < ω(0). Thus , ω (t) 主 ω(0)
> 0 for
all t ε [0 ， t O ]; the inequality (8.14) is thereby proved. This
completes the proof of Theorem 8. 1.
口
38.3. 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 但rst-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 [93] is given. In the present section we restrict ourselves to the case of C 1 -s01utions , 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 如st-ord町，
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 缸d
some new uniqueness criteria (posed on the Hamiltonian
global
C 1-so1utions
f
=
f 衍 ， x ， u ， p))
for
of (8.1 )-(8.2). Let us 位rst repeat the defìnition of solutions to
be considered. De 6. nition 1. A function u
= u(t , x) in C 1 (n T ) n C([O , T) x ]Rn) is called a
global
C 1 -s01ution to the Cauchy problem (8.1)-(8.2) if it satisfìes (8.1) everywhere in
nT
ε ]Rn.
and (8.2) for all x
As was shown in the introductory comments of S8.2 , for the uniqueness of global C 1 -so1utions , the following result may be invoked instead of Theorem 1. 8. S叩'pose
Theorem 8.4. exist
nonnegα tive
f = f(t , x , u , p) sαtisfies the following condition: there
numbers L , M such that
If(t , x , u ， p) 一 f(t ， 叭叭 q)1 主 L(1 for all (t ， 叭叭抖 ， (t ， x ， v ， q) εnT x ]Rl
X ]Rn.
+ Ix l) lp 一 ql + Mlu Iful
= Ul(t , X)
vl
and U2
(8.20)
=
U2(t ， X) αre
global Cl-so11山ons to the problem (8.1)-(8.2) , then Ul(t ， X) 三句 (t ， x) in
Proof. Consider the function u
=
nT.
def
u(t , x) ~ 叫 (t ， x) - U2(t , x). Then 叫 O ， x) 三 O.
Furthermore, by (8.20) and the defìnition of global C 1 -so1utions, we have
|各t， x)1 = 阳 I fκ 附(t川 ， 凯川 x，， Ul 川尝叫一→f(仰式 川 M 川， 句叫 Z叽M U2叫2纠川( 18θU2 /.
'\ I
主 L(l+|z|)|11(tJ) 一一 (t ， x) 1+ Mlul (t , x) - U2(t , x)1 |θzθx'-'-'I
=L(1+|Z|)12叫 + Mlu(t , x)1 for all
(t ， x) 巳 n T .
proves the
Now it follows from Theorem 8.1 that u (t , x)
三 o
theorem.
in
nT .
This 口
The next sharpening (and its cor伽y) of Theorem 8 .4 wiU give some useful uniqueness criteria for global C 1 -so1utions with bounded derivatives. Theorem 8.5.
S叩'pose
f
any compact sets K 1 C ]Rl
f(t , x , u , p) satisfies the following condition: for
, K2
and a nonnegative function μKt ， K2
C ]Rn there exist a nonnegative number L K2 μ Kl> K2(X)
locally bounded on IR n such thαt
(8.20) with L K2 and μ 町 .K2(X) in place of L α nd M , respectively, holds for all
~8.3.
UNIQUENESS OF GLOBAL CLASSICAL SOLUTIONS
93
(t , x , u ， 时 ， (t ， x ， v ， q) ε ÛT X K 1 X K 2. Iful = Ul(t , X) and u2 = U2(t , X) are global C 1-so1utions to the problem (8.1 )-(8.2) 四th I 二三千 (t ， x)1
sup
。，" )εOT I u ;c
then Ul(t , x) Proof. Let
三 U2(t ， X) U
in
< +∞
(j = 1, 2) ,
ÛT.
= u( t , x) be as in the proof of Theorem 8 .4, and let
dmx
sup
|号子(机)1 < +∞， K2 苦苦 C JR. , L 乞fLK2' I n
]=>，~ (t ，.，) εOT
(8.21 )
U品|
X Ic乞! (-k , k) x 11
x (-k， 的 c JR. n
…
(k=1 , 2,...).
(8.22)
times
For an arbitrarily fixed T' E (0 , T) , we consider the sequence {plc}t~ of the following parallelepi peds:
plc 乞!(O ， T')xX Ic ={(t ， x): O
c
p2
C ... C
plc
C . .. and
+∞
1
.'Ü plc = ÛT'.
1c =1
Next , t a.ke
Slc 乞f mHma平 .IUj(t ， x)l ， 叫苦 [_Slc ， slc] C JR. l.
(8.23)
]=>，~ (t ，.，) εp'
We now define a
functionμ=μ( x)
. . (μ vl
v_
μ (x) ~' <飞'叮
(x)
lμ " h+ l ., (x) 飞
It follows that
""1'
μ=μ (x)
by setting
if x εX1' if x ε X lc+ l \X Ic (for k = 1, 2.. . ).
(8.24)
.^2
is locally bounded on JR.n. Moreover , (8.21)-(8.24) together
with the hypothesis of the theorem imply
|各t， X)1 = If(t， X， Ul川等。， x))-f(巾，咐， x)号叫| |月川 δU2/. \ I 三 L(1+|z|)|-i(tJ) 一一一 (t ， x)1 + μ (X)I U 1(t ， X) 一句 (t ， x) I |θzθx ,-, -'1
=川 Ix 1) I 去叫 +μ(x)1吨， x)1 山 (We may check this inequality fÌr st for (t , x) in p\ and then for (t , x) in each plc+l \ plc .) Theorem 8.1 therefore shows that u (t , x) 三 o in ÛT'. Since T' ε (0 ， T) is arbitrarily chosen , the conclusion
follows.
口
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
94
Corollary 8.6. Let f = f(t , x , u , p) b仇ng to C 1( !l T
X ]R l X ]Rn)
such that the
function ν= 仰 ， p)~'
|θf'J
\11
.,
11
1
1-;'~(t ， x ， u ， p)/(l+lx l) l
sup
1\
(""也)εIR n x IlP 1 up
= Ul(t ， X)
is finite αnd continuoω on [O , T] x ]Rn. Iful
αnd
U2
= U2(t ， X)
αre
global
C 1-solutions to the Cauchy problem (8.1 )-(8.2) with
I 学 (t， x)1 < +∞(j=叫
sup
(t ,"')EOT 1σx
then Ul(t ， X) 三句 (t ， x) in !l T.
Proof. For any convex compact sets K 1 C ]Rl , K 2 C ]Rn we se炬， by assumption , that def
L 2=maxν(t ， p) (t ,p)E[O ,T] xK2
< +∞，
and that the function 阳"K.， ).,....;.,:
缸..KJX)~'. . ~.~.!!:.x __ __ . ...l.,....:l" (tt 毡，p)E[O ，T] xK 1 xK 2
|θf I-;'J (t , x , u , -p)1 (J u . . . 1.
1
"
1
is continuous , and hence locally bounded on ]Rn. It is easy to check that (8.20) with L K2 and μ屿，K2(z)in place of L aad M , respectively, holds for any (t , z , u , p) , (t ， 川 q)
E !l T X
K1
X
K2 .
The corollary thereby follows from Theorem
8.5.
口
We conclude this section with the following result of continuous dependence on initial data for global C 1 -so1utions. (Here the continuity is with respect to the topology of t血form convergence on Theorem 8.7. Suppose f = 8 .4. Let Uj initiα l
= Uj(t , x) (j =
∞mpact
sets.)
f(μt ， 凯 x， 飞 u，叫 p ) sαtis拼 :fies
1，刽 2) 加 b e global
the condition
C 1 -SOl1山on旧s
但 ( 8.2 却 0)
in
Theω or， 陀em nb
切 to 伪 the 呵 e q阳ti ωiω on (8.1斗)ψ 饥ith 伪 t he
conditions Uj(O , x) = 句 (x)
where 句 =φj(x)
(j =
on
{t
= 0,
1 ， 2)α陀 given functions of class C O on ]R n. Then
|问 (t ， x) 一句 (t ， x)1 主 exp(Mt) .
sup Iyl 三(1+ 1 "，1)
for all (t , x)
x ξ ]Rn} ，
exp(Lt)-l
ε !lT.
The proof of this theorem will be left to the reader.
白 (y) 一向 (y) 1
S8.4. GENERALIZATIONS TO WEAKLY-COUPLED SYSTEMS
~8 .4.
95
Generalizations to the case of weakly-coupled systems ex缸nine
We now
how the case of systems of first-order partial
di能rential
inequal-
ities or equations can be treated by the preceding method. Let m be a positive integer. Consider the class
V m (ll T ) 彗 V(ll T )
X …
X
V(ll T ).
m times Each element of V m (llT) is therefore a vector function , namely U = u(t , x)
from ll T C j ε{1 ，..
into
]Rn+1
]R m
= (U1(t , X) ,..., Um (t , X)) Uj(t , x) belongs to V(ll T ) for every
such that Uj
., 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( nT ). 1f there exist a nonnegative l
= l(t)
function μ=μ (x)
locally bounded on ]Rn and a nonnegative function
in L 1(0 , T) such thαt
|θUj(t ， x)jθtl 三 l(t). [(1+lx l) lθUj(t ， x)jâxl+ μ(z)h2?m|uh(tJ)|](J=1 ，...， m)
(8.25) for almost every t ε (0 ， T) αnd for all x ε ]Rn ， then
32吃m|uj(tJ)| 三 explC(x) 叫 ere
Ll(r)drJ '"/，..，，， s~Yrt .，，~_
.i=rr.~mluj(O， y)l ，
1111 三(1+ 1"' 1) ex~ J~ l(T)dT-1 j=1 ,..., m
JO
C(x) is given by the formula (8.7).
Proof. For an arbitrary point (to, x O ) ε llT ， we must prove that Z
、
队yc
ι1
<-oz
-BEE--
au
l
4zw
., J
I=
m
u, ( nu Nwu)( d
9u
、‘，，，，
( ) .d4tu
唱A
，，、
。
唱A
〈一
H
ρ0
、
"'
sa + ao ) ‘
rL 唱A
JU ATLW
X 叮 a 户
a?b
P晶 p ud 司
az
t--nδ
flo
、 lj
)
，，-z飞
「 llL
e
叩
,, ••
4ιu
叫<-
m
O
YA 唱A
缸川
I=
.,
J
Let us continue using the notations 1 ~f [0 , t O] , EI(t O, x O ) , Z(. , t O ， 肉 ， h(.) , G 1 introduced in the proof of Theorem 8.1 and then define
90)tf h丘芝m gk(t)
(8.27)
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
96
for t
ε 1，
where
gk(t) 乞fmax{luk(t， x)1
x ε Z(t ， tO ， X O )}
(k=1 ,..., m).
(8.28)
It follows from Lemma 8.2 that for any number ()ε (0 ， t O) each function gk = gk(t) is absolutely continuous on [(), t O] and so is the function 9 = g(t). Moreover , they are all continuous on the whole 1. 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
ω (t) 乞fg可 (t) - g(t) , withη> 0 temporarily fixed and
向 (t) 乞f [g(O) + η] we need only claim
. exp {[C(x O )
thatω (t) 主 ω(0) (=η>
+ η1 . [h( t) + 叶， ε 1.
0) for t
On the contrary, suppose
that there exists t' ε (0 ， t O] with ω (t') < ω(0). By the hypothesis of the theorem ,
one 鱼nds
a set G 2 C (0 , T) of Lebesgue
measure 0 such that 。T\ (G2 X ]R n) C nk=lDif(Uk)
(8.29)
ε (0 ， T) \岛 ， x ε ]Rn.
From the above , it follows
and that (8.25) is satisfied for any t def
that (8.15) still holds where G ~' G 1 U G2 ; hence , there is a number ). with max{O ， ω (t')} < λ<ω(0)
andλεω [0 ， t'] \w (G n [0 , t O]).
Now take
(0 , T) \ G3 九乞f inf {t ε [0， t']
ω (t) = λ}
and 1 ::; j ::; m such that
g(九) = g3(九) = IUj(t* , x*)1 =
ê.
Uj(t* ， 叫，
ε 彗 sign Uj(儿 ， x.)
(8.30)
for some x* ε Z(t. ， tO ， X O ). Next , choose a unit vector e ε ]Rn with
〈矶 ε 尝川)) =一|尝(川) I Finally, let y ]R l such that
t
ε [0 ， T].
= ν (s)
(8.31)
be an ]Rn-valued function continuously differentiable on
y(h(九)) = x* and dyjds = (1
+ I创) . e, and
let x(t) 哲 ν (h(t)) for
Analysis similar to that in the proof of Theorem 8.1 shows ε .
Uj(t , x(t))
三 IUj(t ， x(t))1 三 g(t)
= g.，， (t)
ω (t)
< 9可 (t)
th时
- λ
~8 .4.
GENERALIZATIONS TO WEAKLY-COUPLED SYSTEMS
97
for all t ε[0 ，川， and that ε 'Uj(t* ， x(九))
= IUj (t*, x*)1 = g(儿 )=gη(儿)一 ω(九 )=g可(九)一 λ.
Consequently,
~[ε Uj(机(t))] It=乌兰等(九) This would give |广沟仇U;
•
#沪 川 (tt ,
I
..,
r,
.,
I âu
三νt川 [(扣扣 们 (1l忏+刊忡训|忡队 州* x* a contradiction with
(8却).
口
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 offirst-order partial differential equations of the form θUj/δt
in
+ fj(t , x , u ， θUj/θx)=O
(j =l ,..., m)
nT • These systems are of special hyperbolic type because each equation contains
岛的 order
derivatives of only one unknown function. Since (classical) solutions of
elliptic equations do not depend continuously (with respect to the topology of un 江皿 n form convergence on compact sets) on initial data , theorems of the non-stα tionary type that we have studied in this chapter cannot be expected to apply to partial differential equations or ineq叫ities of elliptic type. [First resu1t s on second-order partial differential inequalities of parabolic and hyperbolic types were obtained by Nagumo a叫 Simoda [108] and We即hal [158].] For our next discussions , we need to extend the notion of comparison equations given in Szarski [128] to the Carathéodory case. Consider an ordinary differential equatíon ω，= p(t ， 叫，
(8.32)
where the function ρ= 仰， ω) is defi.ned on D+ ~f (0 , +∞)
t>
0 ， ω 2:
X
[0 ，+∞) = {(t ， ω) :
O}. The following Carathéodory conditions are always assumed.
(1) For αlmost every t ε(0 ，+∞) the function [0 , +∞) 3ωHρ ( t ， ω) is continuous. (2) For each
ωε[0 ，+∞)
the function (0 , +∞) 3
it---+ ρ ( t ， ω)
is measurable.
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
98
(3) For any rε(0 ，+∞) there exists a function m r = mr (t) in Ltoc(O , + ∞) with |ρ(t ， ω) I 三 mr(t)
Vωε [O ， r]
for almost every t ε(0 ，+∞) .
In this situation we call (8.32) a
C，αrathéodorν differential
equation on D+. A solu-
tion of it on an intervalI C (0 , +∞)， with intl 并 ø ， means a function ω=ω (t) 主 O
absolutely continuous on each compact interval J C 1 (abso11巾 ly continuous on 1 for short) such that ω勺)
= p(t ， ω (t))
almost everywhere in 1. We refer to Coddington and Levinson [31] for what concerns the local existence of a solution of (8.32) through any given point ,(t o, w O ) εintD+. 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 D+ and
ρ (t ， O)
ω=ω (t)
ρ (t ， 叫主 o
on
= 0 for almost all t > 0, will be called a comparison equation if
== 0 is in every interval
(0 ， γ)
the only solution satisfying the condition
!!现 ω (t) = O.
Remark. Let l
l(t) be a nonnegative function Lebesgue integrable on each
bounded interval (0 ,,) C Iæ, and σ=σ(ω) a function of class C[O , + ∞) such that 6
σ(0) = 0 ， σ(ω) > 0 asω> 0 , and 儿 (1/σ(ω ) )dω=+∞ for every /i > O. Then (cf.
[128 , Example 14.2]) ω，
= l(t) σ(ω)
(8.33)
is a comparison equation. In fact , assume the contrary that (8.33) admits a nonzero solution ω=ω (t) on some interval (0 ， γ) with 且mω (t) = O. Letting ω(0)qEfo， t--+O
from this we easily find a n 阳 on 阳 empty su 由 bin阳'val (t泸1 ， 沪 t22] of (仰0 ，卅 忖) such that ω γ 叫(μt泸内 1η)=0 and ω 叫(t吟) > 0 for al1t ε (t泸l， 俨 t 2]. 叫 It follows that
l W (飞 a contradiction. Therefore , (8.33) must be a comparison equation. Motivated by this fact , we propose the following:
~8 .4.
Proposition
GENERALIZATIONS TO
8 且 Let σ=σ( 叫 be
is α comparison
SYSTEMS
c
lR with Jo+∞ e(t)dt = +∞
equation , then so is the equation ωσ(ω)
Converse旬，
(i i)
99
of class C[O , + ∞)， and e = e(t) 主 o be Lebesgue
integrable on each bounded interval (0 ,,)
(i) 1f (8.33)
WEAKLY回 COUPLED
under the condition essinfe(t)
(8.34)
> 0,
ifmo陀over
(8.34) is a com-
tE(O ，+∞}
pα rison eq1川ion，
then so is (8.33). 二 w 1 (t)
Proof. (i) Let w 1
be a solution of (8.34) on some interval (0 ， γ1) with
li问 ω l(t) = O. Find a number γ2
t -t u
>0
γ1 Setting w 2(t) ~f w 1 (0 ， γ2) with li吨 t -t u
,
w2
u:
=
such that
1'Y叫 r)dr
(8.35)
e( r)的， we see that w 2 = w 2(t) is a solution of (8.33) on
(t) = O. Byassumptio凡 ω 2(t) 三 o on (0 ， γ2). Hence 1川 (t) 三 O
on (0 , 1). This shows that (8.34) is a comparison equation. (ii) Let (0 , +∞ )?d 叶1.( t) be the inverse of (0 , +∞ )?d 叶 J: e( r)击， and ω2 w 2 (t) be a solution of (8.33) on some 时 in 时te 衍rva 叫 al (仰0 ，刁 γ2)with J!坦 r;rtuωv汽 骂 2 y
,
defi肘 a number 1 > 0 by (8.35). Then setting ω10)tf tu2(t(t)) , we also see that ωω1 (t) is a solution of
(8.34) on (0 ， γ1) with ~iIl! W1(t) = 0 (cf. [40 , Proposition t 一+0
3 .4( c)]). The rest of the proof runs as
before.
口
In the sequel , for each function 9 = g(t) defined and continuous in a certain interval (0 , t勺， let 马 denote the open 附 {t ε (0 ， t O )
g(t)
> O}. Here is an
elementary property of comparison equations: Proposition 8.10.
Let (8.32) be a comparison eq1川ton a叫 9
=
g(t) α gwen
function α bso l1山 ly cor阳uo旧 on some mt 例αl(OY)such thdj!59(t) 三 o and that g'(t) 三 ρ (t ， g(t)) αlmost everywhere in 马 . Theng(t) 三 o for all t E (0 , t勺，
Proof. On the contrary, suppose that there exists t 1 ι (0 ， t O ) with w 1 ~f g(t 1) > O. Setting g(O) 哩 !!59(t)and t2 智 sup{tε [0 ， t 1 ) g(t) = O} , we 附 that 0 三 t 2 < 沪 ， g(t 2 ) = 0 and (户 ， t 1 ) C 马. Hence , by assumptio凡 g'(t) 三 ρ (t ， g( t) )
almost everyw here in (沪 ， t 1 ).
(8.36)
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
100
Now take
t2 < t < t O ，
ω 主 max{O ， g(t)} ，
fρ(t ， max{O ， g(t)})
if
1ρ(t ， ω)
if t 2 < t < t O ， 。三 ω< max{O , g(t)}.
þ(t ， ω)~. ~
(8.37)
The above-mentioned Carathéodory conditions (1)-(3) are clearly satisfied for
Þ=
仰， ω) on (户 ， t O ) x [0 , +∞). Let ω=ω (t) be a solution through 肘， ω1) of (8.32)
with Þin place ofρ ， and let (t 3 , t 1 ] C (沪 ， t 1 ] be its left maximal interval of existence. We next claim that Vtε (t3 ， t 1 ].
(0 三 )ω (t) 三 g(t)
(8.38)
Assume (8.38) is false. Then one would find a nonempty interval (沪 ， t 5 ) C
W,t 1 )
such that ω (t)
> g(t)
Vt E (t\ t 5 ) ,
(8.39)
with ω (t 5 )=g(俨) .
It fo l1ows from (8.36)-(8.37) and (8.39) that
(8 .40)
g' 但)三 ρ (t ， g(t))
= þ(t ， ω (t)) = ω'( t)
almost everywhere in (沪，俨). Thus (8 削) implies that g( t) 主 ω( t) for all t E (沪，俨) , which contradicts (8.39). So (8.38) must hold. We proceed to show that t 3 = t 2 • In fact , if (0 三) t2 <沪， then (8.37) together def
with Carath层odory's condition (3) , where T'~' max{g(t)
t
r.q .1 ε 肘 ， t 1 ]},
proves that
the limit li吨 ω (t) exists and is 缸lÏte. Hence， ω=ω (t) could be extended (as a t-+t
O
solution of (8.32) with
Þ in
place of ρ) over an interval 川 ， t 1 ] :::> [t 3 , t 1 ], which is
impossible. Finally, (8.37)-(8.38) shows that ω=ω (t) is indeed a solution through (沪， ω1) of (8.32) on (t 2 , t 1 ] with lim. ω (t) = g( t2) = O. Settingω (t) ~f 0 for t ε[0，叫， we t 一斗，t 2
obtain a nonzero solution of (8.32) on (0 , t 1 ) which tends to 0 as t goes to O. Thus we arrive at a contradiction. This completes the
proof.口
We can now combine the method of 38.2 with the technique of Carathéodory comparison equations and prove the following. Theorern 8.11. Let U = u(t , x) be
α vecto T'
function in V"' (n T )
(j 1 ，...， m) ， αnd (8.32) a compa T' ison equation. 1f function R. = R.( t) in L 1 (0 , T) suth that
山th Uj(O ， x) 三 O
the 陀 exists
a nonnegative
iθUj(t ， x)jθtl :S; R. (t)(l + Ix l) 'lδUj(t ， x)jθxl+ρ (t ， k=Uf芒m|uh(t ， z)|)(J=1 ，...， m)
(8 .4 1)
~8 .4.
GENERALIZATIONS TO WEAKLY-COUPLED SYSTEMS
for almo8t every t
ε (0 ， T) αnd
ε ]Rn!
for all x
then Uj( t , x)
三 o
101
in
nT
(j =
1,... , m).
Proof. For an arbitrary point (t O , X O ) E 岛， it suffices to prove that
(8 .4 2)
jJ?fm|uj(toj)|=0.
We shal1 continue using the notations 1 生r [O , t o], ~1(tO ， x勺， Z(. ， tO ， x勺， h(.) , G 1 introduced in the proof of Theorem 8.1 (and , also , of Theorem 8.8) and 1etting g 二 g(吟，
gk = gk(t) be as in (8.27)-(8.28). Obvious1y, (8 .42) will be obtained if
one can verify that g(t O) = O. Since 9 = g(t) is a nonnegative function abso1ute1y continuous on (0 , tOL with!说 g(t) = g(O) = 0 (by assumption) , Proposition 8.10 shows that we need on1y claim that g' (t) 三 ρ (t ， g(t))
a 1most everywhere in (0 , t O).
(8 .43)
By the hypothesis of the theorem, one finds a set G 2 C (0 , T) of Lebesgu measure 0 such that (8.29) and
(8 .4 1) 町e
fulfilled for any t E (0 , T) \ 白 ，
x 巳]Rn.
Assume without 10ss of generality that 9 = g(t) is differentiab1e at any point of
(0 , t O) \G , where G 乞r G 1 U G2 • Now fix an arbitrary point 九巳 (0 ， t O ) \G and take 1 三 j 三 m such that (8.30) ho1ds for some x* 巳 Z(t* ， t O, XO). Next , choose a unit vector e E ]Rn satisfying (8.31). Let
y
ν(8)
be an ]Rn-valued function
continuous1y differentiab1e on ]Rl such that y(h(t*)) = x* and dy/ds = (1
+ Iν 1) . e,
and 1et x(t) 乞r y(h(t)) for t ε [0 ， T]. Of course (see the proof of Theorem 8.1) ,
x = x(t) is abso1ute1y continuous on [0 , T] , x(t*) = X* , and
去。)才(t) . (1 + Ix(t )l) . e
'V t E (0 , T)\G
(8 .44)
Moreover , x(t) ε Z(t ， t O, XO)
'V t 巳 [0， t*].
This together with (8.27)-(8.28) imp1ies ε . Uj(t ， x(t)) 三 IUj(t ， x(t))1 三 gJ(t) 三 g( t)
for al1 t ε[0 ，九).
(8 .45)
Besides that , by (8.30) , ε . Uj(ι ， x(九)) = IUj(t* , x*)1 = gJ(九 )=g(t*).
(8 届)
8. DIFFERENTIAL INEQUALITIES OF HAAR TYPE
102
Therefore , since 九 ε (0 ， t O ) \G , it may be deduced from (8.45)-(8.46) that
g' (t*)
斗 [ε Uj(t， x(t))] It=t
Consequently, by (8.30)-(8.31) , (8.41) , and (8.44) , we conclude that
g'( 儿 ) 三
月\
/ dx
é' ( θ 仇u 勺 伺蚓 j川 驯(t九川*
三|阳 θu 盯j(队 t儿*， 岛 x *)ν/θ 街t叫|卜一 t仰 (t儿 ω*)(1
+怡 1z 叫*1)仆)卜.1θ 仇 Ujρ (t* ， x*)ν/θ âx 叫|
三 ρ(t~2?m|uh(thh)|)=ρ(t们 IUj (t* ， 问)1) =ρ(儿 ， g(九)). Finally, because G has measure 0 and t* ε (O ， t O ) \G is arbitrarily chosen , (8.43) must hold. This completes the
proof.
口
Theorem 8.12. Let U = u(t , x) be a vector function in Vm(!"h) with
(j = 1,... , m) ,
a叫 (8.34)
function μ=μ (x)
L1
Uj(O ， x) 三 O
be a comparison equation. If there exist a nonnegative
locally bounded on ]Rn
αnd
a nonnegative function l = l( t) in
(0 , T) suth that
lâuj(t ， x)j街 |μ(t)[(l+1叫) . 1 仇j(t ， x)jâxl + 州
= 1, . . . , m) for almost !"h (j = 1,. . . , m). (j
every t ε (0 ， T) αnd for all x ε ]Rn ， then Uj(t , x) 三 o in
Proof. For an arbitrary point (t口 ， x O ) ε ÜT ， it suffices to prove (8.42). Let us continue using the method (and notations) introduced in the proof of Theorem 8.1 1. We m町 extend the function l = l(t) over the whole (0 , +∞) and assume essinf l(t)
> O.
Then by (8.47) [instead of (8.41)] we get
<E(O ，+∞)
g' (t) 三 Cl(t) σ (g(t))
almost everywhere in (0 , t O )
[instead of (8.43)] for some positive constant C. By Proposition 8.9(ii) , the Carathéodory differential equation ω ，= CR(t)σ(ω)
is also a comparison equation. Thus (8.42) is straightforward as
before.
口
Chapter 9 Hopf's Formulas for Global Solutions of Hamilton-Jacobi Equations 39. 1. Introduction The aim of this chapter is to present some formulas for exp 1i cit global solutions of the Cauchy problem for Hamilton-Jacobi equations of the form θu/θt
+ f(t ， θu/θx)
u(O , x) =
(x)
=
0
on
in
{t > 0 , x
{t=O , x
巳 ]Rn} ，
(9.1)
ε ]Rn}.
(9.2)
According to Theorem 1. 4 , the Cauchy problem (9.1)-(9.2) has locally a
Ull ique
C 2 -so1ution ifthe Hamiltonian f = f(t , p) and initial function 功 = <1>( x) are of class
C 2 . However , there is general1 y no hope
to 缸ld a global classical solution; there-
fore , 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
Ull iqueness
of (generalized) solutions for convex Hamilton-Jacobi
equations were well-studied by several approaches (see 34.3 for the uniqueness of such a solution).
If the Hamiltonian f = f (p) is continuous and if the initial function <1> = <1>( x) is globally Lipschitz continuous and convex with the Fenchel conjugate
旷=旷 (p) ，
E. Hopf [64] proved in 1965 that the formula
u(t , x)
= 亘古 {(p ， x) 一旷 (p)
- 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 local1y Theorem
1. 18 剖]
问 L i怡 psch巾 i
shows that (9.1) is then satisfied almost everywhere
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
104
If the Hamiltonian f if the initial function
= f(p) is strictly convex with ,,lim f(p)/Ipl = +∞ and Ipl →+∞ = (x) is globally Lipschitz continuous , E. Hopf [64] also
established the following formula for a (generalized) solution of (9.1)-(9.2):
u(tJ)=JEE{ 的)
+t. r((x - y)/t)}.
(9 .4)
The above formulas are often associated with the name of Hopf, although (9 .4) W出 actually
first discovered for n
= 1 by P.D. Lax [97] in 1957.
Step by step , certain more general
c臼es
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 f ~9.2
f(t , p).
Section
concerns the case of convex (but not necessarily globally Lipschitz continuous)
initial data. In Section data:
盘时 for
~9.3
we consider the Cauchy problem with nonconvex initial
the case where
二( x)
can be
represented 回 the
minimum of a
family of convex functions , and second for the case where
= ( x) is a d.c. function (i.e. , it can be represented as the difIerence of two convex functions). Finally,
Section
~9.4
discusses Hopf's formula (9.4) in case = ( x) is just continuous.
Most of the resu1t s presented here were originally published in [142]-[145] and [152]. (For other results in the field , see for example [11]-[12 ], [21 ], [33 ], [52 ], [100 ], and [124].) Our method is based on some techniques of mt山 ifunctions and convex functions.
The relevant material on these subjects from [8] and [117] (without
proofs) can be found in Appendix 11 given at the end of the book. This makes our exposition self-contained. We shal1 continue using the notation of Chapter 8. Moreover , in accordance with Chapter 2 , when T
=
+∞，
we use both Ð and n+∞ to denote the set {O
+∞ ， x ξ IR.n}. Further , for any G c IR., put Ða ~f ((0 ，+∞)\G) x IR.n = {(t ， x) ε Ð t ~ G}. Consequently, Ð = Ðø. Let us recall that Lip([O , T) x IR.n ) 智 Lip(nT) 门 C ([0 , T) x IR. n); accordingly, Lip(否) = Lip(Ð) n C(否)， where Lip(Ð) is the set of alllocal1y Lipschitz continuous functions u = u(t , x) defined on Ð
二。+∞·
Deflnition. A function u = u(t , x) in Lip([O , T) x IR.n ) is called a global 8011巾on of the Cauchy problem (8.1)-(8.2) if it satisfies (8.1) almost everywhere in n T and if u(O ， x) =φ(x)
for all x E IR. n.
99.2. THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA
105
99.2. The Cauchy problem with convex initial data In this section we consider the Cauchy problem (9.1)-(9.2) , with <þ = convex function on Rn. Denote by
<Þ* 口旷 (p)
<Þ (x) a finite
the Fenchel conjugate function of
φ = <þ(叶，
扩 (p)~f sup{ 巾， x) 一 <þ(x)} ., EIR" and by
E
the effective domain of <Þ*
for
p ιRn，
= 伊(时，
E 乞f dom 旷 = {p 巳 ]Rn
旷 (p) < +∞}.
We assume the following two hypotheses:
(E .I) The Hamiltonian f = f(t , p) is continuous in {(t , p) t ξ(0 ，+∞ )\G， p ε ]R n} for some closed set G c ]R of Lebesgue measure O. Moreover , to each N E (0，+∞) there corresponds a function 9N = 9N(t) in Lk二(]R) such that
sup If(t , p)1 ::; 9N(t)
for alrr川 t all
t ι(0 ，+∞) .
Ipl 三 N
(E. II) For
eve叩 bounded
subset V of Ð , there exists α positive number N (V) so
that
(川内)ffMdT<|q肌川旷 (q)-l tf (r， q)dr} 叫 enever (t , x) ε V，
(9.5)
Ipl > N(V).
Hypothesis (E .I) implies the
仁 t -mea 出，su 盯rabil肚 i忧ty
{书t> 0, p ξ ]Rn}. Moreover , since <þ
<Þ( x)
and
p- ∞ cO ∞ 时tinuit n
is 直nite on ]R n , this hypothesis allows
us to define an upper semicontinuous function
ψ=ψ (t ， x ， p)
from Ð x ]R n into
i一∞，+∞) by
ψ (t ， x ， p) ~f (p， x)
旷 (p) -
I
f(r , p)dr ,
JQ
which is , for each p ε E ， actually finite and continuous in (t , x) on 否. The next theorem will be fundamental in this section.
(9.6)
106
9. HOPF‘ 'S FORMULAS FOR GLOBAL SOLUTIONS
Theorem 9. 1. Let <þ
(E. II). Then
α globα1
= <Þ (x) be
a
fin山 convex
solution u = u(t , x) ofthe
function on R. n. Assume (E .I)-
C，α uchy
problem (9.1 )-( 9.2) is given
by
叫州巾川 t式冽 ，机川 Z叫 x) ~f (9.7) Remark 1. Requirement (E. II) could in a sense be regarded as a compatible condition between Hanùltonian and initial data for the existence of global solutions of the Cauchy problem (9.1)-(9.2). To see this , we first rewrite (E.II) in an alternative form that is essentially equivalent (by a standard compactness argument) but seemingly more amenable to verification:
(E.II)' For every (t O, X O ) ε 否， the阿 existpositive numbers r(tO , x O) and N(tO , x O) so that ψ (t ， x , p) <,
maxψ (t ， x ， q)
叫 enever
Iql 三 N(tO ,;,; 0)
(t , x) ξ V(t O ， x勺， Ipl
> N(t O, X O) (9.5')
四here V(t O, x O ) 乞r {(t ， x) ε 否
It - tOI + Ix - xOI < r(tO , x O)}.
We proceed now to consider , for example , the Cauchy problem
仇/a-; 向加 )2 = 0 u(O , x) = x 2 /2
on
in
{t > 0 ,
x E
R.},
{t = 0 , x ε R.}
Then the method of characteristics in Chapter 1 gives the unique classical solution
=U(t， X)~f 一一王二 2(t - 1) in {O 三 t the time t
< 1, = 1.
(9.8)
x 巳R.}. This solution can not be extended continuously beyond
We should note th剖 in this case the set of all (t o, X O ) ε 否 such
> 0 and N(tO , x O) > 0 is precisely the domain {O :=::: t < 1, x ζ R. }. Moreover , if we t叮 to apply Hopf's formula (9.3) , ignoring the fact that the initial function here , <þ = x 2 /2 , is not globally Lipschitz continuous , that (9.5') holds for some r(tO , x O)
then we also obtain the sarne solution as (9.8) in
{O 引<
1, x E R.}.
~9.2.
THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA
Remark 2. Let cþ (E.II) is satisfied tends to
(+∞)，
(E.II)" For
number
= cþ(x) be a finite convex function on lRn . Assurne (E .I). Then
ifψ (t ，
x , p) tends to
(一∞)
locally uniformly in (t , x)
εD 出 Ipl
i.e. , if the following holds:
α叼 λε lR and α叼 bounded
N( λ ，
107
subset V of Ð, there exists a positive
V) so that ψ (t ， x ， p) < 入 whenever (t , x) ε 贝 Ipl
> N( λ ， V).
Indeed，包x an arbitrary qO ξE 乞f dom cþ. 并 ø [117 , Thrn. 12勾. Since the finite function 否 3 (t , x) 叶 ψ (t ， x , qO) is continuous , it follows that
λvqf id ψ (t ， x , qO) > 一∞ (t ,z)EV
for any bounded subset V of Ð. Under Hypothesis (E.II)'\we certainly find a nurnber Nv 主 IqOI (岛r each such V) so that ψ (t ， x ， p) < λv 主 ψ (t ， x , qO) 主 maxψ (t ， x ， q) Iql 三 Nv
as(t ， x)εVand
Ipl > Nv , thus getting the validity of (E. II).
Remark 3. Condition "gN
=
gN(t) ε L~c"
in the hypothesis (E .I) could not
be replaced by "gN = gN(t) ε Lfoc'" To see this , consider the following Cauchy problem θu/θz θu/街+一一一一τ= 01t - 111/2
u(O , x)=x
on
in ...
、
{t > 0, x E lR} l - , -, - ~ -- J
{t=O , x
ξ lR}.
1. Here. ere. nn ~f 二1.
f(t， p)tf-l一丁 It - 11 11
and
cþ(x) 乞f Z
for x E lR, p ε lR and t > O. Then
旷 (p)
=
~丁∞ if p ::f 1 l U 1I P = 1;
hence (E.II) holds when N(V) 主1. All the assurnptions of Theorem 9.1 with Lfoc in place of L~c for Hypothesis (E .I) are therefore satisfied here. However , in this case , (9.7) gives the function
u=u(t ， x) 乞f X
_
2 - 21t - 111/2sign(t - 1) ,
108
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
which
Lipschitz continuous in any neighbourhood of a point (1 , X O).
is 且ot
For the proof of Theorem 9.1 , we need some preparations. We first rec a11 that a direction a1 derivative is of ç near a point ÇO E
defined 出 follows.
]RTn
Let
e 巳]RTn.
and let
ψ=ψ(.;)
be
a 缸lite-valued
function
Denote
可 ψ(çO) ~f i叫 sup {[ψ(çO + O"ë) 一 ψ(俨 )]ó- 1 } ;>UO< 占<.
lë-el<. and θJψ(çO) 哇f sup _ iI}f {[1þ (çO + .>0 u 飞 0<..
óe) _ ψ(çO )]ó -1 }.
lë-el<.
The quantities 叮 ψ( 俨) and 可 ψ(çO) are called , respectively, the upper and lower Dini semiderivatives of ψ=ψ(.;) at the point ÇO in the direction e. If +∞> θ': 1þ (çO) =可 ψ(çO) >∞ for all e ε ]RTn， thenψ=ψ(.;) is said to be directionally differentiα ble
αt
n
at ÇU;
J/Þn , def
t""\..L
J , ...n 'l.
^_II...n
andθ'e 1þ (çO) ~θ': 1þ W) =ιψ(çU) "II"'\.
will be called its derivative
ÇO in the direction e. It can be shown that , when ψzψ(.;) is Lipschitz continuous
near ÇO , its upper and lower Dini semiderivatives may also be defined equiva1 ently by
θfψ(çO) 乞f limsup{[ψ(çO 十 óe) 一 ψ(çO)Jó- 1 } and
可 ψ(俨) ~f li引nf{[ψ(çO+óe) 一 ψ(çO)Jó- 1 } ， respectively. Lemma 9.2. Let 0 be
αη open
tinuous function from 0 x IRn to
subset of
]RTn a叫 ω=ω( 己 ， p) α n
[一∞，+∞ )
(i) There exists a nonempty set E
c
]Rn
with the following two properties:
such ~~
that ω=ω( 己 ， p) is βnite
def
and that ω(己 ， p)1 伏，p)EOxEC 三∞切here EC ~. subset V of 0 there ω( 己 ， p)
(i i) For
corresponds α positive
<.
maxω (ç ， q)
Iql~N(V)
every βxed p ε E ，
Besides that , the
upper semzcon-
]R n\ E. Moreo阴 r，
on 0 x E
to each bounded
number N(V) so that
whenever ç E V , Ipl
> N(V);
the function ω=ω(巳 p) is differentiable in ç εο
gradie 时 θω/θç= 仇J(ç ， p)jθç
is
continωω on
0 x E.
1'hen one has:
a)ψ=ψ(ç) 乞f SUp {ω (ç， p) : p ε lR n} is a locallν Lipsch巾 continuous function in the domain 0;
39.2. THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA b)ψ=ψ( Ç)
is
directionallν differentiable δeψ( 己)
109
in 0 with
= m~.(δω (ç ， p)j饨 ， e)
(çε0 ，
pEL (e)
e ε ]Rm)
ω here
L(己)乞f {p ε ]Rnω(己 ， p) = ψ(ç)} c E.
(9.9)
We sh a11 a1so need the fo l1owing. Lemma 9.3.
Let Condition (i) in Lemma 9.2 hold for a given
ω (ç , p) 叫 ich ω E α ssume
to be co η时tin ω 旧 mη
semz化 ωCωOη 时tμtntωω 叨 wt纣th 陀 r esψ pect 切 to
the
Eε
t础。le ( 己 ， p)
a nonempty-valued, closed and locally bounded
ο (u呻 en 附E 肘 rp ε
on 0
X
]R n.
mult仙nction
function ωz 「 α η d 包叩 ppe旷
E)
Then (9.9)
deten旧 nes
L = L(O of ç E O.
Remark. Lemma 9.3 implies that L = L(ç) is a compact-va1 ued and upper semicontinuous multifunction. In this lemma , the set 0 C ]Rm is not necessarily open. Proof of Lemma 9.3.For any bounded subset V of (see (i)). Then bounded.
L( 己)
c B(O , N v )
Moreover ， ω=ω(ç ， p)
as ç
ε V.
c , denote
lVv
qf NW)
This means that L = L( ç) is loc al1 y
being upper
ser旧 continuous ，
we can deduce (a1so
from (i)) that the supremum 'lþ(Ç) = sup{ω (ç ， p) : p ε ]Rn} should always be finite and attained. Consequently, L = L( Ç) C E is a
noner叩ty-va1ued mt山 ifunction
of
EεO.
To complete the proof, one needs only check the closedness of L = L(O. For this pu叩ose ， let {(俨 ， pk)} t~ be a sequence convergent to a point (çO , pO) in 0 x ]Rn such
that pk ι L( 俨) as k = 1, 2 , 3,.... By the definitions of ψ=ψ (Ç) and L = L( 日， we have ω(俨 ， pk) 主 ω (çk ， p) Sinceω=ω(己 ， p) is upper se皿icontinuous in
for a11 p , k.
(9.10)
(ç , p) and continuous in ç , (9.10) shows
that ω (çO ， pO) 主 limsupω (e ， pk) 三 h→+∞
limω(çk ， p) 口 w(çO ， p)
for a11 p.
k→+∞
Thus pO ε L(çO); and the multifunction L = L(ç) is therefore closed.
口
Proof of Lemma 9.2. a) Let V be an arbitrary compact convex subset of 0 and let N ~f N(V). For any two points e , eε V ， pick up an element pl in the nonempty set L( e) c E n B(O , N) (cf. Lemma 9.3). Then ψ (C)
ψ (e) 三 ω(C ， pl) 一 ω (e ， pl);
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
110
hence the mean-value theorem gives ψ (e) 一 ψ (e) 三 |θω( Ç* ， pl)/θ~Î .Ie - el 三人 .Ie-el
where Ç* εw ， el c V and λis a (finite) upper bound of Iδω( 己 ， p)/θçl over (己 ， p) ε V x B(O , N). Analogously， ψ (e) ψ=ψ( Ç)
ψ (e) 三 λ. le - el. The function
is thus locally Lipschitz continuous in O.
b) For any çOξOande ε ]Rm ， we find two sequences {α ，， }t二， {βKMZofpositive numbers convergent to zero such that 1þ (çO +α "e) 一 ψ(çO)
δJψ(çO) = .lim
，，--++由3αh
and
。(俨 +β"e)
司tψ(50)=.Hm
Let us take an arbitrary
p E L(çO)
ψ(çO)
a
M →+∞
ρk
C E and apply the mean-value theorem to
obtain ψ(çO +α "e) 一 ψ(çO)ω(çO +α "e ， p) 一 ω(çO , p) αhαh
where
a" 巳 (0 ， α ，，).
A passage to the limit as k
=(δω(çO +否"e ， p)/θç ， ε)
→+∞ shows
that
θJψ(çO) 三 (âw(çO ， p)/饨， e)
for any p 巳 L(俨). Hence θJψ(çO) 主
sup (θω(çO ， p)/θ己 ， e).
(9.11)
pEL(俨)
Now choose an element p" ε L(俨+ ß"e) for each
k = 1, 2, . . ..
Since the mt山ifunc
tion L = L( Ç) is closed and locally bounded (Lemma 9.3) , by taking a subsequence if necessary , we can assume that p" → pO 巳 L(çO). Therefore , a passage to the ("→+∞)
limit (similar to the above) in the inequality ψ(çO +β"e) 一 ψ(çO)ω(çO +β"e ， p") 一 ω(çO ， p")
3
β"βh
一
二 (θω(çO +β "e ， p")/θ己， ε)
(where βh 巳 (0 ， β，，)) gives θfψ(çO) 三 (δω (çO ， pO)j饨，冯主
sup (θω(çO ， p)/饨， ε). pEL(俨)
(9.12)
~9.2.
THE CAUCHY PROBLEM WITH CONVEX INITIAL DATA
111
Finally, combining (9.11 )-(9.12) yields δJψ(çO) =δfψ(çO) =
Il!ax__ (δω (çO ， p)j饨 ， e) pEL({ υ)
for any ÇOε 0 ， e ξ ]RTn. This implies the directional differentiability of ψ=ψ(ç) and completes the
proof.
口
We are now in a position to prove Theorem 9. 1. def
Proof of Theorem 9. 1. It can be verified that the function ω=ω(ç ， p) ~
叫t ， x , p) (see (9.6)) satisfies all the assumptions ofLemma 9.3 where E ~f dom 矿并 def • def ø[117 , Theorem 12.2] andm~' n+1 , ç~' (t , x). Here , weput O~. Vandconclude that the definition
L(tl)qF{pε E
ψ(巾 ， p) = u(t , x)}
determines a nonempty-valued , locally bounded (and closed)
mt山 ifunction
L
L(t , x) of (t , x) E V. Our proof starts with the claim that u
u(t , x) is in Lip(V). To this end , def
r
take arbitrarily an rε(0 ， +∞) and denote 民~. {(t ， x) ε V def
Nr~'N( 民)
(cf. (E.II)). Let
1.
gNr = g凡 (t) be as in Hypothesis
two points
W, x 1 ) , (沪，泸 )ε 民 we may choose an element pl
L(t I, x 1 )
B(O ， N俨) and get
c
t + Ixl
<叶，
(E .I). Then for any
of the nonempty set
1均 叫(υ u 肘队 (t 呻咐叭 ♂扎 t 飞〉 γ ，均 X 1可) 一 仙 x
三 Nr 'lx 1
where
STtf
-
x 2 1 十句 .
Itl - t 2
1
esss叩 g N r (t). Analogousl)飞 tE(O ,r)
u(沪 ， X2)
_
u( t 1 ， xl) 三 Nr 'lx 1
x 2 1 + Sr 'Itl - t 2 1.
Thus , u = u(t , x) belongs to Lip(否) . Next , let eO ~f (1 ， 0 ， 0 ，...，时 ， e 1 ~f (0 , 1, 0 ,..., 0) ,..., e n ~f (0 ， 0 ，...， 0 ， 1)ε ]R n+l.
It follows from Lemma 9.2 (we now replace 0 ~f V G , the set G being as in
(E .I)) that u = u(t , x) is directionally differentiable in V G with θleou(t ， x)
= max{-f(t , p)
θ_eou(t ， x) =
max {f (t , p)
: p ε L(t ， x)} ， p ε L(t ，
x)}
(9.13)
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
112
and (for
1 三 i 主 n)
θ'eiU(t ， x) θ_eiU(t ， X)
= max{Pi
P ε L(t ， x)} ，
= max{-Pi
(9.14)
P ε L(t ， x)}.
On the other hand , according to Rademacher's theorem [116 , Theorem 1. 18], there e臼臼创 对 xcists a set Q i沁s
(totally)
c
ρ of ((归 n+1斗)-dimens剑 ωional 叫) Lebe咱
dωiffi 岱 er 白 en 时州 tia 由 ble 圳 w it出 h
仇 (t ， x)/δt=θeou(t ， z)= 一 θ_eou(t ， 吟，
(9.15)
δu(t ， x)/δ问 =δ'eiU(t ， x) = 一δ_eiU(t ， x)
at any point (t , x)
ε Ð\Q.
Hence , (9.14)-(9.15) show that the equalities (for
1 三
i 三 n)
。u(t ， x)/δxi
= m皿{pi
hold outside the null set {ðu(t ， x)/ θx}
def
P~'
: P ε L(t ， x)}
= min{民
P ε L(t ， x)}
(G x ]Rn) U Qj i.e. , L(t , x) is precisely the singleton
except on P. Consequently, (9.13) together with (9.15) implies that
(9.1) must be satisfied almost everywhere in Ð. Further , by a well-known property of Fenchel conjugate functions [117 , Theorem
12.2] , (9.7) gives u(O , x) = Il!~{怡 ， x) p t::ll<..
旷 (p)} = 旷· ( x)
= 4>( x )
for all
x ε ]Rn.
From what has already been proved , we conclude that u = u( t , x) is a global solution of
(9.1)-(9.2).
口
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 rnay fail to be differ-
entiable] at a given point (to, x O ) ε ÐG if L(沪， x O ) is just a singleton. We shall investigate the srnoothness of u = u(t , x) in greater detail in g13.2. Corollary 9.4. Let 4> = 4>( x) be a finite convex function on ]Rn. Under Hypothesis
(E .I), s叩pose that 垣Lf(t ， p)/(1 ，仨 IRn
in t ε[0 ，+∞).
(9.1)-(9.2).
+ Ip l)
is locallν essentially bounded from below
Then (9.7) determines a global solution
01
the Cauchy problem
~9.3.
THE CASE OF NONCONVEX INITIAL DATA
113
Proof. On1y (E.II) needs verifying. Given any r ε(0 ，+∞)， denote 民 ~f {(t ， X) ε
否
t + IXI < r} and s.. ~f ~~~jnf ip.f.J(t , p)/(1 + Ip l). Then tE(O ,..)pE Jæ n
ψ川) = (p , x) 一扩(p) 一 l t f(r ,p)dr 三 rlpl 一旷 (p) - ts 俨 (1
for al1
that
4>* (p) / Ipl
主 r(1
.1im
Ipl →+∞
扩 (p)/Ipl = +∞，
+ 21 句1)十 1
ψ (t ， x ， p) 三一 Ipl ，
Final1y, since
r 巳 (0 ，+∞)
+ Ip l)三 r(1 十 21s.. 1) . max{1 , Ipl} 一旷 (p)
On the other hand , it being known (cf. Remark 2 of
(t ， x) ε 孔 ， p ξ IR n .
Lemma 9.13 1ater) that .
(9.16)
there exists a number
N.. 主 1
such
whenever Ipl 主 N俨. Thus , (9.16) implies provided
(t ， x) ε 孔， Ipl
"2 N俨·
is arbitrari1y chosen , Condition (E. II)" ho1ds; hence so
does (E.I1).口 Corollary 9.5. Let f
=
f(t , p) be continωω on V and 1et
4> = 4>( x) be convex and
globally Lψschitz continuous on lR饵 . Then (9.7) determines a globa1 solution Cαuchy
0/ the
prob1em (9.1)-(9.2).
Proof. Since φ=
4>( x) is convex and globally Lipschitz continuous , E ~ dom 4>*
should be bounded [117 , 313.3] (and nonempty). Independently of (t ， x) ε V ， it follows that 叫巾 t式， 机 凯 x 州，
whenever Ipl is 1arge enough. Hypothesis (E.I1) therefore ho1ds whi1e (E .I) is trivially
satisfied.
口
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 f
=
f(p) depends on1y on p.
39.3. The case of nonconvex initial data In this section we consider the Cauchy problem (9.1)-(9.2) under the more general assumptions that f = f(t , p) is still t-measurab1e and p- continuous as in 39.2 , whi1e
4> = 4>(x) is now a d.c. function , i.e. , the difference of two finite convex functions on
]R n.
The class DC( lR n ) of d.c. functions plays an important ro1e in the theory
of global optimization. This class is rather large: It contains all the semiconvex
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
114
or semiconcave functions. We emphasize that it also contains all functions of class
c2
(of the whole Jæ n) with second derivatives bounded either from below or from [66] 岛r
above. The reader is referred to [6] and
a sufficiently complete study of d.c.
functions. We first prove the following result. Theorem 9.6. Let 1 be an arbitrary
solution of the
Cαuchy
δujδt initiα1
with the
+ f(t ， θujδx) = 0
each αε 1.
u
on
{t
> 0, x ε
{t = 0 , x
0 ， α nonnegative
of 1 such that all thefunctions u'" with α common
V\ W(V)
in
= u ", (t , x) be a globa1 equation
Jæn} ，
(9.1)
ε Jæn}
(9.2 ",)
Suppose that to each bounded subset V of Ð there correspond
W(V) C V of Lebesgue measure in V
set and 1et u'"
H，α milton-Jαcobi
condition u(O ， x) = 如 (x)
for
none1叩ty
prob1em for the same
= u ", (t , x) for αε
Lipschitz constant
and that il!(u ", (t , x) = "'EI
number M(V) ,
M(V) αnd
mj~_， u "，(t， x) "'EJ(V)
J(V) αre
satisfy
and α subset
Lipschitz
(9.1) αt
α set
J(V)
continuoω
every point of
for (t ， x) ε V. Then the function
= u(t ， x) 哇f il!(u ", (t , x) is a globα1 solution of the C.αt包tchy prob1em 仰 (川 9..1盯)-(仰9.2 a仨I
where cþ
= cþ( x ) 哇f i l!(CÞ'" (x). a
•I
Proof. By assumption , u(t , x)
=
mj~_， u"， (t ， x) "'EJ(V)
on each bounded subset V of Ð.
Moreover , lu "， 肘 ， Xl) - u"，(沪，泸) I 三 M(V). (jt1- t 21 + Ix 1 - x 2 1) for any αε J(V) and any fixed (t l, x 1) , ( 沪，泸 )ε V. Assume u(沪 ， x 1 ) 主 u(沪，泸) = u"，o(沪，泸) for some α。 ε J(V). Then 。三 U(t 1 ， Xl) -u( t2， x2) 主 u"，o(t 1 , Xl) _ u"，o(t 2 ， x2) 三 M(V) . (lt 1 - t 21 + Ix 1
This means that u
= u(t , x)
u
=
x 21).
is in Lip(Ð).
Now denote 凡乞f ((t ， x) ε 否 for each k
-
t
+ Ixl <的 ， Jk 乞f
J(Vk) , Wk ~f W(Vk)
1, 2 , . .. and let W o C Ð be a set of Lebesgue measure 0 such that
u( t , x) is differentiable at every point of Ð \ W o (Rademacher's theorem). It
will be shown that u = u(t , x) satisfies (9.1) except on the null set Q 乞f Ut~Wk.
~9.3.
THE CASE OF NONCONVEX INITIAL DATA
lndeed , given any (t O, x O)
115
E Ð\Q , we choose a positive 川 i nte 唔 ge 衍r k > 俨 t 0+ 川 Ix ♂01 Z
皿 az且1
O ). Obviously, some index αa OεJ t ha 时t 叫 u(俨 t O ， X O ) = u臼"o(川 t俨O ， X 勺 占k so 由
u(t , x) - u(tO ， x O ) 三 u"，o(t ， x) - u ", o (to, x O)
(9.17)
for all (t , x) close enough to (to, x O). Since u 口 u(t ， x) and u"o = u ", o(t , x) are both differentiable at (t O, x勺， (9.17) implies θu(t O ， x O )/街口仇。o(tO ， X O )/θt
仇(tO ， X O )/θx= δu"'o (t O, x O)/ åx.
and
But u"'o 二 u"o(t ， x) satis鱼es (9.1) at (to , X O); so does u = u(t , x). On the other hand , it is clear from the hypotheses that u(O , x)
The function
= il!t:.u ", (O , x) = il!t:.4>", (x) = 4>(x)
u = u( t , x)
N ow suppose that
"'EI
。 EI
4>
x 巳 IR n .
for all
is thus a global solution of (9.1 )-(9.2). =
口
4>( x) is given in the form
4>(叫苦 LEVa(z)for z εRn， with 4>", = 4>",( x)
a 鱼nite
convex
functωn
for every
αε 1.
(9.18) Combining Theorems
9.1 and 9.6 , we obtain the following first results for the representation of global solutions in the case of nonconvex initial data. Corollary 9.7. Assume (E .I)-(E. II) for eachproblem (9.1)-(9.2 ",), with φ。 = 4>", (x) function， αε 1.
a finite convex
Assume , furthermore , that all the hypotheses of
Th wnm9.6~U~r~ew~~m
u'"
= u"，(t， x) 与古 {(p， x)
problem (9.1 )-(9.2) 仙 ere
t
"'EI
4> = 4> (x) is defined by (9.18).
Corollary 9.8. Let 4> 二功(z)qzf con时x
-l f(T， p)命}
def .
oft伪 hose pro 呻 协 b lem πn 盹 I
some
ω)
min 白 (x) ， with φ
iE{l ,..., k}
and globally Lipschitz continuous
白 (x) ，...， <Þ k = 何 (x)
functions. 扩 f
=
f(t , p) is continu-
ous on Ð , then a globα1 solution u = u(t , x) of the Cauchy problem (9.1 )-(9.2) can be found in the form
u(t ， x) 哩f
miIl
ma在 {(p， x)- 4>i 但) -
iE{l ,..., k }pE皿
I
Jo
f(T , p)dT}
for
(t， x) ε 否
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
116
Proof. Since 1
def
{1 ,.. . , k} is
~'
Corollary 9.5 and Theorem
a 岳出te
set , the conclusion is straightforward from
9.6.
口
Example 1. Consider the Cauchy prob1em
+ I(δu/δx)2-11=0
δu/θt
{O
in
u(O , x) = exp( 一|叫) = min{exp(x) , exp( -x)}
on
{t = 0 , x ε Jæ}.
By Corollary 9.8 , a globa1 solution of the prob1em is
u=u(t ， x) 彗皿i吨m~{px - hi(p) - tlp2 - 11} =.l， ~p t::皿
where
r p1np -
if p> 0 ,
p
h 1 (p) 哩~ 0
if p if p
1+∞
and
if p> 0 ,
1+∞
h 2 (p) 乞r ~ 0
l
= 0, < 0,
-pln( -p)
if p if p
+p
= 0, < O.
The solution can a1so be rewritten as u(t ， x) 二 min{m_~{px - p1np p主o
-
+p -
tlp2 一 1 日， m缸 {px p 三。
+ p1n( -p) -
p - tlp2 一 11}}
= m_~{ -plxl- plnp + p - tlp2 一 1 日， P主 U
in which we adopt the convention that p 1n p = 0 if p = Example 2. Let h =
h(α)
o.
be a finite-valued continuous function of αon a given
compact set K C Jæ n. We put
(x) 乞f322忡忡忡 |α1.lxl} for x ε Jæn If
f
=
f(t ,p)
be10ngs to C( 剑， depends on1y on t and Ipl , and is decreasing with
respect to Ipl , then it follows from Corollary 9.7 that a global solution u = u( t , x) of the Cauchy prob1em (9.1)-(9.2) can be found by the formu1a u(t , x) ~f mjg{h(α) aεK 飞
= ~g{h( α) 。E且
+ .~?:X .{(p， x) --- . Ipl 三 lal
+ Iα1.lxl- I
JO
I
Jo
f(r , p)dr}}
f(r , a)dr}
for
(t ， x) ε Ð.
~9.3.
THE CASE OF NONCONVEX INITIAL DATA
117
We now consider the Cauchy problem (9.1)-(9.2) in the main case of this section where 功 = <Þ (x) belongs to the class DC( ]R n); i.e. , it has a representation of the form <þ (x) ==σ1 (x) 一 σ2(X) for some finite convex functions
representation of
<þ
仲).
σ
σ l(X)
on
and
(9.19)
]R n
σσ 2(X).
We call (9.19) a d.c.
(Of course , there are always an infinite 丑umber of
such representations for each d.c. function <þ = 功 (x ).) The notations σ;=σ; 但) and σ;=σ2" (p) will signify the Fenchel conjugate functions of σ1 二 σl(X) and σσ2(X).
The effective domains of these conjugates are denoted by E1 and E2'
respectively. Besides Hypothesis (E.I) , we shall also assume the following ones (E.III) To any bounded sets V C Ð and E
c
]Rn
there corresponds a pos巾ve number
N(V, E) so that 'P a( t , x , p)
< . ，!l..l~~_ _, 'P a( t , x , q) Iql 三 N(V， E)
αs
(t , x) ε V， αε E ，
Ipl > N(V, E).
Here , 'P a(t , x ， p)
哇f (p， 叫一 σ:(p + α) -
I
f(r , p)dr.
(9.20)
JO
(E .IV)
All the multifunctions La
= La( t , x)
Lu(tJ)qzf{pε ]Rn are actually single-valued in
of (t , x) E
Ð defined by
'Pa(t ， x， P)=~~'Pa(t ， x ， q)}
(9.21)
qε ]R n
Ð\Qωhere
Q is a certain closed set of (( n
+ 1)-
dimensiona0 Lebesgue measure 0 and is independent of α 巳]Rn. Remark. Let σσ l(X) be a finite convex function on ]Rn and f = f(t , p) be a
p-convex function on {t
> 0,
p E ]R n}. Assume (E .I) and (E.III). Then it can be def
proved that (E .IV) is satisfied with Q ~. G x ]Rn , the set G being as in
(E .I盯)， 汪 i fon
of the following two conditions holds:
(E .IV)' f
=
f(t , p) is strictly p-convex; more prec川 ly， that is to say , it ís strictly
convex with respect to p on ]Rn for almost every fixed t E (0 , +∞) .
(E .IV)" 咛 =σ; (p) is strictly convex on its effective domain E1 哇f dornσ;
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
118
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
11. Theorem 9.9.
Let <þ
功 (x)
be ín the class DC( ]R n) wíth a d.c.
σ2σ2 (x)
tíon (9.19) such that
is globally Lípschítz
contin ωus
representa-
on ]R n.
Under
Hypotheses (E.I) , (E. III) ， α nd (E .lV) , the formula
u(tJ)qf min{σ;(α) 十 ~'.:l;15cp，， (t ， x , p)} E2
'
-,
on
pE rn:. n
ín which E2 乞! dom 吨 ， determines α global solution u
Ð,
(9.22)
u( t , x) of the Cauchy
problem (9.1 )-(9.2)
Proof. Let 如 (x) 乞fσl(X) 一 (α ， x) + σ;(α) as are obviously convex functions. For each
x ε ]Rn ， αε ]Rn.
αξ E2'
Then <þ" = 如 (x)
consider the Cauchy problem
(9.1)-(9.2 ,,). By (E .I) and (E. III) , it follows from Theorem 9.1 that the formula def
u ，， (t ， 叫出 σ;(α) 十 m~cp ，， (t ， x , p)
on
pE rn:. n
Ð
(9.23)
u" (t , x) of this problem. Moreover , we may
determines a global solution u"
assume that Q => G x ]R n , the sets Q and G being as in (E .lV) and (E .I), and then see that all the solutions u"
= u ,, (t , x)
the smoothness of such u" =
u，， (t ， 吟，
Now , since
σ2σ2 (x)
satisfy (9.1) at every point of Ð \Q. (For
see Remark after Proof of Theorem 9. 1.)
is globally Lipschitz continuous on ]R n , the (nonempty)
set E 2 = dom 咛 should be bounded [117 , 313 叫. Given any rξ(0 ， +∞)， denote
1牛哩{( t , x) ε 否十 Ixl < r} and N r ~f N(孔 ， E2 ) (cf. (E. III)). For any (t I, x 1 ) , (t 2 ， ♂) in 凡， we can then choose p" ε L，，(tI， x 1 ) C B(O , N r ) and deduce from (9.20)-(9.21) and u ,, (t I,
x1 )
(9 却)
-
that
U ,,(t 2 , x 2 )
三 cp，， (t 1 ，
x1 , p") -
cp ，，(沪， z2 ， f)
=川 _ x 2 ) 十 f 川。)dT 三 Nr . Ix 1
where
def Sr ~.
x 2 1 + Sr • I 泸州
ess s叩 9N俨 (t) (cf. (E.I)). The solutions
u臼 =
tE(O ,r)
a Lipschitz condition αε E 2 .
-
on 民 with
constants N r and
Sr
u ,,(t, x) therefore satisfy
, which are independent of
39.3. THE CASE OF NONCONVEX INITIAL DATA
119
Next , rewrite (9.23) as u ", (t , x) = σ;(α) + Il!.~~i.p"， (t ， x ， p 一 α)
(9.24)
pc ll萃"
and fix temporarily (t , x) is continuous in
By (9.20) and Hypothesis (E .I), i.p", (t , X ， p
ι V.
Hence (see [117]) , the right side of (9.24) ,
αε ~n.
一 α)
bei吨 the
supremum of a family of continuous functions , actually determines a lower
se口11-
continuous function ofαfrom the whole ~n into (一∞，+∞] whose effective domain is precisely the nonempty bounded set E 2 C
~n.
It follows that
十∞> )31 u", (t , x) = in.! {σ;(α) + n;~?!i.p"， (t ， X ， p 一 α)} "'EE2
。 '="'2
pt工血 M
m坦 {σ;(α)+ 吨~~i.p"， (t ， x ， p 一 α)}
自仨 E 2
pt二血忡
=;也 {σ;(α)+ 去萨i.p"， (t ， x ， p 一 α)}=~汪叫t ， x) ( >一∞). Finally, sinceσ2(X) = Il!.10{ 忡， α) 一 σ;(α)} (see [64 , p. 964日， one has "'EE2
吨坦白 (x) = σ1 (x) 十吨i且 {-(α ， x) + σ;(α)} O:
c .l!.i 2
a:t::.ö 2
:σ1 (x) 一旦32{(z ， α)
σ;(α)}
=σl(X) 一 σ2(X) = φ(x)
As a consequence of Theorem 9.6 , global solution of the Cauchy problem Corollary 9.10.
u
for all def
州 ， x)
x
ε ~n.
出口比( t , x)
o:t:=L:!i2
(9.1)-(9.2).
口
Let f 口 f(t ， p) be of class C O on V and 份
d.c. r，、宅叩 epr附 ese时α ωZωO ti η (9.19) such th α t σ 叫
is therefore a
σ 1(x) , σ 2
<þ(x) have a
σ 2(μ x) α T 陀e globα 11ν Lipsch 材 it.t
Cω or η!tin 川 b
i必Sα global solution
01 Problem
(9.1)-(9.2).
Proof. Since the nonempty set E 1 d~f domσr is bounded [117，但 3.3] ， Hypothesis (E. III) must hold while (巳 1) is trivially satisfied. Hence , the conclusion is lmmediate from Theorem 9.9.
口
Example. Consider the Cauchy problem θujδt 十 f(θujδx)
u(O , x)
=
=0
<þ(x)
in on
{O
{t=O ,
< 十∞ ， x ε 町，
x 巳~}，
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
120
with f(p) 彗 (1
+ IpI3)1/3
(for p εIR) and
仰)营{泸 /3 l x - (2/3)signx
if x if x
ε[-1 ， 1 ]，
1. [-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
∞nvex.
Though the
present Hamiltonian f = 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 lim f(p)/Ipl = 1 手+∞.
Ipl →+∞
In this case , however , it is easy to check the validity of d.c. representation (9.19) where
r0
if x < 0 , if x ε[0， 1] ，
l
if x> 1
σσ1(X) 乞r ~ x 3/3 x 一 2/3
and σ2σ2(X) 乞fσ1 ( -x) are globally Lipschitz continuous on IR. Further , we may invoke either (E .lV)" or (E .lV)' to deduce that (E .lV) holds. Therefore , by Corollary 9.10 , a global solution u = u(t , x) of the problem can be found in the form u(tJ)qzf min
m缸 {x(p 一 α)-2(|p|S/2 一 |α13/2) -
。 ε[-1 ，0]pε[0 ， 1]'
3
"
t(l
+ Ip 一 α1 3 )附}.
As we have seen , Theorem 9.9 and its Corollary concern the Cauchy problem (9.1)-(9.2) in case the initial function cþ = cþ(x) has a d.c. representation σ1(X) 一 σ2 (x)
such that
the case where
domσ;
cþ(x) 三
is bounded in lR". The following will be devoted to
dom 付 = IR n .
Theorem 9.11. Let cþ = cþ(x) be in the class DC( lR n ) with a
(9.19) such that . limσ2(x)/lxl = +∞ . 1"'1 →+∞
Under Hypotheses (E .I), (E .l II) , and
(E.lV) , s叩'pose thαt there exists α function 9
that sup{f(t , p)
p ε
IR n }
三 g(t)
for
d.c. 陀presentatìon
= g(t) in Ltoc( lR) with the property
α almo ωst αII
t
ε(仰 0 ，+∞).
Then
仰 ( 9.22)
dete 旷 仰mt T
Proof. Since
σ2σ2(z)is
a azlite convex function on Rn with
limσ2(x)/lxl 1"'1 →+∞
=
+∞， so is its Fenchel conjugate function σ; 二付 (p); in particular, E2 乞f domσ;= IRn (cf. Remarks 1-2 of Lemma 9.131ater).
!ì 9.4. EQUATIONS WITH CONVEX HAMILTONIANS f
= f(p)
121
We shall continue using the notation ua (t , x) introduced in the proof of Theorem def " .
9.9. Let rε(0 ，+∞)，民~. {(t ， x) ε Ð
t + Ixl <叶，向
sup 1σl(X )1 ,
1 I
and
Sr 彗 J: Ig(r)ldr. Since , ，lim 付 (p)/Ipl = +∞， to any M ε(0，+∞) there Ipl →+∞
corresponds a number
Nr，M 主 1
so that
σ;(α)/1α| 主 T 十句+向 +M
出
|αI>N俨，M.
(9.25)
By (9.20) , it follows that , if (t , x) ε 孔， then ψa (t， x ， p一 α) 三 (p， x)- σ;(p)-rl α| 句 Therefore , (9.24) and (9.25) imply
Ua(t , x)
主 σ;(α) 十吨但 {(p， x)
-
O'
;(p)} -
rlα 1-
Sr
p t::皿"
=σ;(α)+σl(X)
-
rlα I-sr
主 (r +句+向 +M)'Iα 1- J.l r -rlαI-s俨 >M，
provided (t , x)
ε 几 and 1α 1> N.但.
This means that
lim ua(t , x) =
locally
|α| →+∞
+∞
uniformly in
(t , x)
ε Ð.
Hence (cf. Remark 2 after the formulation of Theorem 9.1) , we may find a positive number N r for each
rε(0 ，+∞)
ig!>a(t , x)
。 ε]Rn
=
such that
皿!Iél_
lal 三 Nr
ua(t , x)
whenever
(t ， x) ε 凡
(It should be noted that U a (t , x) is lower semÍ continuous in αon the whole ]Rn.) Moreover , analysis simi1ar to that in the proof of Theorem 9.9 shows that the solutions Ua = ua(t , x) satisfy a Lipschitz condition on 民 with constants depending on r but independent of
αfor 1α| 三 Nr ;
and that they satisfy (9.1) except on a
cOmmon set of Lebesgue measure O. The proof is thus complete in view of Theorem 9.6.
口
39.4. Equations with convex Hamiltonians f = f(p) We now consider the Cauchy problem θU/θt
+ f( θU/δx) = 0
in
Ð = {t > 0, x εR勺
(9.26)
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
122
u(O , x) = fþ(X)
{t = 0 , x
on
巳]Rn}
(9.27)
under the following two hypotheses.
(F .I) The initial function fþ strictly ωnvex
on
]Rπ with
=
fþ(X) is of class
. .lim f(p)flpl Ipl →+∞
C Oα nd
the
Hamiltoniαnf=f(p)is
= +∞.
(F.II) For every bounded subset V of Ð , there exists a positive number N(V) so thαt
~~..，{ fþ( ω )+t. f* ((x 一 ω )ft)}
|四 I S; N(V)
ωhenever
(t , x)
ε V， IνI
> N(V). Here , f*
=
< fþ( ν) + t. f* ((x - y)ft)
f* (z)
denotes the Fenchel
conj叼αte
function of f = f(p). In the sequel , we use the notation
((t ， x ， y) 乞f fþ(y) where (t , x)
ε Ð，
巳]Rn ，
y
+ t. f* ((x -
y)ft) ,
(9.28)
and shall prove the following theorem.
Theorem 9.12. Assume (F .I )-(F.II). Then the formula
U(t ， X)~f ipL((t , x , y) = ipfJfþ(y)+t. f* ((x-y)ft)} yE lR n
determines α globα1
for
(t ， x) ε Ð (9.29)
solution of the Cauchy problem (9.26)-(9.27).
The next auxiliary lemrna is known ([64] , [117 , Theorems 23.5 , 25.5 , and 26.3]) , but what we would like to insist here is on its simple proofby the use of our Lemmas 9.2-9.3. Lemma 9.13. Let f
Then
f* = f* (z)
= f(p) be strictly
con肥x
on
]Rn
with . .lim f(p)flpl = Ipl →+∞
+∞，
is et
f* (z) = (z ， δf* (z)fδz) -
f(θf* (z)fδz)
Proof of Lemma 9.13. The strict convexity on
]R n
for α11 z ξ ]Rn. ofthe function f
(9.30)
= f(p)
says
that this function is everywhere finite and that f(λpl
+ (1 一 λ )p2)
三 λf(pl)
+ (1 一 λ )f(p2)
for any pl , p2 εRn ， λε[0 ， 1]; the sign of equality holding if and only if pl = 泸 or λε{0 ， 1}.
Accordingly, f
= f(p)
is continuous.
~9.4.
EQUATIONS WITH CONVEX HAMILTONIANS f
= f(p)
123
It will now take a simp1e matter to check thatω=ω (z ， p) ~f (z ， 的一 f(p) satisfies all the conditions of Lemmas 9.2-9.3 where we put E def z , ()~. IRm = IRn and shal1 deal with the function
ψ=ψ (z) 乞f SUp {ω (z ， p) : p 巳 IR n } lndeed , since . lim f(p)/Ipl = Ipl →+∞
+∞，
~.
IRn , m
def
~
def
~π ， 5=
= f* (z).
Condition (i) ofthese Lemmas ho1ds whi1e the
others are almost ready. As f = f(p) is strict1y convex , it can be verified that the multifunction L = L(z) defined by
L(z) 生f {p εIR nω (z ， p) = f气 z)} is actually sing1e-valued on the who1e IRn. Therefore , by Part b of Lemma 9.2 , all the partia1 derivatives 叮叮 z)/θzi exist , and L(z) = {θf气功 /θz}. Property (9.30) is thus
∞ming
from the definitions of f*
= 户 (z)
and L = L(z). Further , Lemma
9.3 and its Remark imp1y the continuity of δf*/θz=
δ尸 (z)/δz.
口
Remark 1. Consider a convex and 10wer semicontinuous function f = f(p) on IRn. Assume domf 弄。 and imf
c
(一∞，+∞J (the function
f = f(p) is then called
proper). It will be shown that 1im f(p)/Ipl = +∞
Ipl →+∞
In fact , if
if and on1y if
dom f* = IR n .
1im f(p)/Ipl = +∞， then for each z εIRn the supremum
Ipl →+∞
f* (z)
= sup {(z , p) - f(p)} pE lR n
c
IRn , and hence finite. Converse1y, 1et there exist an M εIR and nonzero points pl ,p2 , . .. in IRn such that f(pk) 三 Mlpkl for k = 1, 2,... and that Ipkl →+∞出 k →+∞. Since IRn is lS
essential1y taken over al1 e1ements p of just a compact set Kz
10cally compact , we m可 suppose that pk /Ipkl → ZOεIRn. Putting z ~f (M + l)zO , we thus get f* (z) 主 BIP{(z ， ph)-f(ph)} 主 ?{|ph|·[(M+l)(zo ， ph/|ph|)-Ml}
主
lim Ipkl = +∞·
k-+ • -00
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
124
Remark 2. Consider a finite convex function iþ conjugate
Let
旷 二旷 (p).
implies
that
lim
Ipl →+∞
旷 iþb俨**
on
= iþ*(p) is 严 p ro 叩 pe 吭 r，
with the Fenchel
∞ c on 盯.ve 叽 乓， 皿d X 刮10 仰， we 盯 r 优 s e皿血lCO ∞ on 时t 川inuo ∞ u
= iþ. Accordingly, dom 旷* = domiþ =
旷 (p)jlpl
]R n
旷** iþ 口 ￠旷旷旷斗丰中叮"叶.
Then it is known [64J that 旷 O且]Rηand
=功(x)
]Rπhence ，
Remark 1
= +∞.
Proof of Theorem 9.12. By (F .I )-(F.II) and Lemma 9.13 , (9.28) determines a continuous function ( = (( t , x , y) whose derivatives θ((t ， x ， y)j挠， θ((t ， x ， y)jθXl ，... ， 何 (t ，
exist and are continuous on the whole Ð x
]R n;
x, y)j åX n
moreover , one may apply Lemma
9.2 to the function ω=ω (~ ， p) 哩仰，吼叫 where p 哲 ν ， E qzf IRnand mtf n+l , ztf(t , z) , Ctf D.Comeqmntly, the function
U
=u(t , z)dehed by (929)
is loca11y Lipschitz continuous and is directionally differentiable in Ð , withθleu(t ， x) equal to ~n ，((1气 (x-y)jt) 一 ((x - y)jt ， θf 气 (x
ν )jt)jθ斗， θf气 (x - y)jt)j 剑 ， e).
yEL(t ,æ)
(9.31 ) Here ,
...L 1
_
]R n+l '3
e,
'\
L( t , x)
def
~. {νε ]Rn
((t ， x ， ν)
accordi 吨 to Rademacher's theorem , p Ol时 outside
a nu11 set
Qc
= u(t ， x)} 并。 (Lemma 9.3). But ,
u = u( t , x)
is (μto 创t阳 ω叫 a11坊y 川圳 ) differen 时由 tia 由 ble 挝 at 阳 aI且1
Ð. Therefore , suitable choices of e in (9.31) give
θu(t ， x)j战 =uzjfa){f*((z-u)/t) 一 ((x - y)jt ， åf气 (x 一的 jt) jθz)} =田 ~~æ) {J气 (x
ν )jt) - ((x - y)jt ， θf* ((x - y)jt)jθz)}
(9.32)
and θu(t ， x)jθXi =
min θf* ((x - y)jt)jδzi =
YEL(t.æ) -
provided (t , x) E Ð\ Q and Now , given any (t , x)
"
iε{1 ，
ε Ð\ Q ，
maxθf* ((x
yε L(t ， æ)
- y)jt)jθZi ，
(9.33)
2,..., n}.
we pick up some y
ε L(t ， x).
Then it follows from
(9.30) and (9.32)-(9.33) that θu(t ， x)jθt 出广 ((x - y)jt) 一( (x 一 ν ) jt ， δ广 ((x - y)jt)jδz)
= -f(θ尸 ((x
一 ν )jt) jθz) = 一 f(θu(t ， x)j θx ).
The equation (9.26) is thus satisfied almost everywhere in Ð.
= f(p)
39 .4. EQUATIONS WITH CONVEX HAMILTONIANS f
125
As the next step , we claim that
Ð3(t
^, u(t , x)
1im
,.,)• (0 ,., 0)
= (x O )
(9.34)
for each fixed X O 巳 Jæn. Indeed , on the one hand , the definition (9.29) clearly forces u(t ， x) 三 (x)+t.f气。)，
hence 1ims叩 Ð3(t
,.,)• (0 ,., 0)
u(t ， x) 三 (X O ).
(9.35)
On the other hand , 1et us 自rst take a sequence {(沪， x k )} t~
c
D converging to
(O , XO) such that __ )iII?- i~~ ^, u(t , x) = .lim u(沪 ， Xk) and second choose arbitrary Ð3(t ,.,) • (0 ,., 0) " k →+∞ points yk 巳 L(沪 ， Xk) (for k = 1, 2 ,...). Then it will be shown that yk 一-t XO. (k →+∞)
In the contrary case , suppose without 10ss of generality that yk
一→俨 ξRn ，
(k→+∞)
where yO 并 X O . (We emphasize here that the sequence {yk}t~ C Jæn is bounded by Lemma 9.3.) Since
lim j* (z)/Izl = 十∞ (cf. Remark 2 of Lemma 9.13) ,
Izl →+∞
(9.35) and a passage to the 1imit as k →+∞ in the equality
U(tk , Xk) = (泸)十 t k . j* ((x k _ yk)/t k )
(9.36)
wou1d yie1d ( x O )
主
1iminf ^, u(t , x)
Ð3(t
,.,)•
(0 ,., 0)
,
a contradiction. This shows that
= .1im u(沪 ， Xk) = (yO) + (+∞) = +∞， k→+∞
1im
'U k h →+∞v
f* = 户 (z) is bounded from be10w since again
a passage to the 1imit as k
→+∞ also
1iII?- i~~
^, u(t , x)
V3(t ， 叫→ (0 ，.， 0)
Finally,
combini吨 (9.35)
,
X O• ,
But the continuous function
üm f气 z)/Izl
Izl →+∞
=
+∞. Therefore,
in (9.36) imp1ies
=
.lim u(沪 ， Xk) 2: (x O).
k →+∞一
and (9.37) gives (9.34) , which says that
(9.37) u 工 u( t , x)
has a (unique) continuous extension over the who1e 否 satisfying (9.27). The proof is thus
comp1ete.
Remark. Assume (F .I). Then (F.II) is satisfied if in (t , x) on each bounded subset of D.
口
,
lim
Iyl →+∞
仰 ， x ， y) = +∞ uniformly
9. HOPF'S FORMULAS FOR GLOBAL SOLUTIONS
126
In fact , let V C Ð be bounded , Put M ~f r.lf气。)1 + ((t ， x ， 叫 =
1> (x) +
+∞ uniformly
in
日町，
V C (O , r) x B(O , r) for some r
rn ax 1> (x) < +∞It 1"'1 三 r
ξ (0 ，+∞).
follows from (9.28) that 皿in((t， x ， ω) 三 四|三 T
t. f* (0) 三 M whenever (t , x) ε V. Hence , if. lim ((t ， x ， ν)= |纠→+∞
(t , x)
on each such V , then for a suitable number N(V)
~
r we
have mm
仰，吼叫三皿in((t ， x ， ω):::; M
|四|三 N(V).
叫
一
< ((t , x , y)
as
> N(V)j
(t ， x) εV， Iν1
i.e. , (F.II) is satisfied. Corollary 9.14.
U叫 er
Hypothesis (F .I), suppose that
lüninf 1> (x)/lxl >
∞.
1"'1 →+∞
(9.38)
Then (9.29) determines a global sol1山 on of the Cauchy problem (9.26)-(9.27)
Proof. By Remark above , it su ffi. ces to prove that. lim ((t , x , y) = |创→+∞
+∞ uniformly
in (t , x) on each bounded su bset V of Ð. To this end , let V C Ð be bounded , say,
V
c
(0 , r) x B(O , r) for some r E (0 , +∞) and let M ξ(0 ，十∞) be arbitrarily given
Condition (9.38) says that there exist numbers 1>(y) 三
λ|ν1
λ ， N ε(0 ， +∞)
whenever
But we certainly find a positive number
νwith
f* (z)/Izl 主 2(M 十 λ)
such that
Iyl ~ N.
the property that as
Izl 三 ν.
Putting N(V) ~f max{l ， N，衍， r(l + ν)} ， w可 tl即可fore deduce from (9.28) that , if
(t , x)
ε V and 1ν1 ~ N(V) , then \
,
f*((x - y)/t)
Ix - yll
I(x - y)/tl
ν1
仰 ， x ， y) 三|一 λ+'-..--'-'-----'--'-~一一一 I .Iyl
L
'"
J
主 [-λ 十阳+入) . ~] .Iyl 三 M because I(x - y)/tl 兰卡 (1 +ν) - r l/ r =ν， Ix - yl/lyl 三 (Iyl- r)/Iyl ~ 1/2.
If 1> = 1>( x) is globally Lipschitz
continuou日
口
on IR n , then (9.38) clearly holds. The
following result of Hopf [64] can thus be considered as a consequence of Corollary 9.14. Coroll盯Y 9.15. 厅 the
and 矿 the 十∞ ，
initial
function 功=
1>( x)
is globally Lipschitz continuous
Hamiltonian f = f(p) is strictly convex on IR n with
then (9.29) determines a global solution of (9.26)-(9.27).
,
.lim
Ipl →+∞
f(p)/Ipl =
Chapter 10 Hopf-Type Formulas for Global Solutions i n the case of Concave-Convex Hamiltonians !ì 10. 1.
Introduction
Consider the Cauchy problem for the simplest Hamilton-Jacobi equation , namely, θujδt
+ f(θujθx)=O
u(O , x) = <þ (x)
on
in
{t>O , x
{t=O , x
ε ]Rn} ，
ε ]Rn}.
(10.1) (10.2)
In the previous chapter , a global solution of this problem was given by explicit formulas in some cases a little more general than the following two of Hopf: (a)
f =
f(p) convex (or concave) and <þ = <þ (x) largely arbitrarYj and (b) <þ = <þ (x) convex (or concave) and f either on f
= f(p)
= f(p) or on <þ
largely arbitrary. It is unlikely that such restrictions ,
= <þ (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
concαve-convex
func-
tions 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.
128
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS
Let us continue using the notation of Chapters 8-9. n
that
x
def
η1
_, , x") _"\ , (x'
~f f
~.
vector in and
Rn2
Rn
+ n2 def
p~'
and that the variables x , p
(p' , p") with x' , p'
will be 0
=
εR叫 ，
x" , p"
shall often suppose
are separated into two as
ε Rn 2 •
Accordingly, the zero-
(0' , 0") , where 0' and 0" stand for the zero-vectors in
R叫
, respectively.
Deflnition 1. [117 , p. 349] A function f a concave function of p' for each
ε Rn
飞叫le
ξ Rn ，
for each
= f(p' ， p叫
is
p" εR时 and
concα时句 convex
if it is
a convex function of p"
巳 Rn 2
called
p' ε R n ，.
In the next section , conjugate concave-convex functions and their smoothness properties are investigated. Section
!ì1O .3
is devoted to the study of Hopf-type
formulas in the case of concave-convex Hamiltonians.
!ì l0.2. Conjugate concave-convex functions Let f
= f(p)
be a differentiable real-valued function on an open nonempty subset n A of R . The Legendre conj叼ate of the pair (A , J) is de直ned to be the pair (B , g) ,
where B is the image of A under the gradient mapping z = θf(p)/ θ'p ， and 9 = g(z) is the function on B given by the formula
g(z) ~f 巾， (θf/θ'p)-l(z)ì - f((δf/θ'p)-l(Z))
It is not actually necessary to have z θf(p) 用'p one-to-one on A in order that 9 = g(z) be well-defined (i.e. , single-valued). It suffices if (Z , pl) - f(pl) whenever δf(的 /δ'p = θf(p2)jθ'p
=
z.
(z , p2) - f(l) Then the value g(z) can be obtained
unambiguously from the formula by replacing the set (θf/θ'p) → (z) by any of the vectors it contains. Passing from (A , J) to the Legendre conjugate (B ， 剖， if the latter is well-defined , is called the Legendre transformation. 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
=
f(p) to be a lower
ser时 continuous
f(p) and A are convex , we can extend convex function on all of Rn with A 缸
310.2. CONJUGATE CONCAVE- CONVEX FUNCTIONS
129
If this extended f f(p) is proper , then the Legendre conjugate (B , g) of (A , f) is well-defined. Moreover , B is a subset of dom f* (namely the range of θflθIp ) ， and 9 g(z) is the restriction of the Fenchel co丑.j ugate f* = f气 z) to B. (See [117 , Theorem 26 .4]; cf. also Lemma 9.13 in the previous chapte r.) Chapter 9 has thereby proved an important role of the the interior of its effective domain.
global Legendre transformation in the global theory of first-order partial differential equations For a class of C 2 -concave-convex functions , H.T. Ngoan [111] has studied the global Legendre transformation and used it to give an explicit global solution to the Cauchy problem (10.1)-(10.2) with f = f(p) = f(p' ， pηin this class. (Fenchel-type) 叩'per
shows that in his class the in symbols
T
T (zυ勺
conj吨ateg=g(z' ， z勺 of
and lower
conj叼αtes
He
[117 , p. 389] ,
and [* = [*(zυ") ， are the same as the Legendre
f = 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
conjugα tes.
(Applications will be taken up in 310.3.) All concave-convex functions f = f(p' ， pη under our consideration are assumed to be finite and to satisfy the fo l1 owing two "growth conditions." f(p'飞， p川
一一一一=+∞ Ip叫→+∞ Ip叫
Ip'l →+∞
Let
f 句 =f 叮p' ，
f(p'飞， p" 一一-一=
∞
Ip'l
z") [resp.
f*
1
=
f*
1
for each p' ε Jæn1.
(10.3)
for each p" ξ Jæn2.
(10叫
(z' , p")] be , for each fì.x ed
p' ε Jæn1
[resp.
p" εR叫， the Fenchel conjugate of a given p"-convex [resp. p'-concave] function
f = f(p' , p"). In other words ,
f句 (p' ， z勺 def sup {(z飞 p勺
f(p' ， p勺}
(10.5)
p" EIR n 2
[resp. f* 1(Z' ， p勺 ~f .i.!!L {(z' , p') - f(p' , p勺}] d • IÆ n 1 for (p' , z川 ε Jæn1
X
Jæ n2 [resp. (z' ， p勺 εR叫 X Jæ n2].
If f =
(10.6)
f(p' ， pηlS concave-
convex , then the definition (10.5) [resp. (10.6)] actually implies the ∞nvexity [resp. concavity] of f 句 =f 句 (p' ， z叫 [resp.
z" E Jæ n2 [resp.
z' ε Jæn1]
[resp. (z' ， p川 ε Jæn1
X
1* 1
f* 1(Z' , p")]
not only in the variable
but also in the whole variable (p' , z叫
ξ Jæn1 X Jæ n2
Jæ n2]. Moreover , under the condition (10.3) [resp. (10 .4)],
130
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS
the finiteness of f
f* '(z' , p")]
=
f (p' ， p叫
yields that of
f*
f句 =f句 (p' ， z川 [resp.
1
=
(cf. Remark 1 ofLemma 9.13) with f*句叮 2气(p'飞， z川
|μz叫
clearly
• +∞
|扣 俨H z 叫川|
locally uniformly in p'
εRnl
,,__ f*叫叮 1叫(z'飞， p" li口 1m 日 m 旧丑1 一一一一 |z ' | +∞ Iz'l
r_
二十∞
[忡 resp.
l
∞
•
.
(10.7η)
R向]. To see this , fix any 0 < rl , r2 < +∞ = f(p' , p") is continuous on R n l x Rn [117 ,
[resp. p" E
As a finite concave-convex function , f
2
Theorem 35.1]; hence ,
, sup If(p' , p")1
def
I
Crl ，r2~'
t'/
< +∞.
(10.8)
Ip'l 三俨1 Ip"l 三 r2
, def
80 , with p"~' r2z" /lz"l [resp. p'~' -rlz' /lz'l ], (10.5) [resp. (10.6)] together with (10.8) implies
f 忖叫(p'飞， z"
inf 一-一一一三 T巧2- 二旦辽旦主一→今 T巧2 叫| H Ip' 圳，什| 三?俨 Iz 叫 Iz 俨川 [resp.
田
f 叫 (z' ， p勺 Cr1ι 一一一一一三 -rl 十→4一→ -rl Ip"l 三俨 Iz'l -"" Iz'l
p
|归 川 J ♂ z 叫，斗|一→-++∞ H
as
Iz'l →+∞].
8ince rl , r2 are arbitrary, (10.7) must hold locally uniformly in p" 巳 Rn 2 ]
p' ε ]Rn 1 [r呻
as required.
Remark. If (10 .4) [resp. (10.3)] is satisfied , then (10.5) [resp (10.6)] gives f*句2(p'飞，卢 ， z'η
|怡 p'l
f(p'飞， 0"
>一一一一一→十∞ 一
Ip'什|
f*叫叮 1叫(z'飞， pη
f(O'飞， p"
Ip"l
Ip"
一一一一一<一一-一→一∞
uniformly in z" E R n2 [resp. Now let f = "partial by ]R nl
7* ':)
|怡 p'l →+∞
出
|怡旷 p'叫'1忏→十∞叫l
(10.9)
(10.10)
z' ε ]Rn 1 ].
f(p' ， pηbe
a
conc盯e-convex
conjugates" 户2 口 f*2(p飞 z 勺 and
following two "total
臼
conj 吨ates"
f*
1
=
function on R n l x
f* '(z' , p") , we
]Rn2
•
Besides
shall consider the
of f = f(p' , p"). The first one , which we denote
7* (z' ， z勺， is de缸ed as the Fenchel conjugate of the concave function p' 叶一 f 句 (p' ，
z"); more precisely,
了 (z飞 z") 生f .~l1f_. {(z' , p') + 尸 2 (p' , z")} p.
t:血川 A
(10.11 )
!ì 10.2. CONJUGATE CONCAV E- CONVEX FUNCTIONS
for each (z' , z叮巳]Rnl
X
]R n2. The second ,
conjugate of the co且vex function ]R n2 '" p"
131
f* = 尸 (z' ， Zll) , is de鱼ned as the Fenchel
•
f
f* l(Z' , p")j i.e. ,
f*(z' ， z勺 qzf sup{(z"， p勺 + j* l(Z' , p")}
(10.12)
pl'E 1R n2
for (z' , z勺 ε ]Rn 1
X ]R n2.
7* (Z' , Z")
By (10.5)-(10.6) and (10.11)-(10.12) , we have
.Lrif__
=
sup {(zγ)
+ (Z" ， p") 一 f(pγ')} ，
(10.13)
p. t;:血山 ， p吨IR n 2
j* (z' ， 川江
.iEL {(z' , p')
sup
p吨 IR n 2 p'EIR n
l
+ (Zll ， p勺
f(p' , p")}.
(10.14)
..
Therefore , in accordance with [117 , p. 389] ,7* = 7气 z' ， z叫 and
j*
= f*(z' ， z勺 will
be called the upper and lower conjugates , respectively, of f
= f (p' ,p勺.
(10.13)-(10.14) imply 7* (z' , z勺三 f 气 z' ， Zll).] For any z' E
]R nl , the function
]R n 1 X ]Rn 2 '"
(p' , Zll) 叶 h(p' ， z勺乞f(z' ， p')+f叮p' ， zH)
is convex. Thus (10.11) shows that r * ]R n 2
'"
[Of course ,
= --=* r(z' ， z叮
z" 叶 (Ah)(z勺 ~f inf{h(p' , z勺
as a function of
Zll is the image
A(p' ， z勺 = z"
ofh = h(p' , z叫 under the (1 inear) projection ]R叫 X ]Rn2 '" (p' , z叫叶 A(p' ， z叫哇f z"· -一*
It follows that r
---=-*
r (z' , z") is convex in z" ε ]Rn2 [117 , Theorem 5.7]. On
the other hand , by definition ,
r = r (z' , Zll) is necessarily concave in
---=*
-=*
This upper conjugate is hence a concave-convex function on ]R n conclusion may dually be drawn for the lower co叫吨ate f*
=
1
X
]R n2. The same
f*(z' , z勺.
We have previously seen that if the concave-convex function f 岛lÏ te on the whole ]R nl f*2
X
尸2(p' ， Z勺 and 尸 1
z' ε ]Rnl.
f(p' , p") is
]R n2 and satis鱼es (10.3)-(10 叫， its partial conj 吨ates
=
f 叫 (z' ， p叫 must both be finite with
holding. Therefo毗 Remarks 1-2 of Lemma 9.13 show that T
(10.9)-(10.10) 产 (z' ， Zll) 阻d
[* = j* (Z' , Z") are then also finite , and hence coincide by [117 , Corollary 37. 1. 2]. In this situation , the conjugate
j*= 尸 (z' ， z勺乞fT(z' ， z川 = [*(z' , z勺
(10.15)
132
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS
of f = f(p' ， p勺 will simultaneously has the properties: 1*气(z'飞， z"
Iz"l →+∞ Iz'l →+∞
一一一一=+∞ Iz"l
f阳 哑，r each ♂ O z ，巳 IR n "
(10.3*)
1*气(z'飞， z" 一一一一∞
for each z" 巳 IR n 2.
(10 .4 *)
Iz'l
For the next discussions , the following technical preparations will be needed. Lemma 10. 1. Let f =
f(p' ， p勺 be α finite conc，αve-convex
function on IRn l x IRn 2
with the property (10.3) [resp. (10 .4)] holding. Then f(p飞 p" 一-一一=+∞ local α aU仿 y un旷 份 if' fμ orm 喃 mly in p' 巳 IR nπ 卫
p |怡P叫→+∞|怡 扩 川 fH叫，叫|
(10.16)
f(p'飞， p"
n 2吁]. 一一一一=一∞ lωocal αaU仿y uniJ. 份 formlωyznp 旷 p" 巳 IR向叫 |怡p'吁1-+ →+∞ |怡 扩州，什| p
(10.17)
Proof. First , assume (10.3). According to the above discussior盹 (10.5) determines a finite convex function
1*
f叮pυ").
2
Further , [117 , Corollaries 10. 1. 1 and
12.2.1] shows that f(p' ， p勺
sup {(z" ， p勺
f叮p' ， z勺}
z"E IR: n 2
for any (p' ,p勺 εIR n l x IRn 2. Let 0
< r , M < +∞
be
(10.18)
arbitrarily fixed. As a finite
convex function , f*2 = f句 (p' ， z勺 is continuous [117 , Corollary 10. 1. 1] , and hence locally bounded. It follows that def
*.... /
:14=sup|f 叮p' ， z")1 I
l'
I
"
Ip'l 三 T
< +∞.
Iz"l 三 M
So , with z" ~f Mp" /lp"l , (1 0.18)-(10.19) imply that f(p' , p " ) C ; K M
J
nf 一一一一 >M 一一一→ M
Ip叫王俨
as Ip叫→+∞. Since
M >0
Ip叫
Ip叫
is arbitrary, we have
f(p' ， p勺， 日m inf 一一一-一=+ Ip叫工斗∞怡'I:$r Ip叫 '
一
(10.19)
~10.2.
CONJUGATE CONCAVE- CONVEX FUNCTIONS
133
foranyr E (0 ，+∞). Thus (10 .16) holds. Analogously, (10 .17) can be ded uced from (10 均.口
De fl. nition 2. A finite concave-convex function f to be strict if its concavity in
p' ε ]Rnl
= f(p' ， pηon
and convexity in
p" ε ]Rn2
It will then also be called a strictly concave-convex function on Lemma 10.2. Let f
= f(p' ， pηbe
is said
]R nl X ]R n2
are both strict.
]R nl X ]R n2.
a strictly concave-convex function on ]Rnl
X ]R n2
with (10.3) [resp. (10 .4)] holding. Then its partial conj叼ate f句 =f叮p' ， z叫 [resp.
f* 1 = f* 1(Z' , p")]
defined by (10.5) [resp. (10.6)] is strictly convex [resp. concave]
in p' ε ]Rnl [resp. p" E ]R n 2 ] and everywhere differentiable in z" εlrh [y-ESp.z'ε ]Rn 1 ]. Besides that , the gradient mapping ]R叫 X ]R n2 3 (p' ， z 勺叶。f* 2 (p' , Zll) /θz"
[resp.
]R nl X ]R n2 3 (z' ， p勺叶。f* 1 (z' ， p")/θz'] is continuous and satisfies the
identity f 句叮(p'飞， z'勺三 (z'飞 θf叮p'飞，♂ ， 扩，勺 z' /θzη 一 f(p'飞， θf叮pυ 勺 /θz"
[resp. f* 1(Z' ， p勺== (z' ， θf* l(Z' ， p勺 /θz') - f(θf叫 (z' ， p勺 /θZ' , p")].
Proof. For
any 缸lÏte
concave-convex function f
=
(10.20)
f(p' , p") satisfying the property
(10.3) [resp. (10 .4)], the partial conj 略目ef句 =f句 (p' ， z叫 [resp.
f* 1 = f* 1(Z' , p")]
is finite and convex [resp. concave] as has previously been proved.
f
Now , assume that ]Rn2
=
f (p' , p")
is a strictly
conc町e-convex
with (10.3) holding. Then Lemma 9.13 shows that
be differentiable in z" ε ]Rn2 and satisfy (10.20). the gradient mapping
]R nl X ]R n2
3 (p' , z勺叶
function on
]Rnl X
f句 =f 句 (p' ， z叫 must
To obtain the continuity of
θf叮p' ， z")/θz" ，
let us go back to
Lemmas 9.2-9.3 and introduce the temporary notations: n ~f 吨 ， E 乞~ ]R n2 , m 乞f nl
~f ][])n , +n?
r def
+ 吨， O~. ]Rnl+时， ç~'
{_, , z") _"\ , and __ -' (p'
def
p~'
p". It follows from (10.16) that the
continuous function
ω=ω(ç ， p) =ω (p' ， Zll ， p勺 ~f (Zll ， p勺 - f(p' ， p勺
(10.21 )
meets Condition (i) of Lemma 9.2. Therefore , by Lemma 9.3 and its Remark , the nonempty-valued multifunction L
=
L( 己)
=
II\.
def
L(p' ， z勺~' {p'l ε ]Rn 2ω (p' ， Zll ， p勺 =f句 (p' ， z勺}
134
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS
should be upper semicontinuous. However , since f = f(p' , p") is strictly
co且vex
(10.21) implies that L = L(p' , Zll) is single-valued , and hence continuous in IRn , x IR n2. But L(p' , z叫 ={θf 气p' ， Zll)/θz 吁， which may be p" ξR町，
in the variable
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 IR叫 x IR n2 :3 (p' , z 叮叶 θf 句 (p' ， z") /δz" has accordingly been established. Next , let us claim that the convexity in p' εIRn ， of 1* 2 = f 句 (p' ， Z叫 is strict. To this
end ，缸。 <λ<
1,
z" εIR n 2
and p' , q' E IRn ,. Of cour肥， (1 0.5) and (10.21)
yield f句 (λp'
+ (1λ )q' ， z勺
maxω(λp'
p"ElR n2
+ (1λ )q' ， z飞 p勺
主 ~~::ç- {λω(p' ， z" ， p勺+ (1 一 λ)ω(q' ， z" ， p")} p" t;;:凰"王
三 λm缸 Aω(旷 旷， 'zr p ，H， 'Jjp 旷，勺 +(υl 一 λ 均 P"\ 乞=且川溢
ma: 础 xω(旷 qd，， z ♂， H，，jpjf ，H， p" t;;:皿 n:.::
=λf句 (p' ， z 勺+ (1λ )f 句 (q' ， z").
If all the equalities simultaneously occur , then there must exist a point p" E L( λp' 十 (1 一 λ )q' ， z 勺们 L(p' ， z勺 nL(q' ， z"
with ω(λp'
+ (1 一 λ)q' ， z" ， p勺 =λω (p' ， Zll ,P勺+ (1 一 λ)ω(q' ， z" ， p");
hence (10.21) implies f( λp'
+ (1 一 λ )q' ， p勺
This would give p'
=λ f(p' ， p勺+ (1-λ )f( q' , p").
q' , and the convexity in p' E IRn1 of
f* 2
f 句 (p' ，
Zll) is
thereby strict By duality, one easily proves the remainder of the lemma.
口
We are now in a position to extend Lemma 9.13 to the case of conjugate concaveconvex functions. Proposition 10.3. R叫 xIR时 with
βned by
Let f
f(p' , p叫
be
a strictly concave-convex function on
both (10.3) and (10 .4) holding. Then 山 co叼ugate
f*
=
f* (z' , z叫
de
(10.11)-(10.15) is also αω ncα时 -convex function satisfying (10.3仆-( 10.扩) .
Moreover , f*
= 尸 (z' ，
Zll)
is 凹 E 叩where
/ -'θf* / _.1 __"\ f 勺 ， z") 三飞 Z. ， 百 (z' ， z") )
continuously differentiable with
..11 θf* f_' \/θf* θf* .,1\\ + \/ Z" ， 否，， (Z' ， ZIl)) -f t 百 (z' ， z13277(z' ， z')) (..1
(10.22)
~10.2.
CONJUGATE CONCAVE- CONVEX FUNCTIONS
135
Proof. For reasons explained just prior to Lemma 10.1 , we see that 7* (z' , z 叫三
t*(z' , z") , hence that (10.11)-(10.15) compatibly determine the conjugate f* 广 (z' ， z") , which is a (finite) conc即e-convex function on Iæ. n 1 x Iæ.时 with (10.3*)(10.扩)
holding.
We now claim that f*
户 (z' ， z叫 =
f* (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 __..L temporary notations: n ~吨， E~' Iæ.吨， m 叫+吨， O~' Iæ.n 1 +吨， E =(z' , z") , aM ptfIJ气 Since (10.10) has previously been deduced from (10.3) , we can verify that the function
ω=ω(ç， p) =ω(z' ， z"， p勺乞f(z" ， p勺十 f叮z' ， p勺
(10.23)
meets Condition (i) of Lemma 9.2 , while the other conditions are almost ready. In fact , as a finite concave function , f* 1 =
10. 1.1]) and so is
ω=ω (z' ， z" ， p")
f* 1(Z' , p") is continuous (cf.
[117 , Corollary
(cf. (10.23)); moreover , Condition (ii) follows
from (10.23) and Lemma 10 立 Therefore， by (10.2) and (10.15) , this Lemma shows that
f*
=
f* (z' , z勺
= f*(z' ， z勺 should
be directionally differentiable in Iæ.叫 x Iæ.时
with
δ(e' ，e们
(10.24)
for (z' , z叫 ε Iæ. n 1 x Iæ.时， (e' ， ε勺 εR叫 x Iæ.吨， where
L = L(ç) = L(z' , z勺乞f{p"ε Iæ. n 2
ω (z' ， z" ， p勺 =f*(z'， z勺 =
f* (z' , z")} (10岛)
lS an upper semicontinuous
ml山 ifunction
(see Lemma 9.3 and its Remark). Howev-
er , because f*1 = f*1(Z' , p") is strictly concave in p" ε Iæ. n 2 (Lemma 10.2) , it may be concluded from (10.23) 皿d (10.25) that L = L(z' , z") is single-valued, and thus co叫 nuous
in Iæ. n 1 x Iæ. n 2. Consequently, according to (10.24) and the continuity
of the gradient mapping Iæ. n 1 x Iæ.n 2 3 (z' , p勺叶 δf* 1(Z' ， p")jθz' (Lemma 10.2) , the maximum theorem [8 , Theorem 1.4.16] implies that all the first-order partial derivatives of f* = j* (z' , z") exist and are continuous in Iæ.n 1 x Iæ. n 2 (cf. also [128 , Corollary 2.2]). The conjugate
f*
= f* (z' ， z叫 is hence everywhere continuously
136
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS
=
differentiable. In particular , since L
L(z' , z勺 1S S1吨le-valued ， it follows from
(10.24) that
L川)三(实川)} , and therefore that 户
"δ1* 1
， θf拿，\
(
去7(z〉勺三亏了( z' ， 王万 (z' ， z")) uz 一
Thus , (10.23) and
(10 岛)
‘/Z 二飞
I
也lZ.'
combined give
f* (z' , z'叫z"，实川)+户
(z名川))
Finally, we can invoke (10.20) to deduce that
f* (z' , z") == (z" , ~f.~(z'， z")) + (z' ， 华 I z'， 立(z'， z"))I )I + 飞
ιlZ..
飞
uz
飞
ιlZ..
一 f(苦 (z" 各川))， ~二川)) / .. âf*
âf* ,. ." \ \/ z' ， 古 (zγη -f~f 古 (z' ， z") ，孟Ii ( z' , z") ) 月俨
\
三飞 z" ，步 (zγη+
â户
\
The identity (10.22) has thereby been proved. This completes the
S10.3.
Hopιtype
proof.
口
Jæ n2} ,
(10.26)
formulas
We now consider the Cauchy problem
θujθt + f(θujθx)=O in Ð 乞~ {t > 0, x u(O , x)
= <þ (x)
{t
on
An explicit global solution u
= (x' ， x") ε Jæn，
= 0, x = (x' ， x")
= u(t , x) =
X
ε Jæn， X Jæ n2}.
u(t ， x' ， x ηof
(10.27)
the problem will be found
under the following three standing hypotheses.
(G .I) The initial function <þ f
= f(p) =
= <þ (x) = <þ( x' , x") is of class C O and the Hamiltonian
f(p' ， p勺 is strictly conoα旧-convex on Jæ n , X Jæ n2 with (10.3)-(10 叫
holding.
(G.II) The
eqωi句
sup
)n.L ((t , x , y) =
y'EIRn ，霄 "EIÆ"2
、
)gL
sup ((t ， 矶的
y" εIR n 2 11' εIRn ，
~10.3.
zs
sα tisfied
137
HOPF-TYPE FORMULAS
in Ð , where ((t , x ， ν) 乞f ø(ν)
+ t. f* ((x -
(10.28)
y)jt)
for (t , x) = (t , x' , x") ε Ð， Y ~f (νν")ε R n l x R n 2. Here , f* = f*(z) = f*(z' , z勺 denotes the co叼ugαte defined by (10.11)-(10.15) of f
(G .l II) To each bounded subset V of Ð there so
=
f(p' , p").
corresponds α positive
number N(V)
thαt
ml旦
sup ((t ， x ， ωω勺<
四"1:::皿"三四，
EIRn 1
|四 "1 三 N(V)
sup ((t ， x ， ων") ,
w' EIRn 1
m缸!!.lL_ ((归t ， 叽x， ω 旷 ω 旷，勺>， imndf
四， εIR n l
四 "εIR n 2
四 'EIR n 2
〈(归t ， 叽x， Uj，， ω 旷"
|四 '1 三 N(V)
def
叫 enever (t , x) ε V， ν;;. (νν")ε R n l
x Rn 2 with min {Iν'1 ，
1ν"I}
> N(V).
The main result of this section reads as follows. Theorem 10.4. Assume (G .I )-(G.III). Then the formula
u(t ， x) 乞f sup
y' εIRnl
，jnfC(t， z ， ν) = ..i!!f__ sup ((t ， 吼叫 for (t , x) ε Ð
y" EIR n 2
霄 "εIR n 2 霄， EIRnl
(10.29) determines a global sol山on of the Cauchy problem (10.26)-(10.27).
For the proof of Theorem 10 .4, we need further generalizations of Lemmas 9.29.3 to the mixed sup-inf c田es. Lemma 10.5. Let 0 be an open subset of Rm and χ=χ( 己 ， y) = χ(ç ， y' ， ν")α continuoω function of ( ιν) = (ινν")εo x Rn l x Rn 2 with the following three properties:
(i) The identity sup
infχ(ιν) = ..ip.L
霄 'ε lR R1y" εIR n 2' - ,_. - •
sup x(已 ν ) holds in O.
y"EIRn 2 y 'EIRnl
(ii) For every bounded subset V ofO , there exists a positive number N(V) such thαt supχ(ç ， ωω勺<
!,l!.Î,!l__
四"t::凰"三四，
EIR n l
sup X(ç ， ων")
四，
EIRn l
|四 "1 ~二 N(V)
αnd
ma 缸茎旦-
W. I::: 皿血('. ~
indff"U-lχ(优Eι， ω 旷 ω，勺
钮1"t: 且皿‘
苟 4
>
infι- χ 刘(çι， ν 旷 ω 旷"
钮，" ε 尬腮 ("2
|四 '1 三 N(V)
叫 enever
ç ε V， ν= (ννηε R n l x R时， min{1ν'1 ， 1ν咐 > N(V). (iii) The function X = χ(ιν ) is differentiable in ç εo with continω us gradient mα;pping 0 x R叫 x R n 2 :;) (ινν勺叶 δχ( 已 y' ， ν")/δι
138
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS hαs:
Then one
(a)ψ= 1/; (ç) ~f sup
.in.L X(ç , y)
inf
霄'ε1R "1 霄 "E1R n 2
supχ(ç ， y) is α locαlly Lψschitz
霄 11 E1Rn2 霄， EIR"l
continuous function in O. (b)ψ=ψ(ç)
is
directionallν differentiable
θeψ(0 =
maxmjn.. (θχ(ç ， νν")/θ己， ε)
霄 'EL'(e) 霄 11
where
YIEL"( e)
.m~..(θχ(ç ， y' , y") /饨， ε) EL" (e)霄 'EL'( e)
min
L'(O ~f {y' ε lR. n1
..in.Lx(ç ， y飞俨 )=ψ(ç)} ， EIRn2
χ=χ(ç ， y' ， ν") αnd locαlly
(i)-(ii) in
EIR"l
Lemmα 10.5
on 0 x lR. n1 x lR. n2 • Then (10.30)
bounded multifunction L
(10.30)
sup x( ç , yγ') =ψ(ç)}. 霄，
Cond巾ons
(çε 0 ， e ε lR. m )
霄 11
L" (ç) ~f {y" ε lR. n2
Lemma 10.6. Let
in 0 with
hold for α
continuous
function
gives α nonempty-valued，
def ~""
= L(O ~. L'( 己)
Proof of Lemma 10.6. For any bounded subset
closed
x L"( 己 ) of çεO.
V of 0 , denote Nv ~f N(V)
(see (ii)). The function
o x lR.
n1
:1 (ç ， y') 叶，，intx(5 ， νν") Y"EIR"2
is upper semÌ continuous as the infimum of a family of continuous ones. It follows from (ii) that for ç
ε V
the supremum 圳(臼 ψ ω 巳0=
su 叩 p
..i!lLχ 刘(çι， Uj，，JUH
1宙"ε 1R "1 Y'川.t:皿"孟
should essentially be taken over all y' of just the compact set Jf" l
(0' , N v ), and
hence be attained. This means ø 并 L'(O C Jf" l(O' , N v ) when ç ε V. Analogously, 。手 L叮o C Jf" 2(0" , N v ) if çε V. Thus L = L(ç) is nonempty-valued and locally
bounded. To complete the proof, we need only check the closedness of L
L( ç). For
this , let {(ç\y ， k ， y"k)}t~ be a sequence convergent to a point (çO ， ν ，O ， y"O) in
o
X ]Rnl X ]R n2
such that (ν'hJHh) 巳 L( 俨) = L'( t,k) x L"( t,k) as k = 1, 2, 3,...
~10.3.
HOPF-TYPE FORMULAS
Then one has χ(EhJ'hJ"h)>id χ(俨， udJ勺 =ψ(俨)= -
y"E ]R n2'" -
139
suPX(ç" ， y' ， ν ，，")
y'EIRn
,
and therefore gets
x(çO ， y'O ， ν'叫 =.lim X(ç" , y''' , y'''') ~ l.im~nf supχ(ç"， y' , y"") h →+∞- "→+∞霄'ε]R nl
主 supχ(çO ， y' ， ν川) . 霄，
E]R nl
infχ(çO ， y ,O, Y叫主 χ(çO ， ν，0ν，，0). It follows from (i)
By duality, one also gets
霄 "ε]R n2
and the definition of ψ=ψ( Ç) that
ψ(çO) = sup
..i!lL _ x(çO ， y' ， 矿')主 id"j(EOJ，。，俨)
霄 'ε ]R n ， y"t二皿
y" t::ll<"三
主 χ(50 ， u'。， u川)
>
SUp
X(çO ， y' ， y川)
霄'ε ]Rnl
> ..i!lL . supχ(çO ， y' ， ν") =ψ(çO) ， y" t::凰 '6 "J. 霄 'ε ]R nl
hence that (y ,O, y"O) ε L'(俨) x L"(俨) = L(俨). The multifunction L = L( 己) is thus
closed.
口
Remark. The above proof also shows thatχ(çO ， ν'。 J，，。 )=ψ(çO). Since ÇOεC is arbitrary, we have χ(ç ， y' ， ν") =ψ( Ç) whenever (νν勺 ε L (Ç). By the way, the function
ψ=ψ( Ç)
takes finite values only.
Proof of Lemma 10.5. (a) Let V be an arbitrary
compact ∞nvex
subset of 0
andN 哲 N (V). For any two points çt , eε V ， pick up elements ν， 1ε L'(çt) c F气。'， N) ， u"2ξ L"(e)
c
]J 2(0" , N) [cf. the proof of Lemma 10.6]. Then
ψ(任￡♂1) 一 ψ(任￡♂2) = ..i!lLχ 刘(ç♂1 ，JU，1 ， Uj 俨")一 supχ(任E♂2 ， ν旷 y' νj 俨"ρ2 1掣，"丘I凰且π2
~ε]R nl
主 χ(çt ， ν， 1νρ) 一 χ (e ， ν， 1νρ)
=1δχ (Ç*， y， l ， ν11 2 )/δ己 l'lçt-el 三 λ'Ie-el where Ç* ε w ， e]
c
Va叫 λis a finite upper bound of 1 句(巳 ν 俨 )/θ己 1 over
(ç ， y' ， y叮 ξ VxB叫 (0' ， N) X ]J 2 (0" , N). Analogously， ψ(Ç2)一 ψ(çt) 三 λ'1 Ç2 -çtl.
The function
ψ=ψ(ç)
is therefore local1 y Lipschitz continuous in O.
140
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS
(b) For any f, 0 E 0 and eξ ]R77毡， we find two 叫uences {α dt~ and {ßdt~ ofp。此 ive numbers convergent to zero such that 。
θJψ( 俨
lþ ( f，0 + α Ic e ) 一 ψ ( f，0)
lim
_T_\_"_ _~"，
"→十∞
and
Ulc
￠ (f，0 十 βIc e ) 一 ψ ( f，0)
可 ψ(50)=.Hma 1< →+∞
ρh
Let us arbitrarily take an element y' 。 ξ L'( 俨 ) and choose a seq 伊uenc白e {切旷U俨"川，Ic勺} J2 ∞R 盯ve吨 r唔 ge删 时t tωo som n 阳eν俨，，0ξE L"叮(
choice is always possible in view of Lemma 10.6.] Then ψ ( f，0 + α Ic e ) 一 ψ ( f，0)
sup χ (f，俨0 十 α Ic 飞e' ν ，飞， νy 川 )
古 'EIRnl
αk
, indf χ (f，俨0飞， ν旷俨， ρo ν 川
y" 巨ε=IÆ n 2
αh
>χ ( f，0 + α lc e ， y'O ， y"lc) _ X( f，o ， y ，O ， y川) αk
=(θω (f，0 十 ak e ， ν，0ν1I 1c )/δf" ε) where alc 巳 (0 ， α Ic)' A passage to the limit as k →+∞ shows that θJψ (f，0) 主 (θω(EOJ，。， ν ，，0)/θf" ε)
for any y'O E L' 但勺， hence that θJψ (f，0) 主
sup
(θω ( f，0 ， νν，，0)/θ己， e)
y' EL' ({O)
for some y"O E L"( 俨). Consequently, 。Jψ ( f，0) 去
inf
(θω ( f，0 ， y' , y勺 /θf"
e).
(10.31)
.. ipr ,_. (θω ( f，O ， y' ， y")/ 饵 ， e).
(10.32)
sup
y" EL" ({O) 型 'EL'( 俨)
It may dually be concluded that θfψ ( f，0) 三
sup
y' EL'({O) y" εL 气 {O)
Finally, combining (10.31)-(10.32) yields ipr ._.
Y"EL"(俨)霄 I
sup EL'({O)
(δω ( f，0 ， y' ， ν勺 /θιe) =θJψ ( f，0) = θIψ ( f，0)
sup
inf.. (θω(俨， νν勺 /θ己， e)
苗 'EU'({O) y"EL"({O)
~10.3.
HOPF-TYPE FORMULAS
141
for any f，o ξ 0 ， e ε ]Rm. All "inf" and "sup" are actually attained because the nonempty set L( 俨) ]Rn 1 X ]R n2 :;)
= L'( 俨) x L"( f,O) is compact (Lemma 10.6) and the mapping
(y' ,y")
f→ (θω(俨 ， y' ， ν")1饨 ， e) is continuous. The proof is thereby
complete.
口
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 θ〈 θ〈 θ〈 θCθ〈 一 (t ， 叫，→ (t ， x ， y) ,... ，... ，-;-一 (t ， x ， y) 一 (t ， x ， y) 王~(t ， x ， y) ， θzL"'θ 'x~
exist 创ld are continuous on the whole Ð x ]Rnl X ]R n2; moreover , one may apply Lem-
ma 10.5 to the function χ=χ( 己 ， y) def ~.
and 0
Ð. Thus u
def .,.
~.
((t ,x ,y) where m
def _
~.
• def
1 +n1 十 T弘 f， ~'(t ， x) ，
= u(t , x) defined by (10.29) is locally Lipschitz continuous
皿 d directionally differentiable in Ð , withθ'eu(t ， x) equal to either of the following
two
qu 缸ltítíes:
j* 叫*(怦平)川一-(平，芸芸(怦斗平旦) )，芸芸妇(斗平旦)沙) (' 毛孔，'" )掣， zm Jμ 揽凯市范忆♂叫"')耐((伶伶刷(←←仙 x ~Y)川一-( 平'芸芸 (怦平) )，芸芸但(平)沙) 牛 f叫*(怦 宙础，吕硝硝航 驯揽执批斗尼 z捎揽凯市范忆'卢叫ω 训a纣们咐)阳宙川jy'f，H s ιι鸟阳时'卢.，) ((仲刷(布←仙 {l2认 ιe)3扫4) Here, e
ø,
ε ]R1+叫+吨 ， L'(t ,x) ~f {y' ε ]Rn 1
L"(t , x) ~f {ν"ε ]Rn2
..i!lL ((巾 ， y' ， 俨) =吨 ， x)} 并
霄 "El盟 "2
sup ((t ， x ， y' ， ν 勺 = u(t ， x)} 并。 (Lemma 10.6).
霄 'ε ]R nl
But , according to Rad町肌her's theorem , u = 吨， x) is totally differentiable at any point outside a null set Q c Ð. It follows that 8(1 ,0' ,0") u( t ,x) θu(t ， x)1街=
,
,
,
-θ'(_1 ，O' ， 0")U(t ， x) hence by (10.33) that
争， z)=JL) 霄，羽毛) {j* (平)一(平等(平))} y♂号，.，) Y'E~~ ，.，) {j* ( 平)一悍，苦(平))} for (t , x) ε Ð \Q. This would imply the existence of an element fj"ε L"(t ， x) so that (X' -
y' x" 一百''\ / x' - y' θ f* (X' - y' x" 一扩'、\ t " )一〔一厂'否(一厂 '-t"))
百(机 )=f 飞一厂
一(斗主 'θUU z" 飞 t
Z" 百" t
J
tI O.35)
142
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS y' ε L'(t ，
for all
飞 lrj
EE
lfJ 、
，/ 一
E
NH-wud
、、、，，，、、飞， f'
、、，，/、、
一
/'''飞、/'''飞、
z-Z
uu-ν-
fzf-b
t no-dtno一，
一
lt
，，，飞飞飞，，，、、飞
z-z
-t
、、 ，/、、‘，，/
气
nhvv 山hv
rl 〈ILrIJ
z-zNWU-MudmGmu E'·1' -mgmgI'JIIU 仄归vnhv
仇-a
z
x). Analogously, it can be deduced from (10.34) that
,
for (t x) εD\Q ， a叫 that there exists 百'ε L'(t ， x) with
去(t， x)= 1*(斗豆，与旦)一(斗豆，笋(斗豆，斗主))
一(斗旦，;二(斗豆，斗旦))pO.36)
,
,
for a l1 y" ε L'勺 ， x). Finally by (10.35)-(10.36) we get 一 (t ， x)
for all (t
,
=
fx - y 飞 /x - Y θ1* fx - y \\ 1*←一卜(一一一卜一) ) 飞
t
J
def , •
\
x) ε 'D\Q ， ν~' (νγ')ε L(t ，
t
θz 飞
, def
x) -;;'
~".
L勺 ， x)
t
J/
(10.37)
,
x L"(t x). In the same
manner one may see that θ1*
fx -
y飞
一 (t ， x) = 一(一~) θz 飞
t
(1 0.38)
J
, = (t ,x' ,x") ε 'D\Q ， y = (ν" y") ε L(t， x). Now ,given any (t ， x) ε 'D\Q ， we pick up some y ε L(t ， x) ,then use (10.22) 皿d
for a l1 (t x)
(10.37)-(10.38) to obtain
仇 (t ，
x) j街
= -f( 仇。 ， x)jθx).
The equation (10.26) is
thus satisfied almost everywhere in 'D. As the next step , we claim that lims叩 1> 3(t
,.,)• ,.,
u(t ， x)
= 4> (x o )
(10.39)
(0 0)
,
for ead axed zoqf(z，oJ，，。 )ε IR n1 x IR n2 • Indeed let us first take a sequence
,x气 X" t~ c 'D converging to (0, , , such that lim u(t k ,x ,k,x"k) and second choose arbitrary points
{(t k
k)}
lim s叩
X O X"O)
u(t ， x) =
1> 3 衍，对→ (0 ，.， 0)
h →+∞
10 def I ,Ie ，，1c品'" UK=(ν' ， ν"~)ε L(t舱， z' ， z")for
Then it will be shown that yk
一→
(k→+∞)
x O in
, ,.
k =1 2
IR n1 x IR n2 • On the contrary, since
the sequence {yk}t~ is bounded (Lemma 10.6) , we can suppose without loss of
~10.3.
generality that yk
→
(k →+∞)
HOPF‘-TYPE FORMULAS
143
yO 哲 (y勺，，0)ε IR n1 x IR n2 with yO 并 X O . It is clear
= L(t , x) that
from (10.28)-(10.29) and the above definition of L
U(tk , x ,k , X"k) = ..i!lL_ ((tk ， x气 z川，沪，旷')三 ((tk ， x气 x'气 jhJ"h) 霄"仨凰 n2
__, k
,
三 φ(川
__， k
、
and
u(沪 ， X， k ， x"k)
= sup ((沪， thJ"KJ'， u"h) 主〈(thJh ， z"hjh ， ν"k) y' ElRn
l
主￠内
(10 .4 1)
We need only consider the following two cases. Case 1: y'。并 x ，O. Then (10 .4*) and (10ω) show that
lim u(沪， z' ， z") 三 φ(u ， z)+|z-u|-hm
f气 (X ， k _y ,k)/tk , O") =一∞. I(X ， k - y ,k)/tkl
So , (10 .41) implies .lim tk f* (O' , (x"k _ y"k)/t k )
一∞， a contraruction with
,0 _ ,,0 \
,
I __
,0
,
_ 0
+∞
I
"_
h→+∞
k 一t+o。、，
(10.3*). Case 2: y"。并 x"O. Analogously, (10.3*) and (10 .4 1) show that
lim
/1k
,k
"k 、自
u(t'".x'-.x"-)=+ ∞-
h→+∞、
hence thatAT∞tkf* ((X， k _ y,k)/tk , O")
= +∞， which contradicts (10 .4*).
The contradictions , which we have got in both Cases 1 and 2 , prove that AT∞yk
x O • Therefore , a passage to the limit as k →+∞ in (10ω)-(10 .4 1) would give (10.39). It may dually be concluded that liminf ~， u(t ， x)
1> 3(t ,.,)
• (0 ,.,0)
= <þ (x O)
for each fixed x O = (x'o , X"O) ε lR叫 x R n2. Thus u
u(t , x) has a continuous
extension over the whole Ð satisfying (10.27). The proof is thereby
complete.
口
144
10. HOPF-TYPE FORMULAS FOR GLOBAL SOLUTIONS
Remark 1. Hypothesis (G.II) says that a "saddle-point" function ]R n ,
X
]R n2 :1 ν=(νν叫叶 ((t ， x ， ν)
(y' ， ν叮 ε L(t ，
x) of the
[with respect to maximizing over ]R n,
and minimizing over Jæ n2] exists. If f = f(p' ， p叫 has a special d.c. representation f(p' ， p勺三 g2(p勺 -g1(p')on
Rn1
×
Rn2 ,
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 nl = 0 , (lO .29) gives the Hopfformula (9 .4) and the formula (9.29).
If n2 = 0 , the Hopf formula for concave Hamiltonians [64] will also be obtained. Corollary 10.7. Unde 1" Hypotheses (G .I )-(G.II) , suppose that
4>( x'飞， x" liminf 一一一一→> |怡E叫→+∞ |μ x"
∞
(lO.4 2)
4> (x'飞， 旷x俨，川，
limsup 一一一一一<+∞
|忡x'吁1-+ →+∞
l扣 ocωαIIνt包Lnz 归iJo 1"ml仿 U in Zt， ι Jæ叫'
|阳Z扩叫，川|
l忱 倪Cα O all仿 ν unη伪 iJ， 10 1"冒 m 凡11 句 l旬古 zn ηz 旷"ε Jæn 町
( 10 .43)
Then (lO .29) dete 1"mines a globα 1 sol时ion oJ the Cauchν p 1"oblem (lO .26)-( 10.27).
Proof. Let V C D be bounded , say, V C (0 ,1")
X
Bnl(O' , r) x Bn2(0" , r) for
some 1" ε(0 ，十∞). By (lO .3*) and (lO.4 2) , analysis similar to that in the proof of Corollary 9.14 shows that
|，y"lEm 巾， zV")=!im{￠川)+t. j* (O'， 斗旦)} = +∞ →+∞ Iy" →+∞ t 飞/ uniformly in (t , x) = (t ， x' ， x勺 ε V. Hence , we m叮 deduce that
。三 η 生fmi口{ 0 ，一 1+ inf
inf ((t ， x ， x' ， ω勺} >一∞·
(t ， x)EV 四 11
EIR"2 -
,
J
Of course , ~~~_ .!~LJ(t ， x ， ω 旷') 2':
四'ε]R Rl 四 "EIR n 2
.!~L_((t ， x ， x' ， 旷')主 η+1
一回 "EIR n 2
(lO.4 4)
|恤 'l :S: r
出 (t ， x)
E V. Fu rther , in view of (lO.43) , there exist
numbersλ ， N ε(0 ， +∞)
that 功(ν' ， z勺三 λ|ν'1
whenever
Ix叮主 1"， ly'l 主 N.
such
~10.3.
HOPF-TYPE FORMULAS
Finally, by (10 .4*), we certainly find a positive f* (z'飞， 0"
一二→一三 2 均(归 η一λ 川)
Iz'l
numberνwith
臼
145
the property that
|扫z'l怆主 ν.
Putting N(V) ~f max{l , N， 针，1'(1 + ν)}， we therefore conclude from (10.28) that , if (t , x) εVand 1 旷|三 N(V) ， then
四'" u
id 耳总n叼J爪川〈 JJ2J 仰札 α (t优t式，叫 ω旷 州川勺')三红((阳式阳Z川旷♂川，斗)=仰 J (ων归 μ γγ ，飞〉 ， 旷♂")川+村 x" t. 叫f *气(斗豆 乌句， ι 厅0σ0 ， 仨 t =且
r\
,
f*((x' -y')/t ， O叮
Ix' - y'll
::;1λ+ .:. . '---'0'-:----=-":"":"""，---'-一一一 1 . ly'l l" I (x' - y') / t I 扩1 J
I 三 [λ+ 2(η 一 λ) . ~] . 1ν， 1 三 η
(10 .45)
because I(x' - y')/tl 兰卡 (1 +ν) - 1']/1' = ν， Ix' - y'I/1 旷|主( 1ν'1- 1')/ 切， 1 主 1/2. No啊，
(10 .44)-(10 .45) show that the second condition in (G.III) holds. Analo-
gously for the first one. Thus the corollary follows from Theorem
10 .4.口
Chapter 11 GJobal Semiclassical Solutions of First-Order Partial Differential Equations 31 1.1. Introduction The
prese且t
chapter is in principle a continuation of the previous three. However ,
it was actually originated in the following problem posed by S.N. Kr田hkov [93]. def Let a smooth [i.e. , of class C 1] function ω=ω (t ， x) satisfy in the strip IIT ~
[0 , T]
X
Jæl the inequality |δω (t ， x)jθtl 三 NIθω (t ， x) jθ叫
N
= const. 主 0 ，
(1 1.1)
and the initial condition ω (O ， x) 三 o
Then it is easy to show (cf.
on
{t
Ha町、 Theorem
= 0,
(1 1. 2)
x E Jæ.I}
1. 5) that
ω (t ， x) 三 o
in II T
.
Therefore ,
the Cauchy problem for the fust-order nonlinear equation θujθt
where f
+ f( θujθx) = 0 ,
f(p) is of class C1(IRl) , cannot have more than one solution in IIT ,
say, in the class of smooth functions with bounded derivatives. As Kruzhkov already remarked , the same conclusion may be drawn without appeal to the differentiabili ty of ω=ω (t ， x) (resp. of the solution u = u(t , x)) or the validity of
(1 1. 1) (resp. of the equation) at the points in any given 缸lÌte union of straight lines {t const. , x εIR} C II T . The following question arises naturally: to what extent can the condition on the smoothness of ω=ω (t ， x) and on the va lidity of inequality (11. 1) in the entire strip IIT be weakened? For example , the Cauchy problem for the equation θuj街 +(θujθx) 2 = 0 with the zero initial condition
u(O ， x) 三 o
has a continuum of piecewise smooth global solutions , such as
U 口比 (t ， x) ~f min{O , alxl _α2叶， αconst. 主 O. Note that each function
~11. 1.
INTRODUCTION
147
u'" (t , x) satisfies the correspondi吨 inequality Iθω/街|三 α|δω/θxl almost ev e可 where in IIT. Therefore , it is interesting to find intermediate classes (as wide as possible) between C 1 (II T ) and Lip( (0 , T) X~1) ， in which only the zero function
ω =
can simultaneously satisfy (1 1.1) and (1 1.2). These questions can be generalized to the multi-dimensional case. The study of this problem suggests that we should single out the widest class between the class of continuously differentiable functions and the class of Lipschitz continuous functions in which the Cauchy problem for a first-order nonlinear partial differential equation has a unique global solution. We shall assume 0 T=+∞)
+∞ (though
most results here still hold in case
and continue using the notation of Chapters 8-9. Our discussions in this
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 (1 1.1) in the entire domains of the corresponding functions under consideration. Moreover , it should be noted (see !ì 8.1) that
C 1 (D T ) n C([O , T) x R. n ) C V(D T ) C Lip([O , T) x R.n ). The smoothness requirement on functions in V(D T ) is really weak enough: roughly speaking , these functions need only be absolutely continuous in time variable. By Chapter 8 , the class V(D T ) would be nominated as best candidate for our discussion concerning the above questions for the Cauchy problem θujδt+j(t ， x ， u ， δujδx)
u(O , x) = (x)
on
=0
{t=O , x
in
DT ,
ε R. n }.
( 11. 3) (1 1. 4)
We therefore arrive at the following de缸ition of generalized solutions: Definition 1. A function u
u(t , x) in V(DT) is called a global semiclassical
solution to (1 1.3)-(1 1.4) if it satisfies (1 1.3) for all x ε R.n and almost all t ε (0 ， T) a叫 ifu(O ， x) =φ(x)
for all x ε ]Rn.
The above definition allows us to deal with the case of time-measurable Hamiltonians j
=
j(t , x , u , p). The next section is devoted to the uniqueness of global
semiclassical solutions in this case. Fu rther , an answer to Kruzhkov's problem wi1l be given. Section 91 1.3 discusses the existence theorems , whose proofs are much based on the resu1ts of Chapter 9.
11. GLOBAL SEMICLASSICAL SOLUTIONS
148
Most of the material presented here was published in [141]-[143] and [150]-[151]. The reader is referred to S2.3 and the references therein for the global existence of classical solutions.
31 1. 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 (1 1. 3)-(1 1. 4). Here , the initial data 4> = 4> (x) is a given continuou日 function on R n . The Hamiltonian f f(t ， 叭叭 p) is always assumed to be measurable in t E (0 , T) and continuous in (x , u ， p) ξR n x R 1 X R n The proof of the following uniqueness criterion is immediate from Theorem 8. 1. S叩pose
Theorem 1 1.1.
that f
f(t , x , u , p) satisfies the following condition:
the陀口时 α nonnegαtive function μ=μ (x)
function I! = I!( t) in
L 1 (0
locally bounded on R n
α nd
a nonnegative
, T) such that
If(t , x , u , p) - f(t ， x ， v ， q)1 三 I! (t) . [(1
+ Ixl)lp -
ql 十 μ (x)lu - vl]
for almost e阴 ry t ε (0 ， T) αηd for all (x ， 矶时，作 ， v , q) εRn x R 1
X
(1 1. 5) Rn. If Ul
Ul(t , x) αnd U2 = U2(t ， X) α阿 global semiclassical so11巾ons to the Cα uchy problem (1 1. 3)-(1 1. 4) , then Ul(t ， X) 三 U2(t ， X) in QT.
Remar k. Condition (1 1. 5) is satisfied if and only if for some positive function
1!=I!(t) i口 U(O ， T) , the function 。T
X
R1
X
R n 3 (t ， x ， u ， p) 叶 f(t ， x , u , p)/[I! (t)(l
+ Ix l)]
is Lipschitz continuous with respect to p unifor llÙ y in (t , x , u) ε QT Lipschitz continuous with respect to u uniformly in (t , x , p) eve叩 compact set
ξ (O ， T)
X
R 1 , and is
x X x Rn for
X C Rn [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 1 1. 2. Suppose Gηy compαct sets K 1 C
thαt
R1
f = f(t , x , u , p)
, K2 C R
n
satis.β es
followi叼 condition:
for
there exist α nonnegαtive function f K2
f K 2 (t) in L (0 , T) and a nonnegative function μKl ， K2 1
the
μ Kl.K2(X) local仿 bounded
91 1. 2. UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS
149
that (1 1. 5) with CK2 and μ Kl. K 2 in place of C αnd μ ， respectively , holds for almost every t ε (0 ， T) α ndfor αII (x ， 凯的 ， (x ， v ， q) ε Jæ.nXKl xK2. Iful = Ul(t , X)
on Jæ.π
such
and U2 = U2(t , x)
α re
global semiclassical solut阳lS to the problem (1 1. 3)-(1 1. 4) with ess sup
眶。， x)1 < +∞ (j =叫
(t ，，，，) εOT'
then Ul(t , X)
UX
== U2(t , X) in nT .
Remark. If f = f(t , p) depends or由 on t , p and is of class C 1 on [0 , T] x Jæ.n , then the condition of Theorem 11. 2 is satisfied. In this case , Theorem 1 1. 2 solves the problem of Kruzhkov (see Corollary 11. 4 later). To prove Theorem 11. 2 we need the following: Lemma 11.3. Let
If it is
ψ=ψ (x)
differentiα ble
be a locally
Lipsch巾 contin包ous
in the whole Jæ.n , then |θψδψi
ess suplτ(x)1 "， E~n IUXi
=
Proof of Lemma 1 1. 3. Fix any i
sup Iτ(x)1
ε{1 ，.
. . ,n}. It suffices to treat the case when
'UXi
"， E~n
= (X' , Xi)
(i=l ,..., n).
'"εRπ IUXi
|δ'Iþ I \ I Si~' esssupl :.二 (x)1
Let us write x
function of x on Jæ. n.
< +∞.
= (Xl'...'X n ), where
instead ofx
, def I
X
、 ~Xl ，...， Xi-l ， Xi+l ，...， Xn)
Then for almost all [with respect to the (n 一 1)吐imensional Lebesgue measure] x' ε Jæ.n-l
we have:
τ (x' ， .) 1 11θ￠ OXi
∞ (~1 )
三 Si
Since the function 'Iþ (x' , .) is absolutely continuous on each bounded segment , it follows that
1["'7δψ\
|ψ (x' ， xn 一仰'， x;)1 = I I
I
J ", l
~~ (x' , Xi)仇|三 silxt - x;1 _1__
aXi
(1 1. 6)
for almost all [with respect to the (n 一 1 )-dimensional Lebesgue measure] x' ε Jæ.n-l and for all x~ ， x; ε Jæ.l. From the continuity of ψ 口 ψ (x) and from (1 1. 6) , we conclude that |ψ (x' ， xn 一 ψ (x' ， xnl 三 84|zj-d|
11. GLOBAL SEMICLASSICAL SOLUTIONS
150
for all (x' ， x t)， (x' ， xn ε Jæn. Therefore , t
ε Jæn.
for all x
E，/ 、、
盯
z
<一
ho-nd
s
This proves the lenuna.
口
Proof of Theorem 11.2. According to the
de丑nition
of V(n T
),
Lenuna 11. 3
shows that
1 学(川 1 = esssupl 轧机 )1 ., E lRn IOXi
sup
., EIRn I OXi
for almost all t
ε
(j =1 , 2; i=l ,..., n)
(0 , T). Taking the essential supremum over t
that
1 华 (t， x)1 = …时学L(t， x)1
ess sup sup tE(O ,T)
ε (O ， 盯 T)， 附 we 负缸缸 I且1
., ElR n I OXi
(t ,., )E {l T Iσxi
Consequently, by assumption , 7·tfmxmsup ,
)=1 2
1.
,
tE(O ,T)
sup| 学(机 )1 < +∞
., EJæ n
def -:;::::;
(1 1. 7)
Iσx
nI
\
def
Let X k be as in (8.22); K 2 ~' B俨 C Jæ n; f(.) ~'fK2(')' For an arbitrarily fixed Tε (0 ， T) , we consider the sequence {pk}t~ of the following parallelepipeds:
pk 乞r (O , T') x X k = {(t , x) Continue using (8.23).
Then the
0
< t < T' , x ε X k }.
functionμ=μ (x)
given by (8.24) is locally
bounded on Jæ n. def
u(t , x) = Ul(t , X)
We now consider the function u u(O ， x) 三 O.
- U2(t , X).
Of course ,
Moreover , in view of (1 1. 7) , the hypothesis of the theorem implies
|Zl 斗机)|=!"， z， 1(t z) 仇1 (机))-f(t， x ， 叫树，旦旦旦(机 ))1 IJ \-,-, ~~\-'-J'
1,.
三 f(t). 1 (1 L'-
θz
θU2 ，. , 1. + Ix，，，l) 1θUl l 一一 (t \ - ， x) 一一一 ôx (t \ -， x)1I +' μ 忡忡 1 (t , x) ,
'-"1θX
7 -
I
J -
I
r
\ -- I
I -- ~ \. - 7
1. 1 U2 (t , x) 11J \1
I I
=仰). [(1 +lxl)I~~(川 1+ 州 lu(t， x)l] for all x
u(t , x)
ξ Jæn
三 o
in
T'). Therefore , Theorem 8.1 shows that (0 ， T) is arbitrarily chosen , the proof is complete. 口
and for almost all t
nT'.
Since
T' ε
ε (0 ，
~11.2.
UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS
151
Corollary 1 1.4. Let j = j(t , x , u , p) be measurable in t E (O , T) , continuous in z ξ ]Rn ， and differentiable in (u , p) ε ]Rl
K
c
X
]Rn such that , for αny compact set
]Rn , the function
tιK ι 川(t) ~f 忡1 +
su叩p
(归 x ，也叽，p 剑) EIR"
is Lebesgue integrable on (0 , T) ，
I/
K=
I/
the function |θ1 ，.
/.\1
K(X , u) ~t esssup sup Iτ (t ， 机 ， p)/eK(t)1 tE(O ,T) pEK 1 V'U
is local旬 bounded on ]Rn clαssical
α nd
主 (t， 叽， x，冉 u飞， p)
xIRl xK I V l'
If Ul
X ]R l.
=
\/I'J
Ul (t , x) and U2
=
U2(t , x) are global semi-
solutions to the Cauchy problem (1 1. 3)-(1 1.4) with
esssup 悔。， x)1 < +∞(j=叫 VX
(t ,X)EOT
I
then Ul(t ， X) 三 U2(t ， X) in ~h.
Proof. Let us introduce the notation def
μK， kjz)=supuk句 (x ， u) 也 EK.
for any convex compact sets K 1 C ]R l , K 2 C ]Rn. Then it is easy to check that
(1 1. 5) with eK2 andμ 町 ， K2 in place of e a叫 μ ， respectively, holds for almost every t E (0 , T) and for all (川 p) ， ( 川 q) ε ]Rn X K 1 X K 2 . The corollary thereby 口
follows from Theorem 1 1. 2.
We leave it to the reader to prove the fol1owing criterion of continuous dependence on initial data for global semiclassical solutions. Theorem 11.5. Suppose j = j(t , x , u , p) satisfies (1 1. 5). Let Uj = Uj(t , x) (j = 1, 2) be global semiclassical solutions to (1 1. 3) with Uj(O ， x) = 白 (x) where 句=内 (x) (j
IUl(t , X) - u2(t ， x)1
= 1, 2)
on
{t
= 0,
x ε ]R n} ，
are givenfunctions continuous on ]R n. Then
主 exp[C(x)
I
JO
e(r)dr].
sup .φ l(y) 一如 (ν) 1,
1111 三(1+ 1 "，1) exp J~ l(T)dT-l
11. GLOBAL SEMICLASSICAL SOLUTIONS
152
C(x) being defined in (8.7).
Remark 1. The example in Remark 2 following Theorem 8.1 shows that the
v(n T )
Lipschitz continuity of functions in the class
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 θu/街=
0 in
n1 ,
u(O , x)=O on {t=O , x
ε ]Rn}.
Remark 2. Consider the Cauchy problem θu/δt
+ (θu/θX)2=0
in
{O
u(O , x)=O on {t=O , x ξ ]Rl}.
(1 1. 8)
(1 1. 9)
×
U>...
x=λ(t-T+ε 〉
t
01
T-ε
x=- λ(t-T+ε 〉
U>..<
Figure 1 1. 1 By definition , if u
= 吨，
x) is a global
semi伽sical
solution to the problem , then
for almost every t ε (0 ， T) , the function u(t ,.) is differentiable on IRl. Obviously,
911.2. UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS
153
(1 1.8)-(1 1.9) has a continuum of piecewise smooth global solutions , such as (see Figure 1 1.1)
t < T - t:, if T 一 ε 三 t < T ,
( 0
e = UÀ，e(机) ~f ~
UÀ ,
l
if 0
min{O ， λIxl 一 λ2(t-T+ t:)}
~
where λ 主 0 ， 0 <ε < T. For λ> 0 the differentiability of the function
UÀ
,e(t ,.)
fails somewhere [at x = 土λ (t-T+ ε) and at x = 0] if and only if t belongs to the interval
(T 一句 T) ，
whose Lebesgue measure is precisely t: (positive but as small
as we ple剖e). Thus , the zero function
UO
,e =
UO
,e(t , x)
[i.e. ， λ=
0] is the unique
global semiclassical solution to (1 1.8)-(1 1.9) in the class offunctions with essentially bounded derivatives (cf. the remark following Theorem 11.2). Remark 3. The requirement that C = C(t) be Lebesgue integrable on (0 , T) is also essential to the above uniqueness theorems. To see this , we consider the Cauchy problem (even for a linear equation): θU
1θ也
一一一·一一一一~ θt 2t 8x 2t 2
=0
in
{t ε (0， T) , x ε ]Rl} ，
U(O , x)=O on {t=O , x ε ]Rl}. Here , for μ (x) 三 1 ， the Hamiltonian
(1 1. 5) with C(t)
三工. 2t
(U+p) 1 f = f(t , u, p) 一一 ==一一一一一;;. 2t 2
clearly satisfi
But we can easily check that the present problem admits the
following two global (阳ni)classical solutions:
U=Ul(t ， X) 营 t and U=U2(t， x)~ft(1+ ♂). The explanation for this event is that the function C 二 C( t) in this case is not Lebesgue integrable on any (0 , T). (We could not choose a Lebesgue integrable
C= C(t) for (1 1. 5) to hold.) By the use of Theorems 8.8 , 8.11 , and 8.12 , the results in this section can be generalized to the case of the Cauchy problem for weakly-coupled systems: θu ;j
at + fí(t , x , U ， θu ;jθx)=O U(O , x)= 1> (x)
Here , the initial data 1>
φ(x)
on
in
nT
(i=1 ,..., m) ,
{t=O , x
ε ]Rn}.
(1 1.1 0) (1 1.1 1)
= (φ1 (x) ,.. . ， φm(x)) is a given vector function
continuous on ]R n. Each Hamiltonian fi
fí(t , x , U, p) is always assumed to be
11. GLOBAL SEMICLASSICAL SOLUTIONS
154
measurable in t
ε (0 ，
T) and continuous in (x , u ， p)
ξ ]Rn X ]R m X ]R n.
First , we give
the definition of global semiclassical solutions for the problem. Definition 2. A vector function
u 二吨 ， x)
in Vm(n T ) is cal1 ed a global semiclas-
sical solution of (1 1.1 0)-(1 1.1 1) if it satisfies (1 1.1 0) for a l1 x
t ε (O ， T) and ifu(O , x) =
q, (x)
ξ ]Rn
and almost a l1
for all x ε ]Rn
We can now formulate some stability results for global semiclassical solutions of the problem (1 1. 10)-(1 1. 11) and leave the proofs to the reader. Theorem 11.6. Suppose fi follo ω
= 1; (t , x , u , p)
(i
= 1,..., m)
there exist α nonnegati 旧 function μ=μ (x)
satisfy the conditions αs
locally bounded on ]Rn and a
nonnegative function C = C(t) in P(O , T) such that
1 1;(巾 ， u ， p) - 1; (t , x , v , q)1 υ(t). [(1 +lxl)lp-ql+μ(z)KJ?rTmMU|](1112) for almost
eve 叩 t ε (0 ， T) α nd
1 ,..., m). Let u
=
u(t , x) , û
=
for all (x , u , p) ,
(x ， v ， q) ε ]Rn X ]R m X ]R n
(i
û(t , x) be global semiclassical solutions to (1 1. 10)
with the following corresponding initial conditions: u(O , x) = <Þ (x) , û(O , x) = <þ (x)
on
{t = 0, x ε ]Rn} ，
where <þ=<þ (x) , <þ = φ(x) αre 9ω en vector functior
rmJJ 卢|扣队 u问i(归川)-û句 ω(ρ( i仪队阳川川 t式冽 ，巾川川 z叫x叫)川1::;三 呻叫[卡 忡 h 印(归叫 C( z x
早ax
sup |古巴 (1+1 "，1) exp J~ l(T)dT-1
.=l.....m
1φ 白i(ω y) 一白 (ωωUω)川1 ，
'
C(x) being defined in (8.7).
Corollary 1 1. 7. Suppose 1; (1 1.12).
= 1; (t , x , u , p) (i = 1,..., m)
satisfy the conditions
lf u = u(t , x) and υ=υ (t ， x) are global semiclassical solutions to the
Cauchy problem (1 1.1 0)-(1 1.1 1) , then u(t ， x) 三 υ (t ， x) in
nT.
Theorem 1 1.8. Let (8.32) be a compαrison equation. Suppose 1; (i = 1,..., m)
sαtisfy
the following
cond巾ons:
there
β (t ， x ， u ， p)
ex时sα nonnegative
function
C = C(t) in L1(0 , T) such thαt lfï(t , x , u ， p) 一 fï(t ， x , v , q)1 :S: C(t)(l + Ix l) lp - ql + ρ(t， h2?mlu-u|)
~11.2.
UNIQUENESS OF GLOBAL SEMICLASSICAL SOLUTIONS
everνt ε (0 ， T) αnd
for a1most
1,..., m). Ifu=u(t , X)
for
αII (x ， 矶时， (X ， v ， q) ε ]Rn X ]Rm X ]Rn
andv 二 υ (t ， x) αre
Cauchy prob1em (1 1.1 0)-(1 1.1 1) , then u(t , x)
globa1 semiclassica1
== v(t , x)
in
so11巾ons
μ=μ (x) 10cally bounded on ]Rn
such
(i to the
nT .
Theorern 11.9. Let (8.34) be a comparison equation. Suppose /;,
(i = 1,..., m) satisfy the following conditions: there
155
=
fi(t , x , U , p)
ex时 a nonnegα tive
function αnd α nonnegative function R. = R. (t) in L 1(0 , T)
th αt
的 (t， x ， u ， p) 训，吼叫|旦(t) [(1+lxIHp 忡(x)σ (k=~.孔 lu…1)] (11 叫 for a1most every t
1, . . . , m). lf u
ε (0 ，
T) and for all (x , u ， 时，怡 ， v ， q)
= u (t , x)
α nd
v
=
v(t ， x) αre
Cauchy prob1em (1 1.1 0)-(1 1.1 1) , then
ε ]Rn X ]Rm X ]Rn
(i
globa1 semiclassica1 solutions to the
u(t ， x) 三 v(t ， x)
in
nT .
In the case of systems , a useful uniqueness criterion for global semiclassical solutions with essentially bounded derivatives is given by the next sharpening. Theorern 1 1. 10. Let (8.34) be a comparison equation.
(i
= 1, . . . , m)
]Rn
function μ Kl ， K2 μ 屿 ， K2
/;,
=
fi(t , x , U , p)
sets K 1 C ]R隅 ， K2 C 1 = R. K2 (t) in L (0 , T) αnd a nonnegative
satisfy the following conditions: for any
there exist α nonnegative function R. K2
S叩'pose
ω mpact
μK112(z)locallu botmded on Rn such that (1113)mth tk2αnd
in plαce of R. αnd μ ， respectzve旬， ho1d for a1most every t ε (0 ， T) and for all
K 1 X K 2 (i = 1,... ， m). 扩 u 1 = u 1 (t ， x) αnd u 2 = u 2(t , x) are global semiclassica1 solutions to the problem (1 1. 10)-(1 1. 11) with (x , u , p) , (x ， v ， q) ε ]Rn
X
m号~ . I!lax
]=1 ，~ 7. =1
then u 1(t , x) 三 u 2 (t ， x) in
ess sup Iθui( t ， x)jθxl
,…, m (t ,")EOT
< +∞，
nT .
With σ(ω) ~f mw in (8.34) [cf. the remark preceding Proposition 8.9], Theorem 11. 10 gives: Coroll盯y
mx
1 1. 1 1. Let fi
εIRnpα nd
= Iï (t , x , u , p)
differentiab1e in
仙 ， p)
be measurable in t E (0 , T) , continωus
E ]Rm x ]R n such
thα t， for α ny compαct
K C ]R n , the function
R. K
= ι (t) 哲 l+FMSup|I 尝 (t， x， u ， p)j(1 + UTJ =l ，...， m( .:c， 毡， p)EIRn x lll1 mν 1<"
IxDI
set
11. GLOBAL SEMICLASSICAL SOLUTIONS
156
is Lebesgue integrable on (0 , T) , and the function VK
= VK(X ， U) 望
m;皿
",'=',,,., m
,
C.αuchy
~<:t~ . I!1ax
,
,."
/n
tE(O T) pEK I uU 1c
is locally bounded on Rn x R m. If u 1 semiclassical solutions to the
Ilâh ,.
esssup sup ~:.' (t , x , u , p)/fK(t)1 u 1 (t ， x) αnd
u2
u 2 (t , x)
=
αre
global
problem (1 1.1 0)-(1 1.1 1) with
ess sup IθuHt ， x)jθxl < 十∞，
)=',:.1 '=',,,., m (t ,"')EOT then u 1 (t , x) 三 u 2 (t ， x) in
nT .
31 1. 3. Existence theorems This section is devoted to the existence of global semiclassical solutions to the Cauchy problem for a single Hamilton-Jacobi equation: θujθt
+ f(t ， θujθx)=o
u(O , x) = (x) with
= (x)
a 缸üte
on
in
{t = 0, x
convex function on Rn.
nT ,
(1 1.1 4)
ε Rn } ，
(1 1.1 5)
Explicit representation of such
solutions wi11 be obtained by the use of Chapter 9. For definiteness , let us rewrite the two sta叫i吨 assumptions (E .I)但 .11) of 39.2 on the Hamiltonian f and initial function
=
( x)
= f(t , p)
in a specified alternative form [when T < 十∞] as
follows. (The Fenchel conjugate function of
=仰)
is here denoted
by 旷=旷 (p)
as usual.)
(E.I) The Hamiltonian f = f(t , p) is continuous in {(t , p) t ε (0 ， T) \G , p ε R n } for some closed set G c [0 , T] of Lebesgue measure O. Moreover, to each N ε (0 ，+∞)
there corresponds a function gN = gN(t) in sup If(t , p)1 :=:; 9N(t) Ipl :-S: N
(E. II) For
e 肥ry ω mpact
L∞ (0 ，
for almost all t
subset V of [0 , T) x
R飞 there
T) such that
ε (0，
T).
exists a positive number
N(V) so that (p， x) 一旷 (p)
-
I •f(r , p)dr < ,Iql，l!l缸 {(q， x) Jo , ,-. :-S: N(V)"."
一扩 (q)
-
I
JO
f(r , q)dr}
~1 1. 3.
创 enever
(t , x) ε V，
157
EXISTENCE THEOREMS
Ipl > N(V).
As we have mentioned in Chapter 9 , (E .I) implies the t-measurability and
< t < T,
continuity of f = f(t , p) on {O
p ε IR. n }.
In addition , since
4>
=
rr
4>( x)
is
finite on IR. n , this hypothesis allows us to define an upper semicontinuous function ψ=ψ (t ， x ， p) from
[O , T) x IR.n x
IR.πinto [一∞，+∞) by taking
巾，卢 (p， x) 一矿 (p) 一 l t f(r , p)dr
(1 1.1 6)
We shall use the notation E ~f dom 4>*手。 and deal with the function u = u(t , x) and multifunction L = L(t , x) of (t ， x) ε [0 ， T) x IR. n determined by the formulas:
u(t ， x) 乞f supψ (t ， 矶时，
(1 1.1 7)
pEE
L(t， z)qzf{pε E def
Lemma 9.3 , with m~' n ψ(巾，时，
• def ,.
+ 1,
ç~'
shows that L = L( t , x) is a
m山 ifunction
of (t , x)
ε [0 ， T)
ψ (t ， x ， p) = u(t , x)}. /~
def
(1 1.1 8)
[O , T) x IR.n
andω=ω (ç ， p) ~
nonempty刊lued ，由sed
and locally bounded
(t , x) ,
x IR. n. This
()~.
mt山ifunction
is therefore upper semi-
continuous. An analogue of Theorem 9.1 says that (1 1. 17) gives a global solution to the Cauchy problem (1 1.14)-(1 1.15). Next , set
512 乞r ((0 , T) \G) x IR.n = {(t ， x) εnT : t
rf.
G}
for any G C [0 , T]. In this section we have: Theorem 1 1. 12. Let
4> = 4> (x)
be a finite con时 x function on IR.n. Assume (E .I)-
(E.II). Suppose that the m川 Z Qg 旦， G being αωs in (E.I月). Then (1 1.1 7) determines α global semiclassical solt巾on to
the problem (1 1.14)-(1 1.1 5). Proof. According to the maximum theorem [8 , Theorem 1. 4.16 ], the formulas (9.13)-(9.14) show that the derivatives 仇/缸，仇/θXl ，... ， θ时 âX n exist and are continuous in n~ (see also [128 , Corollary 2.2]). Hence , the global solution u u(t , x) is continuously differentiable in n~. Since G is of Lebesgue measure 0 , we conclude that u = u (t , x) belongs to V (n T ). An argument similar to that in the proof of Theorem 9.1 now gives us the validity of (1 1. 14) in n~.
口
11. GLOBAL SEMICLASSICAL SOLUTIONS
158
Remark. It can be proved that in u
=
nc.j. all partial derivatives of the global solution
u(t , x) exist [(11. 14) is then satisfied though the solution may fa.il to be dif-
ferentiable] if and only if the
mt山 ifunction
later for the smoothness of u
= u( t , x).
Corollary 1 1. 13.
(E .I)-(E. II). Then
=
Let cþ
L = L( t , x) is
finite ωnvex
cþ( x) be a
si吨le-valued.
See S13.2
function on Jæ n.
(1口1.1 7η) dete旷 付m r 旧zn 旧E臼 sα globα al semi化 ω cl归 αssz化 Cα al ω s 01沁 t包di ωiω on ω to 伪 t he
Assume problem
(1 1.14 钊)-(υ1 1.1 时 5 )可 on 附e of 伪 t he following t~咀 ωt
(ωi) 旷=旷 (ωωp 叫) is strictly con肘x on its effecti肘 domain E ~f dom cþ*. (ii) f
=
f(t , p) is strictly p-convex; more
to p on E
for αlmost
every fixed t
ε (0 ，
precise旬，
it is strictly
con 肘x
with respect
T).
Proof. Each of (i)-(ii) implies the strict concavity of
ψ=ψ (t ， x ， p)
to p on E for every (t , x)
ε
[0 , T) x Jæ n. It follows that the
mu 世 山 11 ltif1 缸 u 山肌 nlctior丑1L 二 L(归t ， 叫 x)
is indeed single-valued , and hence
with respect
叩 u ppe 缸r 配 s em 血lC ∞ on 毗 l此削.tim∞u
continuous.
口
= 1, an existence result is established as fo l1ows:
In case n
Theorem 1 1. 14. Let cþ
=
cþ( x) be a finite ωnvex function on
and let φ*φ*(p) be of class
C2
Jæ, E ~f dom cþ* ,
in int E. Assume:
(i) The Hamiltonian f = f(t , p) belongs to CO(((O , T) \G) x
E)
with
< +∞
esssup If(t , pO)1 tε(O ， T)
for some pO ε E and some closed set G c Jæ of Lebesgue measure 0; it is , moreover, twice continuously differentiable in
p εintE
with
max {esss叩 sup Iδf(t ， p)jδ'pl ， ess sup sup Iδ2 f(t ， p)jθ泸 I} <十∞ tε(O ， T)pEintEε(O ， T)
Ipl 三 N
pEint E ~I 三 N
for any N ε(0 ，十∞) .
(ii) To each
compα ct subset V of [0 , T)
so that ψ (t ， 吼叫<
m哩
x Jæ there corresponds a positive number N(V)
ψ (t ， x ， q) 叫enever (t ， x) ε V， P ζ E ， Ipl
pεE
> N(V).
Here
Iql 三 N(V)
ψ=ψ (t ， x ， p)
is defined by (1 1.1 6) with n = 1, (p , x) =px.
(iii) For every (t ， p) ε ((0 ， T) \G) x int E , it holds that
IV2f
♂旷
百~(T， p)dT 十石~(p)
>0
(1 1.1 9)
~1 1. 3.
159
EXISTENCE THEOREMS
Then (1 1. 17) determines a global semiclassical sol山on to the problem (1 1. 14)-
(1 1.1 5) with n
= 1.
Proof. We first note that E is a nonempty convex set in IR 1 , and is therefore an interval with end-points , say， α ， b ε[ 一∞，十叫， α 三 b. Consequently, [117 , Corollary 7.5.1] implies the continuity of the restriction of 旷=旷 (p) to E. Condition (i) shows that f = f(t , p) is t-measurable and p- continuous on {O
t < T,
p ε E}.
<
Moreover , we conclude from this condition that (1 1. 16) gives a
continuous function ψ =
p ε E
and twice continuously differentiable in
p εintE
with the derivatives
百 (t ， x ， 叫
r t θf
d旷
=x 一石 (p) 一人再(飞 p)dT，
2* 1 \ r t θ 2f 市 (t ， x ， p) = 一石~(p) - In τ (T ， p)dT
h
θp
(1 1. 20)
Under Hypothesis (ii) , the method of Lemma 9.3 proves that the ml山 ifunction L = L(t , x) defined by (1 1. 18) is nonempty-valued and upper semÏ continuous on
[0 ,T) x IR. It may be shown that esssup sup If(t , p)1 tε (O ， T)
<
+∞
pEE Ipl 主N
for all N ε(0 ，+∞)， hence that u = u(t , x) is in Lip([O , T) x IR) (cf.
Proof of
Theorem 9.1). Finally, according to (1 1. 19)-(1 1. 20) , for each (t , x) εn<，j. the maximum set L(t , x) of
is really continuous in n <,j.. Analysis similar to that in the proof of Theorems 9.1 and 11.1 2 shows that u = u(t , x) belongs to V(nT) and that (9.13)-(9.15) [with n = 1] hold in n <,j.. Thus (1 1. 17) determÏ nes a global semÏ classical solution to the
problem
(1 1.1 4)-(1 1.1 5).
Corollary 1 1. 15. Let α叫 let φ*
4>
口
=
4> (x) be α finite
扩 (p) be of class
1 1.1 4 ， αssume
C2
convex function on IR , E ~f dom 旷，
in int E. Under Conditions (i)-(ii) of Theorem
that θ2f
再~(t ， p)
>0
(11. 21)
11. GLOBAL SEMICLASSICAL SOLUTIONS
160
for all
p ξint
E and almost all t
ε (0 ，
T).
Then (11. 17) determines a global
semiclassical solution to the problem (1 1.1 4)-(1 1. 15) with n
=
1.
Proof. Since (1 1. 21) implies (1 1. 19) , the conclusion follows.
口
We conclude this chapter with the following example and remark. Example. Let J
c [0 , 1]
be the Cantor set and
ω=ω (t)
be the Cantor ladder
(see Remark 2 after the formulation of Theorem 8.1). Define g(t) ~f min{lt - sl
Clearly, 9
=
g(t) is
s ξ J}.
Lμip 归sch巾 i
hence differentiable almost everywhere in (0 , 1); its derivative dg/dt
dg(t)/dt
being Lebesgue measurable. Next , consider the Cauchy problem θu da... I θu\ 一+ ~ (t) sin(\ 一) ôt . ...dt' - I ---âx J = 0
in
{O
< t < 1,
州对斗 x on {t=O ,
x
E Jæ. l} ,
XE Jæ. 1 }
Then we may invoke Theorem 11. 1 to deduce that the function u
-
u(t , 叫 x )乞f
仇伽巾 阳 巾 盯n叫眈 s臼sω s叫 sem uc ücl lla臼础 ass olut 1此t 剑仰州 t吟们) + 2÷Z 川e only川global 回
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 11.5-1 1. 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 selniclassical solutions.
Chapter 12 M_inimax Solutions of Partial -Differential Equations with Time-measurable Hamiltonians 312. 1. Introduction We have seen in the previous chapters that the main theorems in the classical theory of first-order nonlinear parlial differential equations are valid only locally, in
su血ciently
restricted domains. The example in the preface shows that the
Cauchy problem might fail to possess a global C 1 -so1ution even if the Hamiltonian and initial data are analytic. Therefore , the need for introducing generalized or weak solutions has arisen in the theory
of 直rst-order
nonlinear equations and
its applications. Such solutions have been investigated by many mathematicians in the past 50-70 ye缸s ([2]- 间， [21] ， [32]-[33 ], [44]-[45 ], [51]-[52 ], [54 ], [63]-[64 ], [87]-
[92 ], [94] , [95] , [97]-[98 ], [112] , [118]). The notion of globα1 solutions in Chapter 9, probably 伍rst used by Hopf [64], affords the existence but not unique-
which was
ness question.
Some other supplementary condition like the semi-concavity or
优 s eml-c ∞ on 盯 回 V 白臼】 e 对〈 X
of a global
叫 s olu 叫 t创10 ∞ n
(in the case of convex or concave
In a similar situation , entropy condition (see
35 .4)
Ham 叩ilto ∞且止叫 -Ja ω∞ 缸 c 呻 O bi 吨 e quatωi 沁 on 叫 s吟)
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 viscos-
ity solutions , which was introduced by
Crandall 缸ld
Lions. Within this theory,
uniqueness and existence theorems have been developed for various types of equations and boundary-value problems , and also some applications to control problems and di侄erential games have been studied (闷， [10]-[20]， [28 ], [35]-[39] , [47]-[50 ], [67]-
[72 ], [79 ], [99]-[101 ], [122]-[123], [131]). The ∞ncept of viscosity solutions is motivated 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
162
12. MINIMAX SOLUTIONS
is assumed to be fiow invariant with respect to the so- called characteristic inclu-
sions. This direction has been suggested by Subbotin in [1 ], [124]-[125], and leads to the notion of minimax solutions. The above term originates from the theory of differential games. It is justified by permanent presence of min-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 [82 ], [105]. To facilitate access to the topics from motives of the characteristic method , let us go back to !ì 1. 1 and consider the following "terminal" Cauchy problem (data
= T of time
being given at the end point t
= 0 in
θujθt+f(t ， x ， θujθx)
u(T, x) = σ (x)
inte凹al): def
f!T~'
on
{t
{O < t < T , x
= T, x
ε ]R"}，
(12.1)
ε ]R"}.口12.2
Here， σ=σ(x) is a given C1-function on ]R". After a change of variables (T , x) 哇f (T - t , x) , from (12.1)-(12.2) we indeed get the usual Cauchy problem (data being given at the initial point
T
= 0 of time interval). However , for the construction in
this chapter , it will be more convenient to consider the terminal condition (12.2).
= f(巾 ， p) is of the form
We first assume that the Hamiltonian f
f(t , x , p) ~f (α(t ， x) ， 时，
(12.3)
with α=α (t ， x)=( αl(t ， X) ，... ， α，， (t ， x)) an ]R"-valued function of class C 1 on f! T. Assume , further , that |α (t ，
for some constant L
x) I 三 L(1+lx l)
ε(0， +∞).
V(t ， x) ε f!T
(12 .4)
The characteristic system (1. 3)-( 1.4) ofthe Cauchy
problem (12.1)-(12.2) in this homogeneous linearity case is reduced to the system
(?=:")
(12.5)
dt
with the condition
x(T)
= ν，
υ (T) = σ (y).
It is known that the problem (12.5)-(12.6) (x , v) =
h田Ulliquely
(x(t ， ν) ， σ(ν))
(12.6) a solution
163
912. 1. INTRODUCTION
on some interval
(T 一 ε ， T]
characteristic curves
y ε ~n.
for each
x 二 x( t , y)
Under the hypothesis (12 .4), the
can be extended over the whole segment
0 三 t 三 T
Appendix 1). Moreover, an easy argument (adapted from [62 , Chapter V ,
(cf.
Peano's Theorem 3.1 , Corollary 3.1 and its Re mark]) shows that the Jacobian
DxjDy of the C 1 -mapping ~n 3ν 叶 Ht(y) 哲 x (t , y) does not vanish for any y ε ~n andO 三 t 主 T. Hence Lemmas 2.3-2.5 imply that x = Ht(y) is a diffeomorphism from ~n to itself for any t ε [0 ， T]. Therefore , the inverse function y = y(t , x) is of class C 1 on 百T and u
= u(t ， x) 哩。(t ， ν(t ， x)) = σ (y( t , x)) is the only global
C 1 -so1ution of (12.1)-(12.2) in the homogeneous linearity case (12.3). The above
de鱼缸缸 I丑 nit
characteristics. In such a homogeneous linearity case , the family {(t ， 吨， ν)) :。三 t 三 T} 证Iæn
intersect in
of characteristic curves covers all of
a given point
(t ， x) ε n T .
and no two of them could
For a nonhomogeneous linear Hamilton-Jacobi equation
θujθt
with 9
nT
nT ; in other words , there exists one unique characteristic curve passing + (α (t ， x) ， δujδx}+g(t ， x)=O ，
(12.7)
= g( t , x) a function in C 1 (n T ), we can continue using the above farnily, ((t , x(t , y)) : 。三 t 三 T}YEIæn ，
of characteristic curves and prove that
u=u(t， x) 巳(y(t， x)) +
l
T
g(T , X(T , y(t , x)))dT
is a unique global C 1 -so1ution of (12.7) and (12.2). Now , turn to the general case. Consider the nonlinear equation (12.1) , with
f = f(t ， 矶时 and σ=σ (x) some twice continuously differentiable functions. In this
c出e ，
the characteristic system (1. 10) may be reduced to
(dz4δf 一= ~_~ (t , x , p) (i = 1,. . . ，叫， dt δD dPi dt
We shall assume that
for 创ly
θf 一~~ (巾 ， p) θXi
(i=l ,… , n).
Y ε ~n there has uniquely a solution
(x , p)
=
(x(t , y) , p(t , y))
(12.8)
‘
12. MINIMAX SOLUTIONS
164
of (12.8) on the whole segment
0 三 t 主 T
satisfying the condition
x(T) 习， pdT)=31(U)(z=LA) 川巾，
Moreover , suppose that the family {(t , x(t , y))
0 三 t 主 T}yElRn
of characteristic
curves covers QT , and that no two of them could intersect in QT. Assume , further , that the inverse y = y( t , x) is of class C 2 on 百T. Then it Can be claimed that rT _
u
=
u(t , x) ~fσ(ν (t ， x))
+I
If(r , x(r , y(t , x)) , p(r , y(t , x)))
一立阳以(咐，Jz));茫?(卡M 圳 ，圳 x (咐， x)) ， 仲p(仆T飞咐， ~l"
is the only global C 1 -so1ution of the Cauchy problem (12.1)-(12.2) [cf. Theorem 2.7]. We must note here that for a general nonlinear equation (12.1) the
above 出
sumptions are not always automatically satisfied as they are in the linearity case (12.3). Su:ffi.cient conditions for them to hold could also hardly be given explicitly because the structure of characteristic curves is then via a projection from Jæn
X
(x ， p) 叶 Z
Jæ n into Jæ n. Without these assumptions the preceding method of finding
a global C 1 -so1ution breaks down. A relevant solution , which is expected to exist , should be understood in some generalized sense; say, one needs to relax the smoothness condition on it as usual. Subbotin's method of de丑ning 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-Bellman equation). The u- and v-stability properties Can be expressed in various ways and used to introduce the notion of minimax 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 dete臼r口 r日皿血I11山 mine the characteristics x
= x(t , y)
can be slight1y separated into two character时ic di.fferential 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.
165
DEFINITION OF MINIMAX SOLUTIONS
Our aim here is to extend
S由botir内 notion
(given in [1] and [124]) ofminimax
solutions offirst-order p缸tial differential equations with continuous Hamiltonians to the case of time-measurable Harniltonians 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
Har山onians.
Almost of them were published in [132]-
[133] and [146]. The outline ofthe chapter is as follows. In Section 912.2 we give the definition of minimax solutions to the (terrninal) Cauchy problem for a general nonlinear evolution partial differential equation with time-measurable Harniltonian. We investigate some properties of multivalued mappings which play a decisive role in the definition of minimax solutions. Section 912.3 is devoted to the relations between minimax solutions and global (serni)classical ones. We prove that a global semiclassical solution is also a rninimax solution , and conversely, a minimax solution satisfies the equation in the classical sense at the points of differentiability. Further , in Section 912 .4 we discuss the invariance of the definition with respect to the choice of concrete multivalued mappings. In Section 912.5 we establish the main theorem of this chapter on the uniqueness and existence of minimax solutions. Finally, Section 912.6 concerns some generalizations to the case of monotone systems of first-order partial differential equations. Our method is based on the
theo巧r
of multifunctions
and differential inclusions , and on a sharpening of a well-known theorem on the Lebesgue sets for functions with
p缸缸neters.
Gronwall' s inequality. Throughout , 0 def
r
Br~' {p ε ]Rn
:
We also use an implicit version of def
+∞， S ={pε ]Rn
Ipl = 1} ,
、
Ipl 三 r} (r>O) ， B~'Bl.
912.2. Definition of minimax solutions 1 0 Formulation of the Cauchy problem
Let us consider the Cauchy problem of the form
去+牛川去) u(T , x)
=0
= σ (x)
Assume that the terrninal data
in on
σ=σ (x)
nT ~f {O < t < T , x εR勺， {t=T , x
(12.10)
is of class C O on ]R n and that the Harnil-
tonian f = f(t , x , u , p) depends on (t , x , u ， p) properties.
ε ]Rn}.
(1 2.9)
εn T
x ]R x ]R n with the following
12. MINIMAX SOLUTIONS
166
a) C.α rathéodory's Conditions: a1) For almost eυery (in the se附 e of Leb仇es叩gue meaαs拍包旷r陀E件叫) βxedt ε (0 , T) , 伪 the ft也mct玩 ωiω on
Rn x R x Rn
'3 (归 x， u， 叫 p )叶 f(仪t ， 叽 x，叽 u，叫 p) 必 i s cω o时 n tìn包 ωO旧.
a2) For each (x , U ， p) ε Rn x R x S, the function (0 , T)
'3 t 叶 f(t ， x ， u ， p)
is
measurable.
b) For a叼 bounded sets D C Rn and E
c R , there exists afunction AD ,E =
AD ,E(t)
in L 1 (0 , T) with If(t , x , u , p) - f(t ， x' ， u ， p)1 三 AD ，E(t) .IX - x'l for α lmost
(12.11)
all t ε (0 ， T) αnd for all x , x' ε D ， u E E , p εS.
c) There exists a function f = f(t) in
P(O , T)
sup{lf(t , x , u , p) - f(t , x , u , q)l-f(t). (1 for almost all t
ε( 0, T)
d) For almost all t
and for all (x , u)
ε (0 ， T) αnd
such that
+ Ix l) .Ip-ql
ζR饥 x
:
p ， q ι B} 三 o
(12.12)
R.
for α II (x , p) E R n x S the f包nction R '3 u 叶
f(t , x , u , p) is decreasing.
e) f = f( t , x , u , p) is positively homogeneous in
p ε ]Rn;
f(t , x , u , s.p)=s.f(t , x , u , p) for almost all t ε (0 ， T) αnd for α II 忡，包 ， p) ε ]Rn
Note that Conditions
a3) For all
a匀，叶，
(x ， 矶时 εRπx
X
i.e. ,
(12.13)
\:/s;:::O
R x S.
and e) together imply:
R x Rn the function
t 叶 f(t ， x ， u ， p)
is Lebesgue
ìntegrable on (0 , T)
2 Differential inclusions for supersolutions and subsolutions 0
In the
prese丑t
section and the next one we shall often use the notations:
町，叫住f
for (t , x) E
~h ，
V2f(t) . (1十
Ixl). B ,
Fu(t ， x ， u ， α) d~f {z ε F(t ， x)
(z ， α) 主 f(t ， x ， u ， α)} ，
FL(t ， x ， 肌肉哇f {z ε F(t ， x)
(z ， 的主 f(t ， x ， u ， ß)}
u
ε ]R， α ， βεs.
(12.14)
~12.2.
167
DEFINITION OF MINIMAX SOLUTIONS
Remark. For any multifunction S1 T ~ (t ， x) 叶 G(t ， x) C ~n and any (儿， x*) ε S1 T ， denote by XG(t* , x*) the set of all absolutely continuous functions x = x(t) from
[0 , T] into
~n
which satisfy almost everywhere in (0 , T) the differential inclusion
生。)巳 G(t， x(t)) dt that G
x(儿) =
x *. We shall always assume G(t , x) is nonempty convex compact valued , measurable in t , upper subject to the constraint
semicontinuous in x , and that
IG(t ， x)1 告fsup{|z|:z ε G(t ， x)} 三 c(t) . (1 on S1 T , with c
= c(t)
+ Ix l)
(1 2.15)
a function in Ll(O , T). From [29 , Theorems 11 刻， 111.15 , and
VI.1 3] (see also [40 , Theorems 5.2 and 7.1]) , it follows that XG (t*, x*) is then a def
U
nonempty ∞mp配t subset of C([O ， 町 ， ~n). Now let XG( ð.)~'
XG(t ， x) ， 。并
(t ,., )E .o.
ð. C S1 T . If ð.
c
[0 , T] x Bro is compact , then by Lemma 8.3 , XG( ð.) C X G({O} x
Br) with r 彗 (r O
1. The compactness of XG({O} x B山 which
+ 1) JoT c(t)dt -
follows from the upper semicontinuity of the compact valued multifunction Z 叶 XG(O ， x)
~n ~
C C([O ， T]， ~n) (see [40 , Theorem 7.1] or [29 , Theorems 11. 25 and
V I.1 3]) , therefore implies that of XG( ð.). Now it can be seen that the multifunctions 。T ~ (t ， x) 叶 F(t ， x)
。T X ~ X
S
~
X ~ X
S
~ (t ， x ， u ， ß) 叶 FL(t ，
S1 T
in (12.14) are nonempty
(t , x , u ， α)
c
~n ，
叶 Fu(t ， x ， u ， α)C
x , u ， β) C
Rn , ~n
compact valued , measurable in t , upper semicon-
co盯ex
tinuous in x. (It can be shown that they are indeed continuous in x.) As was mentioned in the ab ove , the sets
Xu(仇 t儿.， X. , 叽U， α 叫) ~哇主f X岛 t儿.， x.) , F1'u(ι叮.'川.叮'冉a叫) (队
(12.16)
X川 L (t儿.， X.扪， u ， 肉ß) ~哇主f XF托L(ι.叮，.，1t 缸e alwa 叮 ys nonempt句 y
and
∞ c ompact
for all
(t* ， x*) ε S1 T .
So we may conclude the
following: Deftnition 1. A
supersolt巾 on
tinuous function u
= u(t , x)
nU
(
9" 4EE-
、 EE，，，
u(A'kv Z
用4
Z(T
E 咱 E晶
u( T
、、.，，，
a a)
<一
-u4··
、BtJ
mMW
、BtJ
-ua
XU
丁'
a(
F』
uε
saps
of Problem (12.9)-(12.10) is a finite lower semicon-
on S1 T which satisfies the condition
12. MINIMAX SOLUTIONS
168
for all
0 三 t
and also the condition
u(T， x) 主 σ (x)
\f x εIR n .
(12.18)
Definition 2. A subsolution of Problern (12.9)-(12.10) is a finite upper sernicontinωus
function
u 二 u( t , x)
i~t
on
nT
rp.ax
, , _ __
which satisfies the condition ,
_Ju( T, x( T)) - u(t , x)] 三 o
(12.19)
β ES :r :.)εXdt ，，，，冉 (t ，，，，) ，β)
for all 0
~
t<
T ~
T, x
ξIR n ，
and also the condition
u(T， x) 三 σ(x)
\f x εIR n .
(12.20)
The sets of all supersolutions and subsolutions of (12.9)-(12.10) will be denoted by Solu and SolL , respectively. Definition 3. A function u
= u(t , x) in Solu n SolL is called a minimax solution
of the Cauchy problern (12.9)-(12.10). 3 0 Fu rlher properties of Fu
= Fu(t , x , U , 0:)
and FL
= FL(t , x , u ， β)
It will be shown (see [124 , p. 16]) that Fu(t ， x ， 包， α) 门 FL(t ， x ， u ， β) 弄。
In fact , it follows frorn (12.13) that f(t , x , u , 0) If(t ， x ， u ， p)1 三 C(t).
Therefore , if
(α ，的=
1, then
α =ß
for
all α ， βε s.
(12.21)
= 0, hence frorn (12.12) that
(1 + Ix l)
\f p ε s.
(12.22)
and
z qzf f(t， ZJ ， α) .α =f(t ， x ， u ， ß).ß ε Fu(t， x , u ， α) 门 Fdt， x ， u ， β). Further , let 0 三 (α ，的 <1.Setting
E
qzf(1-hJ)2)-附. ((α，的 ·α - ß) , we
get lel = 1 ，作， α) = 0 ，作 ， ß) = 一 (1 一 (α ， ß)2) 1/2 < O. Next , choose z 乞f f(t ， x ， u ， α) .α + C(t) . (1
+ Ix l) . e and conclude frorn
(12.23) (12.22)-
(12.23) that Izl2 [J( t , x , u ， α )J2 + [C(t) . (1 + Ixl )J2三 2[C(t) . (1 + Ix l)户， i.e. , that z ε F(t ， x). On the other hand , (12.23) implies (z ， α) = f(t ， x ， u ， α) ， thus
~12.2.
DEFINITION OF MINIMAX SOLUTIONS
169
z ζ Fu(t ， x ， u ， α). In addition , since (α ， β) 2: 0, it follows from (12.13) that (z , ß) =
j(t , x , u , (α ， β) .α)
+ e(t) . (1 + Ix l). (e , ß).
But , according to (12.12) and (12.23) ,
1(α， β) .α- ßI = (1 一 (α ， ß)2)1/2 = 一 (e， β) ，
j(t , x , u , (α ， ß)
. α) 三 j(t ，
x , u , ß) - e(t) . (1
+ Ix l) . (e , ß).
Therefore , (z , ß) :三 j(t ， x ， u ， ß)j i.e. , z ε FL(t ， x ， u ， ß). Hence , z ε Fu(t ， x ， u ， α) n FL(t ， 叭叭的.
Finally, let (α， ß)
< O.
Setting F(t , x) 3 z 哲 J2e(t). (1
+ Ix l) . (α-ß)/Iα 一 β1 ， 2: e(t) . (1 + Ix l), because 1α-ßI =
= e(t) . (1 + Ix l) . (1 一 (α， β) )1/2 J2(1 一 (α ， ß))1/2 (as 1α1 = 1β1 = 1). Thus , (12.22) gives (z , a) 2:月，川， α) j i.e. , z ε Fu(t ， x ， u ， α). Analogously, z ε FL(t ， x ， U ， ß) ， and in consequence , z ζ Fu(t , x , u , a) n FL(t , x , u ， β). The equality (12.21) is thereby proved [for almost all we obtain (z ， α)
t
ε (O ， T)
and for all
j(t , x , u , p)
(x ， u) εIRn
= sup
.(z , p)=!I!(
min
a. ES zEFu(t ，..，饵 ，0.)
ε (O ， T)
for almost all t
x IR]. It follows that
and for all
max
_.(z , p)
(12.24)
βESzξFdt ，..，毡，β)
(x ， u ， p) εIR n
x IR x IRn. Indeed , let
p ε S.
By
(12.14) , we easily see that sup
min
,(z , p) 2: ,-, -
aε 8 z E Fu(t冉饵 ，0.)'
min
. (z ， 的主 j(t ， x ， u ， p).
zEFu(t ，吼叫p)
Moreover , by (12.21) and again (12.14) , we also have ~n
. (z , p) :三
zEFu(t 冉饵 ，0.)
for any
αζ S.
.
血n
. (z ， 的主 j(t ， x ， u ， p)
Z ε Fu(t ，.. 冉 ，a)nFL(t 向饵 ，p)
Thus , in view of (12.13) , the first equality in (12.24) is satisfied , no
matter whether
p ε S
or not. Similarly for the second one.
From the monotonicity and continuity in u of j
=
j(t ， x ， u ， 抖，
it may also be
deduced that
QFu(t ， z ， u ， α) = Fu(t ， x ， v ， α) ， for almost all t
ε (0 ， T)
of the multifunctions Fu following.
and for all
r) Fdt , x , u ， β)
也
= Fdt , x , v , ß)
αJε S， v ζIR， x εIRn.
= Fu(t , x , u , a) and F L
(12.25)
Another property
FL(t ， x ， U ， β)
is given in the
12. MINIMAX SOLUTIONS
170
Proposition 12. 1. One has 唱
up 1i rn inf
。 ES
înf
，. t+ó
二 I
ó、o x( 托X-;(t，x) ðλ
_ mî~ ，
,
zEFu(r，x(r) ， u ，a)
(z , p)dr
~
f(t , x , u , p) , (12.26)
rt+ó
inf 1îms叩
βε s
/0 1' almost
ó、o
sup
x(.) ε X~(t ， x)
α llt ε (0 ，
号 I ð Jt
ma工作 ， p)dr 三 f(t ， x , u , p)
zEFdγ ， x(r) ，也，β)
T) and /0 1' all
x εR'毡 ， u 巳 R， p ε R n .
The proof of Proposition 12.1 will be based on the next two 1emmas , which are sharpenings of a well-known theorem on the Lebesgue sets (see [119 , p. 158]) for functions with parameters. Lemma 12.2. For α ny
measure 0
s包 ch
E (0 , +∞)， there exísts a set A(r) C (0 , T)
l'
0/ Lebesg盯
that rt+ó
Mift
fhZAP)dT
f(川
( 12.27)
/0 1' all t ε AC(r) ~f (O ， T) \ A( 叶 ， x εB俨 C Rn , u εE俨彗 (-1'，1')
C R , P 巳 Rn .
Proof of Lemma 12.2. Accordîng to b) and e) , we find a function A
= A(t) ~f
AB"E俨 (t) in L 1(0 , T) and a null set A1(r) C (0 , T) so that
If(r , x , u , p) -
f(r ， x' ， u ， p)1 三 Ipl.lx - x'l.
A(r)
(12.28)
def
for 811 x , x' ε Br' u ε Er' p εRn ， r E AHr) ~. (0 , T) \ A 1(r). By a1) , there is a nul1 set A 2 C (O , T) such that the function Rn x R x Rn :1
(x ， u ， p) 叶 f(t ， x ， u ， p)
is
everywhere continuous for each t 巳 A~ ~f (0 , T) \ A 2 • In vîrtue of
Lebesg町、 theorem
(see , for înstance , [119 , p. 158]) , we see that
for any 9 = g( t) in Lloc (0 , T) the set Leb(g) of all t ε (0 ， T) satîsfyî吨
iuf~ is of measure T. Let Q be the countab1e set of all rationa1 rea1 numbers and take
A(r)
~f A1(r) UA 2 U L. \(",'
. "il_~~_Leb (J(.， x' ， u' ， p')) 门 Leb(f) 叫eb(A)r x iQI x iQI
，也'， p')E iQl n
n
/
~12.2.
DEFINITION OF MINIMAX SOLUTIONS
= O. The only point remaining is to prove
[cf. a3) and c)]. Then mes(A(r)) (12.27) for any fixed t
ε AC(r) ，
ε Br ，
x
171
u
ε Er ， p ξ ]Rn.
To this end , let
ε>
0 be
an arbitrary number. Since t ξ AC(r) C Leb(A) n Leb(町， we have 唱
M~‘ max ~
~
sup
I óε (O ， T-t) ο
"t+ ó
I
"t+ó
f(r)dr ,
Jt
Because t ε A~ ， there must be
sup
~
I
A(r)dr
~
+ 1 < +∞.
(12.29)
óE(O ， T-t)ο Jt
0(1)
> 0 such that
If(t , x' , u' , p') - f(t , x , u , p)1 < ε/3 whenever max{lx - x'l , lu - u'l , Ip - p'l}
(12.30)
< 0(1). Now choose
p' εQn ， z'εQn n Br , u(l) , U(2) εQnE r
so that max{lx - x'l , Ip - p'l}
< o(匀 ，
u - 0(2)
< U(l) < u < u(2) < u + 0(2) , where
6(2)tfmin{6(1) ， ε/[6M(r 十 1月， ε/[6M(lpl Since t εLeb (J(.， x' , u 间 ， p')) (k
=
1 ，匀， we could find
0(3)
+ 1)]}.
(12.31)
> 0 with the following
property:
创k) ,， 汕内阳，竹切怡 p' 阶 )dd古T←←川川… 川叫一→卅 - 贝附(归 f(tt 旷h 川川川 川u(川叫(伪例 ρ (k)川k) ,， 旷，p' I~~ii + 川(仙例(k)州川川))》川叫 t
ó
(12.32) Finally, setting 0(0) 乞r min{o(刻，以3)} > 0, we have only to show that
|ifs 川的dr 一川 p)1 < εh 川 For k
= 1, 2 one writes "t+ ó
8Jt
f(r， 川 p)dr - f(巾 ， u , p)
=iJfU川， 叫 p )一川叫 ， 叽w 旷 u，， p'刀 p' +:Ifff 忡时飞+刊s队归旷州)一川
+[卡: Iff忡+刊飞 s + [f(μt ， 旷x， , u (价例h的) , pj，斗) -
f(μt ， 凯 x， 叽 u ， p)川].
(口 33)
172
12. MINIMAX SOLUTIONS
Let 15 E (0 ， 15<的). The first summand in the right-hand side of (12.34) is estimated as 行
，。
4·
+ε-6
14 亏。
A= d, 7 + ε-3
、、
ε-6
fljt 十
一古
ρι
AV
l-d
一
P
..,,
Z
门l<
<-fjt
μ忡
P
z l up
』刊
古
m川
叮十 T
''，、、，，，
一+
U DA
'''飞、.
IJ T Z
feiυ
tiτ‘。
+eo fIIA
by (12.12)-(12.13) , (12.28)-(12.29) , (12.31) and by the choice of x' , p' , 15(0). The third and
:r.岛 ou 盯rt由 h su皿 nu
Therefore , taking k
= 1 in
(12.34) , we get
ijt飞山，附一 f(川，p)<3;=ε，
(12.35)
because the second summand in the right-hand side of (12.34) is then nonpositive by Condition d). Analogously, letting k = 2 implies
;f~
(12.36)
since the second summand in the
咆ht-hand
side of (12.34) is then nonnegative.
Combining (12.35) and (12.36) gives (12.33) , which completes the Lemma 12.3. There
exists α set
r t +6
115运人 for all t
f(r , x(r) , u , p)dr
ε AC ，
x
ε IR n ，
u
A
c
proof.
口
(0 , T) of Lebesgue measure 0 such that
= f( 式川 p) 也niformly
in x(.) E XF(t , X)
( 川 7)
εIR ， p ε IR n .
Proof of Lemma 12.3. It su伍 ces to take A
def -1:∞ ~. .U _A( 的， k=l
where the sets A(k) 町e
constructed in Lemma 12.2 and its proof. In fact , it follows easily from (12.27) (12.29) that (12.37) must hold for all t ε AC ， x ε function far均
IRn ， u εIR ， p ε IRn ，
because the
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 312 .4 later). By the positive homogeneity in p of f = f(t , x , u , p) (s白 (12.13)) andof(z ，抖， we need only prove
!ì 12.3. RELATIONS WITH SEMICLASSICAL SOLUTIONS (12却) for almost all
t
173
ε (0 ， T) and for all x ξ lR. n ， u ξ lR.， p ε S. Indeed , according
to (12.14) , one then obtains rt+6
inf 二， min ， (z ， 的 dr? aES 6、o "(.)EXp(阳)Ó λzε FU(T，"(T) 向a)
p li1p. inf
咽
r t+ 6
> liminf
inf 二， 6、o .，(托:X;'(t ，.，) Ó λ
__
,miIf, , (z , p)dr zεFU(T卢萨)，u ，p)
rt+6
二 ， 八o "(.)EXp(阳 )Ó λ
= liminf rt+6
inf lims叩
β ES
sup
号，
6 、 0- "(.)EX;'(t ,., )ò Jt
f(r， x(r) ， 矶的齿，
inf
_ ,m 8.?',
_, (z ， p)dr 三
zEFdT，"(T) 冉，β)
三 lims叩 s、0-
sup
r t +6 ~，
sup
~，
.,(.)EX;'(t ,.,) Ò 1t
ma?C,
, (z , p)dr
z ε FdT， "(T) ， U ♂)
rt+6
=limsup 6、o
"(.)EXp(t♂) 0
f(r， x( 叶， u , p)dr.
Jt
Therefore , by Lemma 12.3 , the proof is complete.
口
Corollary 12.4. We have 咽
rt+6
.
J_
‘
二， (工(叶 ， p)dr 兰州， x , u , p) , aES 6、o "(.)EX u (阳，钮，a) Ó λ\ dt""r/
p li1p. inf
inf
inf lim sup
sup
pc;:'
for
6\..0
r
1 tH / dx 飞 ~ ，一 (r) ， p }dr 三月， x , u , p)
也 )Ó λ\ dt ，-，>r/ "(.)EXdt ，矶，β Jt 、，
almost αllt ε (0 ， T)
and for all x
ξ ]Rn ， U ξ lR.， p ε lR. n .
Proof. The proof is immediate from (12.26).
口
s12.3. 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)叫 .
Then
i) every global semiclassical solution u = u( t , x) of (12.9)-( 12.10) so that the gradient mapping lR.n 3 x 叶 θu(t ， x)jθx is continuous for almost all t ε (0 ， T) must αlso be a mznzmαx solution of the same problem ,
12. MINIMAX SOLUTIONS
174
ii) there 口时s a null set A c (0 , T) such thαt， at each point (t , x) ε(( 0, T) \ A) x ]Rπ 叫 ere α certain mmimαx
0/
solution
(12.9)-(12.10) is differentiable , the equation
(12.9) must be satisfied.
u(t , x) be a global semiclassical solution of Problem (12.9)
Proof. i) Let u (12.10) such that ]Rn
"
X 叶仇 (t ，
x) /θx is a continuous function for almost every
t ε (0 ， T). Based on (12.14) , for (t , x ， α)εn T x 5 , we defin
r(t , x , a) ~f {ZO E Fu(t ， x ， u(t ， x) ， α) : (ZO ， θu(t ， x)/θx)
Since
nT
"
m山 ifunction
(x , u)
(t , x)
Fu
E ]Rπ 岖，
=
min
zEFu(t ， æ ， 也 (t ， æ) ， a)
忡， θu(t ， x)/ θx)}
(12.38)
is measurable in t , continuous in x and the
Fu(t , x , u ， α) is indeed measurable in t
it follows from [8 , Theorems
theorem [22 , p. F。 =F。 (t ， 2 ， α)
叶 θu(t ， x)/δx
= __
8.2 札 8.2.11]
ε (0 ，
T) , continuous in
and Berge's maximum
123] that the nonempty convex compact valued multifunction is measurable in t
ε (0 ，
T) and upper semicontinuous in x
ε ]Rn
Given any αε5 ， a certain estimate ofthe form (12.15) holds for G(. ,.) ~f F O (. ，.， α) F O (. ，.， α)
because 2γ。，
C F(. , .).
for arbitrarily fixed
He旧e 飞， 挝 a s wa 础s
0 三 t
<
r 三 T，
x
mentior
ε ]Rn
we could find at least one element
X(l) (.)ε XFO(. ，.，a)(t ， x). By (12.24) and (12.38) , it is then easy to see that the continuous function [0 , T] " ç 叶 u(ç ， x (1 )(ç)) is Lipschitz on every segment [E:, T -t:] (0
<ε <
T /2) with the derivative I
dx {l}
a川\
护， x(1 )(ç)) =去(己， z(1)(E)) 叫专(巳)去 (μ (1)(.;)))
三主任， x (1) (巳))+/(ç， x(咐 ， u(ι x(1) (ç almost everywhere in (0 , T). It is therefore decreasing on [0 , T]. Thus , according to (12.2日， if a function x(.) on [0 ，叫 is chosen so that x(.) I[明= x (1 )(.)I[伺皿d
x(.)I[o ,t] =x(川.) I[O ,t] for some X(2) (.)巳 Xu(t ， x ， u(t ， x) ， α) ， then x(. ) ε Xu(t ， x ， u(t , x) ， α). Moreover , u(r , x(r)) - u(t , x) initio凡 u =
u (t , x) is a
u(r ， x( 川 r)) - u(t , X(l)(t)) 三 O. Hence , by deι
s叩ersolution
of (12.9)- (1 2.10). 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 AC C Leb例， with R = 仰) the function
~12.3.
RELATIONS WITH SEMICLASSICAL SOLUTIONS
mentioned in Condition c). Let u
175
= U(t , X) be a minimax solution, differentiable at
some (t(O) , X(O)) E A C x IR n , of (12.9)-(12.10). Firstly, choose αE S and 0 三 s E Jæ so that 加 (t(O) ， X(O))!θx u
= u(t , x) is in Solu , for any 15 E (0 , T -
s. α. Since
t(O)) , there exists an
Xð(.) ε Xu(t(O) ， X(O) , u( t(时， z(0)) ， α)
such that
+ Ó, Xð(t(O) + 15)) -
u(t(O)
u(t(O) , x(O))::::
Let Y6 ~{ Xð(t(O)
+ 15) -
o.
(12.39)
rt(O)+ð .J 一
I
=
X(O)
旦旦 (r)dr.
Jt(O)
dt
Because t(O) E AC is a Lebesgue point of R = R(t) and the function family {Xð(.) }6
c
XF(t(的 ， X(O)) is uniformly bounded , one has
+∞ >C 主r
sup
sε (O ， T-t(O))
rt(O)+ð
子 I υ
R(r)(l
Jd~
+ I句(r)l)dr 主
supl|US|(1240)
ðε (O ，T-t(O)) υ
In accordance with (12 .40) and with the differentiability at (t(的 ， X(O)) ofu = u(t , x) , it follows from (12.39) that
江主 (t(O) ， X(O))
/θu + (νs 一 (t(O) ， X(O))) \'θx ,1/ ,
/n\
In\ , \
:::: (1
+ C) .15. 也
E吨句= O.
一、
(12 .4 1)
By the definitions (1臼2立.川14叫) of the mu 山 1 (札12.16 创) of 也 t he set Xu(t(O) , X(O) , u(t(O) , x(O)) ， α) ， we now see that 一
1
(~例， α)= 圳(守 (r)， a)dr 主~
L:)
rt(O)+ð .TU
f(r， xð(训叩创) ， α)巾，
hence , by (12.13) ,
(~Yð' -丁s 卫句 ~~ (t(O) (t川, X(O))) 叫 >.::.川 一;汇。)+6 ~r .-f(r f( 叫 ， 句叫咐叫 Xð 叫 s川仲(行例 T讨)， u(川叽去 (t(仰 0叩 δ Jt刮(仰例 飞 l o的) Ó"U'
This together with (1 2.37) and (1 2.4 1) implies
ZWM
12. MINIMAX SOLUTIONS
176
Analogously, we have
去 (t町的) + f(川(O) ， u(t时 provided that u = U(t ,X) is in SolL. The equation (12.9) must therefore be satisfied at (t(时， x(O)). Theorem 12.5 is completely proved.
口
312 .4. Invariance of definitions There is a certain indefiniteness in the definitions (12.14) of Fu
Fu(t , x , u ， α)
and F L = FL(t , x ， 叽 ß). For example , we have a large choice of C = C(t). However , this indefiniteness does not influence the basic
De直nitions
in 312.2. As will be
shown in the remainder of the chapter , instead of (12.14) one can , in fact , use any
Fu = Fu(t , x , u , a) [resp. FL = FL( t , x , u , ß)] in the following common families of multifunctions. Let P and Q be arbitrary nonempty sets. We consider any multifunctions
nT
X
IR.
X
P 3
(t ， x ， u ， α) 叶 Fu(t ， x ， u ， α)C
Rn ,
。T
X
IR.
X
Q3
(t ， x ， u ， ß) 叶 FL(t ， x ， u ， 的 C
IR. n
satisfying the general conditions below.
(i) The multifunctions Fu = Fu( t , x , u ， α)αnd F L = FL(t , x ， 也， β)αre nonempty convex compact valued, measurable in t and upper semicontinuous in x. Moreover,
Fu(t ， x ， u ， α ) UFL(t , x , u , ß) c F(t ， x) 乞! Bc(t)(叫.，1) for almost all t E (0 , T) and for all x E IR.n, t呻 ere c = c(t) in L 1 (0 , T) is independent of α， β ， u. (ii) For almost αII t E (0 , T) and for all x E IR.n.， αε P， βE Q, v E 也已。Fu(t ， x ， u ， α) = Fu(t , x , v ， α) ，
IR. one has
Q FL(t, z , uJ)=FL(t , ZJJ)·
U<.V
(iii) For almost all t ε (0， T) and for all 怡，包 ， p) ε IR.n
X
IR. x IR.n the
follo饥ng
inequalities hold: sup
min
aEP zEFu (t ，." 毡 ， a)
(z ， p) 三 f(t ， x ， u ， p)::;
i I![
__II}ax
_， (z ， 的
一 βEQzε FL(t ，." 饵，β)
(12 .42)
912 .4. INVARIANCE OF DEFINITIONS rt+6 , J_
inf
sup li rp. inf
aεÞ
叫 ere，
sup
\
dt'""'/
r t +c5 / dx
1
~ I
6 ',, 0 - .，(托 XL(阳向β)ð
‘
(二(叶 ， p )dr 主 f(t ， x ，
6、o .， (.)ε Xu(t冉也 ρ)ð Jt
inf lims叩 εQ
二 I
177
u , p) , (12.43)
飞
一 (r) ， p )dr 三 f(t ， x ， u ， p) ，
Jt
飞 dt'"' >r/
still as in (12.16) , we let
Xu(t ， x ， u ， α) 乞r XFu(. ，.，'Uρ)(t ， x) ，
XL(t ， x ， u ， β)qEfXFL(-v向β)(t ， x).
The family of all multifunctions Fu = Fu(t , x , u ,o:) (resp. FL = FL(巾 ， u , ß)) satisfying Conditions (i)-(iii) wi11 be denoted by Fu (f) (resp. FL (f)). Ifthe Hamil-
= f(t , x , u , p) satisfies
tonian f
a)叶， then Fu (f) 并 ø ， FL (f) 弄。. In fact , under
such hypotheses , we can choose any P , Q
c
IRn with the property that
{s. α:αε P， s 主 O}={s.ß:
ß
ε Q ， s 主 O}
= IR n
and then use (12.14) to define a concrete pair of multifunctions Fu = Fu(t , x , u ， α) and FL
= FL(t , x , u , ß). This pair would satisfy (i)-(iii). (See Part 3 of ~12.2 for 0
the case where P 乞r S, Q 乞r S.) Remark 1. By the Hahn-Banach theorem, it follows from (i) and (12.42) that
Fu(t , x , u ， α) n FL(t , x , u ， ß)
并。
(12.44)
for almost all t E (0 , T) and for allαε P， ß ε Q ， (x ， u) εIR n x R Remark 2. Condition (12.43) holds if Fu
= Fu(t , x , u ， α) and FL =
FL(t ， x ， u ， 的
satisfy the inequalities 唱
由εÞ
inf
唱
inf lims叩
βεQ
rt+6
二 l 6、o .，(托X~(t，.，) ð Jt
p li rp. inf
6、o
for almost all t
sup
~
;::: f(t , x ， 矶时，
rt+6
I
"， (.)ε XF(t 叫。 Jt
ε (0 ， T)
国~. . (z , p)dr Zε FU(T卢(T) 冉冉)
.m皿
Zε FL( T，.，(T) ，也，β)
and for all x
(12.45)
(z ， p)dr 三 f(t ， x ， u ， p)
巳 IRn ， u εIR， p 巳 IRn.
Remark 3. In the case where f = f(t , x , u , p) is a continuous function of its arguments , S由botin [124] and Adiatullina and S由botin [1] do not assume (1 2.43)
Fu(t , x , u ， α) ， FL FL( t , x , u , ß); they assume the upper semicontinuity in (t , x) of these multifunctions (instead of the measurability in t and upper
for Fu
12. MINIMAX SOLUTIONS
178
semicontinuity in x as we do) and assume the equalities in (12 .4 2). However , it pro叫 that
can be ε>
0,
(12 .4 3) holds then. In fact , under such assumptions , for all
(t ， x) ε il T ， U 仨lR， p ε lR n
there exists αε P so that . (z , p) ;三 f(t ， x ， u ， p) 一 ε/2.
f9.in
zε Pu(t ， x ，也 ， a)
Given any x(.) ε Xp(t ， 功， the multifunction (0 , T) :3 r 叶 Fu( r , x( 叶 ， u ， α) is upper semicontinuous. From this and the maximum theorem , it follows that the function (0 , T) :3 8
> 0 is
r 叶
min
,
zEPu(r ， x(r) ，也 ， a)
(z ,p) is lower semicontinuous. Hence , when
small enough , one has rt+ð
云 I
ò Jt
唱
__
，mi~ ，
.
rt+ð 萨『
(z ， p)dr 去 ~I
zEPu(r卢 (r) ，毡， α)'
__JIli~ ，
=
(z , p)-c/2I dr .
- .
.
J
. (z ， p) 一 ε/2 三 f(t ， x ， u ， p) 一 ε
min
zEPu(t 冉也 ， a)
ε X p ( t , x)
(i ndependently of the choice of x(.)
,
LzEPu(t ， x( 吟，也，α)
Ò Jt
-.
because the function family Xp( t , x)
is equicontinuous at t). In other words , the first inequality of (12 届) must be true Dual 吨uments
give the second one. Therefore , by Re mark 2 , (12 .4 3) holds.
Remark 4. Let Fu = Fu(t , x , u ， α) ， FL = FL(t , x , u , ß) be Carathéodoryin (t ， x) ε
il T [i.e. , measurable in t ε (0 ， T) and continuous in x εR叮 satisfying Condition (i). Given any x(.) ε Xp(t ， 叫， by [8 , Theorem 8.2.14] (cf. [29 , Theorem II I.1 5]) , the measurability in r of the multifunctions (O ， T) :3 r 叶 Fu(r， x(叶 ， u ， α) ，
(O , T) :3
r 叶 FL(r， x(r) ， u ， β)
(which follows from [8 , Theorem 8.2.8]) implies that of the functions
(O , T) :3 r 叶
mi~.
. (z ， 抖，
zEPu(r ， x(r) ， 也，由)'
(0 , T) :3 r 叶p1a;x:，
.. (z , p).
zEPdr ，x(r) ，也，β)
Therefore , the integrals in (12 .4 5) exist. Moreover , if (12 .4 2) and (12 .45) hold , then one has in fact the equalities in (12 .45) , because by Lemma 12.3 , (12 .4 2) clearly forces rt+ð
inf 二 I 。εp 八o x( 托 X F (阳)8λ
p li:rp. inf
__
，mi~.
. (z ， p)dr 二
zEPu(r，x(r) ，毡，由)
r t +ó 二 I sup _ min . (z , p)dr 6 、o x( 托 :X;'(t ，x) J Jt "'EPzEPu(r ,x(r) ,u ,a)
< liminf
inf
< liminf
inf
rt+ð
s、o x( 托 XF(t 卢)
二 I
J
Jt
f仆， x(叶 ， u ， p)dr =
f(t , x , u , p) ,
~12.5.
UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS
179
r t+ 6
inf lims叩
βEQ
6 、0-
sup
~
"'(.)EX:;"(t ,,,,)()
I
_.(z ， p}dr 主
max
1t
zEFdT ， "'(T) 冉，β)
r t +6
~ limsup
sup
二 I
> lims叩
sup
三 I
8、o "'(托 XF(t♂)<5 λ
l~
__
_.(z ， 的 dr
m 8.?C.
,
βEQzEFdT，"'(T) 冉的
rt+6 6 、o
"'(.)EXF(t 卢) U
for almost all t E (0 , T) and for all x E Rn , u For any Fu
f(r , x(r) , u , p)dr
= f(t , x , u , p)
Jt
ε R， p ε Rn.
= Fu(t , x , u ， α) in Fu (J), FL = FL(t , x , u , ß) in FL (J), we have:
Deflnition 4. The set Solu (Fu) of
s叩ersolutions
(relative to Fu) of the Cauchy
problem (12.9)-(12.10) is defined to be the set of all finite lower semicontinuous functions u = u(t , x) on
which satisfy (12.18) and the condition u
, Z)
?tu
nu
••,, 9"
A吐
( )
T Z T
飞
ε Rn.
u
咱EA
x
a a)
<一
for 创ly 0 三 t
e-
，，l 飞
E XU
、.，J
a(
，，.‘飞
‘P
L 也
-uz
帆 mh
sa p UE
nT
)
RU
Deflnition 5. The set SolL(FL) of subsolutions (relative to F L ) of the Cauchy problem (12.9)-(12.10) is defined to be the set of functions u
inf
0 三 t
max. _.[u(r , x(r)) - u(t , x)]
",(.) EXdt ，，，，，也 (t ，剖，β)
x
se皿icontinuous
~
(12 .47)
0
ε Rn.
In Section 5 we shall prove that , for any Fu FL
upper
= u(t , x) on nT which satisfy (12.20) and the condition βEQ
for any
all 鱼nite
Fu(t , x , u ， α) in
Fu (J) 皿d
= FL(t ， x ， 矶 ß) in FL (J), the intersection Solu(Fu) n SolL (FL) consists of a u-
nique function , which is independent of Fu = Fu(t , x ， 包， α) and FL = FL(t , x , u , ß). Therefore , in complete concord with the previous definitions , we may also give the following. Deflnition 6. A function u solt巾on
= u(t , x) in Solu(Fu) n SolL(FL) is called a minimax
(relative to Fu and F L ) of the Cauchy problem (12.9)-(12.10).
!ì 12.5. Uniqueness and existence of minimax solutions The aim of this section is to establish the following main theorem.
12. MINIMAX SOLUTIONS
180
Theorem 12.6. Let all
Cond山 ons
σ=σ (x)
be
coηtinuous
on
a )-e). Then there exists one unique
IR n α nd
f = f(t , x , u , p) satisfy
r旧时mαx solt巾 on
of the
Cα1叫 g
problem (12.9)-(12.10).
For the proof of Theorem 12.6 we need to make some preparations. Given any locally bounded function u = u( t , x) on fl T , we define: U 一 (t ， x)~'
_
liminf
u(r , y)
and
.l.,. , def u十 (t ， x)~'
lims叩
u(r， y)
OT3 竹，由)→衍，" )
。 T3(T， y) → (t ，.，)
for (t ， x) ε fl T . Obviously, u- = u 一位 ， x) [resp. u+ = u+(t , x)] is then the largest lower semicontinuous [resp. the smallest upper
semicor巾lUOUS]
function on fl T
which is at most [resp. at least] u = u(t , x). From now on , fix any Fu
:FL(f) and let c = Fu(t ， x ， 包， α) ，
c(吟 ，
Fu(t , x , u ， α) in :Fu (f) , FL
F L( t , x , u , ß) in
F = F(t , x) be as in Condition (i) relative to these Fu =
F L = FL(t , x , u , ß). For (儿 ， X*) ε fl T ， r ξ [0 ， T ], u 巳 IR ， αξ P， ß ε
Q we set D(t. , x. , r) ~f {x(r)
x(.) ε Xp(t. ， x. 汁，
Du(t. , X" u , r， α) 哇f{z(T):z(.)ξ Xu(t. ，
X"
u ， α)} ，
DL(t. ， x. ， u ， r， ß) 乞fh(T):z(.)ξ XL(t .， x扪 u ， ß)} ， and shall be concerned in the sequel with the following two functions:
百T 3 (t， x) 叶 ψ (l)(t ， X) 乞fmax{σ(ν) : y ξ D(t ， x ， T)} , 百T
3
(t ， x) ←+ψ(2)(t ， z)tf mia{σ (y) : y ε D(t， x , T)}.
(12 .48)
Lemma 12.7.ψ (1) =ψ (1 )(t ， x) is α supersolution (relative to Fu) of the Cauchy problem (12.9)-(12.10).
Moreover飞 both the functions 7þ(1)
ψ (l)(t ， 叫， ψ(2) 工
ψ (2)(t ， X) αre contin包。也s zn 百T 山th ψ (l)(T， x) 三 ψ (2)(T， x) 三 σ (x ).
Proof of Lemma 12.7. By the Remark in Part 2 0 of 312.2 , one can easily prove the upper semicontinuity of the multifunction fl T 3 (t , x) 叶 D(t ， x ， T) c IR n . 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
7þ (1)
= ψ(1) (t ， 叫， ψ(2) =
ψ (2)(t ， X) are in C(百T). All that is left to show is that ψ(1) =ψ(1) (t , x) satis鱼es
(12 .46) [with u replaced by ψ(1)]. To this e时， let 0 三 t
c D(t , x , T) , (12 .48)
~12.5.
UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS
181
implies ψ (1)(r， x(r)) 三 ψ (1)(t ， X)j i.e. , (12 .46) withψ(1) in place of u holds. Lemma
12.7 is
proved.
口
= u(t , x)
Lemma 12.8. Ifu
is α locally
bounded function on f! T satisfying (12.18)
and the condition
C，αuchy
哇
thenu- =u-(t , x) is problem (12.9)-(12.10).
(t ， x) ε f!T ， r ε [t ， T]，
9"
A
u( z)l <- plw
))
nud 、‘.， J
T
A 咱』
T Z
，，，‘、
u
a'ι
+e a
，，，‘、
pIu t s)
nR 1'
X U ( 4·
'''飞飞
for all ε>0 ， to Fu) of the
z )
ε
uε
sapp
αs叩ersolution ( 陀lαtive
Proof of Lemma 12.8. Let x εR饨， ε(1) >ε(2) > ... >ε (k) → O. Then there exists a sequence {( t(的 ， X(k))}t~ c 百T converging to (t , x) so that u 一 (T， x) = hliT U(t(h) ，川的). By (12ω) ， for any 直xedαε P one can find functions -今
xl
X(k)(.) ε Xu(t(的 ， x( 的 ， u(t(的 ， X(k)) + ε( 的， α)
with σ (x(k)(T)) 三 u(T， x(k) (T)) 三叫别的 ， x(k))+2ε( 的 ，
k
= 1, 2 ,. . .
(12.50)
Since ~ ~f {(T， 叫 ， (t(I) ， x(I)) ,... ， (t俐， z(h)) ， ...}is compact and since {X(k)(.)}t~ C XF(~) ，
we may (by the remark in Part 2 0 of 312.2) assume that x( 的(. )→ x(.) ε XF(T， x) in C([O ， 罚， IRn).
Letting k →+∞ in (12.50) , we get σ (x) 三 u-(T， x). Thus ,
u- = u- (t, x) satisfies (12.18) [with u replaced by u-j. Fix now
0 三 t
x
εRn ， αε P.
Then there exists a sequence
{(t(的，川的 )}t~ C [0 , r)
converging to (t , x) so that u- (t, x)
= . lim
h→+∞
x IRn
u(t俐，川的). We choose ε(1) >ε(2) >
... >ε(k) → o and find , by (12 .49) , functions x( 的(. )ε Xu(t(的，川的 ， u(t间，川的)+ ê(的， α) such that
u(r , x(k)(r)) 三 u(t(的 ， x(k))+2ε(的 ， Foreveryε> 0 , since . lim [u(t俐，川的 )+ε(k)j h →+∞
k
= 1, 2,. . .
(12.51)
= u- (t, x) , it follows from Condition
(ii) that x(k)(.) ε Xu(t(的，川的 ， u- (t， x) + ê ， α) when k is large enough. Therefore ,
182
12. MINIMAX SOLUTIONS
again by (i i) and the Remark in Part 2 of ~12.2 ， we may assume that x{ 的(. )• 0
x(.) εeQOXu(t ， z ， u-(t ， z)+ε ， α) = Xu(t ， x ， u- (t， x) ， α).
Letting k →+∞ ln
(12.51) , we get u-(r， x(r)) 三 u- (t， x). Hence , u- = u- (t, x) satis直es (12 .46) [with u- in place of u]; i.e. , by definition, it is a s叩ersolution (relative to Fu) of the
Cauchy problem (12.9)-(12.10). The proof is then Now let us define a function À =人 (t ， x) on
λ (t ， x) ~f inf{u(t ， x)
complete.
口
nT by the following formula:
吨， .)εSolu(Fu)}.
(12.52)
Since Du(t , x , u , T， α) C D(t , x , T) for all (t ， x) ε n T ， u ε IR.， αε P ， it is easily seen that u(t ， x) 三 ψ (2)(t ， X) on
nT
for all s叩ersolutions (relative to Fu) u = u(t , x)
of the Cauchy problem (12.9)-(12.10). Therefore， ψ (1)(t ， X) 三 λ (t ， x) 三 ψ (2)(t ， X) on
nT
[by Lemma 12.7 ， ψ(1)ψ (1)(t ， X) is a supersolution (relative to Fu) of
the Cauchy problem (12.9)-(12.10)]. This together with the continuity of 1/; (1) = ψ (1 )(t ， 功， ψ(2) =ψ (2)(t ， X) implies that
ψ (1 )(t ， x) 主入 +(t， x) 三 λ (t ， x) 主人 -(t， x) 主 ψ(2) (t , x)
on
nT .
(12.53)
Moreover, we have:
Lemma
12.9. 人 =λ (t ， x) is αl扣 ou
01 伪 t he Cauchy problem (12.9)-(12.10). λ 一=入 - (t,
x) is a supersolution (relative to Fu) ofthe Cauchy problem (12.9)-(12.10). Moreover , slnce λ=λ- (t , x) is a lower semicontinuous function equal to σ=σ(x) on {t = T , x ε IR.n} (cf. (12.53) and Lemma 12.7) , all th剖 is left to prove is that Proof of Lemma 12.9. By (12.52)-(12.53) we need only show that
(12 届)
with u replaced by
Fix now 0 三 t
<
λ-
holds.
r 三 T， x ε IR.n ， αε P and ε(1) >ε(2) > ... >ε( A:)→
O. Choose a sequence {(t(的，川的 )}t~ c [O , r) x IR. n converging to (t , x) so that λ (t ， x) = .limλ ( t{ 的，川的). Then by (12.52) , there exists a sequence of functions &---++0。
川的=川的 (t ， x) in Solu(Fu) with λ ( t{ 的， x{ A:))三 U{k)(t{ 时， x{ A:))三 λ (t(k) ， x(k)) + ε例
(12.54)
hence , by (12 .46) and (12.52) , there exist x{ A:)(.)ε Xu(t俐 ， x 俐，川的 ( t( 的， z(h)) ， α) such that λ 一 (r， x{ A:)(r)) 三 λ (r， x{k)(r)) 三川的 (r， x(k)(r)) 三 U{k)(t(的 ， X{k)) ，
k=1 , 2 ,...
(12.55)
~12.5.
Since
UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS
183
lim u( 的 (t( 的 ， X(k)) 二 λ 一 (t ， x) , we may assume , in the same way as at the 1C--t +O。
end of the proof of Lemma 12.8, that the sequence {x( 的(.)} t:;. converges to some x(.) εecoxu(t ， z ， λ 一 (t ，
+ ε ， α)
x)
= Xu(t ， x ， λ - (t， x) ， α). Letting k →+∞ ln
(12.54)-(12.55) , one gets λ -(r， x(r)) 三 liminfλ 一 (T， x(k)( r)) 三 λ -(t ， x)j '←~+o。
i.e. ， λλ- (t,
x)
satis鱼es
[withλ-
(12 .46)
in place of u]. Lemma 12.9 is proved.
口
For any 8ε [O ， T] and ß ε Q ， let us now consider a function 阳，β=ρ9，ß( 巾， ψ) ， which is given on
nT
x IR by:
阳，β (t ， x， ψ)tf sup{λ(8， y) : y ε DL(t， x ， ψ ， 8， ß)} 一 ψ(12.56) It is then easy from (12.53) that min{ψ(2)(8 ， y)
y ε D(t ， 矶的}一 ψ 三 ρ9，ß(t ， x ， ψ) 三 三 max{ψ(1) (8 ， ν) : y 巳 D(t ， x , 8)} 一 ψ ，
(12.57) and from Condition (ii) that the function 句，β (t ，
ρ9，β=ρ9，β (t ， x ， ψ)
P9，β=ρ9，ß(t ， 矶的，
x) defì.n ed on
nT
ψ.
Beside
we shall also consider another function
句，β=
is strictly decreasing in
by:
Ç9 ,ß(t , x)
坦finf{ψεIR:ρ9，ß(t ， x ， ψ) 三 O}.
In virtue of (12.57)-(12.58) and of the monotonicity
(12.58)
inψof P9 ，β= 阳，β (t ， x ， ψ) ， one
must have: min{ψ (2)(8 ， y)
νε D(t ， x ， 8)} ~ Ç9，β (t ， x) 三 max{ψ( 勺。， ν) :νε D(t ， x ， 8)}.
(12.59) Finally, let us investigate an auxiliary function ( Ç9 ,ß(t , X) ed(t , z)={F 1λ (t ， x)
(Note that , by (12.56) and (12.58) ,
阳，β= 阳，β( t , x)
by the formula:
if
(t ， x) ε[0 ， 8]
x IRn ,
if
(t ， x) ε [8 ， T]
x IRn.
(12.60)
ç9，ß(8 ， x) 三 inf{ψεIRλ (8 ， x) 一 ψ 三 O} 三
人 (8 ， x).)
The following lemma gives us an important property
ofμ9，β= 阳，β (t ，
x).
Lernrna
12.10.μ豆，β=μ豆;β (t ， x) is αs叩ersolution (relative to Fu)
01 the
Cauchy
problem (12.9)- (1 2.10).
Proof of Lernrna 12.10. The nonempty compact valued mu1tifunction ÜT
'3
(t , x) 叶 D(t ， 矶的 c
]R" is locally bounded. Therefore by (12.59) , taking account of the continuity of ψ(1) =ψ (l)(t ， 叫， ψ(2) =ψ (2)(t ， X) (Lemma 12.7) , we see that Ç8，β = Ç8 ,ß(t , X) , and hence，阳，β= 阳，β (t ， x) are also locally bounded. Moreover , it follows from (12.53) , (12.60) , and Lemma 12.7 thatμ8，ß(T， x) 三 σ (x). Thus , according to Lemma 12.8 , we have only to prove (12 .49) withμ8，βin place of u. To this end , let us fix any ε> 0 ， αε P， O 三 t
1:
8 三 t.
By (12.60) and Lemma 12.9, there exists
x(.) ε Xu(t ， x ， μ8，ß(t ， X) ， α )CXu(t ， x ， μ8，ß(t ， X) + ε ， α)
so that μ8，ß(r， x(r)) = λ (r， x(r)) 三 λ (t ， x) = μ8，ß(t ， X); (12 ω) [with u replaced by μ8，ßJ
is thus satisfied.
一 Case
2: r 三 8. By (12.60) , we have 阳，β (t ， x) = Ç8 ,ß(t , X) , hence by (12.58) and
by the monotonicity in ψof ρ8，β=ρ8，ß(t ， x ， ψ) ， we get ρ8，ß(t ， x ， μ8，ß(t ， X) +ε)
< o.
Because of (12.44) , there exists at least one x(.) ε Xu(t ， x ， μ8，ß(t ， X) +ε， α) 门 XL(t ， x ， μ8，ß(t ， X) + ε ， ß).
Of course , DL( r , x( r) ， 阳，β (t ， x)+ε ， 8 ， ß) C DL(t ， x ， 阳，β (t ， x)+ε ， 8 , ß) and therefore , by (12.56) , it may be seen that ρ8，ß(r， x(r) ， μ8，ß(t ， X) +ε) 三 ρ8，ß(t ， x ， μ8，ß(t ， X)+ê) <0 ,
This together with (12.58) and (12.60) implies 阳，β (r， x(r)) = μ8，ß(t ， X)
ç8 ,ß(r , x(r))
~
+ê; (12 .49) [with 阳，βin place of uJ is thus satisfied , too.
- Case 3: t
<
8
<
r. By Case 2, there exists X(l)(.) ε Xu(t ， x ， μ8，ß(t ， X)
+ ê ， α)
such that
μ8，ß(8 ， x(川。))三 μ8，ß(t， x) + ε.
( 12.61)
By Case 1, there exists x(2)(.) ε Xu(8 ， x( 川的， μ8，ß(8 ， x(1)(8)) ， α) such that μ8，ß(r， x(2)(r)) 三 μ8，ß(8 ， x(1)(8)).
(12.62)
~12.5.
Choose now
UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS
x(.) ε C([O ， T] ，lR n )
185
with
X(.)I[O ,9j = X(1)(.)I[O ,9j
and
X(.)I[9，到 =
X(2) (.)
1[9，町
By (12.61)-(12.62) and Condition (ii) , we then see that x(.) ε Xu(t ， x ， μ9，ß(t ， X) + ε ， α)
and that μ9，ß(r， x(r)) 三 μ9，ß(t， X) + ε; i. e. , (12 .49) is also satisfied with u ~fμ9，β This completes the
proof.
口
Lemma 12.1 1. λ+λ+ (t, x) is a subsolution (relative to Fd 01 the Cauchy
problem (12.9)-(12.10). Proof of Lemma 12.11. By (12.53) and Lemma bounded with À(T， x) 三 σ (x).
12.7 ，人 =λ 衍，
x) is locally
Let 0 主 t 三。三 T， x εIRn ， βε Q ， ε> O.
We consider the auxiliary function
阳，β= 阳，β (t ， x)
given in (12.60).
Accord-
ing to Lemma 12.10 ，时，β=μ9，β (t ， x) is a supersolution (relative to Fu) of the Cauchy problem (12.9)- (1 2.10); hence by (12.52) and (12.60) , one has μ9，ß(t ， x) 三阳，β (t ， x)
λ (t ， x) 三
= ~9，ß(t ， X). Thus , by (12.58) and the monotonicity in ψof
P9，β=ρ9，ß(t ， x ， ψ) ， one also has ρ(t ， x ， À(t ， x)
- é)
> O.
SO , it follows from (12.56)
that there exists x(.) ε XL(t ， x ， λ (t ， x) 一 ε ， β) such that λ (B ， x(B)) > λ (t ， x) 一 ε. In so doing , we are led to another property of λ=
!l!f.
supλ (B ， x(B))
βεQ "， (.)ε XL(t ，，，，，λ(t ，，，，)-e ，β)
for all
ε>
0,
(t ， x) ε f!T ， B ε [t ， T].
人 (t ， x) ，
that is ,
λ (t ， x)] ~三
ε
Therefore , by arguments dual to the ones in
the proof of Lemma 12.8 , we conclude that À+ =λ+ (t， x) is a subsolution (relative to FL) of the Cauchy problem
(12.9)-(12.10).
口
The next lemma will play a crucial role in proving our main theorem. Adiatullina and
Subbotir内 method
of proof ([1] and [124]) , based rather directly on Gronwall's
inequality, seems to break down in the case of time-measurable Hamiltonians. Our road to this result here is devious (by some "perturbation technique" on sets of Lebesgue measure 0) , and proceeds via an implicit version of Gronwall's inequality. Lemma 12.12. Let u
= 百(t ， x) and 旦=旦(t ，
x) be super- and sub-solutions (relative to Fu and FL ) 01 the Cauchy problem (12.9)-(12.10) , respectively. Then u(t ， x) 主 旦(t ， x) on f! T.
12. MINIMAX SOLUTIONS
186
Proof of Lemma 12.12. Assume the contrary, that 百(t(O) ， x(O)) < 坐( t(的， x(O))
for some (t(O) ， x(O)) ξ 否T.
(12.63)
Let D ~f Br C Jæn be such that x(t) ε D for all
x(.) ε XF(t(O) ， x(O)) , t ε [0 ， T]; and A = A(t) ~f AD.E(t) where
AD.E
=
AD.E(t)
is
the function (in L 1 (0 , T)) existing in Condition b) corresponding to the bounded sets D and E qfh( 斗，以2)] with
一∞ < m(l) 乞r
,II).ÌI!... _u(t， x)<m(2)~f ，
(t ，，，， )E[O.到 xD
"
，m.a~ _!!.(t ， x)<+∞.
(t."')E[O.叫 xD一
,
(12.64)
There exists a set A of measure 0 such that (12.13) and (12.43) hold for all U
ε Jæ， p ξ Jæn ，
Xε
the function Jæ 3
Jæn ，
ε AC.
t
u 叶 f(t ，
x , u , p) is decreasing for all
that (12.11) is true whenever u For every
E
We may assume that AC C Leb(c)
> 0, the set
ε E ， p ε S， x ， y ξ D ，
n Leb(A) , that
p ε Jæn ， X ε Jæn ，
t
t
ξ AC ，
and
ε AC.
A can be contained in an open set
U~
C (0 , T) with
产(1 +巾(帧 <ε
(12.65)
Let A(~)(t) 乞f 2A(t) +εand let γ(~) =γ(~)(t) be the function given by M=
rIJIL ε8
(
'A 咽
T
+T ) C
，，，‘、
b
IJ 、、
γ
e) (,,
)
AEU
+
plv
if t
ε U~ ，
if t
ε U~
(12.66)
.
For any t ε[t(O) , T] we consider the following (compact) set:
N(~)(样{(言(.) ，主(.) )εx川盯 (0)) x |主(t)
一 五(t )|2
4 >)(
XF(t(O) , x(O))
T) dT
优叼叫 P叫(L:) 介扣)片 yAM(υ例叫川 e功纠句川)川忻炯) 仕例 (T)d 讨训 沛 )d古T
百(t旧坷言页印(归 ， 问仰 x t均归)
Of course , N(ε)(t(O)) 并回. Setting t(~) ~f sup{ tε [t(O) ， T]
N(~)(t) :兴的， we
see that N(~)(t(~)) 兴。 because XF(t(O) , x(O)) is compact and 旦=旦(t ， x) (resp. u= 百(t ， x))
is upper (resp.
Let us prove that t(ε)
lower) 町nicontinuous.
= T To this e时， find (町，主(.) )ε N(功(户)) and assume again the contrary, that t(~) < T , We need only consider the following two cases: ,
312.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS Case 1: t{ ε)ε U~. Take
J > 0 small eno鸣h so that [t{~) ， t{ε) + J)
tion , for any fixed αε P， βε Q there exist XL( t{ 吟，主(O) ， ~(O) ， ß) with
x {1)(.)
c
187
U~. By defini
E Xu(t{吟，王(的，甜的， α) ，主(1) (.)ε
U( t{~) + J，王(1 )(t{~) + J)) 三百( t{ 吟，王(O)) <旦( t{ 吟，主(O) )三旦(t{~) + J，主(1) (t{~) 十 J)) ，
(12.68) where 王(O) ~f x(t(勺，主(0) 智主(t(~)) ， 否(0) ~f u(t( 吟，裂的) <坐(0)tfHt(e) ，王(0)) .
(12.69)
Let r
(王(t) if 0
:s t 三 t(e)(
xC 2 )(t) ~f ~川，、 l x \l l(t) if t\~1
主(t) if 0 三 t 三 t( 吟，
and 主(到 (t) 生f ~川，、
三 t 主 T，
l 至1 盯 (t)
if tl~1
三 t 三 T.
(12.70) Then (王(2)(.) ， ;!y)(.)) ε XF(t(O) ， x(O)) XXF(t(的 ， x(O)). In addition , since (x(.) ，;r.(.)) ε N(~)(t(勺， it follows from (12.67) and (1 2.70) that |主(2)(t(~)
+ J)
王(2) (t(~)
+ JW r t (< )十 ð
= I ;r.(t{~))
王(户)W+ 儿
rt(ε叶
，
三 jγ(吟。 )dt. exp(
I
r t(叶
、
2(住 但川主;i: ω (2)σ 伺 2)旧 (μt←) 一 主封俨俨 l归z勾 川I (例 t均 t) ，川 ;r.(主;r.( 纠~川 t均) 一 -x( 王拦沪户(但问川 2功)
rtμ(< )、
I
rt μ(ε )+ð
A(吟。 )dt) +
I
r t (<) 十S\
三 I I
γ(ε)(t)dt + I
γ(εl( t)dt
rt(<)+ð
,
三 /γ(e)(t)dt. exp(
，
r t (叫 H
I .exp( I
r t(叶 H
I
ì(e)(t)dt 、
A(叫 (t )dt )
、
A(ε )(t)dt)
(12.71) (Here , the first inequality follows also from the
Bunhiakovskii而 Schwartz
inequality
and the fact that Idx(t)jdtl 三 c(t)(l + Ix(t) 1) 三 (1 + r)c(t) a.e. in (0 , T) for all x(.) ε XF(t(的， x(O)).) Thus , since again (言(.)，;r.(.)) E N{e)(t(勺， we see , by (12.67) (12.7月， that ø 手 N(~) (t(e) +
J) :7 (王{2)(.) ，主(2)(.)) ， a contrad时lOn.
一 Case 2: t(~) ε U'}. Let us use (12.69) and p(O) ~f ;r.(0) 一言(0)
one has m = S U P ðE(O ,T-t(<)) 节
sup
ðE(O ,T-t(.)) ", (.)EX F (t( 叫 ，，，， (0))
Jt(<)
.
Since t(e) E Leb(c) ,
r t (ε)+8
E jih}C(t)dt<+∞， and hence
rt(<) 十 8 I -1_
~ I 0
1
rt(<)+ð
I
亏 (t) dt 三
sup
~
I UιðE(O ， T-t(叶 )ο
I
c(t) (1 + Ix(t)l)dt
Jt(叫
"'(.)EXF(t( 时，.， (0))
三 (l+r)m.
(12.72)
12. MINIMAX SOLUTIONS
188
Moreover , because t(e) E Leb(A) , there exists 0(1) > 0 so that , for 811 0 ε(0 ，以 1) ]， we have
叫;J:762A叫> ex川(e))一 ε) ,
l. e.
,
优P(lfsAe(t)dt)>exp(26A(t(e)阳 + 20 . A(t(e))
(12.73)
On the other hand , the ∞mp配tness of Xp(t(的 ， x(O)) implies the existence of 0(2) with the property that Ix(t) -
x(t(功 )I<~三
whenever x(.) ε Xp(t(O) ， x(O)) ， It - t(e)1 t( e) ,
one can find αε P， βε Q ， and 1
"，(， )EXF(t(叶，王{的) 0
1
~
sup
rt(<) +cS
I
0(3)
< 0(2). Fin81ly, since (12 .4 3) holds at t 苦 > 0 so that
min ，_，
(z ， p(O))dt 主 f(t(吟，至(时，百例 ， p(O) )一 ε/8 ，
max
(z , p(O))dt ~ f(t(吟，主(O) ，y'(O) ， p(O)) + ε/8
zEPu(阳(趴阴阳)
Jt(<) rt(<)+ cS
I
"， (.)EXF(t(<) ，主(0)) ð Jt(<)
(12.74)
,_, _,
zε Pdt ,'" (吟，:!!.(O) ，β)
一
(12.75) as 0
< 0 < 0(3).
If we choose now
0
def min {0(1) ， 以2) ， 6(S) ， T-t(ε)} > 0 , then by definition ,
there exist 否(1) (.)ε Xu(t(吟，至(0) ，而0) ， α) ， ~(1)(.)ε XL(t(吟， 40) ，且(时 ， ß) such that (12.68) is fulfilled. Let (沪l(.)， ~(2) (.))ε Xp(t(O) ， x(O)) x Xp(t(时 ， x(O)) be 韶山 (12.70). It follows from (12.75) that rt(<)+ cS ,Æ.;;{ 2)
土 I
o
Jt(<)
\
rt(<) +cS I
oI
~
Jt(<)
‘
( 旦二 (t) ， p(O) ) dt 主 f(t(e) ， 裂的，而0) ， p(O)) 一 ε/8 ， dt
dx(2)
,-".
I
( 二一 (t) ， p(O) )dt 三 f(t(e) ，主(0) ,y.(0) , p(O)) + ε/8. \ dt ,-,,,I
By (12.76) , setting p(t) 乞f ~(2)(t) ~ o
rt(<)+ cS ,.1_
I
Jt(<)
(12.76)
\
X< 2)(t)
for t 巳 [0 ， T ], we have
‘
( 哩。 )， p(O) )dt 三 f(t(吟，主(0) ，型的 ， p(O)) _ f(t(吟，拦的，甜的 ， p(O)) + ε/4. 飞 α/;
(12.77) Combining (12.72) , (12.74) , and (12.77) gives rt(<)+ cS ,.1_
i L<)
‘
惯例 ， p( t) ì 们 f(t(e) ，~
312.5. UNIQUENESS AND EXISTENCE OF MINIMAX SOLUTIONS
189
and therefore ,
二 (Ip(t(~) + oW -lp(t(~)W) 2ð
三 f(t(吟，主(时，而时 ， p(O)) _ f(t(吟，否(0) ，否(0) , p(O)) + ε/2 三 A( t(~))
by
(12.1 月，
. Ip(O) 1 . 1 ;r.(0) 一五(0)1 +ε/2
= A(t(~))
.lp(t(~)W + ε/2
(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 that (否( .), ;r.( .))ε N(~)(t(~)) ， we see that Ip(t(~) + oW 三 [1 + 2ðA (t(~))].lp(t(~)) 户 +ε6
三到约剑州 i怡 Ip(t( 川 W 纠 以州 (t( t川(μ 州 μ 例叫 户e叫勺)与) 州
三~ (L: LU:: 汇汇 二〉 〉γ钊(例叫叩 )V e吟纠)川(州 帧 t均M 灿 )ddt 优呻叫 p叫(汇汇汇 Lρ.) 川〉}〉〉AM州川 (μω叫阳 e吟纠训)川怡(t +fue)(帧 4)+\(~) (t)dt 旺P(fsA(e)(t)a) (12.78) Again , by (12.67)-(12.70) and (12.78) , we get ø 手 N(~) (t(~) + 的 3 (言(2}(.) ，主(2)(.)) ， a contradiction. The contradictions we have got in both Cases 1 and 2 show that
t~
= T , i.e. ,
that for any ε> 0 there exists at least one (王(~)(.)，主(圳.) )ε N(卅 T). It then follows from (12.67) that ~T
|主(~) (T) 一王伊)(TW 三 l
~T
γ (~)(t)dt. exp(
I
Since XF(t(的，川的) is compact , one can find a sequence
A(功。 )dt). E( 1) > ε(2)
(12.79) > ... >ε (k) → o
such that 王(川的)(.)→言(. )，主(~(川)(.)→主(.) in C( [O ，町，]Rn). Letting k →+∞ ln (12.79) with
E(k)
in place of ε ， we obtain 1;r.(T) - x(T)1 = 0 (cf. (12.65)-(12.66)).
Hence , a passage to the limit as k
→+∞ in
the inequality
u(T， 王。仙)) (T)) 三 u( t(O) , x(O)) < 旦(t(O) ， x(O)) 三坐(Td(e(的 )(T))
[notice that
u 二百(t ， x)
(resp. 旦=旦(t ，
x )) is lower (resp. upper) semicontin∞us]
glves σ (x) 三百(T， x) 三 u(t(的， z(0))< 旦(t(O) ， x(O)) 三旦(T， x) 三 σ (x) ,
where x 哇f ~(T) = x(T). This is a contradiction , which says that (12.63) must be wrong. Lemma 12.12 is so completely
proved.
口
We are now in a position to prove Theorem 12.6. Proof of Theorem 12.6. Since the Hamiltonian f = f(t , x , u , p) satisfies
a) 叫，
Part 3 of Section g12.2 shows that :Fu (J) 手 ø ， :FL (J) 手。. Let Fu = Fu(t ， x ， u ， α) 0
and FL
FL(t , x , U , ß) be in :Fu (J) and
be the function defined on
nT
玩 (J)，
respectively, and λ=λ (t ， x)
by (12.52). In virtue of Lemmas 12.9 and 12.11 ,
λ=λ (t ， x) 三 λ - (t， x) is in Solu (Fu) ，人 +λ+(t ， x) is in SoldFL). Therefore ,
Lemma 12.12 together with (12.53) implies λ (t ， x) 三 λ + (t， x) 三 λ (t ， x).
Thusλ=λ (t ， x) belongs to Solu(Fu )nSolL(FL). Moreover , if FbO) = FbO)(t , x , u , 0:) and Ft) = Ft) (巾 ， u， ß) are also in :Fu (J) and :FL (J), respectively, and λ(0) = λ (O)(t， x) belongs to Solu (FbO)) n Sold可以 then Lemma 12.12 , applied to Fu = FWJ , tw)and Fio)=Fi川机 ， u ， 的， givesλ (t ， x) 主 λ (O)(t ， x) on 百T. Another application of Lemma 12.12 to F~) = FbO )( 式川 α) and FL = FL(t ， 川 β) gives 人 (O)(t ， x) 三 λ (t ， x) , hence λ (O)(t ， x) 三人 (t ， x). The proof is complete.
口
Remark. Condition e) in Part 10 of Section g12.2 (positive homogeneity of the Hamiltonian f
f(t , x , u , p) with respect to p) can be omitted slightly by the
method proposed in [1 ], [124] as follows. Let -, .ç
f(t ， x ， 川 PJ)=‘ Here , y E JR. and q ε
JR.
(
i
Iqlf(t , x , u - y , p/lq l) for q 并 0 ， liIl! [q. f( 巾 ， u - y , p/q)] for q = 0
\. q 、。
are two supplementary variables , and the limit is assumed
to exist. It should be noted that the auxiliary Hamiltonian f = f(t , x ， ν ，凯 p ， q) is positively homogeneous with respect to the variable (p , q) E JR. n+l. We also assume that the Hamiltonian f = f(t ， 川 ， p) satisfies conditions under which the positively homogeneous auxili町y Hamiltonian f = f(t , x , y , u , p , q) has the properties indicat ed in Part 10 of Section g12.2. Then it can be shown that the minimax solution of the auxiliary Cauchy problem 一 /θEθ石、，
一 + f(t ， 咐，否一一) =0 飞 'δz'θyJ
in
{O
u(T , x , y) = σ (x)+y on {t=T , xE JR. n , yE JR.}
~12.6.
THE CASE OF MONOTONE SYSTEMS
satisfies the identity u:(阳， ν) 三百(巾， 0) mir山nax
can be considered as the
+ y.
191
The function u = u(t , x) 哲 u:( 机， 0)
solution of (12.9)-(12.10) with non-homogeneous
Hamiltonian f = f(t , x , u , p). For this
c回e，
an existence and uniqueness theorem
(like Theorem 12.6) can also be established. !ì 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)classical solutions. The detailed proofs can be found in Thai Son ,
N.且，
Liem ,
N. 且，
and
Van , T.D. [133]. Our method here is to combine that of the previous four sections with some other techniques of monotonicity.
,. . . , u m ) , V = (Vl , . . . ， V m ) ε for all i = 1,..., m. Moreover , if u 主 vand
In the sequel , m is a positive integer. Given u = ]R隅， Uj
=
we shall write u
Vj 岛r
ψ=ψ (t ， x)
~ V
some index
if Ui
~ Vi
1 三 j 三 m， then
(Ul
we write U 2U An IRm-valued function
= (ψ1 (t ， 叫，... ， ψ隅。 ， x )) is called lower (resp. upper) semicontinuous if
its components 叭=叭。 ， x)
(1 主 i 三 m)
are alllower (resp. upper) semicontinuous.
Similarly, it is differentiable if its components are all differentiable. 1 0 Formulation of the Cauchy problem
Let us consider the Cauchy problem for a weakly-coupled system of the form θUi/ &t
+ fï(t , x , U ， θUi/θx)=O u(T, x)
= σ (x)
Assume that the terminal data
in
on
σ=σ (x)
!l T
(i=1 ,..., m) ,
{t
= T, x
=
(σ1(叫，... ， σm ( x))
ε ]Rn}.
(1 2.80) (12.81)
is a gi ven vector
function continuous on ]Rn , and that the Hamiltonians fï = fï(t , x , u , p) depend on (t ， x ， u ， p) ε !lT X
]Rm
X
]Rn with the following properties.
(a) Carathéodory's Conditions:
(a1) For almost all (in the sense of Lebesgue measure) t
]Rn
X
]Rm
X
]Rn 3 (x , u ， p)
叶 fï(t ， x ， u ， p)
are continuous
ε (0 ， T) ,
(1 三 z 三 m).
the functions
192
αre
12. MINIMAX SOLUTIONS
(a2) For αny (X ， U ， p) εIR.n x IRm x S, the functions (0 , T) 3 t 叶 Iï (t ， x ， u ， p) measurable (1 三 t 三 m).
(b) For any bounded sets D C Jæn and E C
Jæ'飞 there
exists a function An.E
AD.E(t) in L 1 (0 , T) with IIï (t , x , u , p) -1ï (t ， x' ， u ， p)1 三 AD.E(t) .
Ix - x'l
for almost all t ε (0 ， T) αnd for all x , x' ε D ， u ξ E ， p ξS， i ε{1 ，... , m}.
(c) There
ex时s a function R = R(t) in L 1 (0 , T) such that
sup {If;(t , x , u , p) -lï(t , x , u , q)l- R(t) . (1 for almost all t ε (0 ， T) αnd for all (x , u) ε Jæn
(d) Monotonicity Condition (cf. αnd
[72 ， 但，
+ Ix l) . Ip - ql : X
Jæm,
p , q ε B} 三 0
ε{1 ，...
, m}.
(A.1) & (A.3)]): For almost all
t 巳 (O ， T)
for all jε{1 ，... , m} , (x ， p) ε JænxS， U ， V ξ Jæm ， ifvj-uj =_早早(闪一问)主 1
<..，气 m
!i (t , x ， 凯的主 !i (t ， x ， V ， p).
0 , then
(e) The functions Jæn 3 p 叶 Iï (t ， x ， u ， p) are positively homogeneous; i.e. ,
J; (t , x , U, s . p) = s. J; (t , x , u , p) for almost all t ε (0 ， T) αnd for αII (x , u ， p) ε Jæn
X
ì:/ s 主 0
Jæ m
X
S (i = 1,..., m).
Note that the monotonicity condition here implies: ( d1)
Quasi-monotonic句 Cond巾on
(cf. [72 , Definition 2.2 , Lemma 4.8]): For
almost all t ε (0 ， T) αnd for all j ε{1 ，...， m} , (x ， p) ε Jæn
U 兰 v，
then
X
S , u ， v ξ Jæm ， if
!i C机 ， u， p) 主 !i(巾 ， v , p).
(d2) The functions Jæ 3 Ui 叶 lï(t ， x ， u 1，...， u← 1 ，问，问+1，...， Um ， p) αre decreasing for almost all t 巳 (0 ， T) and for αII (U 1,. . . ，问 -1 ， U;+ 1,..., U m ) εRm-1 ， (x ， p) ε Jæn X
S (i = 1,. . . , m).
2 0 Differential inclusions for supersolutions and subsolutions First , for (t ， x) ε nT ，
U ε ]Rm ， α ， ß ε S， 1 三 i 三 m ， let
F(t ， x) 乞r
V2R(t) . (1 + Ix l) . B ,
(Fu);(t ， 叭叭 α) 乞r {z ξ F(t ， x)
(z ， α) ~三 Iï (t， x , U ， α)} ，
(FL)i(t ， x ， U ， ß) 乞f {z ε F(t ， x) : (z ， 的三 β (t， x , u , ß)}.
(12.82)
912.6. THE CASE OF MONOTONE SYSTEMS
Further , for any IR"'-valued function c.p on
= ψ (t ， x)
=
193
(c.p l(t ， X) ，...，c.p"， 衍 ，
x )) defined
nT , we set (Fu) f(机，町， α) ~f (Fu)队 zJO)(tJ) ， α) ，
(12.83)
(FL)f( t ， 吼叫 ， ß) 哩 (FL)i(t ， 民币。 (t， x) ， 的， where ui 巳1R1 and <þ(i)(t , x) ~f (ψ l(t ， X) ，...， 白一 l(t ， X) ， U们白+1 (t ， x) ，... ，c.p"， (t ， x)) ξIR m .
Then it can be seen that the multifunctions
(1 三 t 三 m)
。T '3 (t ， X) 叶 F(t ， x) C
nT nT are
noner即ty
the other
(12.84)
IR n ,
x IR'" x S
'3 (t ， x ， u ， α) 叶 (FuMt ， x ， u ， α)C
Rn ,
x IR'" x S
'3 (t ， x ， u ， ß) 叶 (FL);(t ， x ， U ， ß) C
IR n
convex compact valued , measurable in t and continuous in (x , u). On
ha时， it
(Fu);(t ， 叭叭 α)
followsfrom (d1)-(d2) that (FuMt ， x ， u ， α) C (FU)i(t ， x ， v ， α)
if u ~ v ,
(FL)i(t , x , U, ß)
if u 主 υ ，
c
:J
(FL)i(t , x , V , ß)
(FU)i(t ， x ， Ul ，...， Ui-l ， Ui+ ε ， Ui+l ，.
(FL)i(t , x , u , ß) c (FLMt , x , Ub"'
. . ， u隅， α)
， 问 -l ， Ui-e ， Ui+ l ，...， Um ， ß)
Therefore , by [133 , Proposition 4.1] , if c.p
= c.p (t , x)
tinuous , one can check the measurability in
t 缸ld
ifε> 0 , jf
e>O.
is lower (resp. upper) semicon the upper semicontinuity in x
of the multifunctions (Fu) f(.,., Ui ， α) (resp. (FL)f(.,., Ui ， β)) for 1 三 t 主 m ，向巳 R1 ， α ， ß ε S. And hence , according to the Remark of ~12.2-Part 2 0 , the sets
(Xu) f(儿 ， X. ， 间， α) ~f X(Fu )i(. ，.，u; 叫 (t问 x*) (resp.
(XL) f(儿， hJhmqEfx(FL)了( .川剧，β) (t们 x*))
(12.85)
are all nonempty and compact for (t* ， x*) ε n T . So , we can make the followi吨: Definition 7. A
s 叩 ersolution
JR"'-valued function U = u(t , x)
of Problem (12.80)-( 12.81) is a lower semicontinuous
= (Ul (t , x) , . . . , um (t, x))
on
nT
which satisfies the
condition I!l.a]C sup ,, ___p ,!l}Ín , • JUi(T， X(T))-Ui(t ， X)] 三 0 aES"( .)E(Xu n(t ，." 叫 (t ，.，)， a)
1 至 i<二'"
12. MINIMAX SOLUTIONS
194
for all
0 三 t
and also the condition
u(T， x) 三 σ (x)
Definition 8. A subsolution of Problem ]R rn-valued
Vx
ξ ]Rn.
(12 朋)-( 12.81)
(12.86) is an upper semicontinuous
function u = u(t , x) = (Ul (t , x) , . . . , urn(t , x)) on
nT
which satisfies the
condition
.lI!a x . . _JUi(T , X(T)) - Ui(t ， X)] 主 0
min inf
l::;i::;rn βES :c (.)E(XL) 了 ( t ，:c，问 (t ，:c) ，β)
for all
0 三 t
<
T 三 T，
x 巴]R飞 and also the condition
u(T , x)
三 σ (x)
The sets of all supersolutions and
(12.87)
Vx E ]R n.
s由solutions
of (12.80)-(12.81) will be denoted
by Solu and SolL , respectively. Definition 9. An
]R rn-valued
function u = u( t , x) in Solu n SolL is called a minimax
solution of the Cauchy problem (12.80)-(12.81). 3 Invariance of definitions 0
Just as the invariance of definitions that has already been discussed in 312 .4 for the case of single equations , instead of (12.82) , we can now use
a叮 t叩le
Fu
=
((Fuh ,..., (Fu)rn) [resp. FL = ((Fdl"'" (FL)rn)] in the following common families of tuples of multifunctions. Let P and Q be arbitrary nonempty sets. We consider any multifunctions
satisfyi吨 the
。T
x
]R m X
。T
x
]R m
P :7
x Q :7
(t ， x ， u ， α) 叶 (FU)i(t ， x ， U ， α) C ]R飞 (t ， x ， u ， ß) 叶 (FL);(t ， x ， u ， ß) C ]Rn
general conditions below (1 三 Z 三 m).
(FU)i(t , x , u , a) and ( 且 )í = (FL) í(t , x , u , ßl are nonempty convex compact valued, measurable in t and 叩'per semicontinuous in
i) The multifunctions (FU)i (x , u). Moreover ,
(FU)í(t ， x ， U ， α) U (FL)í(t , x , U, ß) C F(t ， x) 乞f B巾 )(1+ 1"'1)
~12.6.
THE CASE OF MONOTONE SYSTEMS
for almost α llt ε(0 ， T)αndfor αll(x ， u) ε ]Rn
X
195
c = c(t) iη L 1 (O ， T) is
]Rm ，叫 ere
independent of i ， α ， β.
ii) For almost αII t E (0 , T) and for αllx ε ]Rn ， αε P， βι Q ， u (Vll""V m ) ε ]R7n
= (Ull""u m ), v
one has (FuMt ， x ， u ， α) C (FuMt ， x ， v ， α)
if u 主 υ ，
(FL) i(t , x , u , ß)
if u 主 v ，
:J
(FL )ï( t , x , v , ß)
， u 711o ' α)
可 ε> 0 ,
(FL )ï( t , x , u , ß) C (FL Mt ， x ， Ul ，...， Ui- 1， 问- ê , Ui +1,"', u m , ß)
if ε> O.
(FuMt ， x ， 包， α) C (FU)i(t ， X ， Ul ，"'， U←1，问 +ε ， uí+ 1，
iii) For
αlmost
=
all t E (0 , T)
αnd
...
αII (x ， u ， p) ε ]Rn X ]R m X ]R n
for
the following
inequalities hold:
sup
min
,
0
由 εÞ zE(Fu) (t''''. 1L .a)
(z , p) :S h(t ， 民 u ， p) :S i~f
max
_0
一 βEQ zE(Fd ， (t. "，.毡，β)
ιhuιμ
PAPA
.， b''ι
>一<
( ( ( ( ) ) )) TT TT zz uu pp Ju
fj'
， α
i?0
唱
、飞 3，，，、飞飞
uu 制
叩A ，。
/，{飞飞，，，，飞飞
‘ tiτ O
eoeo t tJu-JUJu-lu ++ fIhfjt z-ae--uz-t
XEL zz
SE
aaμ
εE
γAeAU
4』 PEM
门Q
川、四、
uι 、
pp
fdpd JP n)U) SO-nmiEouo
(z , p) ,
where
(Xu);(t， x ， u， α) ~f X (Fu) ,(.,..1L
The family of all t 叩les Fu
=
((Fuh ,..., (Fu)m) (resp. FL
=
((FLh ,..., (FL)711o))
satisfying Conditions i)-iii) will be denoted by :Fu (f) (resp. :FL(J)). If the Hamiltonians 力 = h(t , x , u , p) (1 三 z 三 m) satisfy (a)-(e) , then :Fu (f) 并 ø ， :FL(J) 并自
In fact , under such hypotheses , we can choose any P , Q
c
]R n
with the property
that {s. α:αξ P， s 主 O}
= {s. ß
: βE
Q, s
主 O} =]R n
and then use (1 2.82) to define a concrete pair of tuples Fu = (( Fu h , . . . , (Fu )7110) and F L = ((FLh γ.
•
, (FL)m)' This pair would satisfy i)-iii).
From now on , let us fix any (Fu , FL) ε :Fu (f) x :FL (f) and continue using the notations (12.83)-(12.85). It is [133 , Proposition 4.1] (see also the Remark of S12.2-Part 2 0 ) that makes the following de缸山ions 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 ]R m_ valued functions u = u(t , x) = (Ul(t , X) ,..., u m (t , x)) on 百T which satisfy (12.86) and the condition
1P.a;.x sup ,,____,??-Ï n
"Jui(r， x(r))-ui(t ， X)] 三 0
l~i 主 m a EP"'(.)E(Xu )7 (t 冉钮 i (t ，吟， a)
for any
0 三 t
Definition 1 1. The set SolL (FL) of s包bsolt巾ons (relative to F L ) of the Cauchy problem (12.80)- (1 2.81) is defined to be the set of all upper semicontinuous ]R m_ valued functions u
= u(t , x) = (Ul(t , X) ,... , um(t , x)) on nT which satisfy (12.87)
and the condition min inf ,,_,._ ， ~ax
, , .Jui(r, x(r)) - Ui(t , X)]
1 主 i~m βε Q "， (.)ε (Xdr (t，吼叫 (t ，吟，β)
for any 0 :::; t
三 T，
x E ]R n.
It can be proved that , for any (Fu , FL ) Solu(Fu) 门 SolL (FL)
~ 0
ε Fu (f)
x FL (f), the intersection
consists of a unique (Rm-valued) function , which is independent
of (Fu , FL)' Therefore , in complete concord with the previous definitions , we may also give the following. Definition 12. An ]R m-valued function
u 二 u(t ，
x) in Solu(Fu) nSolL( F L) is called
a minimax solution (relative to Fu and FL) ofthe Cauchy problem (12.80)-(12.81). 4 0 Main results In this part , we formulate the main results on the existence and uniqueness of mllllmax solutions for monotone systems
of 如st-order
nonlinear partial differen-
tial equations with time-measurable Hamiltonians and their relations with the (semi)classical solutions (see Definition 2 in Chapter 11) Theorem 12.13. Assume (a)卡). Then
(i)
e 阳'Y
global semiclassical solution u
gradient mappings ]Rn 3 all t
ε (0 ， T)
u(t , x) of (1 2.80)-( 12.81) so that the
x 叶 δUi(t ， X)/θx (1 三 t 三 m)α陀 continuous for αlmost
must also be a minimax solution of the same problem
~12.6.
THE CASE OF MONOTONE SYSTEMS
197
exists α null set
(ii) there
ωhere α certαin
A C (0 , T) such thαt， at eαch point (t , x) ι ((O ， T) \ A) x lR n minimax solution of (12.80)-(12.81) is differentiable , the eq包αtions
(12.80) must be satisfied.
Theorern 12.14. Let the Hamiltoniαns fï
=
in巾α1
data
σ=σ (x)
be continuous on IR n and the
fï(t , x ， 矶时， 1 三 t 三 m ， satisfy all Conditions (a)-(e). Then
there exists one unique
mir山nax
solution of the Cauchy problem (12.80)-(12.81).
Exarnple. Now we present an example of monotone systems (12.80) of first-order nonlinear
pa此 ial
differential equations with time-measurable Hami1t onians. (The
case of single equations is obtained when m
= 1.) It
is left to the reader to check
all the hypotheses (a)-( e) for those systems. Let G i = Gi(t , x ， λ) be of class C 1 on ÜT x lR such that lR õ:) λ 叶 Gi(t ， x ， À) is increasing for all t ε (0 ， T) , x ε IRn (1 三 i 三 m). Fur ther , let hi bounded , decreasing, and globally Lipschitz continuous function of
=
hi(S) be a
s 巳lR
for any
def
i ε {1 ，...， m} (we can take , for instance , hi(s) ~的·町ctan( s) with a certain negative constant
Vi
for each index i).
Define 力 (t ， x ,
for (t , x , u ， p) ε ÜT Cij
= Cij(t , X)
X
IR m x ]R n and i ε {1 ，...， m}. Herel
= l(t)
is in L 1 (0 , T) and
are given C 叮linctions satisfying on 百T the conditions
2二 Cij(t ， X) ~ 0 ,
and , Cij(t， X) 三 o if
笋j
Chapter 13 Mishmash In this final chapter we present various things , more or less "for completeness."
313. 1. Hopf's formulas and construction of global solutions via characteristics First , consider the Cauchy problem θujδt
+ f(θujθx) = 0 u(O , x) = <þ (x)
Ð def ~. {t > 0 , x
in on
{t = 0, x
ε ]Rn} ，
(13.1) (13 功
巳]Rn} ，
under the following two hypotheses.
(F .I) The initial function <þ = <þ( x) is of class strictly ω nvex
on
]R n
with
lim f(p)jlpl =
Ipl →+∞
C Oα nd
the
H，α7旧ltoniαnf=f(p)is
+∞.
(F.II) For every bounded subset V of Ð , there exists a positive number N(V) so that ~~..， {<þ( ω )+t. f* ((x 一 ω )jt)}
|叫三 N(V)
ωhenever
(t , x)
ε V， IνI
> N(V). Here , 户
=
< <þ( ν )+t. f* ((x-y)jt)
f* (z)
denotes the Fenchel
conj叼αte
function of f = f (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
furr由hes
only its restriction to a
neighborhood of {t = O} in which it is smooth if the Hamiltonian and initial data are supposed to be sufficiently smooth. In fact , as we have seen in Chapter 1, if
f = f(p) and <þ = <þ (x) are of class C 2 on the solution by
fitti吨 characteristics
characteristics form a hypersurface x spa臼 such
that x(O， y)
ν ， v(O ， y)
]R n ,
Cauchy's classical theory furnishes
of (13.1) to the initial data
<þ 二 <þ( x ).
These
= x(t , y) , v = v(t , y) in txv-space above tx= <þ (y). For sufficiently small t > 0 , this
313. 1. CONSTRUCTION OF GLOBAL SOLUTIONS VIA CHARACTERISTICS hypers旧face
199
has a simple projection upon tx-space (the tx-projections x = x (t , y)
of the characteristics issuing from different points
νat
t = 0 do not intersect). Then
one can eliminate y from ( 13.3) and the hypersurface appears in the forrn v
= u( t , x)
and smooth. This is the classical solution.
with
u 工 u( t , x)
however , the hypersurface has , in general , multiple projection other words , v for all t
= u( t , x)
single-valued
Outside of the classical tx-domain , upo日 tx-space
or , in
becomes multivalued. This multi-valued function exists
> O. It is natural to ask [64] whether our single-valued global solution
of (13.1)-(13.2) can be deterrnined directly from the several values of
υin
(13.3)
above the same point (t , x). In the first case ofHopf's formulas , f = f(p) convex (or
co日cave) ,
cþ = CÞ(x) 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 depending on the For
co盯exity
definitene风 let
character of f
us suppose f
u(t , x) ,
= f(p).
= f(p)
is convex. Recall that the characteris
tic 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-J acobi equations) , the char acteristic strips are defined as
(… (t ， y)=y+t.f 仲 (y) ) v = v(t ， ν) = cþ(ν )+t.(( 旷 (y) ， f'(矿 (ν)) )一 f( 矿(的)) , p = p(t , y) = 矿 (ν) . Here and subsequently, we write
x， ν， p,
f'
df __ -'.LI
(1 3 .4)
dcþ
= .:;and cþ' = 一'- as horizontal vectors dp ---T dx
in ]æ. n; and (.,.) denotes the (E山1i dean) inner product in ]æ. n. As was mentioned in the above , in general , we could not completely eliminate y from (1 3 .4) to get a single-valued function v
u( t , x) (i.e. , in this forrn ,
u
吨，
x) might become
multi-valued). A町way， if we try to do 队 we may rewrite v = 咐， ν) in (13 .4)出
v = ((t , x(t , y) , y) where
((t , x , y) ~f 圳的十 t. 尸 ((x - y)/t)
(13.5)
200
13. MISHMASH
Then the hypersurface (13 叫 projects upon the entire half {t ::::: O} oftx-space , and the global solution (9却) of (13.1)-(13 勾 equals ， at each point (t , x) of {t
> O} , the
smal1est v-coordinate of the hypersurface in the following sense. Theorem 13. 1. Let
4>
=φ (x) be of class C 1 in
convex and of class C 1 in Iæn with
,
Iæ n , and let f = f (p) be strictly
lim f(p)/Ipl = +∞ . Assume (F.II). Then the
Ipl →+∞
白alue u(t(的 ， x(O)) at each (t(O) , x(O)) , t(O)
> 0 , ofthe global solution u =
by Hopf' s formula (9.29) of the
problem (13.1 )-( 13.2) can be
the smallest values of v(t(的， ν)
Cαuchy
=
u(t , x) given
determined αs
((t(O) , x(t(O) ， 时， ν); the minimum bei叼 taken over
all y such that the characteristic c旷ves {(t , x(t , y))
t 主 O} starting from
(0 , y)
meet each other α t (t(O) , x(O)). Proof.
By hypotheses ,
f*
f* (z) is the same as the Legendre conjugate of
f = f(p) (see Lemma 9.13 and [117 , Theorem 26.6]); moreover , both f' and one-to-o日e mappings from
from (13.5) that (=
f*'
Iæ onto itself with n
((t ， x ， ν)
f*'
are
= (1 ')-1. Therefore , it follows
is differentiable in y with
θ〈
石。， x， ν)= 矿(y)
-
f*'(斗旦)
It has been shown in the proof of Theorem 9.12 that (9.29)
( 13.6) deterrr山es
a global
solution u = u(t , x) of (13.1)-(13.2) , and that the infimum
叫t叭 z(0))=AC(t 叭 z 叭 y) is attained at some y(O) ι Iæ n .
(13.7)
Clearly, y(O) must be a stationary point of the
function ((t(的，川， .); i.e. ,
手 (t(的， x(O) ， y(O)) uy
= 0
Thus (13.6) implies 扩 (y(O)) = fμ ((x(O) - y(O))/t(O)). 8ince fμ= (1 ')-1 , we have ",
(0) _ ,, (0)
气而~ = f'(4)'( ν(0))) ，盹
x(O) = y(O)
+ t(O忡忡， (ν(0))) =
x(t(O) , Y叫
for every such stationary point y(O). 80 , the characteristic curve {( t , x( t , y(O))) t 主 O} ， which is starting from (O , y(O)) , must p出s through (t(O) ，川的). Because the
minimum in (13.7) is not affected if it is taken not for all y but only for those stationary y , the proof is then
complete.
口
313. 1. CONSTRUCTION OF GLOBAL SOLUTIONS VIA CHARACTERISTICS
We now turn to second case of Hopf's formulas , <þ
= φ (x)
201
convex (or concave) ,
f = f(t , p) largely arbitrary, and continue investigating the construction of global solutions via characteristics. Consider the Cauchy problem θujθt
+ f(t ， θujθx) 口 o u(O , x)
= <þ (X)
in
{t
on
Ð=
{t
> 0,
= 0, X ξ
x εR吁(1 3.8)
Jæn} ，
(13.9)
as in Chapter 9 , under the following standing hypotheses:
(E .I) The Hamiltonian f = f(t , p) is continuous in {(t , p) Jæn} for some closed set G
c
Jæ of Lebesgue measure
(0 ，+∞ ) there corresponds α function 9N
sup If(t ， p)1 三 9N(t)
= 9N(t)
ε(0 ， +∞)\ G ， p ε
O. Moreover , to each
Nε
in L在(Jæ) such that
for α lmost all
t ε(0 ，+∞) .
Ipl 三 N
(E.II) For every bounded subset V of Ð , there
exists α positive
number N(V) so
that
(川们p) -l t 川 dT
-l
t
f(r , q)dr}
Ipl > N(V).
It was shown in the proof of Theorem 9.1 that , if <þ = <þ (x) is a finite convex function on Jæ n , 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 constn时 ed by means of characteristics. For this , suppose the Hamiltonian and initial data are of class C 2 . The characteristic strips for (1 3.8)-(13.9) i 口 this case are defined (cf (1.1 0)-( 1.1 1)) as
z 二叫 t ， y) = ν + l t 号叫忡， {t(
θf
υ=υ (t ， y) = 的)+儿 l (功， (y) ， 再 (r ，<þ '(y))) -
飞
f(r , <þ'(y)) )d飞
(13.10)
p = p(t , y) = 功， (ν) . 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 m町 rewriteυ=υ (t ， ν) in
(13.10) as U 二伽
From now on , let us assume 却 2 3.5
and
药 2 5.1可]，
to be convex on ]Rn. U sing [117 ,
φ=φ (x)
v= ψ (t ，
The 凹 or 四 em
(13.11) 臼
we can then further rewrite
x(t , y) ， 矿 (ν))
where 叫t ， 叫
(13.12)
It will be proved in Theorem 13.3 that the hypersurface (13.10) projects upon the entire half {t 主 O} of tx-space , and the global solution (9.7) of (13.8)-(13.9) equals , at each point (t , x) of
{t 主 O} ，
the largest v-coordinate of the hypersurface.
First , let
il(t ， x) 彗f supψ(t， x ， 矿 (ν)) ，
(13.13)
yEIRn
u(t， z)45supψ(巾 ， p)
(13.14)
pεdom .
for(t ， x) ε Ð.
We have:
Proposition 13.2. Proof. Let E
il(t ， x) 三 u(t ， x)
def ~' {矿 (y)
: yε
]Rn}.
on Ð. It follows from Corollary 26 .4 .1 in [117] that
ri (dom 旷) C E C domφ飞
(13.15)
where ri (dom 4>*) is the relative interior of dom 旷. Therefore , (13.13)-(13.14) yield 仰 ， x) 三 u(t ， x).
and choose
We now prove u(t ， x) 三仰， x). Fix any q εri (domφ*) ， ε> 0 ,
q εdom
4>*
such that
u( t , x) On setting p(叫乞r (1 - l/m)q
三 ψ (t ， x ， q) + ε.
+ (l/m泪，
we see , by (13.15) and [117 , Theorem
6.1 ], that p(m) εri (dom 旷) C E for m = 1, 2, 3,... Since the Fenchel conjugate 伊 =
4>*(p)
Ís a lower semicontinωus proper convex function , [117 , Corollary 7.5.1]
shows that 旷 (q) = lim 旷 (p(m)). Hence , it follows from (13.12) that m---+c白
+
ε
h7ε
阳+
01J 川飞
)ttεPA +zl
目
、叶 J
飞乙
4问」
ψz ET P‘
'忖峰，，，飞、lJ
、PU-uε
/1
ZJ
#川区中
E
Z
、、 ，，，
ATb
Ymspmu 川 <一一一<一一一
u
l
叫
~13. 1.
Because
CONSTRUCTION OF GLOBAL SOLUTIONS VIA CHARACTERISTICS
ε>
0 is arbitrarily chosen , the proof is then complete.
203 口
Remark. Let 4> =φ(x) be convex and of class C 1 on ]R n. Assume (E .I )-(E.II). It has been proved in Chapter 9 that (9.7) then coincides with (13.14) , and that the supremum is always attained (at some point
p εdom 4>*).
The situation becomes
different for the supremum in (13.13). We show this by the following example. Let n
def ~.
1,
叫 i:一向 and
p> 1, pε[0 ， 1] ，
p
创J(;11，
Then
*(p) =
< 0,
Z 三 0，
> O.
x
pε[0 ， 1 ]，
{士 P
p~[O ， l ]，
and 位 (t ， x) =s叩 {x. e宙一旷 (el')
- tf(e ll )}.
F 三。
It is easily seen that the
value 位 (1 ，一 2)
= sup {-2e ll } = 0 can not
be attained in
F 三。
(13.13) at any point y ε ]Rl. In the remainder of this section , we make the following assumption: (E.V) For each
(t ， x) ε Ð，
the
s叩 remum
in (13.13) is attained at some point
yεRπ.
Remark. Let 4> =仰) be convex and of class C 1 on ]R n. Assume (E .I )-(E.II). Then (E. V) is fulfilled , for example , if dom 旷 is an open set (cf. (13.15) and the Remark above). In
par 叫.
convex function , i.e. when it is strictly convex with . lirp. [4>(入 x)/ λ]= +∞ for all Z
ε ]Rn\{O}.
See [117 , Theorems 26.5-26.6].
λ→+∞
= f(t , p) and 4> = 4> (x) be of class C 2 on {t ~ 0 , p ε ]Rn} r叩ectively， such that 4>" 乞r (θ2 4>/(θ的θXj ))ω=1 ，2 ，...，n 切sαJ灿 ωαν伊s pos必iti肘
Theorem 13.3. Let f
αnd
]Rn ,
deβηite. Assume (E. I1)αnd
(E. V). Then the value u(t{的， x{O)) α t each (t{的， x{O)) ε
13. MISHMASH
204
Ð
0/ the global solution u
=
U(t , x) given
bν Hopf's
/ormula (9.7)
0/ the
Cα也chy
problem (13.8)-(13.9) can be determined as the largest values 0/ u(t(0) ， ν)=ψ (t(O) ， x(t(的， ν) ，矿 (ν) );
the m α xz阳 阳 m ~1包L川mb 川 阮ez饥 叼 n gt归 α ken
0
t 主 0 叶} st切 αr付ting from 仰 ( 0 ， νω) me倪et each other at (t(O) , x(O)).
Proof.
<Þ"
Since
is positive-definite , <þ
<þ( x) is convex on Jæ n.
Notice that
(E .I) is satisfied trivial1y. From Proposition 13.2 and its Remark , we also see that u(t ， x) 三岳 (t ， x)
on Ð. Moreover , it has been known (c f. (13.11)-(13.12)) that
仰，巾)) =仰)
+ (州
hence ,
r t θf
一 ψ( 巾，矿 (ν ))=(x-y- I 一 (7，矿 (ν ) )d7 )<Þ" ( ν). 句 A 句
(13.16)
In view of (E. V) , the supremum
u(t(O) , x(O)) = 岳 (t(的， z(0))=supψ (t(时 ， x(的，矿 (ν))
(13.17)
掣 E lR n
is attained at some ν(0)ε Jæn ， which must be a stationary point of 叫 t(0) ，川的，扩(.)) . 飞fVe
then have
主ψ(t(O) , x(O) , <Þ' (y)) I
θνly=y(O)
= 0
Because <þ" (y(O)) is positive-defini旬， it follows from (13.16) that
x(O) 一 ν(0) 一 fZM(U(0)))dT 口 0; therefore , (13.10) implies x(t(O) , y(O)) = x(O). Notice that the maximum in (13.17) is not affected if it is taken not for al1 y but only for those stationary y. This 口
comp1etes the proof.
Remark. To our know1edge , Hopf [64] was one of the first to ask whether the globa1 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 cou1d be determined direct1y from the severa1 values of v in
(13码 and
(13.10) , respective1y. The answer
~13.2.
SMOOTHNESS OF GLOBAL SOLUTIONS
205
(essentially Theorem 13.1) was in the former case considered by Hopf [64] himself while he had no answer in the latter.
S13.2. Smoothness of global solutions Asw 晌 'e have 时 e
seen (仙 岛 f orexamp 卢 le ， by 咀 T Theo 阳 回 e ，倪 or 四 r err
classical solutions to the Cauchy problem for nonlinear pa町xtial exist
∞ 0nl恃 y
diffe 町 'ren 时 1尬t 创 削ial
equations
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
regt血xity
condition is relaxed significantly. (A minimax
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 如mulas. Consider the Cauchy problem (13.8)-(13.9) , = (x) being a convex function on R.... 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. Re call that the definition
L(tJ)qf{pεR'"ψ (t， x ， p) = u(t , x)} C dom 扩 determines a nonempty-valued upper semicontinuous multifunction L = L( t , x) of
(t , x)
ε Ð. Here ， ψ=ψ (t ， x ， p)
is defined in (13.12). We now prove:
Theorem 13.4. Let = ( x) be α convex function on R.... Assume (E. I)-( E. II). Then the global solution u
=
u(t , x) given by Hopf' s formula (9.7) of the Cauchy
problem (1 3.8)-( 13.9) is contin包ously differentiable in αn 叩En SEt U C DGtf {(t ， x) ε Ð
t fj. G} if and only if L( t , x) is a singleton for αII (t ， x) ε Ui
the set G being as in (E .I).
Proof. By the proofofTheorem 9.1 , ifu = u(t , x) is differentiable at (拟的， x(o)) ε
ÐG , then (9.14)-(9.15) give (for 1 主 t 三 n)
p(t叭 x(O)) = max{Pi aXi
: p E
L(t叭 x(O))} = min{阳:
pE
L(t叭 x(O)) }
13. MISHMASH
206
So , L( t(O) , X(O)) is precisely the singleton {θu(t(时 ， X(O))/θx}. (Consequently, (9.13) together with (9.15) implies that (13.8) must be satisfied at (t(O) , x(O)).) Conversely, suppose now that L( t , x) is a singleton for a l1 (t , x) in an open set
•
U C Ð G . Being single-valued , the mapping U 3 (t , x) L( t , x) is continuous. Hence , according to the maximum theorem [8 , Theorem 1. 4.16] , (9.13)-(9.14) show that the derivatives θu/挠， δu/θXl ，...， ðu/ θx n exist and are conti且uous in U (see also [128 , Corol1ary 2.2]); i.e. , the global solution u = u(t , x) is continuously differentiable in
U.
口
Remark. A special case of Theorem 13 .4, when n equals
1 ，功=功 (x)
is global1 y
Lipschitz continuous , was considered by Kruzhkov , S.N. and Petrosyan , N.S. [94 , Lemr丑臼 1
and 3]. There , instead of the multi-valued function L
used the single-valued p-
=
L(t ， 叫，
they
= p- (t, x) 乞finf L(t , x)andp+ =p+ (t， x) 哲 sup L(t , x) ,
and established some interesting properties of these single-valued functions.
In
particular , we have: Proposition 13.5. [94 , Lemma 5] For n
globally Lipschitz continωus， f
= 1,
s叩'pose
<þ
= <þ( x) is convex and
f(p) is of class C 1 on R
(0 ，∞) x lR, p(O) ε L(t(的 ， x(O)) , C ~f {(t , x)
x
Let (t(O) ， x(O)) ξ
= x(O) + (t - t(O)) f' (p(O)) , 0 三 t 三
t(O)}. Then p(O) ε L(t ， x) for αII (t , x) εC. F~旷ther， 矿 L( t(O) , x(O)) is a singleton , so is L(t , x) for
(t ， x) εC.
Corollary 13.6.
solution u
Under the hypotheses of Proposition 13.5 ,
s叩pose
the global
= u(t , x) ~f m~2' {px - <þ*(p) - tf(p)} of (13.8)-(13.9) is diJfere时iable at pEIR
(t(O) , x(O)). Then it is also diJfere时zα ble at (t ， x) εC for αllt ε (0 ， t(O)). Proof.
Since u
u(t , x) is differentiable at (t(的 ， x(O)) ， the proof of Theorem
13 .4 shows that L( t(O) , x(O)) is a si吨leton. By Proposition 13.5 , L(t , x) is also a SI吨leton
for a l1 (t , x) E C. Therefore , by the proofofTheorem 9.1 and its Remark ,
the partial derivatives of the solution exist at (t , x) εC for all t 巳 (0 ， t(O)). Notice that u = 州 ， x)
= ffié皿 {px
- <þ *(p) - tf(p)} is convex , because it is the maximum
pcu思
of a family of affine functions.
Hence , according to [117 , Theorem 25.2 ], it is
differentiable at the above-mentioned points (t , x).
口
Example 1. Investigate the smoothness of the global solution defined by (9.7) of the problem δu
_1 δu 、
( :,- )
战飞 δxJ
=
0 in
{t
> 0,
x ξ lR}，
~13.2.
SMOOTHNESS OF GLOBAL SOLUTIONS
u(O , x) = x 2f2 where
ε lR}，
{t = 0 , x
on
r 0,
207
p 三 0，
f(p) 营 { p ,
O
p> 1. Note that <þ
=
x2
/2
is convex , but not globally Lipschitz continuousj and we can
not use Hop f' s forrnula (9.3) for n
=
1. In this c回e ， 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(t ， x) 乞fm阻 {px - p2/2 - tf(p)}. p t:皿
More clearly, we have
u(t , x) = max {~~{px - p2 /2} , l!l;~，{px - p2/2 - tp} , m~{px - p2/2 - t}} , - _. 主o
pε[0 ， 1]'-
,-
or
p~l
x2
if (t , x) ε Ð1'
2
if (t , x) ε Ð2'
。
u(t , x) = ~ (x _t)2
if
2
x2
一-
2
(t ， x) ε Ð3 ，
if (t , x) ε Ð4 .
-t
Here ,
Ð1 ~主! {t 主 0， x 三 O} ， Ð2 彗 {t~max{x ， x2/2} ， x~O} ，
Ð3 哲 {max{0， 2(x - 1)} 三 t 三 x ， 0 三 Z 主 2} ， Ð4 雪 {t 主 0， x ε 1R}\ (Ð1 U Ð2 U Ð3). This solution is continuously differentiable in {t "si吨ularity"
x ε 1R}\ (C) ， where the
curve (C) is given by 问 9
I/-2 z2
x ~ 2, 、、，，，
飞
、
唱'-占
一-
』《 tt
AEU
2'''
rt
( C)
1<x
This fact can be foreseen by noticing that L(t , x) and L(t , x)
> 0,
= {x -t , x}
ift
= 2(x
< 2. {O , x} if t
泸 /2， x 主 2 ，
-1) , 1 三 Z 主 2， while L( t , x) is a singleton
13. MISHMASH
208
if (t , X)
1.
(C).
(Direct computation shows that L(t , x)
{x} for (t ， x) ε Ðl，
L(t , x) = {O} for (t ， x) ε Ð2 \ (C) ， L(t , x) = {x - t} for (t ， x) εÐs\ (C) ， and L(t , x) = {x} for (t ， x) ε Ð4 \(C).)
Example 2. We consider the Cauchy problem
去-
[1
+ (去)2l1/2=0in{t>0， zd}， _2
u(O ， x) = 亏
on
{t=O , x ζIR}
A global solution of this problem is: U
===u(tJ)tfm肌 {px 一之 + t(l + l)附} p t:皿
L
By computing and relying on Theorem 13 .4 we recognize that continuously differentiable in {t > 0 , x
巳 IR}\ {(t ， O)
:
characteristics , we see that when t > 1, characteristic the two curves {(t , x(t , 1))
t 主 O}
t 三 1}. cu凹es
u
u(t , x) is
Using the method of intersect. Concretely,
and {(t , x(t , 2)) : t 主 O} ， where
x(t， 归一 7:」 y' l+ν4
starting from (0 , 1) and (0 , 2) , respectively, meet each other at the point
(一ιMJL) 2V2 - J5' 2V2 - J5 Nevertheless , the differentiability of the solution is not broken down in some neighborhood of this point.
1313.3. 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 闷，
~13.3.
RELATIONSHIP BETWEEN MINIMAX AND VISCOSITY SOLUTIONS
209
[10]-[20] , [28] , [35]-[39 ], [47]-[50 ], [67]-[72] , [79] , [99]-[101] , [122]-[123] , [131] , and the references
therei叫
These
contributions make great progress in non1inear 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 prin ciple which distinguishes it from other defi.n itions 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 , a l1 of wruch are used in work with nonsmooth functions. For equations with continuous HamiltonÌans , 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 definÌtion of viscosity solutions. We now examine the relationship between minimax and viscosity solutions in the case of equations with time-measurable HamiltonÌ ans. 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 ψ=ψ(ç) be afinite哨ùued function of ç near a point ç(O)ε ]Rm and let dIRm ， γε ]Rm.
Re call
th剖 the
upper and lower Dini semiderivatives of ψ=ψ( Ç)
at the point ç(O) in the direction e are defi.ned as:
可 ψ(ç<0) )哩 i!lf sup {[ψ( ç<0) +缸)一 ψ(çυO<ð<~
le-el<e
(13.18)
θJψ(ç<0) )乞f sup ~ ~rf_ {[ψ(ç<0) +缸)-ψ(ç<0))1/ <5}. ~>Oυ 气。气
le-el<~
We introduce the following notation: A:ψ(ç(O)) 乞f i!lf sup {[ψ(ç)ψ(ç(O)) _ (-y, ç _ ~ .>U O< I( _(<0) I<~
ç< O))l/lç _ ç(O)I} ,
AJψ(ç<0)) ~f sup_.. i~L . {[ψ(ç) 一附(0)) 一衍， ç _ ç(O))l/lç _ 7 e> 00 < |f-d(0)| < e
D+ψ( ç<0)) ~f {γεIRm:A:ψ(ç<0))
D- ψ( ç<0) )乞f{γε ]Rm
::;
O} ,
ß:;ψ(ç<0) )主 O}.
ç< O)I} ,
(13.19)
(13 却)
Themappi鸣]Rm '3 e 叶。fψ(ç(O) )ε[ ∞，∞1 is of course positive-homogeneous ,
i.e. ， δ;二 ψ (Ç 0). For e 二 0， the relation δJψ 任何))ε{O ，∞}
13. MISHMASH
210
holds. It also follows directly from (13.18) that this mapping is upper semicon tinuous. Analogous properties are e时 oyed by the mapping
]R"' '3 e 叶 θJψ( ç- (O) );
it is lower semicontinuous and satisfies the relations θlij 1/;(巳 (0) )巳 {O ，∞}皿d θ二1/;(ç- (0)) =λ·δ'; 1/; (çtO)) (人> 0).
The sets D+ ψ( 己 (0)) and D 一 ψ( ç- (O)) are called the superdifferential and the subdifferential , respectively, of 1/; = 1/;(.;) at the point ç- (O). The elements of these sets are called supergradients and subgradients , respectively. These sets are also called the upper and lower Dini semidifferentials. If ψ=ψ( 己) is differentiable at ç-(时， then D+ ψ( ç- (O)) = D 一 ψ 任何)) = {θψ( ç- (O))/θ.;}; in this case ,
ψ( 己 )=ψ 任何))
+ (θψ(己 (0)) /θç-， ç- _ ç- (O)) + 0( Ç-
ç- (O)) ,
where o( ç- _ ç- (O)) / Iç- _ ç- (O) 1• Oas ç- •ç- (O). According to (13.19)-(13.20) ,
1/;( ç-)三 ψ( ç- (O))
+ (r,ç- _ ç- (O)) 十 0( Ç-_ç- (0))
for any s叩ergradientγε D+ ψ( 己 (0)). A s由gradientγεD ψ( çt 0 )) also satisfies a one-sided but opposite estimate: 1/;(刮去 ψ(çtO))
+ (γ ，ç- _ çtO)) + 0( Ç- _ ç- (O)).
We can verify that D+ ψ( ç- (O)) = {γE ]R"'θfψ(çtO) )三 (γ ， ε) Ve E ]R"'}, D 一 ψ(çtO)) = {γε ]R"'θJψ( ç- (O)) 三 (γ ， ε) Veζ ]æ."' }.
(13.21)
In fact , let γE D+ ψ( ç- (O)) ， e ε ]R"'\{O}. Consider the function ç-己 (8， e) 哩f
ç- (O)
+ 8e for 8 > 0 and eε ]R"'.
出 le
Of ∞旧se ， 381e 1/2 主 I Ç-( 8, e) _ çtO) 1 =二 81el
2:
81ε1/2
- el 三 lel/2. Consequently, (13.18)-(13.20) imply 0 主 A:ψ( ç- (O))
2:
i~f sup
ψ(Ç- (8， e)) 一 ψ(çtO) )一 (γ ，Ç- (8， e)-çtO))
;:;õ 0~8~ ，
1';( 8, e) - çtO) 1
lë-el<'
主 inf sup {2[ψ任何)+缸)一 ψ(çtO) )一竹，缸)l! (381εI)} .>u 0<6< , lë-el<'
zih「1[1号f s叩 {[ψ( ç-(O) + 8e) Ù
~/υ O<ð< ，
lë-el<'
出 ;|e|1 附ψ川一川，
ψ(Ç-(O))l/J} 一(r， e) 1
~13.3.
RELATIONSHIP BETWEEN MINIMAX AND VISCOSITY SOLUTIONS
211
from which we get θfψ(ç(O) )三 (γ， ε) Ve E ]R1叫 {O}. This still ho1ds even for e = 0 , since otherwise ， θJψ(ç(O) )手 0 ， one would have i~ sup {[ψ(ç(O)
+ Je) 一 ψ(ç(O)) ]j J}
=得 ψ(己 (0)) =∞;
二>UO<ó<~
lel<~
hence , inf sup {[ψ(ç(O) .>U O<Ó<~
+ Je) 一 ψ(ç(O))l/(Jlel)}
=∞
。
It follows that 。三 Atψ 任何) )主 inf sup {[1þ (ç(O) 十 Je) 一 ψ(己 (0)) 一竹， Je) 1/ (Jlel)} ~.>U O<Ó<~
。
主 i~ sup {[ψ( 己 (0)
+ Je) 一 ψ( çCO)) ]j (Jlel)) 一 |γ|
~.>U O<Ó<~
。
=∞
|γ1= ∞，
a contradiction. In the converse directio矶 suppose a: 1þ( 己 (0) )三 (γ， ε) Veε ]Rm. U日i吨 (13.19) ，
o as k →∞，
such that
_ ({, çC k) _ çCO))l/1 俨) _ ç(O)I} = .6，.~1þ (ç(的).
(13.22)
we can choose a sequence {ç( 的 h 此{[州的)一州的)
c ]Rm , 0 并
lç(k) _ ç(O)1
•
We may assume without 10ss of generality that e(k) 彗( ç(k)
已 (O))/Iç(k) _ ç(O)1
converges to some e ε ]Rm (Iel = 1) 出 k →∞. Since ç(k) =己 (0) +别的 ε( 的， with
J(k) 哇f lç(k)
巳(0)1 → 0， (13.22) together with (13.18) imp1ies
A:ψ(ç(O) )口.lim {[ψ 任何)
+ J(k) e(k))
ψ( 巳 (0) )
竹 ， J(k)e(k))]/J(k)}
/<-势。。
=
.lim
{[ψ 任何)十 J(k)e(k)) 一 ψ(ç(O)) l! J(k)} - (γ ， e)
，回-→ 00
三 θfψ(ç(O)) 一 (γ ， e) 三; 0 , i.e. ， γξ D+ ψ(ç(O)) by (13.20). So the first equa1ity in (13.21) has been comp1ete1y
proved. The proof of the second is simi1ar. We are now in a position to give the definition of viscosity solutions to the Cauchy prob1em of the form
41 十 I1 t， x ， u， 手) m
飞
ux/
= 0
in
i1 T
~f {O < t < T , x 巳 ]Rn} ，
(13.23)
13. MISHMASH
212
u(T , x) Here , 0 ~n ，
< T
< ∞.
= σ (x)
on
{t = T , x E
Assume that the terminal data
and that the Hamiltonian
continuous in (x , u , p)
f
f (t , x , u , p)
ε ~n X ~ X ~n.
~n}.
σ=σ( x)
(13.24) is of class C O on
is measurable in
ε (0 ，
T) and
In accordance with [35] and [39] , we propose
the following: Definition.
A viscosity solution of (1 3.23)-( 13.24) is defined to be a function
u = u(t , x) continuous on DT satisfying the terminal condition (13.24) and the pair of inequalities
for almost all t
α + f(t ， x ， u(t ， x) ， 的主 o
V' (α ， b) εD十 u(t ， 叶
(13.25)
α + f(t ， x ， u(t ， x) ， b) 三 o
V' (α ，的 ε D-u(t ， x)
(13.26)
ξ (0 ，
T) and for all x ε ~n.
If the Hamiltonian f
f(t , x , u , p) satisfies the conditions
a)叫 indicated
in
Part 10 of 312.2 , then it is known (Theorems 12.5-12.6) that a minimax solution of (13.23)-(13.24) exists and is unique. For almost all
t ε (0 ，
T) , the minimax
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
叩ln m 血山 i日
solution. In this section , the main result on the relationship between minimax and VlSCO均 solutions
(in the case of equations with time-measurable Hamiltonians) is
as follows. Theorem
13. 忆
(13.23)-(13.24) is
Under
the hypotheses of Theorem 12.6 , the minimax solution
αlso α viscosity
0/
solution.
Proof. Take A C (0 , T) to be a null set satisfying (12.37) of Lemma 12.3. By [119 , def
p. 158], we may assume that AC ~' (0 , T) \ A
c
Leb(哟，
with C 口 C( t) the function
mentioned in Condition c) (Part 1 of 312. 匀， and Leb(C) the set of all t ε (0 ， T) I' t+ c5 satisfying !im 二 I C(r)dr-C(t) =0. 6、Õ J λ Let u = u(t , x) be the minimax solution of (13.23)-(13.24) that exists and is 0
unique by Theorem 12.6. Actually, we have shown more , namely that u does not depend on the choice of the multivalued mappings Fu and F L
= u( t , x)
Fu(t ， x ， 也， α)
Fdt , x ， 包， β) (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.
313.3. RELATIONSHIP BETWEEN MINIMAX AND VISCOSITY SOLUTIONS
Let us first prove (1 3.26) for t E A C , x AC , x(O) εR飞 (α ， b) ε D- u( t(O) , X(O)).
b = s. α. Since u
=
ε ]Rn.
To this
e时，
suppose
213
t(O) ε
Choose αεS and 0 三 sε ]R so that
u(t , x) is in Solu , for any 5ε (0 ， T - t(O)) , there exists an x ó(.) E Xu(t(O) ，川的 ， u(t(O) ， x(O)) ， α)
with u(t(O)
Let Yó ~f xó(t(O)
+ 5, xó(t(O) 十 J)) -
u( t(O) , X(O)) ::;
o.
(13.27)
rt(O) 十Ó J 一
+ J) -
X(O)
=
I
旦旦 (T )dT. Because t(O) E A is a Lebe咱1e C
dt
Jt(O)
point of f = 仰) and the function family {xó(.)}ó C XF(t(O) , x(O)) is uniformly bounded , (12.14) yields ~;.可
sup ~IYól 三 óE(O ,T-t(O)) U
sup vx~ ÓE(O ,T-t(O)) U
rt(O) 十 S
1.
f(T)(l
Jt( 的
+ IXó(T)I)dT < ∞-
Consequently , (13.28)
AJtTYru for some Y E ]Rn and 5(1)
> J(2)
>…>拟的→ o. But , by (12.16) and (12.14) , we
now see that
(抖， α)=ijrs(守川)dT 主 ;jrsh hence , by (12.1 剖， that
(~例， b) 三 ikJ~ This together with (12.37) and (13.28) implies (ν ， b) ~三 f(t(O) ， x(的 ， u(t(O) ， x(O)) , b). Therefore , it follows from (13.27)-(13.28) and (13.18)-(13.21) that 。三日叩^ ~I}f/_ {[u(t(O)
+ J, X(O) 十 5(νó/5))
- u(t(时 ， x(O))]!5}
e>OU 气。飞 Z
主 θ马 u( t(O) , x(O)) 去 1.α + (y , b) 主 α+ f(t(O) , x(O) , u(t(的 ， x(O)) ， b)
The inequality (13.26) has been proved for almost all t ε (0 ， T) and for all x ε ]Rn. Similarly for the inequality (13.25). The proof is thus complete.
口
Appendix I Global Existence of Characteristic Curves In this appendix , we will give
su伍cient
conditions which guarantee the global solvIn [43], B.
(2. 四)-(2.16).
ability of the Cauchy problem (2.7)-(2.8) and
D。由nov
gave sufficient conditions for the global solvability of Hamilton's equation. First , we report his problem and results. Let f
= f(t , x , p) be a C 2 -function defined on
R吭 ， p ε lR n }.
lR知+1
= {(t , x , p) : t
ε lR， x 仨
Consider the Cauchy problem:
(dztθf 一=→ (t ， x ， p) (i = 1, 2 ,... , n) , dt dPi
θp‘ δf 一=一一(巾 ， p) dt
θ町
(A I.1) (i
= 1 ， 2 ，...，叫，
x(O) = xO, p(O) = pO.
(A I. 2)
∞
一切
一句
→、 Lf
δ
J-η
/''飞 t、
f
θaa
，，，‘、
,
hvf
。
俨I
十
): r
.，，
叫
、、
ofJ-P lum-Zi-pa' .. 'P· J =nO 文U
，，.‘、.，
hp 、
I， . ιi
户'叫
、
=
nO玄。
fEEE、 FEEE
with C
OO --f-P ! l
也衍d
Assume the following conditions:
(A I. 3)
C(s) a certain continuous function defined on [0 ，∞)， and rk 三1. In
(A I. 3) , all O-estimations are uniform with respect to t and x.
That is to say,
p = O(f k) for Ipl →∞ if and only if there exist positive constants R and K
independent of (t , x) such that Ipl 主 Klfkl
for
Ipl ~三 R.
B. Doubnov [43] proved that , if (AI. 3) is satisfied, the Cauchy problem (A I.1)(AI. 2) h副 uniquely a global solution on {t 兰的 for all (沪 ， pO) ε ]Rn
X
lRn . We
give here a brief explanation for this result. As it is well-known that there exists a local solution of (AI. 1)-(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 (A I.l )-(A I. 2). Then it holds that 函 f(t ，
Since
rk 三 1
x(t) , p(t))
θf = 在( t ， 咐 ， p( t)).
in (A I. 3) , we get the estimate If(t ， x(吼叫t)) I 三 Mé t
where L and M are constants. As p = 0 (1勺 ， p 二 p(t) does not blow up in a finite time. Using the third inequality in (A I. 3) , we see that x = x(t) does not blow up ma 自n巾 time.
Therefore , we can say that the seminal idea of B.
D。由nov
[43]
comes from the fo l1owing example: The Cauchy problem
(d/dt)x(t) = x(t)k , is global1 y solvable for any
y ε lR
x(O) = ν ，
with
if and only if k
~二1.
Now we consider the Cauchy problem (2.15)-(2.16). By almost the same rea soning 田 in
[43] , we assume the following conditions for (2.15).
(C.1) There exists a constant N > 0 such that
lZ-fZ| 三 Nlfl (C.2) For
anν constant
L >
0，矿 we
, Z U PA
?b
IJ-z
+
noxo rJ-u ?ι
pa
，
z u
J，，，、、
''ι
nO又。
and
~.
{(t , x , u , p)
If(t ， x ， 包， p)1
< L} ,
> 0 such that 盯一句
then there exists a constant M
def
put D L
<-
MZ
z u p p <- M PA
on D L . Proposition A I.1. Suppose (C.1) αnd (C.2). Then the C.αuchy problem (2.15)-
(2.16) has a
globα 1
solution x
data. We leave the proof to the readers.
x(t ， ν) ， υ=υ (t ， y) ，
p
= p(t ， ν )
for
anν initial
216
APPENDIX 1
We conclude this appendix with another remark on [43J. B. that , if f
D。由nov
wrote
= f (t , x , p) was an algebraic function of p such that . l.im f (t ， 矶时=∞， Ipl →∞
Condition (A I. 3) would be satisfied. However , when S. Ouchi , A. Kaneko , and M. Murata translated [43J into Japanese , they already pointed out that this is not true. Their counter example is as follows:
f(t， x ， p)~f(1+t2)(p~+p~)
for
n=2.
Though this example does not satisfy (A I. 3) , the Cauchy proble皿 (A I.1 )-(A I. 2) has a global solution for any initial data. Therefore , we would like to give an example where the Hamiltonian f
= f(x , p) is a polynomial of p with . ~m f(x , p) Ipl →∞
= ∞，
but the corresponding Cauchy problem (A I.1 )-(A I. 2) cannot be solved globally with respect to t.
= 1, and let a = α (x) be of class C∞ on JR such thatα (x) = Ixl 4 on {Ixl 主 1} ， and that a(x) 主 C = constant > 0 on {Ixl 三 1}. Put f(x ， p) 彗 α(x )p2.
Example. Let n
Then the Cauchy problem (A I.1 )-(A I. 2) cannot have a global solution for some initial data. For example , if x O solution.
>
1 and pO
> 0, then
it does not admit a global
Appendix 11 Convex Functions , M u Itifu nctions, and Differential Inclusions In this appendix , for the convenience of the reader , we summarize without proofs the relevant material on convex functions , m tÙ tifunctions , and differential inclusions that we have used in an essential way since Chapter 8. For the proofs we refer the reader to 闷， [22 ], [29 ], [40], [64 ], and [117]
sAII. 1. Convex functions Throughout , IRn is the usual vector
sp缸e
of real
= (Xb...' Xn ).
r时 uples x
The
Euclidean norm and inner product in it are denoted by 1.1 and (., .). A subset D of IRn is called affine (respectively, convex) if anyx
巳 D，
y
(1 一 λ )X+ λuε D
for
ε D andλεIR (respectively， λε(0 ， 1)).
Obviously, the intersection of an arbitrary collection of a而且e sets is again Therefore , given any D
c IRn , there
exists a unique smallest
D , namely, the intersection of the collection of
a血ne
a血ne
a任me.
set containing
sets M such that M :> D.
This set is called the affine hull of D and is denoted by aff D. The relative interior of a convex set D in IRn , which we denote by ri D , is defined 出 the
interior which results when D is regarded as a subset of its
a血ne
hull aff D.
In other words , ri D qzfhε affD
(x
+ êB) n (affD) c
D for some
ê
> O} ,
where B stands for the unit ball (centered at the origin) in IR n . Theorem AII. 1. [117 , Theorem 6.1] Let D be a convex 8et in IRn. Let x αnd y ε D. Then (1 一 λ )x + λuεriD for 。三人<1.
巳 riD
APPENDIX 11
218
Let cþ
= CÞ(x) be a function whose va1ues are real Rn .
a subset D of
or 土∞ and
whose domain is
The set
epi ￠ qzf{(z ， μ is cal1ed the epigraph of cþ
= cþ( x ).
x 巳 D ， μ 巳 R， μ 主制 x)} We define cþ
= cþ( x)
to be a convex function (on
D) if epi cþ is convex as a subset of Rn+1. A concave function is a function whose negative is convex. An
al声ne
function is a function which is finite , convex , and
concave. The effective domain of a convex function cþ = cþ(x) on D , which we denote by dom 仇 is
the projection on Rn of the epigraph of cþ
domcþ 营 {x
= cþ(x):
(x ， μ)εepi cþ for some μεR} = {x : φ(x) < +∞}.
This is a convex set in Rn , since it is the image of the convex set epi cþ under a def
1inear transformation. If imφ~. {cþ(x) then cþ
= φ(x)
tωo 也 t h剖 of
x
ε D} C (一∞，+∞]
is cal1ed proper. Trivial1y, the convexity of cþ
the restriction of cþ
=圳 cþ (x 叫)
tωo
=
and domcþ 弄 ø ，
cþ( x) is
吨 e qu 旧lva 乱ler
dom cþ. All the interest really centers on
this restriction , and D itself has 1ittle ro1e of its own. Moreover , one cou1d limit attention to functions given on al1 of Rn , since a convex function cþ
= cþ( x)
on D
can always be extended to a convex function on all of Rn by setting cþ( x) 乞f+∞ for x
~
D. Therefore , by a "convex function ," we shall henceforth always mean
a "convex function with possibly infinite values which is de:fined throughout the spac巧 Rn ，"
unless otherwise specified. The convexity condition can be expressed in
several different ways. For example , we have: Theorem AII.2. (一∞，+∞]，叫 ere
convex on D
x
= CÞ(x) be α function from D to D is a convex set (for exαmple D = Rn). Then cþ = cþ(x) is
if αnd only 矿
cþ( (1 for α ny
[117 , Theorem 4.1] Let cþ
一 λ )x + λν) 三 (1 一 λ )cþ(x) + λcþ(y)
αnd
叫 enever
0
<λ<1
y in D.
Theorem AII.3.
[117 , Theorem 4.2] Let cþ
卜∞，+∞].
= cþ( x)
Then cþ
= cþ(x)
be α function
is convex if and only if
cþ( (1 一 λ )x + λy) < (1 一 λ)α+λß
for any
λε(0 ， 1)
from R n to
~AII. 1.
whenever <þ( x) < ααnd <þ (y)
CONVEX FUNCTIONS
< ß.
Theorem AII .4. [117 , Theorem 5.7] Let Jæ n formαtion.
Then , for
219
'3
x t-+
Ax ξ Jæm
convex function <þ = <þ( x) on Jæn , the
eα ch
be a
lineαr
trans-
function σ=σ (y)
defined by
σ(ν) ~f inf{ <Þ (x)
Ax = ν}
is convex on Jæ m. σ=σ (y)
The function A , in
symbols ， σ =
in Theorem AI I. 4 is called the image of <þ = <þ( x) under
A <þ. The inequality in Theorem AI I. 2 is often taken as the
definition of the convexity of a function <þ = <þ( x) from a convex set D to (This approach causes difficulties , however , when <þ 一∞ among
its values , since the expression
= <þ( x)
furthermore , if <þ
= <þ (x)
∞一∞ could
can have
(∞，十∞].
both 十∞ and
arise.) In this approach ,
is finite and if the inequality is strict for any two different
points x and y in D , then the function <þ = <þ( x) is called strictly
con归x
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 品[口11口7 ， Corollary 7.5ι5.1日] For α lou f包肌 η cti 仇 iωOη <Þ=<Þ 剑(x) ，
one has 的 )=1如 <þ ((1 - λ )x + λy)
for every x
ξdom <þ
and every y.
Theorem AII.6. [117 , Corollary 10. 1. 1] A convex function finite on all of Jæn is necessarily continuous. The (Fenchel) by 扩=旷 (p) ，
conj叼αte
of a convex function <þ
= <þ( x)
on Jæn , which we denote
is another convex function on Jæn defined by the formula
扩 (p) 乞f sup { 巾， x) 一 <þ(x) : x ε Jæn} ，
p ε Jæn.
(AII.1)
(The conjugate <Þ* = 旷 (p) of an arbitrary function <þ = <þ( x) from Jæn to [一∞，十∞l can be defined by the same formula as above. It is actually a lower semicontinuous
APPENDIX 11
220
convex f1ll1ction.) The
followi吨 main
facts about conjugate convex functions are
from [117 , Theorem 12.2 , Corollaries 12.2.1-12.2.2 , Theorems 23 .4-23. 时， or partly from [64 , Theorem 4.1 and the addition]. Theorem AII. 7. (i) Let <þ 旷
=
φ =
旷 (p)
= <þ( x)
be α convex
function. The conj叼ate function
is then α l必o切阴e旷r semt化Cωo时 n timω us ω c，'onve口 x functi ωiω 01η 叽 Z马， prope肝 T 可 αn 叫 d only ，if叮
<þ( x) is proper.
(ii) The
co叼包gα cy
dence in the class anν <þ =
operation <þ
of α II
clαss ，
<þ (x) in this
dom <þ if p
lower
f-t
<Þ*
induces α symmetric
one-to-one correspon-
semicontinuo ω proper con时 x
suprem包m
the
functions on IR. n. For
in (AII.1) is attained
(maxim也m)
in
εri(dom <þ*).
(iii) For any convex function <þ 旷 (p)
= <þ( x)
= sup{ 巾， x)
on
IR.飞 one α ctually
has
- <þ(x) : x εri (dom 的}，
p εR飞
Further, we have: Theorem AII.8. 扣 I η
order
finite
伪 thα at
[口11口 7，
Corollary
dom 旷 be
everywhere αnd
加 b oun 川川 7ηl由 d ed，
that there
13.3.3 叫 口叫 3.叫 .3 叫] 3
Let <þ
= 仲 )阶 加e b
it is necess α ry
α pr， 叩 叩 O 'pe旷 T ω 削 t口 xf包 unct仇 阳 ω 阳O ω 佣 阳 川 7ηZ
α η d sul声 cient
exist α real number μ;二 o
1<Þ (x) - <þ(y)1 三 μ Ix-yl
that
φ =
<þ (x) be
such that
Vx , y.
Let <þ = <þ( x) be a proper convex f1ll1ction that is (finite and) differentiable at some y
巳IR.n.
Then [117 , Theorems 23.5 and 25.1] implies 旷(矿 (ν))
=
\矿(的， ν )- <þ (y).
This equality suggests that the conjugacy operation
<þ 叶旷 is
closely related to the
classical Legendre tran由rmation in the case of differentiable convex functions. (See 310.2 for the definition of the Legendre
transformatio日.)
In fact , this relationship
is in detail as follows. Theorem AII.9. [117 , Theorem 26 .4] Let <þ = 功(x) be αnν lower semicontinuous def .
proper con时 x function such that D ~. int (dom <þ) is 旧n-empty α nd <þ = <þ( x) is differentiαble
on D. The Legendre co叼叼αte (B ， σ ) of (D , <þ) is then well-defined.
Moreover, B is α subset of dom 旷 (nαmely， the range of the grad阳d mappmg z 叶扩 (x)) ， αnd σ=σ (p)
is the restriction of <Þ* = 伊 (p) to B.
~AII. 1.
CONVEX FUNCTIONS
= <Þ (x) be any differentiable
Corollary AII. lO. (cf. [117 , Corollary 26 .4 .1]) Let <þ
convex function on Rn.
Then the
221
Legend陀 conjugate (B ， σ )
of
(Rn ， φ ) is ω ell
defined. One has ri (dom 旷) C B~f 忡'(x) F旷thermore， σ=σ (p)
x ε Rn } C dom 矿.
is the restriction of
<Þ*
旷 (p)
to B , and
σ=σ (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 spe a.k 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 ， 仿) such that D is a non-empty open convex set and <þ
= <Þ (x)
is a strictly convex function on D satisfying:
= <þ (x) is differentiable throughout D. . lim I<Þ' (x(的 )1 = +∞ whenever x {1), X(2) ,...
(i) <þ (ii)
is a sequence in D converging to
'←~+o。
a boundary point of D. For convenience , a pair
(D ， 的 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 <þ <þ
= <Þ (x)
= <þ( x)
has x 同 <Þ' (x) one-to-one if and only ifthe restriction of
to D ~f int (dom 的 is a convex function of Legendre type.
TheoremAII.ll. [117 , Theorem 26.51 Let <þ
= <þ( x) be a lower semicontinuous
convex function. Let D ~f int (dom 的 and D* 生f int (dom 旷). a convex function of Legendre type
if αnd
only if
(D气旷 ) is α convex
Legendre type. When these conditions hold,
(D飞 <Þ*)
(D ， 的， αnd (D ， 的 is
co叼叼αte
mα.ppmg x 叶 <þ'(x)
in turn the Legendre
Then (D ,<þ) is function of
is the Legendre conjugate of of
(D* ，<þ汀
The
gradient
is then one-to-one from the open convex set D onto the open
convex set D* , continuous in both directions， α叫旷， = (的一 We now describe the case where the Legendre transformation and the Fenchel conjugacy correspondence coincide completely. A fini te convex function <þ
= <þ( x)
on R n is said to be co-finite if epi <þ contains no non-vertical half-lines , and this is equivalent (by [117 , Corollary 8.5.2]) to the condition that λlml￠(λ x)/ 人1=+∞ 3。
for all
x ε R n \ {O} ，
222
APPENDIX II
or (by [117 , Corollary 13.3.1]) , to the condition that
f包nction
on ]R n. In order that
x 叶旷 (x)
]Rn onto itselj, it is necessary a 叫 sufficient that cond巾ons hold， 旷=旷 (p)
co-finite. When these
function on ]Rn which is
strictly ω nvex
as the Legendre conjugate of 1> 旷 (p)
=
功(x) ，
and
=伤'(p)
1> = 1> (x)
Theorem AII.12. [117 , Theorem 26.6] Let convex
1>*
1>
is finite everywhere.
be a (finite) differentiable
be a one-to-one mapping from =
1>( x)
is likewis e
be strictly convex and α differentiable
co-βnite ， and 旷=旷 (p)
convex
is the
sαme
i.e. ,
\p , ( 矿)一 1 (p)) 一1>( (旷 )-l(p))
The Legendre conjugate of 旷口旷 (p) is then in turn
Vpξ ]Rn.
1> = 1> (x).
We conclude this section with the following fact about differentiability of convex functions. Theorem AII.13. ]Rn 飞 αn 叫 d ω 1 et
y be
[口11口 7，
Theorem
α point at 切hich
1>( x)
condition for <þ
汩 2 5.2 到]
Let
1> = 1>( x)
1> =
1>制(叫 x ) be αCω ont 川 tυJ口 e xf 扣 包 t阳 u η n ct
is finite. A
necess αry
and sufficient
to be differentiable at y is that the n two-sided
pα rtial
der Ít
!ì AII.2.
Multifunctions and differential inclusions
Let X 舌的 be a set. Then 2 x is the family of all subsets of X. Given another set 0 并仿， a correspondence 0 :1 ç 叶 L( Ç) ξ2 x will be called a multifunction (a set-valued map , or a
mt山ivalued f也nction).
Sometimes , we permit ourselves to
write briefly L = L( Ç). The sets L( Ç) C X are the values of the allowi吨 L( Ç) =
Talk
of 咄e
with
p( 的 ε L(己)
øis
(very 时domly)
mt山 ifunction;
convenient for purely formalistic reasons only.
single-valued case" means L( Ç) = {p(ç)} on O. A function p = p(ç) on 0 will be called a selection of L = L(ç). We refer to
L(0)qF EEoL(Oand
graph(L)tf{(己 ， p) : ç ξ O ， p ξ L(ç)}
as the rα叼e and the graph of L = L( Ç). def ~~ Throughout the book , X ~. ]R n and ø
• def r
并 OcY 二]Rm;
both ]R n and ]R m being
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
~AII.2.
MULTIFUNCTIONS AND DIFFERENTIAL INCLUSIONS
223
that a multifunction satisfies P if and only if its graph satisfies P. For instance , a m山 ifunction
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 measurable subsets and on X = Jæn the
=
Jæm the a-algebra of all Lebesgue
σalgebra
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,
comμ ct-valued，
def
called locally bounded if L( n) ~.
o.
and so on. Of course , L
= L( ç)
is
.lL L( ç) is bounded for any bounded subset n of
EεQ
We shall consider only nonempty-valued mu1 tifunctions.
Further , as a rule ,
m u1 tifunctions investigated in this book are always compact-valued. For such m u1tifunctions , we can use: Definition 1. A
mt山ifunction
L
= L(ç)
is said to
L- 1 (A) ~f {çε o
be 叩'per semicontinωωif
: L(ç) n A 并 0}
is (relatively) closed in 0 whenever A C Jæ n is closed. Definition 2. A multifunction L
= L(ç)
is said to be
10ωer semzco时inuous
if
L- 1 (V) is (relatively) open in 0 whenever V C Jæn is open. Definition 3. A multifunction L
= L(ç)
is said to be continuous if it is simulta-
neously upper and lower semicontinuous. Evidently, upper (respectively, lower) nuity if L
= L(ç)
町nicontinuity
is nothing else than conti-
is single-valued. Here are usefu1 tests for upper semicontinuity:
Proposition AII.14. (cf. [40 , Propositions 1.1-1. 2]) Let L
= L(ç)
beωmpact
valued. Then (i) 可 L (ii) 可
L
= L(ç) is 叩per semicontinuous and 0 is closed, then L = L(ç) is closed; = L(ç) is closed and loclαlly bounded, then L = L(ç) is upper ser时continu-
ous. Proposition AII.15. (cf. [29 , Theorem 1I. 20]) Let L
values. Then it
is 也pper
= L( ç) have compact convex
semicontinuous if and only if the function
03
ç ~→
sup pε L(Ü
(p , x)
APPENDIX II
224
is upper semicontinuous for every x
ε ]Rn.
Proposition AII.16. (cf. [29 , Theorem 11. 25]) Let L = L( ç) be α n 包叩 :pper semtcontimωω11阳ltift 扣mcti ωtω on ωωt必th cω ompα ct v泪 αl沁 包 t Le 削 E臼s.
subset
n
Then L(n) is
compα ct
for any compact
of O.
Let L = L( ç) be given. We associate with any rea1-va1ued function defined on 0 x
]Rn
ω=ω(ι p)
the following marginal function:
ψ=ψ(ç)~f supω(ç ， p)
for
ç εO.
(AI I. 2)
pEL (e)
The maximum sets wi1l be denoted by
LO (O 彗 {p ε L(巳) :ω(ι p)= ψ(ç)} for 仅 O.
(AI I. 3)
Berge's maximum theorem concerning the above margina1 function can be forrnu1ated (see [22 , p. 123], or [8 , Theorem 1.4.16]) as follows. Theorem AII.17. (Maximum (i) 矿 L
Theor叫 Let
and ω=ω (ç ， p) αre
= L(ç)
lower
L = L( Ç) be
compa由1仇ed.
semi比 ωCωoη 时timω包 ωs鸟，
Then
so is the marginal
functi ωzωoη;
(ii)
可
L = L( 巳 )
αη d ω
= ω (α巳ι， p) α T陀E 也叩 .pper sem 川 lt比 ωCω o时 n timω包 ωs鸟， so is the marginal
functionj
(iii) if L = over, L O
L( 巳)αnd ω=ω(己 ， p) αre contìn包0时，
=
so is the marginal functionj more-
L O (巳 ) defined by (AI I.3) is then αn 也.pper semicontínωω (nonempty
value d) multifunction. We are now concerned with calculus of measurab1e mu1tifunctions. As we have mentioned earlier in this section , a multifunction is measurab1e if and on1y if its graph is measurab1e. Notice that measurab1e multifunctions with closed va1ues can a1so be defined in the following way: Deflnition 4. A closed-valued measurab1e set 0 C whenever V
c
]R n
]Rm
mu 叫 山 11 1tifi 缸 u 山n 配 lction
L = L(ç) on a
∞ n OI丑lν 冈 -empty
Lebesgu
is said to be measurable if L -1(V) is (Lebesgue) measurab1e
is open.
Especial1y, the above definition agrees with the classica1 one for measurab1e functions if L =
L( 巳)
is sing1e-va1ued.
~AII.2.
MULTIFUNCTIONS AND DIFFERENTIAL INCLUSIONS
Theorem AII.18. (cf. [8 , Theorem
=
sing1e-va1ued map , and 1et G
8 立 8])
G(已 z)
Let 03
be a
ç 叶万(Ç)
mu1t可unction
αnd continuous in z ε Jæk (with compact va1ues in Jæ n).
0
'3 ç 叶 G(己，万(0)
ε Jæk
be a measurab1e
measurab1e in
function measurab1e in
çε
o
and
ω=ω(ι p)
continω也smp ξ Jæn ， αnd
be
α rea1-va1ued
1et L = L(ç) be a
measurab1e mu1tifunction on 0 with closed (non-empty) va1ues in Jæ n. function ψ=ψ (0
multifunction L O
ç εC
Then the multifunction
is measurab1e.
TheoremAII.19.(cf.[8.Theorem8.2.11]) Let
margina1
225
= LO 亿 )
Then the
defined by (AII.2) is measurab1e. Furthermore , the
defined by (AI I. 3) is a1so meαsurab1e.
Theorem AI I. 20. (cf. [8 , Theorem 8.2.14] and [29 , Theorem 11 1.1 5]) Let 0 be a Lebesgue measurab1e subset of
o with closed (non-empty)
Jæm. 扩 L
L
=
L(ç) α陀 convex
L( ç) is a measurab1e
mult~卢mction
on
va1ues in Jæn , then the function 03
is measurable for every x
=
ε ]Rn.
ç叶
sup (p , x) pEL (e)
The converse
stα tement
ho1ds true if the va1ues of
and bounded.
In the remainder of trus appendix , given J
def
[0 , T] C ]R, a multifunction
J x Jæn '3 (t , x) • + G(t , x) C Jæn , we are looking for absolutely continuous solutions
of the
d~万erentia1
inclusion
生 (t) ε G(t， x(t)) dt For any
(t* ， x*) ε J
functions x x(九)
=
=
almost everywhere on
(AI I. 4)
J.
x ]Rn , denote by X G( t* , x *) the set of all absolutely continuous
x(t) from J into ]Rn which satisfy (AI I. 4) subject to the constraint
x*. Topological properties of the solution sets
X G (儿，且)
will be investi-
gated in the Banach space C(J, ]R n) of continuous functions given on J with values in Jæ n. (The norm in C(J,]Rn) is the usual "max" one.) We have: Theorem AII.2 1. (cf. [40 , Theorems 5.2 and 7.1]) Let G
=
G(t , x) have non-
empty closed convex va1ues and be measurable in t ε J， upper semicontinuous in Z ε ]Rn
such that
IG(t， x)1 乞fsup{|z|:z ε G(t ， x)} 三 c(t).(l+lx l)
for
t ε J， x ξRn ，
APPENDIX II
226
with c = c(t) α function in L 1 (J). Then XG(t. , x.) is a non-empty compact subset of C(J, R n ) for each (t. ， x.) ε J x R n . Further, the mult可unction Rn :1 x 叶 XG(O , x) C C(J, Rn) is upper semicontinuous.
We conclude this appendix with Filippov's theorem for difIerential inclusions , which
is 出 important
Theorem AII.22.
as Gronwall's lemma for ordinary difIerential equations. (Filippov) [8 , Theorem 10 .4 .1] (cf.
[40 , Lemma 8.3]) Let
= G( t , x) have closed non-empty vα lues and y = y(t) be αn JRn-valued function absolutely continuous on J , αnd let ó > O. Assume: (i) G = G(t ， 叫 is measurable in t ε J; (ii) there exist αn r > 0 and α nonnegati肘 function C = C(t) integrable on J such
G
that G(t , x 1 ) C G(t , x 2 ) +C(t)lx 1 for almost all t (均 the
-
x 2 1B
Vx 1 ， x 2 ε y(t)+rB
ε J;
def.
function t 叶 γ(t)~' inf{ly'(t) - pl : p E G(t , y(t))} is integrable on J.
Set
t μ叫州(柑 t 缸呻叫 p叫(l飞 t叫C仲削(行价T叶材叫)忡古叫) αT叫 叭.， ( 腊 t 叫附削叫 t叶叫)叫(←扣 6仆+ 扑 lt二γ训仰州(扩例T叶)
扩η 刑(T) 三 T飞， then for every x. ε y(O) XG(O ， 且)
+ óB ,
there exists a solution x
of (AII .4) such that Ix(t) - y(t)1 三 η (t)
VtεJ
α nd
Ix'(t) - y'(t)1 三 C(t) η (t) + γ (t)
almost every叫 ere on
J.
= x(t) in
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Index
Cantor function (1 adder) , 86 Cantor set , 86 Carathéodory differential equation , 98 Carathéodory's conditions , 97 , 166 , 191 characteristic curves , 1 equations , 2 , 4 strips , 4 system of differential equations , 4 Ck-function , 1 Ck-solution , 7 collision of characteristic curves , 26 , 28 of singularities , 41 , 63 comparision equation , 98 concave-convex function , 128 conJugate (Fenchel) concave-convex , 131 (Fenchel) convex , 105 , 219-222 Legendre , 128, 220-222 contact discontinuity, 70 cusp point , 58 d.c. function , 113 d.c. representation , 117 differential inclusion , 225 differential inequality of Haar type , 85 Dini semiderivative , 108 , 209 Dini semidifferential , 210 directional derivative , 108 effective domain , 105 , 218 entropy condition , 52 epigraph , 218 equation of Hamilton-Jacobi type , 34
equation of the conservation law , 12 , 45 Fi 1ippov's theorem , 180 , 226 fold point , 57 genera1ized solution , 32 , 55 general PDE of first-order , 4 global C 1 -so1ution , 92 global semiclassical solution , 147 , 154 global solution , 104 Haar's differential inequality, 85 Haar's theorem , 7 日 adamard' s lemma , 19 Hopf's formulas , 104 Jacobian , 2 Jacobi matrix , 2 Lebesg肘 's theorem , 170 Legendre transformation , 128 , 220-222 life span of classical solutions , 12
marginal function , 224 maximum theorem , 224 minÌ max solution , 168 , 179 , 194 , 196 monotonÌ city condition , 192 multifunction , 222 closed , 223 closed-valued , 223 continuous , 223 locally bounded , 223 lower semicontinuous , 223 measurable , 223-224 upper semicontinuous , 223
INDEX
non-degenerate singularity, 58 proper, 123 , 218 quasi-linear PDE of first-order , 1 quasi-monotonicity condition , 192
strict convexity, 118 , 122 , 219 subdifferential , 210 subgradient , 210 subsolution , 168 , 179, 194 , 196 superdifferential , 210 supergradie时， 210
supersolution , 167, 179 , 193 , 196 Rankine-Hugoniot's condition , 41 rarefaction wave , 68 relative interior , 217 semi-concavity, 32 , 39 , 56 shock wave , 53
viscosity solution, 212 Wazewski's theorem , 9 weakly-coupled system , 97 weak solution , 45
237
