Recent Topics in Nonlinear PDE II (North-Holland Mathematics Studies) (v. 2)

RECENT TOPICS IN NONLINEAR PDE I1

This Page Intentionally Left Blank

NORTH-HO LLAND

MATHEMATICS STUDIES

128

Lecture Notes in Numerical and Applied Analysis Vol. 8 General Editors: H. Fujita (University of Tokyo) and M. Yamaguti (Kyoto University)

Recent Topics in Nonlinear PDE I1

Edited by

KYUYA MASUDA (Tohoku University) MASAYASU MIMURA (Hiroshima University)

KlNOKUNlYA COMPANY LTD. TOKYO JAPAN

NORTH-HOLLAND AMSTERDAMeNEW YORK-OXFORD

KlNOKUNlYA COMPANY-TOKYO NORTH-HOLLAND-AMSTERDAMeNWE YORK*OXFORD

@ 1985 by Publishing Committee of Lecture Notes in Numerical and Applied Analysis All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying. recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87938 2

Publishers KINOKUNIYA COMPANY LTD. TOKYO JAPAN

*

*

*

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands

Sole distributors for the U.S.A. and Canada ELSEVIER SCIENCE PUBLISHING COMPANY. INC. 52 VANDERBI1.T AVENUE NEW YORK. N.Y. 10017

Distributed in Japan by KINOKUNIYA COMPANY LTD. Distributed outside Japan by ELSEVIER SCIENCE PUBLISHERS B. V . (NORTH-HOLLAND)

Lecture Notes in Numerical and Applied Analysis Vol. 8 General Editors

H.Fujita University of Tokyo

M. Yamaguti Kyoto Universtiy

Editional Board H . Fujii, Kyoto Sangyo University M. Mimura, Hiroshima University T . Miyoshi, Yamaguchi University M. Mori, The University of Tsukuba T . Nishida, Kyoto University T. Taguti, Konan University S. Ukai, Osaka City University T . Ushijima, The Universtiy of Electro-Communications

PRINTED IN JAPAN

PREFACE This volume is an outgrowth of lectures delivered at the second meeting on the subject of nonlinear partial differential equations, held at Tohoku University, February 27-29, 1984: The first meeting was held at Hiroshima University, 1983. The topics presented at the conference range over various fields in mathematical physics. We would like to take the opportunity to thank all the participants of the meeting, and the contributors to this proceedings. Special thanks should go to Professors T. Muramatsu and J. Kato who helped in many ways to make the conference a success. We are also grateful to the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan for the financial support.

K. MASUDA M. MIMURA

This Page Intentionally Left Blank

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

J. Tohmas BEALE and Takaaki NISHIDA: Large-Time Behavior of Viscous Surface Waves.. .......................... 1 Hitoshi ISHII: On Representation of Solutions of HamiltonJacobi Equations with Convex Hamiltonians . . . . . . . . . . . . 15 Keisuke KIKUCHI: The Existence of Nonstationary Ideal Incompressible Flow in Exterior Domains in R3 . . . . . . . . . . . 53’ ,

Kyiiya MASUDA: Bounds for Solutions of Abstract Nonlinear 73 Evolution Equations ................................. Shin’ya MATSUI and Taira SHIROTA: On Prandtl Boundary Layer Problem ..................................... 81 Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI: On a Free Boundary Problem in Ecology . . . . . . . . 107 Ryiiichi MIZUMACHI: On the Vanishing Viscousity of the Incompressible Fluid in the Whole Plane . . . . . . . . . . . . . . . . . 127 Fumio NAKAJIMA: Index Theorems and Bifucations in Duffing’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Mitsuhiro NAKAO: Global Solutions for Some Nonlinear Parabolic Equations with Non-monotonic Perturbation ....... 163 Yoshihiro SHIBATA and Yoshio TSUTSUMI: On a Global Existence Theorem of Neumann Problem for Some Quasi175 Linear Hyperbolic Equations .........................

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 8 , 1-14 (1985) Recent Topics in Nonlinear PDE 11, Sendai, 1985

Large-Time Behavior o f Viscous Surface Waves

J. Thomas BEALE

*

by

*

and

Duke University Department of Mathematics Durham, NC 27706

Takaaki NISHIDA

Kyoto University Department of Mathematics Kyoto, 606 Japan

Introduction

§ 1

We are concerned with global in time solutions to a free surface problem of the viscous incompressible fluid, which is The motion of the fluid is governed

formulated as follows:

by the Navier-Stokes equation Ut

(1

+

- vau + vp

(U.V)U

0

=

-1 1

in v.u

where

Q(t) =

(

x

6

R

2

,

Q(t)

,

0

=

-b < y < q(t,x) }

occupied by the fluid. The free surface

is the domain

SF : y = q(t,x)

satisfies the kinematic boundary condition (1.2)

rlt +

UlllXl + U2Qx2

- u3

0

=

on

SF

.

The stress tensor satisfies the free boundary condition : (1.3)

pni - v ( u . 1,x

j

+ u

j ,xi

)nj

=

________________________________________--------------

*

Both authors are supported in part by the Mathematics Research Center, The University of Wisconsin-Madison.

1

J . Thomas BEALE and Takaaki NISHIDA

2 where

n

is the outward normal to l3

gravitation constant and

,

SF

g

is the

is the nondimensionalized

coefficient of surface tension.

SB : y = -b

On the bottom

we have the fixed boundary condition u

(1.4)

=

o

on

SB

.

We consider the initial value problem of (1.1)-(1.4) with the data at

t = 0

i

rl

=

Q0(X)

U

=

uO(X,Y)

(1.5)

Ro =

where

X

I

C

R

~

Ro

in

I

nco,.

The local existence theorems for (1.1)-(1.5) are proved for both cases with A

91

without considering the surface tension ([11,[21).

global in time existence problem for (1.1)-(1.5) neglecting the

surface tension ( B = O ) has a difficulty which was pointed out in

1 1 1 . However if the surface tension is taken into account, the following global existence and regularity theorem is proved. Theorem

1 . 1 ([21)

Let

3

r < ?/2

.

Suppose the compatibility condition on the

initial data :

(1.6)

1

8.u 0

=

( (UOIirx

0

in

o

,

Ro

+ ( u ) j ,xi)nj’tan

=

o

on

y

j

uo

There exists

=

0

6o > 0

on

y = -b

.

such that if the initial data

= rlo(x)

I

3

Large-Time Behavior of Viscous Surface Waves

satisfy

then there exists a unique global solution 1 .5)

(

I

q I u I p of (1.1 ) -

which satisfies

T1 > 0

Further given any

k > 0

and any

there exists

61 > 0

such that if

Eo

(1.9)

61

t > T1

then the solution becomes smooth for (1.10)

Q € K -r+k+1'2((Tlrm)xR2 )

Hr(

domain

)

.

k 2 2

the fluid domain

Q(t)

~ ( R + X R ~ )is

2 E Kr((0,T)XR )

q 1 f Kr(R+xR2)

of

I

Ir

on the

is composed of the restriction to

of the functions belonging to

( 1 -11) Kr((TlrT2)XR3 ) = H 0((T1rT2);Hr(R 3)

e

is classical.

is the usual Sobolev space with norm Kr((T,'T2)XQ(t))

r

-

In particular the solution with Here

I

u E K r + k ((TIIm)xn(t)

I

rip€ Kr+k-2f (TIr")xQ(t))

i.e

) A Hr/2((T1rT2)rH0 ( R3 ) )

defined as follows : for any

and

q2

T > 0

and

n

=

n1

+

n2

such that

is the Fourier transform in space-time

L1 function of bounded support.

See [ 2 ] for the detailes of the function spaces. In this summary we give an asymptotic decay rate for the

-

4

J . Thomas BEALE and Takaaki NISHIDA

solution of the above theorem. Theorem

1.2 u0(g L 1 (R2 )

If

then there exists

€i2

> 0

such that if

then the solution has the decay rate :

In 3 2 we transform the free boundary problem (1.1)-(1.5)

to that

on the fixed domain and reduce the components of the stress tensor to zero. The linear decay estimate is discussed in I 3 and the nonlinear one in

§

4

.

Reduction of the Problem

9 2

We remind ourselves some main ideas for the reduction of the free surface problem in [2]. First we use the transformation of the free boundary problem (1.1)-(1.5) to that on the fixed (equilibrium) domain : we extend it for

Q =

y < 0

{x E R

n(trxry)

=

-b < y < 01. Given

q(t,x)

as follows :

(2.1)

L

?-'(

el'ly

;(trc))

r

A

where

q(tr')

is the Fourier transform with respect to

7-lis the inverse. rl(trSr*)belongs to

H

If rl(t,.) s+l/2

denotes the upper surface

(a)

y = 0

belongs to

Hs(SF)

s2

.

For each

and

then

where and hereafter of

x

SF

t > 0

we

5

Large-Time Behavior of Viscous Surface Waves

define the transformation 8

R

on

onto R(t) = (x E R

2

,

-b < y < rl(tlx)1 by

The vector Q

on

R(t) = 0 ( R )

is defined from the vector

v

on

by

(2.3) where

-

u

ui

-

'i1x V j / J j

t

c1 ij V j

I

is the Jacobian determinant of

J

qy(l+y/b)

.

0.v = 0

in

d8 =

(eiIx

) =

l+n/b +

j This map conserves the property of divergence free. $2

iff

0.u = 0

in

.

uilX = clj al(clik vk) j and so on, we can rewrite the free surface

Using the transformation (2.2)(2.3) = (de)-'

R(t) and

1.1)-(1.5) to that on the equilibrium domain

-

'It

v3

=

0

On

F'

R

,

as

I

I

v

=

o

on

SB

I

on

SF

Here we have gathered the linear terms on the left hand side and all the nonlinear terms on the right hand side of the equation. Next we reduce the tangential component of the stress tensor

J . Thomas BEALE and Takaaki NISHIDA

6 Fi

,

i = 1, 2

to zero :

choose the vector

Given

Fi E H

v-w w3 rl

,

v’

The prime of

F

=

Fi

0

=

0

=

v’ = v - w

the replacements

,

,

i

=

i = 1, 2

R

,

on

SF

.

1, 2

on

I

’F

satisfy the system ( 2 . 4 ) - ( 2 . 9 )

q

F4 = F - wt + VAW

by

,

in

and

Fi

,

with

i = 1, 2

‘I,

v, q

with

in the operator form.

,

F = F4 Let

P

Fi

be the

projection on the subspace of solenoidal vectors orthogonal to 0 = the subspace v @ : $ E H ’ ( R ) , 4 = O on sF 1

4

of

H

0

(a) ,

(2.10) Applying (2.11) Here

Pvq

i.e.,

Ho

P

=

0

by

is omitted hereafter.

Last we rewrite the system ( 2 . 4 ) - ( 2 . 9 ) i = 1, 2 , for

(SF)

E Hr+’(.Q) satisfying the condition

z

w 3 = o , wi,x3 + w3,xi

Therefore

r-312

PH

0

to ( 2 . 5 )

@

we have

vt - VPAV + PVq

=

PF4

.

can be decomposed to three parts a s follows :

=

0

,

.

7

Large-Time Behavior of Viscous Surface Waves

where

TI

(i)

i = 1 , 2, 3

I

are defined by

We denote

Using these notations the system (2.4)1(2.11) has the form (2.14) (2.15)

where

'It = vt +

A V

f ( q , v,

(2.6)(2.7)(2.8)

R V

*

+ R ((g-BA)'I)

=

f

I

Vq) = P F ~- VIT ( 3 ) with

Fi = 0

give the domain condition of

v .

5 3

Rates of Decay f o r Linear Problem We investigate the decay rate of the solution of the

linearized equation.

A

on

J . Thomas BEALE and Takaaki NISHIDA

8

(3.3)

Q(0) =

,

q0

u(0)

=

uo

at

t = O .

in

R

These are supplied with the conditions : (3.4) U

(3.5)

irx3

+

u

v*u =

0

=

o

3rxi u

(3.6) Theorem

=

,

i

o

= 1, 2

on SB

, on

,

SF

.

3.1

Then the solution of (3.1)-(3.6) has the decay rate :

(3.7)

The theorem is proved by several steps. Let

.)A

= { v = ( r l , u ) : rl

( p , q ) , = g(p, q),

and

set

W

+

O(Vp,

1 E H (SF) , u

}

. Let

PHo(Q) 1

,

where

0 0 ) ~ is the inner product of

= { v : q E H5j2(SF)

(3.4)(3.5)(3.6)

E

,

2 u G PH ( Q ) and

u

1

H (SF)

satisfies

u s define the operator

and consider its closed extension which will he denoted by

G

again. Lemma 3.2 The operator and

W C D(G)

G

generates a contraction semigroup

.

Consider the resolvent equation :

etG

on

,.c

1

9

Large-Time Behavior of Viscous Surface Waves

(3.10)

The resolvent of

can be extended to the left half plane as

G

lemmas 3.3-3.5.

-Lemma 3.3

-rO > 0

For any

+ i.r

A E { A = u

,

-c~IT/

<

then the solution of ( 3 . 1 0 )

A

We treat the resolvent near (i) The supports of

~ T I>

,

< T~

0

.r0

such that if

,

}

has the estimate

=

A

i.e.,

co > 0

there exists

in two cases separately

0

,

.

A

, f(5,y)

h(5)

.

( ( 5 1 5 C0 1

(ii) The supports belong to

Fourier transform with respect to

x

{I51 2 5 ,

belong to Here

}

means the

.

Lemma 3.4 -For any

E0 > 0

there exists

ro > 0

such that if A

1x1

&

to

{

1 5 ) 2 E0 }

solution (3.12)

< ro 1

and the supports of

,

( q , u)

IU

I

A

,

h(5)

f 5, Y1

then the resolvent equation

3.10)

belong has the

satisfying

lqI5I2

5

c

(

Ih

A

Let

G(S)

be the Fourier transform of

G

with respect to

x

.

Lemma 3.5 -There exist

c1

> 0

and

1x1

r l , r2

2 v ( 1 ~ / 2 b ) > r2 > rl > 0

151 < 5,

,

)

such that if

rl <

(A

exists except for a one-dimensional eigenspace

- ;(5))-’

< r2

(

and

which is analytic with respect to

5

.

then

The eigenvalue and

J. Thomas BEALE and Takaaki NISHIDA

10

eigenvector have the following expansions.

A

=

-(gb3/3v))512 + O 0 E I 4 )

rl

=

1 + O()5l2)

I

r

(3.13)

u

i(gEj/2v)(y

=

j

u3

2

2 -b 1

+ O((5I

3

1

r

j = 1 , 2 ,

(g1EI2/2v)(y3/3-b2y-2b3/ 3 ) + O ( ( E 14 1

=

By using lemmas 3.2-3.5 the decay estimate (3.7) can be proved by the transformation of the integral path of the reprensentation v(t) = e

(3.14)

tG

vo =

I

lim

2Fi

~ + m

o+i.r e a-iT

(A

- G)-l vo dA

to the left half plane.

3 4

Nonlinear Decay Estimates The free surface problem (1.1) - (1.5) was reduced to the

following system in 9 2.

where

f

is nonlinear terms depending on

q r u,

Vp

and their

derivatives. The initial value problem (4.1) - (4.3) has the unique solution (Theorem 1 . 1 ) Namely we know that if

which becomes smooth for Eo < 61

then

t

2

T, > 0.

a > 0

11

Large-Time Behavior of Viscous Surface Waves

Let us define

in 9 3 , this solution satisfies the variation of constants formula: L

' I G f(s)ds

I

The first term of (4.7) has already estimated in

has the decay rate :

(4.8) a ( 8 qo(t)(o

s

COE2 t

(4.9)

3 ,i.e.,

r(

t-(l+a)/2

IDano(t))O

I

ID3riO(t)I0

5

COE2 t -3/2

I

5

COE2 t-'

uo(t)

l2

,

0 s a s 5/2

,

i.e.,

,

a = 0, 1, 2

,

and

I

It is sufficient by (4.4) to prove the decay rate for

t

b

2

.

J . Thomas BEALE and Takaaki NISHIDA

12

Let us decompose the second term of (4.7) into three parts

Let

Fi

,

i = 1 , 2, 3 I

by using

for

rl

rl

-

.

R

5

be the extension of

+

IrlILrn)IUl2

as above. Therefore Lemma

4 .2

For

t L 2

on

-

C(E3)(fDq12

DaFi, a = 1 , 2, 3 ,

Fi

We have

-

\File

(4.12)

on

have the same estimate as

D

a-1 F

f

has the same estimate as (4.11).

,

we have for

i = 1, 2

SF

Large-Time Behavior of Viscous Surface Waves

Also we have for

i = 3

It is proved by Theorem 3.1 Nirenberg's inequality [ 4 1 .

-

q(t)

(4.5)

for any

t

and 2

1

,

13

and by Lemma 4 . 1 and Sovolev-

In particular, we know that since by

u(t) are bounded in

and

H6

we have by ( 4 . 6 ) for any

H5

respectively

t 2 1

Proof of Theorem 1.2 It follows from (4.9) and Lemma 4.2

that

COE2 + C.,CM(Q,u;t) 2

M(~,u;t)

Therefore there exists

62 > 0

such that if

E2 < 6 2

,

Eo < ti1

then M(q,u;t)

C

C E2

.

This proves Theorem 1.2. Remark. G(5)

If the fluid has an infinite depth, the eigenvalue of

has the following expansion:

which is quite different from ( 3 . 1 3 ) .

The details will be

,

14

J . Thomas BEALE and Takaaki NISHIDA

published elsewhere.

References [ l ] J. T. Beale, "The initial value problem for the Navier-Stokes

equations with a free surface", Cornm. Pure Appl. Math.

34

(19801, 359 - 392. [ 2 ] J. T. Beale, "Large-Time Regularity of Viscous surface Waves'',

Arch. Rat. Mech. Anal. =(1984),

307-352.

[ 3 1 T. Kato, "Perturbation Theory for Linear Operator'', Springer-

Verlag, Berlin-Heidelberg-New York, (1976) [ 4 1 L. Nirenberg, "On elliptic partial differential equations",

Annali della Scuola Norm. Sup. Pisa, Q(1959)

115-162

Lecture Notes in Num. Appl. Anal., 8, 15-52 (1985) Recent Topics in Nonlinear PDE I t , Sendai, 1985

On Representation of Solutions of Hamilton-Jacobi Equations with Convex Hamiltonians

Hitoshi ISHII Department of Mathematics Chuo University Tokyo 112 Japan

10.

Introduction Recently Crandall and Lions [5] introduced the notion of viscosity solu-

tion for Hamilton-Jacobi equations to settle the uniqueness problem of generalized solutions of Hamilton-Jacob1 equations [4] and Ishii [El.

-

see also Crandall-Evans-Lions

The existence of viscosity solutions of Hamilton-Jacobi

equations was established also under the same hypotheses on the Hamiltonians as those for the uniqueness of viscosity solutions. See Crandall-Lions [5], Lions [11,12], Souganidis [15], Barles [l] and Ishii [ 9 ] . In [ll] Lions made the observation that the dynamic programming principle implies that the value function of an optimal control problem is the viscosity solution of its Bellman equation.

It was proved by Souganidis [14],

Barron-Evans-Jensen [Z] and Evans-Souganidis [6] that the idea extends to the case of differential games. These results can be regarded that the viscosity solutions of the Bellman equation (0.1)

u

+ max

f-g(x,a)-Du

-

f(x,a))

=

0

N in E

aEA and of the Isaacs equation (0.2)

u

+ max

min f-g(x,a,b).Du a€A b€B

-

f(x,a,b)}

15

= 0

N in R

16

Hitoshi ISHI

are represented as the value functions, respectively, of an optimal control problem and of a differential game.

In (0.1) and (0.2)

Du denotes the

gradient of u, (au/axl,***,au/axN). The representation results have weakness in the generality compared with the existence and uniqueness theorem for the viscosity solution of the stationary problem for the Hamilton-Jacobi equation

the representation theorems require much more on the Hamiltonian H. We pose here a question: Is any viscosity solution the value function of an appropriate optimal control or differential game problem? We give a partial and positive answer to this question. Thus our goal of is to represent the Hamiltonian H(x,p) functions of

this paper

as a "max" or "max-min" of linear

p, i. e. to rewrite (0.3) in the form of (0.1) or (0.2), and

to prove the uniform continuity of the value function of the associated optimal control or differential game problem.

Then the dynamic programming

principle and the uniqueness of the viscosity solution of (0.3) imply the value function is identical to the viscosity solution of (0.3). carry out this program, we assume here that p

+

H(x ,p)

To

is convex. New

difficulties arise when proving the uniform continuity of value functions and rewriting ( 0 . 3 ) in the form of (0.1).

The former difficulty is resolved

by introducing an argument analogous to the proof of the uniform continuity of viscosity solutions of Hamilton-Jacobi equations in [ 9 ] into optimal control theory.

In the argument the continuity of the value function is

derived from the continuity properties of the Hamiltonians with the help of selection lemmas (see e. g . Lemma 3.1).

Its simplified version appears when

we get a bound of the value functions (see 1 2 ) .

Our main tool to resolve

Hamilton-Jacobi Equations

17

the second difficulty is a uniform continuous selection lemma (see Lemma 5.1). In Section 1 we introduce our

The plan of this paper is as follows.

control problem, We prove the uniform continuity of the value function of In Section 4 we observe

the optimal control problem in Sections 2 and 3 .

that the dynamic programing principle implies that the value function satisfies the associated Bellman equation.

In Section 5 we prove a representa-

tion theorem for convex Hamiltonians.

In Section 6 we present a represen-

tation theorem for viscosity solutions of Hamilton-Jacobi equations with convex Hamiltonians. We use the following notation throughout. closed ball in RN of center x

and radius r 2 0. For x, y E R N x-y

denotes the Euclidean inner product in R for r 6 R .

N

UC( R )

and

N B(x,r) = B (x,r) denotes the

N BUC( R )

N

.

We will write r+ = max {r, 0 )

denote the spaces of uniformly contin-

uous functions and of bounded, uniformly continuous functions on R N , respectively. For any set F, P(F)

denotes the set of all subsets of

F.

This work was begun when the author was visiting the Istituto Matematico, Universita di Roma and supported in part by the (Italian) CNR.

The author

wishes to thank Prof. I. Capuzzo Dolcetta for his friendly hospitality.

He

also wishes to thank Prof. L. C. Evans for his criticism regarding the method of proof of the uniform continuity of value functions and Prof. R. Iino for useful comments on selection lemmas.

91.

optimal control problem of infinite horizon

An

Let A g: RN (Al)

x A

f

be a nonempty metric space, and let

+ R N be given.

and

f: EN x A + R

We assume in the following that

g are continuous on RN

x

A.

and

18

Hitoshi ISHI a : [0,-)

A (Lebesgue) measurable function the set of controls is denoted by ;r(t)

=

-t

A

is called a control and

A. We consider the ODE

g(x(t),a(t))

t 20,

for a. a.

(1.1) x(0) = x, where

a €

A

and x €lR

.

N

continuous function x(*)

By a solution of (1.1) we mean an absolutely which satisfies (1.1).

A1 (1.1)

A (x) = {a € g

for x €lR -t

e

N

.

Aad(x)

has a global solution)

A

A control a €

(x) is called admissible at x if t -+ g is integrable on [0,-) for a global solution x(-)

€(x(t>,a(t))+

(1.1).

denotes the set of admissible controls at x.

the set of all solutions of (1.1) for x € E N and a €

Aad(x),

We define

x(-)

€

X(x,a)

a €

A.

is called admissible if x ( * )

of

X(x,a)

denotes

For x € l R N

and

is a global solu-

tion of (1.1) and satisfies the above integrability condition. The set of admissible x(-) 6 X(x,a)

is denoted by Xad(x,a).

X (x,a) g

denotes the set

of global solutions of (1.1). An annoying point of our setting is that (Al) does not ensure in general the uniqueness of solutions of (1.1).

This means that one can not control

the system completely by selecting one of controls. This is, however, inevitable for us to get a general representation formula for viscosity solutions in view of the representation of Hamiltonians in 55 and also clarifies the effectiveness of our method of proof of the continuity of value functions. Now we define the

COSt functional

1

m

J(x,a,x(*))

(1.2)

=

e-t f (x(t) ,a(t))dt

0

for x

N

,a

8 Aad(x),

x(*) E Xad(x,a)

and the value function

19

N

for x € R ,

This is an infinite horizon problem, and the goal is to find V(x) furthermore to find, if exist, a control a 6 Aad(x) for which the infimum in (1.3) is achieved.

and x ( - ) 6 Xad(x,a)

Such a control is called an

optimal control. Our contribution here is to demonstrate that V and V

and

€

UC( WN)

solves (0.1) in the viscosity sense under assumptions (Al), (A2) and

(A3) (see 52 for (A2) and (A3)).

52.

A bound for the value function We will study the value function defined by (1.3) in this and the next

two sections. Throughout these sections we assume (Al) and that the function H: RN

x

IRN

-+

R

defined by H(x,p) = sup I-g(x,a).p aEA

(2.1)

-

f(x,a))

satisfies (A2) and (A3) listed below. (A2)

For each R

0

there is a continuous function aR: [0,2R] + [ 0 , - )

satisfying oR(0) = 0 such that

(A31

There is a continuous function w: such that

[0,-)

+

[0,-)

satisfying w ( 0 ) = 0

20

Hitoshi ISHI

We may assume that [0,2R]

x [0,m)

(r,R)

-+

uR(r)

is nondecreasing in each variable on

and that

for all r 2 0 and some constant C1 > 0. We will write u(r) = ur(r) N r 2 0. Note that p + H(x,p) is convex for x e R

for

.

Remark 2.1.

Conditions on H

like (A2) and (A3) were employed by Crandall-

Lions [5] when they formulateduniqueness results for viscosity solutions of N (0.3) in the class BUC( R ) . It was later observed that conditions (A2) and (A3) are enough to ensure the existence and uniqueness of the viscosity solution of ( 0 . 3 ) in the class UC( RN). See Ishii [8, 9 1 , Lions [12], Souganidis [15] and Barles [l]. Theorem 2.1.

Under assumptions (Al)

-

(A3).

one has

N for x f R Corollary 2.1.

Under assumptions (All

-

.

(A3),

N holds for x, y 8 R ,

Proof. By (A3) and (2.2)

for x, y 6 ElN.

This and (2.3) together yield (2.4).

Q.E.D.

To prove Theorem 2.1, we need the following lemma. Lemma 2.1.

For any

E

> 0,

N R > 0 and x f R , there are a

e

Aad(x)

and

21

Hamilton-Jacobi Equations x ( * ) E Xad(x,a)

such t h a t

+ f(x(t),a(t))

H ( x ( t ) ,O)

(2.5)

<

E,

and

(2.6)

Ix(t)

Proof.

inf

f o r each

r

5 R

(E

+ o(R))

Set h ( x , a ) = H(x,O)

an

t

XI

t 2 0.

for all

Since

-

h(x,a) = 0

aeA

x E RN

> 0

so t h a t

f o r each

(2.7)

x

for

+

f(x,a)

x € RN and

€ A.

x € RN by ( 2 . 1 ) , one can select a n

h(x,ax) < € 1 2 .

e RN

fc

By t h e c o n t i n u i t y of

h

ax 2‘ A t h e r e is

such t h a t

h(y,ax) <

for

E

y 8 B(x,rx).

Setting rx tx = 2 E and l e t t i n g

t i a l data

for

x(t;y)

be any s o l u t i o n of (1.1) w i t h

a(t)

=

ax

and t h e i n i -

y, w e have

y € B(x,rx/2)

yo 8 B(x,rx/2)

and

R

+ o(R)

and

and 0

0 f t ~ t , . Indeed, supposing t h e c o n t r a r y , w e f i n d c

to

t

such t h a t

22

Hitoshi ISHI

(2.10)

x(t;yo)

8

B(x,rx)

It follows from (2.1) that -g(y,a).p

for all 0

+

(H(y,p)

t 2 to.

f(y,a)

for y, p B E N

and

a 8 A, and hence, by (A2),

if

g(y,a) # 0. Thus by (2.7) and (2.10) we find that

Hence

this contradicts ( 2 . 9 ) .

Thus (2.8) holds.

By compactness, for each

subfamily of

{B(x,rx/2)I

mappings: x

a(x)

such that

T

+

N.

and

r

+

T(r)

from

to

[O,m)

(Olm)

is nonincreasing and that

€

RN and, using the mappings a and

above, define a sequence {(aj,

€

Therefore it is possible to define

x 8 B(O,S)}.

from RN to A

Finally we fix x

for j

can be covered with a finite

S > 0, B ( 0 , S )

Then we set

Tj

, xj, yj(.))jjen

T

introduced

by the recursion formula

23

Hamilton-Jacobi Equations

and

We want to show that t* t* <

m

lim

E

j-

t = j

m.

To this end, we suppose

and will obtain a contradiction. We have then

lim

However, since h(y(t),a(t)) Rl?(t)l

5

E

+ o(R)

This shows that contradiction. €

T

< E

05 t

for a. a.

>

j =

for 0 bt < t*

+

~ ( 1 x 1+ t*(e

Thus lim

j-

t =

< t*.

my

j

u(R))/R)

=

T

j-

0.

j

by (2.11), it follows that

Therefore

for all j

€

N, which is a

which proves that a 8 A (x) and g

y(.)

xg (x,a). Since (2.11) implies (2.5) and (2.12) is exactly (2.6),

to check that

t

* e-t f(y(t),a(t))+

(2.5), it is enough to verify that

LO,-).

is integrable on

t + e-t H(y(t),O)

By ( A 3 ) , (2.2) and (2.12) we have

it remains only

[O,m).

In view of

is integrable on

24

Hitoshi ISHI

for t 2 - 0. The right side of this inequality is clearly integrable on as a function of

[O,m)

t

and so is its left side. Thus the proof is

Q.E.D.

completed. Proof of Theorem 2.1.

e Aad(x)

and

Let

E

and x e R

> 0

x(-) e Xad(x,a)

N

.

By Lemma 2.1 there are a

such that (2.5) and (2.6) with R = C1

hold.

We see then that -t

e

f(x(t),a(t))

-t

< e

for t L - 0, and thus J(x,a,x(.))

By (2.1) we have H(x,p) € A.

IE +

H(x(t),O))

C1

< 2~

+

Therefore, fixing x € R

,a

2 0. Using this, we have

€

(E

+ C1+

-g(x,a) - p N

that

for t

-

< e-t(c

-

Aad(x)

+

-

o(Cl))t

o(C,)

f(x,a)

H(x,O)l

- H(x,O).

for x, p

and x(-)

€

Hence

€

EN and

Xad(x,a),

a

we find

25

Hamilton-Jacobi Equations for a. a, t 2 0. Integrating this over

t + e-tf (x(t),a(t))+

Taking into account that find that J(x,a,x(.))

+ H(x,O)

V(x)

[O,T], with

2 - C1 - u(C1).

is integrable on

z - C1 - o(C 1),

+ H(x,O)

T > 0, we get

we

[O,m),

from which we conclude that

This together with (2.13) proves (2.3) for

N

Q.E.D.

X € E .

Remark 2.2. Theorem 2.1 implies that if a measurable function a: [O,T]

* A and a solution x(-) of (1.1) defined on [O,T], with T 1 0 , ~ ) so that

then one can extend these functions to

a

> 0, are given,

e Aad(x)

and x(.)

Xad(X,d.

13. Uniform continuity of the value function The objective of this section is to prove the following Theorem 3.1.

Under assumptions (Al) N

uniformly continuous on R 6: [0,m)

*

[0,m)

.

-

(A3), the value function V

is

More precisely, there is a continuous function

depending only on w

in

(A3) and

uR

in

(A21 such

that 6(0) = 0, 6(r) > 0 for r > 0 and

We need a generalization of Lemma 2.1 for the proof of this theorem.

Lemma 3.1.

Let R > 0, E > 0 and x € R

be a function such that

t

family of functions: (y,n) RN

x

RN.

N

.

Let p: RN x E N x [0,-)

cA

B(0,R)

+

p(y,n,t)

is measurable for y,n €RN and the

*

p(y,n,t),

with

t 2 0, is equicontinuous on

Then there are an absolutely continuous function 5 : [O,-)

sequences {ailieH

-t

and

{Xi)i,,

m

CL

(0,m)

N -+R ,

such that the following

Hitoshi ISHI

26 conditions are satisfied:

(b)

For each T > 0 there exists an nT O z t c T and

i > n T'

hold for a. a.

t 2 0.

€

n

such that X i(t) = 0 for

rm

Remark 3.1.

This lemma is closely related to the theory of relaxed controls.

See e. g . Warga 1161. Remark 3.2.

on

Condition (c) implies the integrability of

[O,m).

We will use the next lemma to prove Lemma 3.1. Lemma 3.2. 3.1.

Let T > 0, R > 0,

> 0 and

x B RN

.

Let p be as in Lemma

Then there exist an absolutely continuous function 5: [ O , - )

{ailie,CA,

{Xi}ieA

m

CL

(O,m>,

functions xn(-) e X(x,an) that

E

+ RN ,

ianlneA CA and a sequence {xn(.)Inen

of

fulfilling (a), (b) and (c) in Lemma 3.1 such

Hamilton-Jacobi Equations

27

uniformly for 0 2 t 5 T and (3.3 1

lim xn(t) = S(t) n-

uniformly on

[O,T].

The following lemma is needed to prove Lemma 3 . 2 . Lemma 3 . 3 . that x n

-+

Let x x

€

as n

R -+

N m

, fxnlnenCRN,

CA, T > 0

and

r > 0. Assume

and that

m

20

Xi(t)

(i = l,---,m) and

1

hi(t)

=

1

for 0

2

t (T.

i=l

{aklkeNCA, a sequence

Then there are an increasing sequence {n(k)}k8Ny {xk(.)lkeN

of functions xk(.)

€

X(X~(~), ak)

defined on

[O,T] and an

absolutely continuous function 5: [O,T] + R N such that

uniformly for 0 ( t 5 T, for a, a.

0 2 t

2 T,

and lim xk(t) = S(t) k-

(3.7)

Remark 3 . 3 . so

As the proof below shows, we can take

that x(t)

Proof. For

uniformly on

€

{al,.-.,aml

for t 2 0 and k

€

I%}en

[O,T]. in this lemma

H.

notational simplicity we assume T = 1. For any n

€

N

28 and

Hitoshi ISHI 1 5 i 2 m, define A?)

€

Lm(O,l)

Xi(s)ds (k-1) /n

by

k-1 if - n2 t

k n

< -

for some k = l,.--,n.

Then

p

(3.8)

for i = l,-.-,m as n

+

-

+ Ii

1 in L (0,l)

and

(3.9)

for 0 2 t 21 and n 8 N.

for n 8 N, 0

zt < 1

Next we set

and define a n

E

A

for n 8 N

by

(3.10)

For each n 8 N

we choose an xn(-) 8 X(x ,a ) n n

defined on

[O,T].

The existence follows from ( 3 . 4 ) together with the standard local existence theorem. Moreover ( 3 . 4 ) guarantees that

{xn(*) lneN

bounded and equicontinuous family of functions on

forms a uniformly [0,1]. In view of the

Ascoli-Arzela theorem we can extract a subsequence 'Xn(k) ('I 'ken {xn( .) lneN

such that lim ~ ~ ( ~ ) ( t )= c(t)

(3.11)

kfor some 5

8

C([O,l], RN). Note here that

uniformly on

[0,11

Of

29

Hamilton-Jacobi Equations

r t

0 2 t 21,where o(1)

for all n

€ A

and

as n

and

[nt] denotes the integral part of

+ m

+

0 uniformly for 0

2

t

51

nt. Therefore, using

(3.8) and (3.11), we have

as n = n(k)

-+

m,

and hence

This proves (3.6) and that 5

is absolutely continuous.

Similarly we see that

uniformly for 0 2 t 5 1 as n = n(k) * CA, { x ~ ( (.)Iken ~ )

Proof of Lemma 3.2.

c Lm(O,m)

and

Thus we know that {an(k))ken

€

a

WN

Q.E.D.

have the required properties.

Step 1: We will choose {ailiel C A

for all x

for x, p € R N and

5

m.

.

€ A.

For simplicity we write

and

{Ai(*;x))iGn

Hitoshi ISHI

30 = 0,

h(x,p,a) Since inf a 8 A f? RN

for each x, p

E

> 0, we can select

such that h(x,p,ax,p) < ~ / 2 . AS

€12) is open for a, p

h(y,q,ax,p) C A

fixing

8 RN,

{(y,q)

8

a XYP RZN

A

8

I

we can choose a sequence

ai ien

such that <

for any

S >

0 and some m = m(S)

Now we fix S > 0, and let m and

8 [0,m)

x

n.

8

8 A

be such that (3.12) holds.

Let

t

We claim first that

8 B(0,S).

there exist A1,...,X m € R m .~. X = 1 such that i i=l

(3.13)

51

satisfying X i => 0 for all

i and

1

where rl =

1"i=l X ig(x,ai).

To see this, we introduce the notation: m A = { A = (Xl,---,Xm)€RmI Xi 2 0

for all i = l,--.,m and

1

Xi

€

A.

=

11,

i=1 m

1

r l ( ~ )=

Xig(x,ai)

for

X

8 A,

i=l m $(A)

= {y 8 A(

1

5;)

vih(x,p(x,O(X),t),ai)

for X

i=l

for X

€ A.

It follows from the continuity of

$: A

P(A)

is upper semicontinuous (see Kakutani [lo]).

.+

Kakutani's fixed point theorem [lo] to X

8 $(A).

IJJ

0, p(x,-,t)

and find X

That is, (3.13) holds.

Now, keeping x

m

0 and is moreover a compact convex subset of

Then, by (3.12), + ( A )

8

B(0.S)

fixed, we show that

8

and

h

R

that

Now we apply A

such that

Hamilton-Jacobi Equations (3.14)

Since A +

p( E?)

there exists a measurable

-+

fi(A,t)

A: [O,-)

is continuous and

-+

A

such that

t + fi(A,t)

is measurable, $:

is measurable in the sense of [3, 53.11.

(3.13), nonempty for

31

$(t)

is closed and, by

t 2 0. Hence by a measurable selection lemma (see, e.

g., [3, Theorem 3.1.11) there exists a measurable function A: such that A(t)

e

[O,m)

$(t)

inequality in (3.14).

[0,-) -+Rm

for all t 2 0. This inclusion is equivalent to the

Thus we conclude (3.14).

The observation (3.141, the continuity of h

and p(*,*,t> and

the standard compactness argument together imply that. for each x 8

RN, there exist r(x)

[O,m)

(3.17)

+ [O,m)

8 (O,l), m(x)

(i = l,..-,m(x))

inf r(x) x8B (0,S)

EI

N

and measurable functions A (-;x): i

having the properties listed below:

>

0

for S > 0.

32

Hitoshi ISHI

(3.18) N Choose a continuous function 'I: E -+ (0,-)

N for x elR

.

S(t) = x

Note that if

+

'1

s

mf)

5

and

N

R )

f C([O,-),

for s

Xi(u;x)g(c(u),ai)du

2

for s

+

t 2s

Step 2: We will construct uous function 5 : LO,-) Lemma 3.1 for x e R

2t 2

s

+

T(x),

N

T(x).

CLm(O,-)

and an absolutely contin-

+ E N satisfying conditions (a), (b) and (c) in

.

To do this, we use a step-by-step argument.

N Let x e R , and define

IT 1

C(O,-),

{Ai}ieNCLm(O,m)y

jam C([O,-), RN) and {5j)jamCRN as follows. Set

5

satisfies

is1

a B(x,r(x))

then c(t)

s 1. 0

such that

j

for 1 2 i

x p

T~ = T(X)

and

2 m(x),

=

for i > m(x) for 0

t < T ~ . Let

5

N ~([O,T~], R )

8

+ fi IT=, xi(s)g(c(s),ai)ds

for 0 5 t 5

be a solution of S(t) = x T ~ .

(Its existence follows from

Lemma 3.3 or the argument used to verify the existence of xn(*) of Lemma 3.3.) 0

2

t

'Il.

Set

5,

=

C(T~).

It follows that c(t)

Hence, by (3.16),

(3.19) for 0

t < T ~ where ,

Next we set

n(t)

T~ = T~

-

+ ~(5,)

Xi(t)g(S(t),ai). and

e B(x,r(x))

in the proof for

Hamilton-Jacobi Equations

if 1 2 i

Ai(t;C1)

33

zm(S1),

=

Ai(t)

if i > m(C1) 1=

< t < T

for

T

S(t)

t = x + 10

= E,(T~).

{o

2'

Extend 5

Ziz1 A i(s)g(5(s),ai)ds m

j

C([O,T*), IR ), where

Notice that if

T*

=

T* =

limj +

m

then

m,

f

N

C([0,r2], R )

holds for 0 5 t

c (0,=), {AilifaC

IT 1

N

5

so that

A s above, we see that (3.19) holds for

procedure to obtain B

[0,-r2]

to

T ~ .

02 t < Lm(O,.r*)

We set

T ~ .

and

Xi(s)g(5(s),ai)ds

m.

This implies

for all t

+

T(cj)

Repeat this

5 < T*.

satisfies conditions (a) and (b) and

Thus, it is enough to show that T* <

52

Then (3.19) holds for 0 (t

'j*

(3.19) is equivalent to the first inequality in (c) since 5(t)

+ 1;

and

=

x

2 0.

T* =

0 as j -+

00.

m.

To the contrary, we suppose Combine the inequality

m

-

H(S(t),q)

1

+ f(5(t),ai)}

Ai(t){s(5(t),ai)*q

i=1 (which follows from the definition of H), with q

-

i(t)/li(t)\,

and the first inequality in (c) to get

Therefore, 15(t) inf T*

=

E: A

-

XI

5

~(5.1= inf J

= p(E(t),i(t),t)

j

(E

+

o(R

T(~(T~))

+ 1))~"

for 0 2 t <

T*,

and hence

> 0. This is a contradiction, which shows

m,

Step 3: For any T > 0 we will choose 5 f C([O,-), CLm(O,m)

satisfying (a), (b) and (c) so that 5

solutions of (1.1) on

[O,T].

EN) and

{AilieA

can be approximated by

34

Hitoshi ISHI

functions x(j)(-) n

lowing manner.

E

~(x,a, )I'(

Set ~ ( l )=

on

[O,.r(j)]

for every j

and

5")

(3.21)

j

E

N

in the fol-

and

T(X)

for 0 2 t < ~(l). By virtue of Lemma 3.3, there exist sequence {xn(1)

E

of functions x(l)(.) n

8

x ( , a : )

{an(1)} , & A , on

a

[O,T(')]

C([O,T(~)]; RN) satisfying the following conditions (3.20)

and (3.22)

j

j'

for j = 1.

uniformly for 0 2 t 2 T (j1

.

A s in Step 2, we have

Applying Lemma 3.3 with replaced by

[T('),T(~)],

{x

1

n nfN

=

and the interval

we find that there exist {a:*'

lneN

[O,T]

c A, a sequence

35

Hamilton-Jacobi Equations Ixn(2) f

(*)IsN

a X(x,aft2))

of functions x(~)(-) n

on

[O,T(~’] and

5 (2)

c(2)(~(1))

=

C([~,T(~)], EN) such that (3.2012, (3.2U2, (3.22)2 and

are satisfied. The inequality

is valid for j = 2, where q(j)(t) Put

c2

=

5

(2)

=

s(j) E

(3.22) j’

1

e N, we can define

cA

f x(x,aLj))

)a()’(.

n hold.

and (3.24)

j

RN 1, {a:J)lneA

c([o,T(j)],

{xn(J 1 ( . ) I ~of~ functions ~ (3.21)

X~’)(t)g(5(’)(t),ai).

Lastly we

(p>

Repeating in this way, for any CLm(O,T(j)),

I;=,

T(j)

> 0, CXij))ifA

and a sequence

on

[o,? (’’1

s o that ( 3 . 2 0 )

j’

It is verified as in Step 2 that

j

lim r ( j ) =

a.

1What we saw in Step 2 is that we can extend

(j),

5”)

to

[0,m)

CAI ( j1IieN

conditions (a), ( b ) and (c) are satisfied for the extended c(j),

Thus we conclude that, for any T > 0, there exist 5

{AilienCLm(O,-),

{a 1 n n€N C A

belonging to X(x,an)

5 and {Ail

€ C([O,m),

that and

EN),

of functions x ( - )

and a sequence

such that

so

satisfy (a), (b) and (c)

and that (3.25) uniformly for 0 2 t 5 T and (3.26)

lim xn(t) = S(t) n-tm

The proof is now completed.

uniformly on

[O,T] Q.E.D.

36

Hitoshi ISHI

(a), (b) and (c).

m

N

Let 5 8 C([O,m),

Proof of Lemma 3.1.

R )

{hiIienCL

and

(0,m)

satisfy

As in Step 2 of the proof of Lemma 3 . 2 , we have

5

1t(t)l

E

+

o(R + 1)

for a. a.

t

'> 0,

and so (3.27)

[S(t)

-

XI

5

(E

+

U(R

+

1))t

for

These,together with (c) and that H(S(t) , O ) L

t

2 0.

m

- liCl

hi(t)f

(S(t) ,ai)

for

t 2 0, yield .

m

< E

+

(E

+ o(R

+

t 1))R

u(R) + C1 + C1(E

+ o(R+l))t

for a. a.

t L 0. This shows that there is a T > 0 independent of the

choice of

6

and

CX,}, ,

such that

Replacing T > 0 by a larger number if necessary, we assume in addition that (3.29)

-T e {IH(O,O)I

+ C1(IxI +

ET + a(R+l)T

By virtue of Lemma 3 . 2 , we can select CLm(O,-),

a 8A

and

x(-) 8 X(x,a)

(c) are satisfied and that

and (3.31)

+

3)

+ o(C,))

6 8 C([O,-=),

defined on

t.

N R ), {Ai)i,n

[O,T] so that (a), (b) and

31

Hamilton-Jacobi Equations hold for 0 2 t & T. Choose

8 Aad(x(~))

JMT) ,B,Y(.I)

and y(.) 8 Xad(x(T),B)

so

that

+ v(x(T)).

<

Then, by Theorem 2.1, (3.27) and ( 3 . 3 1 ) , we get

I +

C1(IxI

+ ET + a(R+l)T +

3)

+

O(C,>.

Now we define for 0 5 t 2 T, for t > T and for 0 2 t L T , for t > T. Note that

8

% ( a >

Xad(xYG)

I

and that

T

~(x,a,a(.))

=

+ e-TJ(x(T),B,y(-)).

e-tf(x(t),a(t))dt

0

Combining this with (3.30)

,

( 3 . 3 2 ) , (3.28) and ( 3 . 2 9 ) yields m

V(X) 2 ~(x,a,%(.))

<

E

+

f e-ti=l Xi(t)f(F(t),ai)dt. 0

Q.E.D.

Thus we have completed the proof. The dynamic programming principle is stated as follows. Proposition 3.1. (3.33)

V(x)

For any t > 0 and x =

inf

{c

8

RN one has

e-’f (x(s) ,a(s))ds

+

e-%(x(t))

1,

38

Hitoshi ISHI

where the infimum is taken over all

CI 8

Aad(x),

x(-)

8

Xad (x,a).

We refer to Lions [ll] for a proof of Proposition 3.1. Let T > 0, R > 0,

Proposition 3.2. Lemma 3.1.

E

N , and let p

> 0, x 81R

be as in

Then there exist an absolutely continuous function 5: [ 0 , m )

{aijisAC A

and

{Ailien

CLm(O,m)

-+

7RN,

satisfying (a), (b) and (c) in Lemma 3.1

such that

I

t

V(x) 2

(3.34)

m

e-'

0

for 0 < t

1

+

Ai(s)f(5(s),ai)ds

e-tV(5(t))

+ e-t(3C1 +

20(C1))

i=1

2 T.

By Lemma 3.2, we can choose an absolutely continuous function s. [Otm)

N

* IR , {ailisR. c A, {Xilie,

where x ( - 1 6 Xad(x,an), n

C Lm(O,m),

{anlneRC Aad(x)

and

6:

{xn( .) Inem,

satisfying (a), (b) and (c) such that

and (3.36)

IXJt)

hold for 0

t

5 T and n

8

A.

- 6(t)(

1

n

By Proposition 3.1, we find that

t V(x)

<

e-sf(xn(s),cxn(s))ds

+ e-tV(xn(t))

for t 2 0.

0 Thus, using (3.35), (3.36) and Corollary 2.1, we have

+ for 0

t

2 T. Letting n

-+

m,

1 + 3c1 e-t(C1 n

+

20(~ 1) 1

we conclude (3.34).

Q.E.D.

39

Hamilton-Jacobi Equations Proof of Theorem 3.1. For x (1xI2

+ u2)li2.

r 2 0 , e(r)

RN and

€

We choose a 8

C”( R )

€

0, we let <x>

p >

!J

denote

e(r) 5 r for

such that 0

r for r 2 1 and e(r) = 0 near r = 0. For v > 0, we set

=

ev(r) = ve(r/v)

for r f : w .

Our goal is to prove that (3.37)

for any

To this end, we fix

E

(3.38)

> 0

=

and choose C

5

w(r)

We define T

> 0, E = E ( E ) > 0 and

y =

(0,l) such that

€

Y(E)

6 = B(E)

0 there exist

E >

E

+

=

C(E) > 0

+

20(C1))

=

y = min {&,

Let x, y e~~

> 0

= Y(E)

by

E,

E = max {2C, 48€eT, 32(3C1

(3.39)

that

for r 2 0.

Cr

T(E) > 0, E = E(E) > 0 and y 2e-T(3C1

so

+

20(C1)),

2C1},

$ 1.

satisfy

(3.40) and choose

!J >

0 so that

<x

(3.41) and then v > 0 (3.42)

so

-

y>y

<

r 16

!J

<

-T 5 16

that VYP

Y-1 eT

We may assume V(x) )V(y);

and

vyEpY-l

<

E.

otherwise we have nothing to prove. We

40

Hitoshi ISHI B E Aad(y)

choose

and

y(.)

e X ad (y,B)

so t h a t

(3.43)

W e a p p l y P r o p o s i t i o n 3.2 w i t h p ( z , q , t ) = YE<^

t o obtain tion

5:

-

{ailienC A , {AilieN C Lm(O,m)

and a n a b s o l u t e l y c o n t i n u o u s func-

-+EN s a t i s f y i n g ( a ) , ( b ) , (c) and ( 3 . 3 4 ) .

[0,-)

Using ( A 3 ) and

(3.381, we c a l c u l a t e t h a t

f o r a . a.

t

-

0.

Next we n o t e t h a t , by ( 3 . 3 4 ) and ( 3 . 4 3 ) ,

for all

0 < t

T.

Thus, by ( 3 . 4 4 ) and C o r o l l a r y 2.1, w e have

> a v II

i-l U

h

I

0

V

m rn 0

w w U

v li

$4

v II 0 $4

w

V

rl

rl rl

f

d

3 U

w h

0

a

9

H

v II

2

U d

d

U

v II

G

U V

2

0 al

c U a

8

n

?-I

s

rl

II

zl n

0

U

h

V

I

0

U

h

*IF V

V

m e m 0 U

h h

U v

h

'h I U

h

V

:4F -

a,

7

V

7

+

V

>A

l-i

U

h V

I

h

m

V

v1 I 0 $4

lu

rl V

a n A U

v II

*

U

w

V

4F

V

* v /I

+ a A n U

V

I

h h

U L P

V

m

U

c, m

m a

U $4

h

n C d

+ a h

m

U

M

X

m

I

U

h

n

rl

U

v I1

X

I

h

* s

al

0 U

V

U

zl 2 7

V

4F

w

U

0

h

I

h

N

9

al

a

V

m

U

4

+

N

b

V

U

rl

n

h

v II

rl

ld

rl

$4

0

d

(0

m

U

zl

m L:

U

V

LAr

v

U

n

I

h

V

n U

A

V

w

V

7

w

W

al

0 U

m 0

rl

U

I

w

>

rl

I

U

w

0

4

w 14

MI4

n

0

3

al

+

N

rl

U

m

V

0 U

I 0) N

+

zl

r

rl

W

aJ A

*zl

N

h

U

V

m

2 I

rl

0,

m

h

V

m m

h

v

sm

n

c

a

W

W

I

U

4

n V

v 4F

U

0

A

h

W

0

>

+

W

I U d c

B

+ m

w

8

a

v II

d

0

m

n

4

m V

w

; rl

0 w

4

$4

11

U d

0 A

al

V

w

U

L) d C

V

rn

+

n U

U

V

9

v

0

d

a

I

h

V

$4

h

C

w

L)

V

V

u

U

m C

d U

P

w

lu

E

*

d

c

H

G U

d U

nr

al

L: U

U

U

0 a,

o

l.l

w

a rl

rl

U

I

h

v

h

A

11

+

\D

+

V

U

n

I

V

rl

4F

U

I

+

U

U v

4F -

V

h

h

-

A

U

0 U

V

rl U

h

h

w

n

m

m V h

m

a

h

n -

m V 'h

h

h

V

4F V

U

W *

+

h

m

W

I h

1

W

7

m

m

>

wV

4F

u

n

I

h

V

n

*a

I

rl

a

-4P -

V

n v1

m

W

Y

Id

0 I

m al O

L_

U

+

W

m v I1 n

h P

V

I

X

n V

I

0

m

a

>

v II

., 5

al

2

a

0

m

m

h

U

c: U

d

m

V

42

Hitoshi ISHI t = T, we now find as above that

Using (3.45) with (3.46)

V(x)

-

V(y)

2 3~ + E<x -

y>'

+

y>'

< 5~ -

After sending p

.C

E<x

-

IJ

+ vyEpY-' +

2e-T(3C1

+

20(C1))

by (3.42) and ( 3 . 3 9 ) .

U

0, this proves assertion (3.38).

(3.37) the existence of a function 6: [0,=)

+ [0,-)

We conclude from such that

6(0) = 0,

6(r) > 0 for r > 0 and (3.1.) holds.

Q.E.D.

84. The Bellman equation In this section we will prove the following Theorem 4.1.

The value function V

is the (unique) viscosity solution of

where H i s defined by (2.1) and satisfies (Al), (A2) and (A3). We refer the reader to [4, 51 for the definition of viscosity solutions. The uniqueness of the viscosity solution of (4.1) follows from [8]. Such uniqueness results were first obtained in [5] in the class BUC( R N ) . The basic idea of our proof is the same as that in [ll]. Proof. Let

1

N

$ 8 C ( R ), and assume that

V

-

$

attains its local minimum

at xo 8RN. We will show that (4.2) Since V

is uniformly continuous by Theorem 3.1, we may assume that

C2 c sup {ID$(x)l

I

N x 8R 1 <

m

and that V

-

$

attains its global minimum

43

Hamilton-Jacobi Equations at x 0' For any a f Aad(xo)

> 0 and

E

and x(-) f Xad(x0,a)

By the definition of

if

s

2

0.

so that

H, we have

+ 0, and

g(x(s),a(s))

for a. a.

t > 0, by virtue of Proposition 3.1, we can select

so

Plug this into (4.3) and use Theorem 2.1 and integration

by parts, to get

(4.4) for some constant C > 0 independent of 3 Let

if

0< 6

g(x(s),a(s))

for a. a. obtain

s

1 and R = C2

+ 0.

+

E

and

t.

1. We see as above that

Hence

2 0. Inserting this into ( 4 . 3 ) and integrating by parts, we

Hitoshi ISHI

44

This inequality, with

6 = 1, and (4.4) show that Ix(t)

for all 0 (4.5) by

< t

t

-

xoI

5 C4t

1 and some C4 independent of

E

and

t.

Thus, dividing

and sending t 4 0, we have

this implies (4.2). Now we assume that V

-

Cp

attains its local maximum at xo

€

RN and

will prove that

To this aim, fix a e A that a(t)

=

a

and choose a

for 0 2 t

for t > 0, and hence

e Aad(xo)

and x(*) e xad(xo,a)

2 1. By Proposition 3.1, we have

so

45

Hamilton-Jacobi Equations for sufficiently small t > 0. Dividing this by

t

and letting t J. 0, we

conclude

Since a

15.

is arbitrary, we see that ( 4 . 6 ) holds.

€ A

Q.E.D.

Representation of Hamiltonians N

Let H: R

x

R

N

* R be a given function satisfying (A2) , (A3) and the

following condition. (A4)

p

+

H(x,p)

is convex for all x f R

If necessary, replacing w (5.1)

w

N

.

by a new one, we may assume in addition:

is increasing and concave on

[O,m).

This implies that

In order to state the main result in this section, we define

where

x:

[O,m)

+ [O,m).

for the function x: r

* Jw(r)+

w(r).

Hitoshi ISHI

46

To prove Theorem 5.1, we need the next selection lemma.

N Lemma 5.1. Let K: R and convex for x € R

N

-+

.

M P ( R ), and assume that K(x)

is nonempty, closed

Assume further that

K(~)

(5.5)

n B~(o,R) +

0

for all x, y € R N and some constant R > 0, where

x:

[O,-)

-+

[0,m)

is an

N increasing, concave, continuous function satisfying ~ ( 0 ) = 0. Let x € R 0

and

Co € RM satisfy Co

€

K(xo).

N Then there exist a function 5: R

and a constant L > 0 depending on 5,

for x, y € R

N

and R

M

-+ R

such that

.

Proof. For any x 8 RN , taking into account that K(x)

is a nonempty, closed,

M R , we set

convex subset of

C(x) = the nearest point of

K(x)

c0.

from

It is obvious that (5.6) and (5.7) hold. To see (5.8), we first note that

1C(x)

- col 2 lCol +

Let x, y € R N and assume

1S(x)

- Col 2

by (5.5).

definiteness. We may assume point from C(y)

n

=

t o + ((C.(y)

1C(x)

-

C o l > 0. Let

1C(y) rl E

on the line determined by two points

-

Co) -v)v, where v = (C(x)

-

Co)/lS(x)

R

-

for x € RN

Col

for

RM be the nearest

C(x)

- Col.

and

C,,

i. e.,

A simple

41

Hamilton-Jacobi Equations calculation or plane geometry shows that 1S(x)

-

S(y)I2

=

-

lS(y)

Sol

-

-

S(Y)~

-

IS(x)

SollS(x)

-

- Sol 2 + 21S(x)

- S011S(X)

2

2 2(1S0[ + R)X([x

- YO. Q.E.D.

=

Jw(r).

E

RN

.

Let N

Define H1: B

rll

rll.

This completes the proof. Proof of Theorem 5.1.

-

by ( 5 . 4 ) , we conclude that

)

[S(x)

2

x

be the function on N

[0,-)

+ R by H1(x,p) = H(x,p)

R

defined by

- H(x,O)

i(r)

for x, p

We will prove H1(x,p) = max

(5.9)

N for x, p 6 R

.

c-

C(X).P

- n(x)l

(TI,<)

e A(i,H1)I

It is clear that (5.3) holds.

By the definition of A(f,H ), we have 1

H1(x,P) 2

SUP { -

G(X).P

- rl(x)l

h , c ) e A(:.H~)I

N for x, p E R , Therefore it is enough to see that there exists A(i,H ) 1

for each x, p

(5.10) +

- 5 (XI 'P - IT (x).

P( RN+l) by setting

{ ( n , ~ ) B R X I R ~ I-r;.p-n(~~(x,p)

=

N for x € R

N

.

It is easy to check that K(x)

N for all x f R

.

6

RN such that

H1(x,p)

We define K: R

K(X)

E

(n,~)

for

N pen}

is nonempty, closed and convex

Hitoshi ISHI

48 We claim that

N for x, y f R so

.

To see this, fix x f y

e RN and choose

E

2 0 and CE

> 0

that

5

2w(r)

E

+ CEr

for all r 2 0 and

(5.12) 2u(r) Let

(rl

for p

,SX>

8 R

N

€

K(x),

+c

E

- yI.

Ix

and observe that

by (A3) and (5.12).

P N is concave on R

(rlyycy)

=

.

+

-

cx'P

-

Note that rl,

-

E

- CE(X - Y((1

(PI)

Therefore, by a standard separation theorem, there exists

f R x RN such that

N for p B R

.

From this we see that

and (Sy for p B RN

.

-

5,bP

+ rly -

Inserting p

=

rlx

= <

E

+ CEIX - Yl(1 + lPl)

0 into this inequality, we have

49

Hamilton-Jacobi Equations

and taking p = t(5

cy f

if

rl

I,

X

dividing by

t

and sending t

+

01

(n,5) 8 K(y)

and

n

< rl',

(n',<)

then

8

K(y).

Thus, replac-

by a larger number if necessary, we conclude that there exists

Y

(ny,Cy)

Cy - 5

-

Sx, we have also

Note that if ing

Y

E

such that ( 5 . 1 4 ) and

K(y)

hold. We have proved (5.11). Next we show that

for x E R N and some constant R > 0 . a separation theorem there exists a

-

5 . p _I H1(x,p) X

for p

IcxI Thus

(o,Cx)

E

K(x)

Fix xo, p0 8 R 50*po

- no.

such that

R

.

If

, and

Cx

9 0,

then

2 o(l) N

for x E R

choose

for x E R N , by

5, € R N for any x € R N such that

2 H1(X,- cx/Icxl)

n BN+'(0,o(l)) N

-

€

N

Since Hl(x,O) = 0

(noyc0)

8

by (A2).

which proves (5.15).

K(xo) N

so

that H1(xOYpO) =

By Lemma 5.1, there exist n: R + R , 5: RN +RN and L > 0

rl(xo)

=

no, s(xo)

=

c0,

Hitoshi ISHI

50 Is(x)

-

L(Y)I

N

5 Li(lx - Y O

for x, y 8 R

.

The proof is now finished.

56. Representation of viscosity solutions As in the previous section we assume here that a function H: RN -f

EN is given and satisfies (AZ), (A3) and (A4).

x

We assume also (5.1).

R

N

Our

purpose is to represent the viscosity solution of

i

u

+

N

in R ,

H(x,Du) = 0

u 8 UC( EN)

as the value function of an appropriate optimal control problem. By Theorem 5.1, we have

for x, p € R

N

, where x is the function: r -+ Jw(r) + w(r)

on

[O,m).

We r e a r d A = A ( x , H ) as t h e m e t r i c sDace equipDed w i t h t h e u s u a l m e t r i c N IRN+l i n C ( IR , ),

for a = (n.5)

€

We d e f i n e f : I R N x A - + I R

A(x,H).

and g : W N x A * I R

It is clear that f and

g

by s e t t i n g

satisfy (Al).

It is

immediate from (6.2) that (6.3)

H(x,p) = max { - g ( x , a ) . p a6A

-

f(x,a)l

for x, p 6 R

N

.

Thus, in view of the uniqueness of the viscosity solution of (6.1), we have the following theorem as a consequence of Theorem 4.1.

51

Hamilton-Jacobi Equations Theorem 6.1.

The viscosity solution u of (6.1) is represented as

{I

m

u(x) = V(x) r inf

(6.4)

e-tf(x(t),a(t))dtl

e Aad(x), x(*)

a

8 Xad(x,a)l.

0

Remark 6.1.

This theorem extends a result of Lions [ll].

Evans-Souganidis

[6] treated the same representation problem without assuming (A4) but requir-

-

ing H

to be represented as a "max-min" of

f and

g satisfy some Lipschitz continuity assumption.

g(x,a,b)-p

-

f(x,a,b),

where

References

-

1.

G. Barles, These de 3e cycle, Univ. Paris IX 1983.

Dauphine, Paris, 1982/

2.

N. E. Barron, L. C. Evans and R. Jensen, Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls, to appear in J. Diff. Eq.

3.

F. H. Clarke, Optimization & Nonsmooth Analysis, Wiley, New York, 1983.

4.

M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutions of Hamliton-Jacobi equations, Trans. AMS. 282 (19841, 487 502.

5.

M. G. Crandall and P . L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. AMS. 277 (1983), 1 - 42.

6.

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, to appear in Indiana Univ. Math. J.

7.

W. H. Fleming and R. W. Rishel, Deterministic Control, Springer-Verlag, New York, 1975.

8.

H. Ishii, Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations, to appear in Indiana Univ. Math. J.

9.

H . Ishii, Remarks on existence of viscosity solutions of Hamilton-Jacobi equations, Bull. Facul. Sci. & Eng. Chuo Univ. (1983), 5 - 24.

and Stochastic Optimal

a

10. S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J., 5 (1941), 457 - 459.

11. P. L. Lions, Generalized Solutions Boston, 1982.

of Hamilton-Jacobi

Equations, Pitman,

52

Hitoshi ISHI

12. P. L. Lions, Existence results for first-order Hamilton-Jacobi equations, Ric. Mat. Napoli, 2 (1983), 3 - 2 3 . 13. E. Michael, Continuous selections. I, Ann. Math.,

63

(1956), 361

-

382.

14. P. E. Souganidis, Thesis, Univ. of Wisconsin, 1983. 15. P. E. Souganidis, Existence of viscosity solutions of Hamilton-Jacobi equations, to appear in J. Diff. Eq.

16. J. Warga, Optimal Control of Differential Academic Press, New York, 1972.

Functional Equations,

Lecture Notes in Num. Appl. Anal., 8, 53-71 (1985) Recent Topics in Nonlinear PDE II, Sendai, 1985

The Existence of Nonstationary I d e a l Incompressible Flow i n E x t e r i o r Domains i n R 3

Keisuke Kikuchi Department of Pure and Applied Sciences College o f General Education Tokyo 153, Japan

Introduction. We c o n s i d e r t h e motion of an i d e a l incompressible f l u i d p a s t a f i n i t e number of i s o l a t e d r i g i d bodies 01,

The v e l o c i t y v

pressure p

=

-

...,0,

i n R3

( v1( x , t ) , v2 ( x , t ) , v3 ( x , t ) ) and t h e ( s c a l a r )

p ( x , t ) of t h e f l u i d motion a r e governed by t h e

m l e r equation i n fi = R ~ , ( O ~ U . . . ~ O , )

av at

(1)

+ (v.p)v + vp

=f,

div v = 0

t

E

CO,T1

s u b j e c t t o t h e following c o n d i t i o n s a t i n f i n i t y and on t h e boundary S

- an

s a t i s f y i n g the i n i t i a l condition

where f

-

f ( x , t ) i s a given e x t e r n a l f o r c e v e c t o r , v o ( x ) i s

a given i n i t i a l v e l o c i t y and v , , i s

53

a given constant v e c t o r .

.

Keisuke KIKUCHI

54

The purpose of t h i s paper i s t o g i v e t h e o u t l i n e o f t h e proof o f t h e e x i s t e n c e o f a s o l u t i o n {v, p \ o f (l), ( 2 ) and

( 3 ) which s a t i s f i e s t h e a s y m p t o t i c c o n d i t i o n a t i n f i n i t y t h a t v converges t o voo f a s t e r t h a n 1x1- (I+') f o r a c e r t a i n 5

2

0.

For f u r t h e r d e t a i l s , s e e 1161. The problem o f e x i s t e n c e and u n i q u e n e s s o f s o l u t i o n s o f t h e E u l e r e q u a t i o n has been c o n s i d e r e d by s e v e r a l a u t h o r s . R e c e n t l y , when Cl.

=

,'R

t h i s problem w a s s t u d i e d by Swann L181,

Kato 1131, S a r d o s and F r i s c h [ 2 ] and C a n t o r [ 53. bounded i n R ' ,

When Q i s

t h e problem w a s s t u d i e d by Ebin and Marsden

181, Swann [193, Bourguignon and R r g z i s 141, Temam 1201 and Kato and L a i [141. I n t h e two-dimensional c a s e , t h e e x i s t e n c e of a g l o b a l s o l u t i o n w a s s t u d i e d by J u d o v i x 1111 and Kato [121 ( i n a bounded c a s e ) , and by Kikuchi case).

1151

( i n a n unbounded

Among t h e above p a p e r s , t h e works of C a n t o r [ 5 J and

Swann (19) e s p e c i a l l y i n s p i r e t h e iderz of our p r o o f .

I n 151,

Cantor c o n s t r u c t e d s o l u t i o n s i n t r o d u c i n g t h e weighted Sobolev space o v e r 2 3 , on which t h e L a p l a c i a n i s a n isomorphism ( s e e Lemma 2.4 i n t h i s p a p e r ) .

T h i s work s u g g e s t e d t o t h e a u t h o r

t o u s e t h e c o r r e s p o n d i n g s p a c e on t h e e x t e r i o r domains.

The

advantage t o t h e p r o p e r t y o f t h e weighted Sobolev space makes possible the construction o f solutions o f l i n e a r e l l i p t i c system ( s e e P r o p o s i t i o n 3.1).

Our proof i s e s s e n t i a l l y a

modified form o f t h a t of Swann 1193, who c o n s t r u c t e d s o l u t i o n s by c o n s i d e r i n g t h e v o r t i c i t y e q u a t i o n , w i t h s u i t a b l e t r e a t m e n t

o f t h e e x t e r i o r problem ( s e e P r o p o s i t i o n 3.2).

55

Nonstationary Ideal Incompressible Flow

Notations and statement o f t h e r e s u l t .

I.

The domainn i s simply connected and R 3\ f i

...,Om;

numbers o f compact components 01,

+...+Sm (S 20

by I

t h e boundary S

=

S1+

= 20 .) i s s u f f i c i e n t l y smooth.

F o r 1< -p s

consists of m

J

c- 00,

t h e norm i n LP(n) i s denoted by I

( i n t e g e r ) and 1 < -p

-

(00,

t h e norm i n WsVp(Q)

For s 1 - 0 ( i n t e g e r ) , CE(fi)

*

b.

For

is denoted

i s the s e t of a l l

u r C s ( n ) such t h a t a l l t h e i r d e r i v a t i v e s up t o o r d e r s a r e bounded.

C:(z), s being non-negative

integer, i s the s e t o f

a l l f u n c t i o n s t h a t belong t o Cs(n)and have compact support

in

Ti. Let d ( x ) = (1+\x\2)1/2. F o r p

ME,x

denotes t h e c l o s u r e o f C:(fi)

2 - 1, h 2- 0 and

s

2 - 0,

w i t h respect t o t h e norm

We w r i t e

For a vector-valued function u, ~ - n i s\ t~h e outward normal component o f u on S, and we w r i t e DUE(=

au

; j = 1,2,3),

j

2 a 2u D u = (- a x . axk' j,k = 1,2,3). J

Now we can s t a t e our main r e s u l t .

-

56

Keisuke KIKUCHI

Theorem 1.1.

(i)

Let p

>3

5

and 0

1

(n),

vo belongs t o

C,(E)

d i v vo = 0 ;

v o * n l s = 0;

C

<

1-3/p.

Assume t h a t

r o t v O tM Y , 6 + 2 i l i m vo = vm, \Xl-+Oo

f belongs t o C ( I O , T ) ; C b ( ~ ) n C 1 ( , ) ) ,

(ii)

r o t f c LO"( ( 0 , ~;M:,~+~). ) Then t h e r e e x i s t s To vw, r o t f and \v,

n,

> 0,

To

LT,

depending only on r o t vo,

such t h a t (11, ( 2 ) and ( 3 ) have a s o l u t i o n

p ) on 10,To] s a t i s f y i n g

Such a s o l u t i o n is unique up t o an a r b i t r a r y f u n c t i o n o f t which may be added t o p.

11.

Preliminaries. P r o p e r t i e s of M:,~.

2.1.

The Htllder i n e q u a l i t y implies Lemma 2.1.

Let

h 2 0.

If & E L P ( n ) ,

t h e n u eLr(Q)

provided t h a t 3p/(hp+3) 4 r G p . Lemma 2.2. p

>3

and h'L_0.

(The Sobolev imbedding theorem.) Then M Z , n C

(2.1)

I n particular, i f

k

> 0,

then

'$-'(a)

and

Let s

2 1,

57

Nonstationary Ideal Incompressible Flow

Lemma 2.3. s

> 3/p, F 2- 0

(Cantor and 0

153: Proposition

<- k <- s ,

1.1.) If p

> 1,

then t h e pointwise multiplica-

t i o n o f functions:

induces a continuous map. Lemma 2.4,

(Cantor 163: Theorem 2.

and Walker 1171 : Theorem 2.1.)

<6

3/Q-3

Let P

Also see Nirenberg

> 3, 2 0

and

Then t h e Laplace o p e r a t o r

(1-3/p.

i s an isomorphism.

Lemma 2.5. and 0

F

(Cantor 17): Theorem 2.1.)

< 1-3/p.

Let p

> 3,

s

20 2U

Then t h e map N defined by N(u) = (-ALL,=)

i s an isomorphism.

2.2.

Some boundary value problem. Lemma 2.6.

Let u t C ( n ) be a v e c t o r f u n c t i o n such that

r o t u = 0 (generalized).

Then t h e r e i s a s c a l a r f u n c t i o n

1 q E C (a)such t h a t u = p q .

( g e n e r a l i z e d ) and

l i m u ( x ) = 0 , t h e n q i s harmonic and IXl+~

satisfies

If, i n addition, div u = 0

l i m q(x) = const. IXH-

58

Keisuke KIKUCHI

Proof. 1

Since

i s simply connected, as i s well known, i f

Q

u € C (O), then r o t u

=

0 implies t h a t t h e following equation

i s well-defined:

where t h e i n t e g r a l of u i s along any p a t h i n n f r o m a f i x e d point xo t o x, and q(xo) i s an a r b i t r a r i l y given constant. Then t h i s q has t h e required p r o p e r t i e s ( s e e [16J). Furthermore, t h e uniqueness theorem f o r t h e e x t e r i o r Neumann problem implies Let u be a harmonic f i e l d ( r o t u = 0 and

C o r o l l a r y 2.7.

div u = O ) Then u

111.

s a t i s f y i n g usn

Is

= 0 and tending t o 0 a t i n f i n i t y .

= 0.

Construction o f s o l u t i o n s . I n t h i s s e c t i o n we s h a l l prove Theorem 1.1.

To t h i s end

we consider t h e v o r t i c i t y equation obtained by t a k i n g t h e r o t a t i o n o f t h e f i r s t equation of (1) and using d i v v = 0 :

(3.1)

rot v

(3.2)

div v = 0 ,

(3.3)

aw + at

=

w,

(V.V)W

-

( w . 9 ) ~= r o t f

w i t h t h e boundary conditions ( 2 ) f o r v and t h e i n i t i a l

condition:

(3.4)

W(X,O)

=

r o t v,(x)

59

Nonstationary Ideal Incompressible Flow

for w.

We construct solutions of the vorticity equation by

means of the following iterative process.

The vectors v,(x)

and wo(x) = rot vo(x), which are the initial velocity and the initial vorticity respectively, are taken as the zeroth approximations. When the n-th approximation for the vorticity wn(x,t) is known, then the n-th approximation for the velocity v,(x,t)

(3.5)

is determined as follows: rot vn = wn,

div vn = 0 ,

vn-n is= 0,

lim vn =.,v IXI+W

(For the zeroth approximation (3.5) is automatically And when the n-th approximation for the velocity

satisfied.)

vn(x,t) is known, the (n+l)-th approximation for the vorticity wn+1(x,t) is a solution of the following equations:

a Wn+l + at

(Vn' Vhn+l

-

(Wn+l' V)Vn = rot f,

(3.6)

div w

~ =+0,~

W ~ + ~ ( X , O=) rot vo(x).

The following Propositions 3.1 and 3.2 imply that (3.5) and

(3.6) are solvable for all n (n = 0,1,2,

...) and Lemma 3 . 3

gives estimates for wn and vn which are uniform in n. We need

Definition.

Let p and 6 be as in Theorem 1.1 and

We define

r

xE,s+l = \vG ME,x+l:

div v = 0, v.nlS = 0 ) .

s

2

1.

60

Keisuke KIKUCHI

Proposition 3.1.

Let w E. C(L0,T];Yy,g+2).

Then t h e r e i s

a unique s o l u t i o n v o f (3.1) and (3.2) under t h e boundary

condition ( 2 ) .

This v s a t i s f i e s v-vw EC(IO,TI;Mz,S+l).

In

a d d i t i o n , t h e r e a r e c o n s t a n t s c2 = c2(Q) and c3 = c3(CL,voD) such t h a t

Proposition 3.2. d i v v = 0 and v . n w EC(k0,T];Yy,x+2)

=O.

Let vEvm+C([O,T];M~

, +

Then t h e r e i s a unique s o l u t i o n

o f (3.3) and ( 3 . 4 ) s a t i s f y i n g

where t h e constant c4 depends o n l y on 6. satisfies

1) be such t h a t

T h i s solution a l s o

61

Nonstationary Ideal Incompressible Flow

Lemma 3.3.

There e x i s t p o s i t i v e c o n s t a n t s K1, K 2 and T1

T) depending only on r o t vo, vpo, r o t f a n d n and

(T1<

sat i sf y i ng

(3.11) (3.12)

3.1.

Sketch of t h e proof o f Proposition 3.1. The following estimate h o l d s ((161). Lemma 3 . 4 .

There i s a p o s i t i v e constant c 5 = c 5 ( f i , s )

such that

Lemma 3.5.

solution V E (3-14) Proof.

2.7.

Let w & Y y , 6 + 2 .

Then t h e r e i s a unique

x ; , ~ + ~o f r o t v =w. The uniqueness follows from ( 2 . 2 )

and Corollary

To prove t h e e x i s t e n c e , we need t o show t h a t w can be

extended t o a solenoidal v e c t o r f u n c t i o n YGMY We c o n s t r u c t v e c t o r f u n c t i o n s w

j

( j = l,...,m)

For each j , t h e r e i s a s o l u t i o n q . E W 2 * p ( O j ) J

problem :

,6+2 (R3).

such that

o f t h e Neumann

62

Keisuke KIKUCHI

( s e e Agmon, Douglis and Nirenberg 113).

Using t h e i n v e r s e

theorem on t r a c e s ( s e e Besov, I l ’ i n and N i k o l ’ s k i i [ 3 : Theorem 2 5 . 2 ] ) ,

we have u . € W 2 ” ( O j ) J t h e boundary c o n d i t i o n :

(j

-

...,m )

Furthermore, t h e r e e x i s t smooth s c a l a r f u n c t i o n s y

v’pj

(3.18)

on S

= n

satisfying

1,

j

such t h a t

(j = I,.. .,m).

j

We put

Then we have

w j = (V1pj*v)uj - (uj-v)v’pj + ( d i v uj)V(pj

-

(dlpj)uj + vqj.

Hence s i n c e i t follows from (3.16) and (3.17) t h a t d i v u j = o on S

j’

we can e a s i l y see t h a t (3.15) holds.

c o n s t r u c t i o n of w

j

(The above

i s suggested by It$ 110: Lemma 2.91.)

Now we d e f i n e a v e c t o r f u n c t i o n G on R 3 by

i n Cl

W

(3.20) W

j

on 0

j

(j = l,...,m).

Then (3.15) i m p l i e s t h a t %EMf,x+,(R3) Lemma 2.4,

- Au = w.

4

solenoidal.

and d i v $ = 0 .

From

i t f o l l o w s that t h e r e i s a s o l u t i o n u€M5,,(R3) d

of

I n a d d i t i o n , we can see t h a t d i v u = 0 s i n c e w i s Writing V

=rot

( u l n ) , we have

63

Nonstationary Ideal Incompressible Flow

Lemma 2.5 y i e l d s t h a t t h e r e i s a s o l u t i o n q 6 M!,F

of the

Neumann problem:

(3.23)

v =T

- pq.

Then i t f o l l o w s from (3.21) and (3.22) t h a t V & X ~ , ~i s+ a~ s o l u t i o n o f (3.14). Definition.

Let w E Yp,F + 2 and l e t v t X;,z+l

s o l u t i o n of ( 3 . 1 4 ) .

(3.24)

be t h e

We d e f i n e t h e o p e r a t o r F as

P ( w ) = V.

Then Lemma 3.4 i m p l i e s t h a t

Furthermore, we have

Corollary 3.6,

F i s continuous from C(LO,T];Y: 9

+2

) to

,

C ( ( 0 , T 1;XE 6+11 with

f o r any w EC([0,T];Yy,b+2). It f o l l o w s f r o m Lemma 3.5 and C o r o l l a r i e s 2.7,

(3.27)

v = P(w) + vo

-

3.6 t h a t

P ( r o t vo)

g i v e s a unique s o l u t i o n o f (3.11, (3.2) and ( 2 ) .

We can

64

Keisuke KIKUCHI

prove t h a t t h i s v s a t i s f i e s (3.7) and (3.8) (1161).

3.2.

Sketch of t h e proof of P r o p o s i t i o n 3.2. We c o n s t r u c t a s o l u t i o n o f (3.6) following Swann’s

argument ( s e e [lg],

a l s o see K a t o

1121 and Judovig [ 113).

Define a family o f curves ( X ( x , t ; s ) , s ) i n A X [ O , T ] X(x,t;s) = v(X(x,t;s),s)

by

t,s EIOIT1

(3.28) X ( x , t ; t ) = x. Then s i n c e v-vW t C ( [O,T] ;M$ (3.29)

, +

1)

c C ( [O,T];C1+e(fi)),

X ( x , t ; s ) E C1([O,Tl;C1+e(~))

(0

< 8 < 1-3/~).

Hence (3.28) g i v e s a unique l o c a l curve i n f i x [ O , T ’ J

f o r each

( x , t ) € ~ x ~ O , T l It . f o l l o w s from v * n l S = 0 t h a t (3.30)

( X ( x , t ; s ) , s ) i n n % [ O , T ] cannot reach S X [ O , T ] .

Hence a l l s o l u t i o n s o f (3.28) i n n % [ O , T J e x i s t g l o b a l l y ( s e e K a t o (12: Lemma 2 . 2 1 ) .

Furthermore t h e d e f i n i t i o n (3.28) o f

stream l i n e s i m p l i e s t h a t (3.31)

X(X(x,t;s),s;T) = X(x,t;t)

We w r i t e t h e Jacobi matrix o f X ( x , t ; s ) as

Then we e a s i l y s e e t h a t (3.33)

det G(x,t;s)

= 1.

s , t , trZO,TI.

Nonstationary Ideal Incompressible Flow

Indeed, d i f f e r e n t i a t i n g (3.28) (3.34)

G(x,t;t)

=

E

in x

j'

65

we see t h a t

( i d e n t i t y matrix)

and t h a t G ( x , t ; s ) i s a fundamental m a t r i x s o l u t i o n o f t h e homogeneous o r d i n a r y d i f f e r e n t i a l equation: (3.35) The t r a c e o f t h e c o e f f i c i e n t matrix o f t h i s equation is Ldiv

V ( ~ , S ) ] J ~ = ~ ( ~ , ~ Hence ; ~ ) .

from d i v v

= 0

we deduce

(3.33). We d e f i n e w ( x , t ) = G ( x, t ; O ) - ' a ( X (

(3.36)

where a = r o t vo and b

=

r o t f.

x,t;O) )

Then u s i n g (3.28)-(3.35)

we can prove t h a t w given by (3.36) s a t i s f i e s t h e required p r o p e r t i e s ( [ 16 I ) .

3.3.

P r o o f of Lemma 3.3.

Let

(3.38)

T1 = min[T, 1 / ~ 4 e ( c l + l ) c 2 c 4 ( K l + ~ r ovolp,1,6+2 t +c

We prove (3.11) by induction.

f o r wo.

)}I.

I t i s c l e a r t h a t (3.11) h o l d s

Assume (3.11) h o l d s f o r wnml.

Then ( 3 . 5 ) , (3.8) and

66

Keisuke KIKUCHI

(3.38) i m p l y t h a t Tlllvn-llll,m

5 1.

Hence by (3.61,

(3.91,

( 3 . 1 0 ) , (3.37) and (3.38) we can o b t a i n

(3.12) f o l l o w s from ( 3 . 5 ) , (3.7) and

T h i s proves (3.11).

(3.11).

3.4.

P r o o f of Theorem 1.1. L e t To = Tl.

Lemma 3.7.

Then w e have There e x i s t s a unique s o l u t i o n \v, w ) of

( 3 . 1 ) ' ~ ( 3 . 4 ) and ( 2 ) on [O,To],

Proof.

We s h a l l

which s a t i s f i e s

f i r s t show t h a t as n

+ 00

(3.40)

w n ( x , t ) converges i n Mg,g+2 uniformly i n t E I O , T o ] ,

(3.41)

v n ( x , t ) converges i n Mp1,6+1 u n i f o r m l y i n t 6[O,T0].

Let

wn = wn-wnB1

and un =

and 3.2 we have (3.42)

C( LO,TOl

(3.43)

r o t un = wXn,

V ~ - V ~ - ~ Then .

by P r o p o s i t i o n s 3.1

67

Nonstationary Ideal Incompressible Flow

(3.44)

a at

where q ( w , u )

+ * ( v n - l ’ ~ ) = ~(wn-l,un-l);

-

w.v)u

-

(u.b)w.

= 0,

It f o l l o w s from (3.13), (3.42)

and (3.43) t h a t f o r a l l n (3.45)

lun(t)

Hence f r o m ( 2 . 1 ) and (3.11) we deduce

Thus (3.441, t o g e t h e r w i t h (3.81, (3.91, (3.46) and Lemma 3.3 imply t h a t

where c6 = 4c 4 {c 1c 5( c 2+1)Kl+c3).

\ITllp,o,6+2

2K1.

I n a d d i t i o n , we have

Hence r e p e a t i n g (3.47) successively, we

obtain

From t h e above estimate we s e e t h a t wn s a t i s f i e s (3.40).

Combining (3.45) and (3.481,

we have (3.41).

Let w = l i m w and v = l i m vn. Then (3.39) follows n--+m n+w from (3.40), (3.41) and Lemma 3.3. I n addition, taking the l i m i t ( 3 . 5 ) and ( 3 . 6 ) ,

we conclude t h a t \v, w ) is a s o l u t i o n

o f t h e v o r t i c i t y equation.

To show t h e uniqueness, suppose t h a t t h e r e i s another

s o l u t i o n (7,F \ s a t i s f y i n g t h e c o n d i t i o n o f t h e lemma.

68

Keisuke KIKUCHI

Writing w*

=

w

- W and

u =v

- F and

r e n e a t i n g t h e arguments

used t o deduce (3.45) and (3.47) f r o m ( 3 . 5 ) and (3.6) we s e e

<- c 5 l p ( t )lp,o,J+2.

(‘(t)‘p,l,s+l

T h i s proves t h e uniqueness.

Let \ v , w ) be obtained i n t h e above lemma. velocity v satisfies (2). t h e other conditions.

Then t h e

We s h a l l prove t h a t v s a t i s f i e s

From ( 3 . 4 ) , (3.27) and C o r o l l a r y 3.6

we s e e t h a t v(x,O) = vo( x ) .

Proof.

xav

e x i s t s and belongs t o L w ( ( O , T O ) ; M T , 6 + d It f o l l o w s f r o m (3.39) and Lemma 2.3 t h a t

Lemma 3.8.

Hence Lemma 2.5 i m p l i e s that t h e r e

( v ~ v ) v c L m( O( , T o )

i s a s o l u t i o n qGLm((0,TO);M5,r) of t h e Neumann problem:

(3.50)

- {(v.v)v).n,S.

- 0 9 = div[(v.V)v),

We put (3.51)

u(t) = v(t)

-

1

t

vo

+

i ( v - p ) v + vq

0

Then i t f o l l o w s f r o m Lemmas 3.5,

Furthermore, we s e e t h a t r o t u

-

-

P(rot f))dt.

3.7 and (3.50) t h a t

0 holds.

and 3.7 imply t h a t f o r any 7ECF(Q),

Indeed, Lemmas 3.5

69

Nonstationary Ideal Incompressible Flow

T h i s , t o g e t h e r with ( r o t u ) l

=

0, proves t h a t r o t u

Hence by (3.52) and C o r o l l a r y 2.7 we have u = O .

(3.53)

ata v

=

-

c(v.V)v

We note t h a t t h e right-hand

This y i e l d s

F ( r o t f)}.

s i d e of (3.53) belongs t o

+

(v*v)v

-

f.

Then we have r o t U = 0 s i n c e

v s a t i s f i e s ( 3 . 1 ) , ( 3 . 2 ) and (3 . 3 ). from Lemmas 2 . 2 ,

0.

This completes t h e proof.

Loo( (O,To) Let U = a v

+ vq -

=

I n addition, i t f o l l o w s

3.7 and 3.8 t h a t U E L m ( ( O , T o ) ; C b ( ~ ) ) .

Hence

from Lemma 2.6 t h e r e i s a s c a l a r f u n c t i o n p such t h a t

T h i s shows t h a t {v, p) s a t i s f i e s (1). The uniqueness f o l l o w s from Lemma 3.7.

References

c11 Agmon, S . , Douglis, A . and Nirenberg, L.: Estimates near t h e boundary f o r s o l u t i o n s o f e l l i p t i c p a r t i a l d i f f e r e n t i a l equations s a t i s f y i n g g e n e r a l boundary c o n d i t i o n s . I , Comm. Pure Appl. Math. 12 ( 1 9 5 9 ) , 623 - 727. [23

Bardos, C. and F r i s c h , U.: P i n i t e - t i m e r e g u l a r i t y f o r bounded and unbounded i d e a l incompressible f l u i d u s i n g Hblder e s t i m a t e s , Turbulence and Navier Stokes equation, Orsay ( 1 9 7 5 ), Lecture N o t e s i n Mathematics 565, Springer-Verlag, 1 1 4 .

-

70

Keisuke KIKUCHI

[3]

Besov, O.V., Il'in, V.P. and Nikol'skii, S.M.: Integral representations of functions and imbedding theorems, vol. 11, Winston-Wiley (1979).

(41

Bourguignon, J.P. and BrQzis, H.: Remark on the N e r equation, J. Funct. Anal. 12 (1974), 3 4 1 363.

151

Cantor, M.: Perfect fluid flows over Rn with asymptotic ( 1 9 7 5 ) , 73 84. conditions, J. Funct. Anal.

L63

Cantor, M.: Spaces of functions with asymptotic conditions 902. on Rn, Ind. Univ. Math. J. 14 ( 1 9 7 5 ) , 897

173

Cantor, 1.: Boundary value problems for asymptotically homogeneous elliptic second order operators, J. Diff. Eq. 34 (19791, 102 113.

-

-

-

-

181

Ebin, D. and Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. lath. 92 (1970)p 1 0 2 153.

-

191

Gttnter, N.M.: Potential theory and applications to basic problems of mathematical physics, Frederick Unger Publishing Co. (1967).

[lo1 It3, S.:

The existence and the uniqueness of regular solution of non-stationary Navier-Stokes equation, J. Fac. Sci., Univ. Tokyo, Sec. I, 2 ( 1 9 6 1 ) , 1 0 3 - 140.

1111 Judovix, V. : Two-dimensional nonstationary problem o f the f l o w of an ideal incompressible fluid through a given region, Math. Sb. N.S. 64 (1964), 562 588 (Russian) = A . M. S. Transl. ( 2 ) 57 (1966), 277 - 304.

-

1121 Kato, T.: On classical solutions o f the two-dimensional non-stationary Ehler equation, Arch. Rat. Mech. Anal.

25

(1967)

188

-

200.

1131 Xato, T.: Nonstationary flows of viscous and ideal fluids 305. in H 3 , J. Funct. Anal. 9 (19721, 296

-

(141 Kato, T. and Lai, C.Y.: Nonlinear evolution equations and 28. the Euler flow, J. Funct. Anal. 56 ( 1 9 8 4 ) , 15

-

Nonstationary Ideal Incompressible Flow

71

1153 Kikuchi, K.: Exterior problem for the two-dimensional Euler equation, J. Fac. Sci., Univ. Tokyo, Sec. IA, (1983) 63 92.

30

-

1163 Kikuchi, K.: The existence and uniqueness of nonstationary ideal incompressible flow in exterior domaions in R 3,

to appear.

[17] Nirenberg, L. and Walker, H.: The null spaces of elliptic partial differential operator in Rn, Appl. 4_2 (19731, 271 - 301.

J. Math. Anal.

(183 Swam, H. S.G. : The convergence with vanishing viscosity of nonstationary Navier Stokes flow to ideal flow in R 3,

Trans. Amer. Math. SOC. 157 (19711, 373

- 397.

[191 Swam, H . S . G . :

The existence and uniqueness of nonstationary ideal incompressible flow in bounded domains in R 3 , Trans. Amer. Math. SOC. -1 (1973), 167 - 180.

R.: On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20 (1975), 32 - 43.

1201 Temam,

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 8, 73-79 (1985) Recent Topics in Nonlinear PDE It, Sendai, 1985

Bounds for Solutions o f Abstract Nonlinear Evolution Equations

Kyfjya

Masuda

Mathematical Institute, Tohoku University Japan

S e n d a i 980,

1.

Introduction

The purpose Of this paper is to give some bounds for solutions of abstract nonlinear evolution equations.

Although

w e only state an abstract theorem without showing any applicat i o n s in t h e p r e s e n t p a p e r , t h e t h e o r e m i s v e r y u s u f u l f o r showing the analyticity o f solutions o f many different kinds of differential equations such as Korteweg-de Vries equations, Euler equations, nonlinear Schrodinger equations, Navier-Stokes equations, nonlinear diffusion equations in t h e t h e o r y of infiltration.

( F o r t h e d e t a i l see. T. K a t o - K . M a s u d a

[5].)

The abstract evolution equation we consider are o f the form (AE 1

dtu

( dt = d/dt ) where

To define the

+

F(t,u)

=

0, O < t < T ;

u(0)

=

uo

F is a nonlinear operator.

F precisely, w e first introduce the notion

of an S-functional.

Let

X,

Z b e t w o real ( o r c o m p l e x ) B a n a c h

73

Kyiiya MASUDA

74 spaces w i t h

C X,

Z

t h e imbedding being continuous.

we s a y t h a t a r e a l - v a l u e d c o n t i n u o u s f u n c t i o n a l i s an S - f u n c t i o n a l

if the right derivative o f

dt+O(v(t))

Then Z x X

Y on

O(v(t))

l i m [ O(v(tth))-@(v(t))]/h

=

h+O+ e x i s t s and s a t i s f i e s t h e i n e q u a l i t y

whenever

v cCt((OyT);Z)~C1+((O,T);X)

@(V) =

Remark.

Ct((O,T);Z)

Y(V,V)

z.

i s t h e s e t of a l l Z-valued r i g h t c o n t i n -

ous f u n c t i o n s on (0,T) X-valued

v E

y

where

;

and

C”((0,T);X)

c o n t i n u o u s f u n c t i o n s o n (O,T),

i s the set o f a l l having the r i g h t

d e r i v a t i v e i n (0,T). We show t w o e x a m p l e s o f S -

functionals.

Examples 1.

I n connection w i t h t h e existence problem f o r a system o f

ordinary d i f f e r e n t i a l equations, n o t i o n o f an S - f u n c t i o n . valued f u n c t i o n

S(x)

a)

S(X)

b)

S(0) = 0 ;

c)

the inequality

on

i s c o n t i n u o u s on

Kamke [ 3 ]

(see a l s o Hukuhara Rn R~

introduced the [2]).

A real-

i s called the S-function ;

if

75

Abstract Nonlinear Evolution Equations

v(t)

holds f o r a l l l e f t ( r i g h t ) d i f f e r e n t i a b e functions w i t h values in

Rn.

Then t h e Y

d e f i n e d by

.

e f t d e r i v a t i ve )

denotes the

(dt-

Y(v,w)

=

S

(v, WCR")

W)

is

a n S - f u n c t i o n a l o n R~ x R ~ . 2.

{X,Y,ZI

Let

be an a d m i s s i b l e t r i p l e t ( o f Banach s p a c e s ) .

( For t h e d e f i n i t i o n s e e

Kato-Lai

Let

[4].)

<.

,

->zxx

be a n o n - n e g a t i v e c o n t i n u o u s b i l i n e a r form a s s o c i a t e d w i t h t h e above t r i p l e t . on

Then t h e

.> zxx

i s an S - f u n c t i o n a l

Z x X.

We n e x t d e f i n e an S - f a m i l y . S-functionals ( =

O,(v)

(

,

<.

E Z)

in

on

Y,,

Yu(v,v) and i f

Z x X

A family {Yul

,

,

O
of

w i l l be c a l l e d t h e S - f a m i l y i f

i s continuous in u ( O < u
)

v E Z

f o r each

(I, and i t s d e r i v a t i v e

is differentiable

O,(v)

i s continuous i n

auOu(v)

u

and

v ( E Z). Now c o n s i d e r a somewhat more g e n e r a l e q u a t i o n t h a n (AE' 1

dtt

u

+

F ( t , u ) = 0,

O
u(0)

=

(AE):

uo.

W i t h above n o t a t i o n s o u r theorem now r e a d s : THEOREM

F(t,v)

Let

{Yul

,

0 < u < u*

be a mapping o f

[O,T)xZ

, be an S - f a m i l y . +

X

satisfying

Let

F =

(

O,(v)

where Let

Y,(v,v))

=

a(s),

uO€Z.

for

B(s)

VCZ,

0

5

t < T,

and 0 < u < u*,

are given continuous functions o f s.

Suppose t h a t

u = u(t)

i s a solution o f (AE’)

with

Then f o r a n y

uo ( 0 < u

u*) t h e r e i s a

<

To ( O < T O < T )

such t h a t

where

z = z(t)

i s t h e maximal s o l u t i o n of t h e o r d i n a r y

d i f f e r e n t i a l equation

(4)

and

a(t) =

Remark

uo e x p [

The u = a ( t )

-

t

J

a(z(s))ds].

0

i s d e f i n e d on t h e m a x i m a l i n t e r v a l

[O, T m a x ) o f e x i s t e n c e o f s o l u t i o n z. t h e i n t e r v a l [O,To)

The To i s s u c h t h a t

i s t h e maximal i n t e r v a l on which

a ( t ) < u*.

77

Abstract Nonlinear Evolution Equations

2.

P r o o f o f Theorem

The p r o o f i s s i m p l e , d o n e b y t h e m e t h o d o f c h a r a c t e r i s t i c s . We f i r s t c o n s i d e r t h e f i r s t o r d e r p a r t i a l d i f f e r e n t i a l inequality o f the form

where and

[O,T)x R

a, b a r e g i v e n c o n t i n u o u s f u n c t i o n s o n at+y

denotes t h e r i g h t p a r t i a l d e r i v a t i v e o f

1

y

,

with

respect t o t :

T h e n we w a n t t o e s t i m a t e y ( - , t ) LEMMA

Let

every

a.

z

where

y = y(a,t)

= z(t)

i n terms o f

y(.,O).

be a s o l u t i o n o f ( 6 ) .

Then f o r

i s t h e maximal s o l u t i o n o f t h e o r d i n a r y

d i f f e r e n t i a l equation

t a ( t ) = oo e x p [ -

(9)

1 a(s,z(s))ds]

;

0 and

To

i s t h e supremum o f a l l

T'

such t h a t

a ( t ) < a*

78

Kydya MASUDA

[O, TI).

on

Remark

By a s o l u t i o n

(6)

of

y

we

mean a c o n t i n u o u s

with i ) t h e r i g h t d e r i v a t i v e at+y aUy

i i )

i n u

and

usual p a r t i a l der vative

b o t h e x i s t and a r e c o n t i n u o u s i n

y = y(u,t)

P r o o f o f Lemma.

> 0,

E

let

t.

(6).

satisfies the inequality Given

0,

z = z,(t) E

be t h e maximal

solution o f the equation dtz

Let

uE= u E ( t )

replaced by t h e

w(t)

= blt,z)

+

E

,

z ( 0 ) = y(uo 0) +

b e t h e f u n c t i o n d e f n e d by ( 9 ) zE =

d e f i n e d above.

-

Y(U,(t),t)

E.

with

z

Then t h e f u n c t i o n

ZE

satisfies the inequality dt+W

5 f(t,W)

-

E,

~ ( 0 )=

-

E

Hence i t e a s i l y f o l l o w s f r o m t h e c o m p a r i s o n t h e o r e m f o r a differential

inequality that

w(tl<

0, and so t h a t

Abstract Nonlinear Evolution Equations

is as T in Lemma.)

( TE

Kamke's theorem ( see [l])

On the other hand it follows from that there is a subsequence{sjly=,

tending to zero, through which

{z,(t)l

converges to a

solution z o o f (8) uniformly on any compact subinterval o f

[O,TO). the

z0

Using the comparison theorem, one can conclude that is the maximal solution.

Thus by ( 1 0 ) we have (7).

This completes the proof of Lemma. Poor of Theorem

It i s now straightforward.

On setting

in the Lemma, the Theorem immediately follows from the Lemma.

REFERENCES

[l]

P.Hartman,

Ordinary Differential Equations, John Wiley

and Sons, 1964. [2]

M. Hukuhara, Sur la fonction S(x) d e M.E,Kamke, Japanese Journal o f Mathematics, 17 (1941 ) , 289-298.

[3]

E.Kamke, Uber die eindeutig Bestimmtheit der Integrale von D i f f e r e n t i a l g l e i c h u n g e n , Sitrunsber.Akad.Heidelberg,

1 1 (1931), 10-15.

[41 T.Kato and C.Y.Lai, Nonlinear evolution equations and the Euler f l o w , J.Funct. Anal., 56 (1984), 15-28. [51

T.Kato and K.Masuda, Nonlinear evolution equations and Analyticity.

( pre.print).

79

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 8, 81-105 (1985) Recent Topics in Nonlinear PDE 11. Sendai, 1985

On P r a n d t l Boundary Layer Problem

S h i n ' y a MATSUI and T a i r a SHIROTA Department o f Mathematics, Hokkaido U n i v e r s i t y Sapporo, 060, Japan

1.

Introduction. I n t h i s paper we study two mathematical bases o f t h e boundary l a y e r

problem f o r two dimensional incompressible l a m i n a r viscous f l o w .

The one o f

them i s t h e existence o f a s e p a r a t i o n p o i n t and t h e o t h e r i s r e l a t e d t o t h e approximation t o t h e Navier-Stokes f l o w b e f o r e such s e p a r a t i o n p o i n t s .

The

equations o f t h i s problem along a r i g i d w a l l a r e uu + vu = vu x y yy ux + v

Y

i n t h e domain DA = {(x,y);

0

< x <

-

pxy

= 0,

A, 0 < y < - 1 .

Here (x,y)

a r e orthogonal

coordinates i n t h e boundary l a y e r w i t h x r e p r e s e n t i n g t h e l e n g t h along t h e w a l l and y t h e perpendicular d i s t a n c e from t h e w a l l (The s u b s c r i p t s i n (1.1) denote t h e p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e corresponding v a r i a ble).

(u(x,y),

v(x,y))

i s t h e unknown v e l o c i t y v e c t o r f i e l d , px = px(x) i s

a p r e s c r i b e d pressure g r a d i e n t and t h e constant v i s t h e k i n e m a t i c v i s c o s i t y . For a given e x t e r i o r f l o w U = U(x), t h e a p p r o p r i a t e boundary c o n d i t i o n s are

81

Shin'ya MATSUI and Taira SHIROTA

82

u = v = O for

y=O,

(1.2) u(x,y)

-+

U(x)

as

y

+

u n i f o r m l y i n x on any

compact subset o f [O,A). I n order t o o b t a i n w e l l - s e t problem, we suppose t h a t a t an i n i t i a l p o s i t i o n , say x = 0, adatum u ( y ) i s assigned t h e v e l o c i t y component u, i.e., 0

Here we assume t h e B e r n o u l l i law and, f o r convenience o f discussion, t h a t t h e i n i t i a l p o s i t i o n i s n o t a s t a g n a t i o n one, i . e . ,

(1.4) U(0) > 0. I n S e c t i o n 2 we s h a l l present how t o o b t a i n t h e s o l u t i o n o f t h e bounda r y l a y e r problem u n t i l i t s s e p a r a t i o n p o i n t , whenever a s t a g n a t i o n p o i n t occurs downstream.

Furthermore we s h a l l a l s o prove t h a t t h e s e p a r a t i o n I n p h y s i c a l and e n g i n e r i n g

p o i n t s appear b e f o r e t h e s t a g n a t i o n p o i n t ([8]).

p o i n t s o f view these r e s u l t s a r e w e l l known as experimental f a c t s ([7]). I n f a c t , i n t h e G o l d s t e i n ' s research he assume t h e r e s u l t s above. ( F o r t h e r e g u l a r i t y assumption mentioned below i n ( 2 . 1 ) ,

(2.2)

and t h e G o l d s t e i n ' s

one see ([61, [161)). I n s e c t i o n 3, as t h e a p p l i c a t i o n s o f t h e above, f i r s t we s h a l l show t h a t i f t h e pressure g r a d i e n t i s p o s i t i v e a t some downstream p o i n t , t h e n there e x i s t s a s o l u t i o n w i t h i t s separation point. w i t h t h e S e r r i n ' s ([15])

and N i c k e l ' s ( [ l o ] ) .

This r e s u l t contrasts

I n S e c t i o n 3.2 we s h a l l t r y

t o o b t a i n some r i g o r o u s connection between t h e Navier-Stokes equations and t h e P r a n d t l equations along t h e F i f e ' s c o n s i d e r a t i o n s

([l])

( i n t h e case

83

On Prandtl Boundary Layer Problem where t h e pressure g r a d i e n t p l

5 -a

< 0 f o r some p o s i t i v e c o n s t a n t a ) .

our case t h e g r a d i e n t p i i s n o t always negative.

In

B u t we must assume t h a t

t h e Navier-Stokes f l o w has no s e p a r a t i o n p o i n t i n a c e r t a i n s t r o n g sense (see ( v ) i n 0 below). Though a l l o f our methods o f c o n s i d e r a t i o n s a r e simple, t h e r e s u l t s

w i l l be i m p o r t a n t as t h e mathematical bases o f t h e boundary l a y e r t h e o r y ( f o r instance, see [2]).

2.

The e x i s t e n c e o f separation p o i n t s . 2.1.

Preliminaries.

I n t h i s s e c t i o n we assume t h e f o l l o w i n g The e x t e r i o r f l o w U(x) i s s u f f i c i e n t l y smooth, vanishes a t

(2.1)

some p o i n t x = Xo ( 0 < Xo

<

m)

and i s p o s i t i v e f o r x

E

[O,Xo).

I f p x ( c ) = 0 a t a p o i n t x = c, then a c e r t a i n N-th d e r i v a t i v e

(2.2)

o f px does n o t vanish a t t h i s p o i n t . We d e f i n e t h e spaces 12+', ration point.

2 P ([O,A))

used i n t h i s paper and t h e sepa-

0 l e t B ([a,A]

f o r an i n t e r v a l [a,A]

space o f u n i f o r m l y continuous f u n c t i o n s i n [a,A] f o r any a(O < a 2 2/31 and yo > 0 l e t C*([a,A]

x x

[0,m))

x

[0,m)

[yo,m))

be t h e Banach w i t h supremum norm. be t h e Banach space

o f Holder continuous f u n c t i o n s , i . e . ,

f o r (xi,yi) (0,m))

=

E

n YO>O

[a,A] C"([a,Al

x

[yo,m) (i=1,2) x

[yo,-)).

and M = M(yO,u),

and l e t C"([a,A]

furthermore we define Ba([a,A]

x

x

(0,m))

=

Shin'ya MATSUI and Taka SHIROTA

84 Bo([a,Al

n C"([a,A]

x [O,m))

(0,m)).

x

We a l s o d e f i n e Bo([O,m)),

Ca((O,m))

and B2+a( (0,m)) by t h e analogous way. Then we d e f i n e t h e space o f t h e i n i t i a l data : 12+U = I

2+a

(v,U)

=

6

B

2+a

((O,m))

;

u(0) = 0, u ( 0 ) > 0, uy(y) '0 Y u * U(0) as y * and vu ( y ) YY

-

pX(o) = o(yL)

as

for y

y

E

[O,m),

* 01.

The c o n d i t i o n i n 12+a

i s a (strong) c o m p a t i b i l i t y condition. The space o f t h e s o l u t i o n s i s given as f o l l o w s : (u,v)

E

2 P ([a,A]) (i)

(ii) (iii) (iv)

means t h a t

u, u u

Bo([a,Al

COYm)),

uxy v, vy

co([a,A1

COYm)),

u(x,y)

in

Y' YY

E

> 0

uy(xyO) > 0

for

[a,A] x

x

E

(0,-),

[a,A].

Then we d e f i n e t h e space b y

Now we d e f i n e the separation point of a s o l u t i o n t o t h e problem (l.l), (1.2)

and (1.3)

:

85

On Prandtl Boundary Layer Problem

t o our problem i n t h e domain D (x,,Y,)

i n [O,s)

x

+

([S])

i n i t i a l data uo(y)

(2.4)

i f (u,v)

E

2

P ([O,s))

and f o r some sequence

(s,O)

u ( x ,y ) Y n n

and

0

+

as

n

m.

-+

Results and Remarks.

Theorem 1.

t h a t (s,O)

SY

[OP)

(xn,yn)

2.2.

i s a separation p o i n t o f a s o l u t i o n (u,v)

A p o i n t (s,O)

Definition.

E

For t h e problem ( l . l ) ,

12"

(1.2) and (1.3) w i t h t h e

t h e r e e x i s t s a s o l u t i o n (u,v)

2 P ([O,s))

such

Furthermore i f we p u t S(uo) = s,

i s i t s separation point. sup{S(u0); uo

E

E

12+Y< xo

holds. Here we n o t e t h a t p h y s i c i s t s c a l l (s,O) = 0 f o r t h e s o l u t i o n (u,v)

a s e p a r a t i o n p o i n t i f u (s,O) Y ( a s y m p t o t i c a l l y ) (see [7]). On t h e o t h e r hand

t h e separation p o i n t (S(uo),O)

i n Theorem 1 i s t h e same as t h e above, i f u

0 and u belong t o C ([O,s] x [0,-)). But t o o u r knowledge, t h e e x i s t e n c e o f Y such an exact s o l u t i o n i s n o t proved under o u r assumptions y e t . Furthermore

t h e separation p o i n t c a l l e d b y p h y s i c i s t s does n o t depend on t h e v i s c o s i t y . Therefore i n a c e r t a i n case t h e p o i n t x = sup{S(uo): uo

E

I

2+a

( 1 , U ) l may be

thought as t h e separation p o i n t p h y s i c i s t s c a l l , For convenience o f s t u d y i n g we consider t h e t r a n s f o r m a t i o n o f t h e independent v a r i a b l e s i n the system (1.1 ) o f t h e form

where

That i s , t h e stream l i n e s o f (1.1) a r e thought as t h e new coordinates.

If

Shin'ya MATSUI and Taira SHIROTA

86 we p u t w(x,$)

u2 (x,y),

reduces the problem ( l . l ) , (1.2) and

then (2.5)

(1.3) t o t h e Von Mises' form : L ( w ) = v J J w J I J I - wx = 2px

(2.6)

in

GA

with the conditions : w(X,O)

= 0,

w(O,JI)

+

(2.7)

where GA = {(x,JI);

2

0 < x

U (x)

c

as

J,

+

uniformly i n

m

Then

~,,,(o)

~($1 >

I?

{W($);

E

w ( i oY u ( t ) d t ) = u2 ( y ) , u

i f and o n l y i f

W,

w

> 0, w ( $ ) 0, ~ ~ ( 90 )f o r IJ-

$

and

+

E

[O,A),

and u o ( i 0Y u o ( t ) d t ) = u2o ( y ) .

A, 0 < $ < m l

H e r e a f t e r we always assume t h a t t h e i n i t i a l data w

IFa =

x

JJ w

2

I F a f o r some E

U ( 0 ) as JI

+

where

I 2+a 1

Ba((O,m)),

JIJI

a,

W ( 0 ) = 0,

and w s a t i s f i e s t h e

compatibility condition :

where p o s i t i v e constants ql, B

<

m y k depend on

W.

Furthermore f o r any B ( 0 <

1/2) t h e p o s i t i v e constant K depends on JI,, 13 and

W.

F i n a l l y we d e f i n e

Here we summarize t h e r e s u l t s and remarks on t h e l o c a l s o l u t i o n s o f

87

On Prandtl Boundary Layer Problem (l.l),

(1.2), (a)

([ll] and [14]).

(1.3) obtained b y O l e i n i k

For some A.

> 0 t h e l o c a l s o l u t i o n w(x,JI)

( 2 . 7 ) and (2.8) i n GA

P2,+"([0,Ao])

E

o f (2.6),

i s given b y t h e f o l l o w i n g way ( c . f . Lemma 4).

For

0 small

E

A, 0 < JI < I / € ) Then . there e x i s t s

> 0, l e t G i = { ( x , ~ ) ) ; 0 < x <

t h e approximate p o s i t i v e s o l u t i o n wE(x,JI) i n GE , which s a t i s f i e s (2.6) w i t h AO boundary c o n d i t i o n s : WE(0'$) = WO(E+9), (2.10)

w E (0) = w0( E) exp t V( E ) WE

x/w0(

E)

( x ,1/ E ) = w0( E+I / E ) expIU(Et1 / E ) x/wo ( E+I /E) 1 .

Furthermore t h e s o l u t i o n w,(x,$)

belongs t o

Pr"(F)

t h e l a s t space i s d e f i n e d as i n PEa([O,Ao]), x

by BCI([O,Aol

(0,-))

1,

x

f o r CI < 2/3, where AO b u t we must r e p l a c e Ba([O,A]

( O , l / ~ l ) . And t h e s e t { w ~ ( x , $ )<< ~ ~ has t h e f o l -

lowing properties :

A l l constants i n

(2.11)

E

E

(i.e.,

K and k ) and supremum

a, m, Jil,

%

o f t h e f u n c t i o n s wE, w E X Y w E , 0 JI do n o t depend on E ( < < 1 ) .

norms over w'-'w

PEa

&w

and

EJII)

X

Then t h e r e e x i s t s a subsequence

such t h a t w E ,

{WE,}

-+

w

0 i n C ([O,AO]

x

JWE~W

1 and wEI , wEI, , converge t o t h e corresponding %J) x j i 0 f u n c t i o n s o f t h e unique l o c a l s o l u t i o n w i n C ([O,Ao] x [l/N,N]) as E ' .+ 0. 2 Here we n o t e t h a t w(x,$) * U ( x ) as $ .+ u n i f o r m l y i n x on [O,Ao]. [O,N])

f o r any

N

>>

-

(b)

O l e i n i k assumes t h a t t h e i n i t i a l data uo belong t o B2+CI([0,a)),

b u t ours belong t o B

2+CI

((0,m)).

Our assumption o f r e g u l a r i t y i s enough f o r

the existence o f t h e l o c a l solution, 2 ( c ) A s o l u t i o n (u,v) E P ([O,Al)

of (1.11,

(1.2) w i t h i n i t i a l data

Shin’ya MATSUI and Taira SHIROTA

88 u0

E

I 2+a e x i s t s , i f a solution w

PP([O,A ofl)(2.6),

E

( 2 . 7 ) and (2.8)

with the corresponding i n i t i a l data w0 by (2.5) e x i s t s .

Now by t h e transformation (2.5), we have w

=

2u

Therefore we define

Y‘ the separation point of the solution w t o the problem (2.6), J,

by the same way as the above, i . e . , a point (s,O) a solution w ( x , + ) , i f w [O,s)

x

E

( 2 . 7 ) and (2.8)

i s the separation one of

P p ( [ O , s ) ) and t h e r e e x i s t s a sequence ( x n Y q n )E

[O,-) such t h a t

We a l s o denote t h i s separation point by S ~ W , ) . Then, in f a c t , we obtain

t h a t S(uo)

Lemmas and Proof of Theorem 1 .

2.3.

Lemma 1 . (a).

f o r uo corresponding t o w0 by ( 2 . 5 ) .

= S(wo)

Let w ( x , $ J ) be t h e local solution mentioned in Section 2.2-

Then there e x i s t positive constants M and

(2.12)

lwXl

Proof.

x

for

0

< x -

zAo

such t h a t

and

0 < J, < A.

Let w E ( x , $ ) be t h e approximate solution given i n Section 2.2-

( a ) . We p u t E constant

:M$

x

=

t(x,$): 0 < x -< 0A ’

0

will be determined l a t e r .

[WE1 w E

1

5 M in

< )I 2 A}, where the samll p o s i t i v e From the f a c t ( a ) , i f we show E

X

f o r some M y A and

E

( < < l ) , then we obtain ( 2 . 1 2 ) .

Hence we will show t h e

above inequality. Since uE

>

0 (see ( a ) ) , we may p u t g E ( x y $ )

=

In (wE(x,$)),

( 2 . 6 ) gE s a t i s f i e s the following equation (we omit subscript

f o r convenience of d i s c r i p t i o n s ) .

E

then from in functions

89

On Prandtl Boundary Layer Problem (2.13)

D i f f e r e n t i a t i n g (2.13) w i t h r e s p e c t t o x, we o b t a i n an e q u a t i o n f o r h = g,

+

2ve3g/'g

h JIJ,

-

egh2

-

eghx = 2pxx.

exp 3912 by i t s value from (2.13), t h e above e q u a t i o n f o r h $4 i s w r i t t e n i n t h e form Replaceing vg

(2.14

where

ve 3g/2h C = 1 heg

+

+ J,J,

Ch t 2 ~ e ~ ~ - / eghx ~ g = ~2pxx, h ~

3px = 1 wX + 3p,.

From (2.11),

f o r some p o s i t i v e c o n s t a n t M1 (independent o f

E).

Therefore (2.14)

implies

t h a t j = h-exp{-(Ml+l)xl s a t i s f i e s

Now i f j a t t a i n s i t s maximum i n then a t t h i s p o i n t ve3g12j

JlJ,

-

E without the l i n e s x

= 0, J, = 0 and J, = A,

egjx < 0 and j = 0 hold.

Hence from (2.15)

and (2.16) we deduce 2pxx.expI-(M1 +1) XI (2.17)

j:

C-M1 - 1

I M2

a t t h e maximum p o i n t , where t h e p o s i t i v e c o n s t a n t M2 does n o t depend on E. By t h e same arguments, a t t h e minimum p o i n t o f j i n t h e same subdomain o f E

:

Shin’ya MATSUI and Taira SHIROTA

90 we have (2.18)

-M2.

j

If

E

and A a r e s u f f i c i e n t l y small, from (2.9) i t f o l l o w s t h a t h(O,$)

w ~ ( E + J , ) - ~ ~ ( E =+ $O(1) J ) and h(x,O)

= W ~ ( E ) - ’ ~ ( E= ) O(1).

=

Furthermore (2.11)

i m p l i e s h(x,A) = O ( 1 ) . Thus from (2.17) and (2.18) we conclude t h a t

where t h e p o s i t i v e constant M3 does n o t depend on

E.

T h i s proves Lemma 1. q.e.d.

Using t h e maximum p r i n c i p l e ( f o r instance, see Theorem 8 o f t h e s e c t i o n 5 i n [5]),

we have

Lemma 2.

(1101 o r [8])

L e t W(X,JI) be a s o l u t i o n o f ( 2 . 6 ) , (2.7) and

(2.8) belonging t o PFa([O,A]).

Then w(x,J,) i s monotone nondecreasing w i t h

respect t o

J,

i n GAS

I t i s easy t o see t h a t Lemma 1 and 2 i m p l y t h e f o l l o w i n g C o r o l l a r y 1. 2.2-(a).

L e t W(X,JI) be t h e l o c a l s o l u t i o n mentioned i n S e c t i o n

Then t h e s e c t i o n ~ ( x , . ) belongs t o

Lemma 3.

There e x i s t s no s o l u t i o n w(x,$)

which belongs t o PFa([O,Xo)) x

<

IF

f o r any x ( 0 < x

zAo).

o f (2.61, (2.7) and (2.81,

and whose s e c t i o n ~ ( x , . ) belongs t o IM 2+a ( 0 <

-

xoi. Proof.

F i r s t we note t h a t from o u r assumptions (2.1) (2.2)

B e r n o u l l i law (1.4) (2.19)

and t h e

t h e r e e x i s t s a constant d ( > 0) such t h a t

t h e pressure g r a d i e n t px(x) i s monotone non i n c r e a s i n g i n [xo-d,xol.

91

On Prandtl Boundary Layer Problem

Next f o r a c e r t a i n A

E

(XO-d,XO) determined l a t e r , l e t w

E

Pp([O,A])

such t h a t

t h e n we see below t h a t

on [XO-d,Al

x

and w(x,O)

f o r Xo

-

0

f o r some p o s i t i v e constants m and Y.

Using (2.20)

= 0, we o b t a i n

d

5 x 2 A and 0 2 J,

have w+(A,O) X

CO,Ja/vl,

5 0.

<< 1.

Therefore i f JI .L 0 and x 4 A, then we

T h i s c o n t r a d i c t s o u r assumption w (x,O) J,

>

0 for 0 5 x <

(see t h e d e f i n i t i o n o f P p ) . Now we show (2.20) i n t h e case where Xo = d = 1.

Since u(O,$)

E

I:+”,

we may f i n d a p o s i t i v e c o n s t a n t m such t h a t

(2.21)

Furthermore (2.21)

implies t h a t

Shin’ya MATSUI and Taka SHIROTA

92

(2.22)

2

w(O,$)

2(m$-m2q2/4)

px(t)dt

0 < $

for

2

2/m.

Here we d e f i n e

= 2(mq-m2q2/4)

H(x,$)

2 < U (x)

Then s i n c e o(x,J,)

:1

p,(t)dt

L(H)

-

i n [O,A]

5 J, 5 2/m.

< H(x,$)

h o l d s on t h e l i n e s

Furthermore by simple c a l c u l a t i o n we o b t a i n

0 and $ = 2/m.

L ( w ) ‘0

0

p x ( t ) d t ( u s i n g m o n o t o n i c i t y o f w and (1.4))

= 21:

i t f o l l o w s from (2.21) t h a t an i n e q u a l i t y w(x,J,)

x = 0, $

for

x

[0,2/m]

f o r any A < 1.

Hence by t h e maximum

p r i n c i p l e we have

(see t h e p r o o f o f Theorem 3 i n [ll]). Now from (2.21) and (2.231,

t h e i n e q u a l i t y (2.20) holds on t h e l i n e s

x = 0, $ = 0 and J, = l / v f o r 2v > m. p,

Furthermore, from t h e m o n o t o n i c i t y o f

and (2.11, we deduce t h a t

L(F) =

uJF F$$

-

F,

< 2cm3 / 2 ~ ’ / 2 ( 1-A) p x ( x ) +

2mpx(x)/vA, 1

for 0

IJ,

zl/v,

0 ‘x

‘A

and any Y ( > m/2), A ( < 11, where c2 = 8 v 2 \ o p x ( t ) d t .

Hence an i n e q u a l i t y L(F) 5 2px(x) holds f o r 0 5 x

i f an i n e q u a l i t y

5 A and 0 5 J, 5 l / v

93

On Prandtl Boundary Layer Problem ~ r n ~ / ~ y ’ / ~ ( l - A+) m/y A

(2.24) holds

< A

.

Therefore c o n s i d e r i n g the quadraic i n e q u a l i t y i n A, we may f i n d such A ( < 1) f o r Y > m.

Thus b y t h e maximum p r i n c i p l e we have t h e i n e q u a l i t y (2.20) i n

t h i s case. F i n a l l y by the coordinate transformation

?=

(2.25)

(x-Xo+d)/d,

=

$/a,

t h e general case reduces t o t h e previous one.

Thus we o b t a i n Lemma 3. q.e.d.

From t h e above argument, e s p e c i a l l y (2.24) and ( 2 . 2 5 ) , we have t h e following C o r o l l a r y 2.

For t h e s e p a r a t i o n p o i n t ( s , O )

we have t h e f o l l o w i n g a

p r i o r i estimate : (2.26)

where

0

R

= 1

-

12px(x)

I

s LA = X

0

-

d(l-z),

D = 1 + 3 3c 2m4 and c2 = 8v

2/(1+D’/‘),

For a p o i n t x = X, max

<

E

(O,Xo)

respectively.

l e t kl and k 2 be min UL(x) and 0~X~X1

Furthermore f o r s u f f i c i e n t l y small p o s i t i v e

OlXlX,

constants

$o

2+a and k , l e t W be t h e subset o f I,,, such t h a t w0(!4)

E

W satisfies

Then we o b t a i n t h e f o l l o w i n g lemma which i s e s s e n t i a l l y g i v e n b y

Shin'ya MATSUI and Taira SHIROTA

94 O l e i n i k [ll]. Lemma 4.

For any w0

chosen dependently o n l y on Proof o f Theorem 1.

W, t h e constant A. $oy

i n S e c t i o n 2.2-(a)

may be

k , kl and k2, b u t independently of w0.

L e t w(x,$)

be t h e l o c a l s o l u t i o n i n GA

0

.

From

C o r o l l a r y 1 and t h e f a c t s (a) and ( b ) i n S e c t i o n 2.2 t h e r e e x i s t s a s o l u t i o n which i s a c o n t i n u a t i o n o f t h e l o c a l s o l u t i o n and belongs t o P2'"([0,A])

M

some A > A.

(By w(x,6)

we a l s o denote t h i s c o n t i n u a t i o n ) .

C o r o l l a r y 1 we see a l s o t h a t i t s s e c t i o n w(x,.) (0 'x

belongs t o

for

Furthermore from

IFa

f o r any x

(A).

L e t s = s u p I A ; t h e above c o n t i n u a t i o n e x i s t s i n G I . Then from Lemma A 3 we see t h a t s < Xo. Assume t h a t (s,O) i s n o t t h e s e p a r a t i o n p o i n t . That i s , f o r sequence An < s w i t h An

+

s (as n *

m)

there e x i s t

stants m q0 and a n a t u r a l number no which do n o t dopend on 0' i n f { w (A yv); 0 ( 6 16 1 > m + n 0 - 0

(2.27)

for

p o s i t i v e conII

and s a t i s f y

n ?no.

For an a r b i t r a r y b u t f i x e d n L n i f we c o n s i d e r the p o i n t x = An as 0' an i n i t i a l p o s i t i o n and t h e s e c t i o n w(x,.) as an i n i t i a l datum, t h e r e e x i s t s t h e s o l u t i o n o f (2.6), (2.7) and (2.8) i n GAntB, An+B1) f o r some B > 0.

which belongs t o P F ( [ A n y

On t h e o t h e r hand from Lemma 4 and ( 2 . 2 7 ) t h e con-

s t a n t B does n o t depend on n 2 no.

Hence, f o r s u f f i c i e n t l y l a r g e n, an

i n e q u a l i t y An +B > s holds. Thus we g e t a s o l u t i o n o f ( 2 . 6 ) , (2.7) and ( 2 . 1 8 ) i n GAn +B as a c o n t i n u a t i o n o f the s o l u t i o n ~ ( x , $ ) E P p ( [ O , s ) ) , which 1 i s c o n t r a r y t o t h e d e f i n i t i o n o f s.

T h i s proves t h e f i r s t p a r t o f Theorem 1.

The second a s s e r t i o n o f Theorem 1, i . e . (2.4), i n g way.

i s proved by t h e follow-

F i r s t we mention t h e r e l a t i o n between t h e B l a s i u s s o l u t i o n and o u r

95

On Prandtl Boundary Layer Problem s o l u t i o n o b t a i n e d above. and

The B l a s i u s s o l u t i o n i s g i v e n as f o l l o w s (see [3]

[41). Let

f

f ( n ) be t h e s o l u t i o n o f f " ' + ff" = O

in

[O,-),

w i t h c o n d i t i o n s f(0) = f l ( 0 ) = 0 and f ' ( n )

-f

1 as

Q

+ -,

We p u t i n (2.5)

where rl = (

U(Xo-d)+e 1/2 y, t h e c o n s t a n t d i s t h e same one i n t h e p r o o f o f ) 2vx

Lemma 3 and e i s a p o s i t i v e constant. Then

satisfies

0,

wB(x,$)

+

{U(XO-d) + e l L

as

$ +

m,

pointwise

in

x

w,fx,$)

+

{U(XO-d) + e l 2

as

x

0, p o i n t w i s e

in

J, > 0.

Now l e t w(x,$)

+

be t h e s o l u t i o n o b t a i n e d above w i t h S(wo)

>

> Xo

- d.

Then we have

We n o t e t h a t from C o r o l l a r y 2 S ( w 0 )

< Xo holds.

The p r o o f o f (2.28) i s

Shin'ya MATSUI and Taka SHIROTA

96

given by t h e maximum principle and f " ( q )

>

0 for

q >

0 ( s e e Theorem 8.1 in

We omit i t s d e t a i l s here (see [ 8 ] ) .

[3]).

To show (2.4) , from Corollary 1 , we may consider t h e point x Xo

-

d

t

E

as an i n i t i a l position f o r s u f f i c i e n t l y small

E

>

0.

=

Furthermore

from (2.28) we obtain V

f o r 0 5 JI 5 2 / m B y with mB tion w(Xo-dtE,.),

>>

Then the constant m with respect t o t h e sec-

1.

in the assumption f o r (2.20), can be chosen independently

of any solution w w i t h S(o0)

>

Xo

-

d.

Hence the constant A in (2.20) can

Then from Corollary 2 t h e -dtE,-). 0 This proves Theorem 1 .

be a l s o chosen independently of w ( X inequality (2.4) holds.

3.

Applications of Theorem 1 . 3.1.

Existence theorem o f separation points f o r a mare general e x t e r i o r flow.

In this section we show the existence of separation points in the case where U(x) does not s a t i s f y ( 2 . 1 ) and ( 2 . 2 ) . (3.1)

the e x t e r i o r flow < x positive f o r 0 -

U(x) <

We assume here t h a t

i s s u f f i c i e n t l y smooth and

m

and (3.2)

the pressure gradient px i s p o s i t i v e f o r some point

x1

E

[0,m)

*

Let m be a positive constant such t h a t

97

On Prandtl Boundary Layer Problem

(3.3)

where u o ( jY0 u o ( t ) d t ) = u0(y) 2

px

IZta, the point

f o r uo

Xo and t h e f u n c t i o n

w i l l be d e f i n e d below. For (1.1 ) , (1.2) and (1.3) w i t h t h e i n i t i a l data uo

Theorem 2 .

E

IZta,

i f t h e constant m i n (3.3) i s s u f f i c i e n t l y small, then t h e r e e x i s t s a s o l u -

t i o n (u,v)

2

c P (10,s))

Lema 5 .

w i t h t h e s e p a r a t i o n p o i n t (s,O),

The pressure g r a d i e n t p,

on soem i n t e r v a l [X,,X3] Proof. px(xn)

-f

c

.+

-.

<

s

a).

i s nionotone decreasing and p o s i t i v e

,a),

we f i n d a sequence { x n l such t h a t

I n f a c t , assume t h a t px 2 a > 0 on [X4,-)

constant a and some p o i n t X,(>> U2 ( x ) = U2 (X,)

<

[+,a).

Even if px > 0 on [X1

0 as n

(0

1).

+ (x-X,)

Then from (1.4)

f o r some

it implies t h a t

1

1 (U2),(X4tt(x-X4))dt

0

= U2 (X,)

-

(x-X,)

1

1

2px(X4+t(x-X4))dt

0

<

0

This i s a contradiction.

for x

>>

1.

Hence, form t h e assumption o f p,

Lemma 5 i s v a l i d . q.e.d.

From L e m a 5 and (1.4) we have t h a t

Shin’ya MATSUI and Taira SHIROTA

98

L e t i ( X ) be t h e f u n c t i o n such t h a t

i ( x ) = U(x)

for

x < X3

and

Furthermore s e t

Then

where t h e p o i n t x = Xo i s t h e unique vanishing one o f G(x). positive for x

continuous, non-increasing,

o f (2.6),

and

ix E C

([O,X,))

t h e s o l u t i o n w(x,$)

U

E

E

2+a

I#

.

Here we n o t e t h a t

s a t i s f y (2.1) and (2.19) w i t h o u t t h e s u f f i c i e n t l y

and

i s the one o f t h e o r i g i n a l problem because o f t h e d e f i n i -

px.

To show t h i s i n e q u a l i t y f o r some p o s i t i v e constant such t h a t

when d = X

= 0.

Now i fwe show

smoothness.

tions o f

is

(2.7) and (2.8) w i t h t h e e x t e r i o r f l o w i ( x ) , t h e

pressure g r a d i e n t p x ( x ) and t h e i n i t i a l data w0

3

and px(X,)

F i r s t we consider a s o l u t i o n u(x,+)

Proof o f Theorem 2.

P?([O,S(W,)))

X[ X, ),

Isx

0

-

X2.

Here we remark t h a t

w0,

even i f S ( w o )

>

X2,

let

6

be a

99

On Prandtl Boundary Layer Problem xO

px(t)dt = JX

[O,Xo)

for x

[ J

xO = i2(x) > 0

(-?),(t)dt X

Then, i f t h e constant m i n (3.3)

and

i.

i s s u f f i c i e n t l y small, by t h e same way

d e r i v e d ( 2 . 2 3 ) i t i m p l i e s t h a t t h e constant Because (2.23)

px

holds from t h e d e f i n i t i o n s o f

E

i s a l s o s u f f i c i e n t l y small.

Hence from (2.26) we see t h e d e s i r e d

i s v a l i d w i t h o u t (2.19).

a p r i o r i i n e q u a l i t y f o r such an i n i t i a l datum u0.

Thus by t h e same way as

i n t h e p r o o f o f Theorem 1 we o b t a i n Theorem 2 . q.e.d. 3.2.

Approximation t o t h e l a m i n a r Navier-Stokes flow.

For t h e sake o f s i m p l i c i t y h e r e a f t e r we a r e concerned w i t h f l o w s p a s t The s t a t i o n a r y Navier-Stokes equations a r e

a f l a t s e c t i o n o f boundary.

uux

(3.4)

uvx ux

f

f

vu = u(uxx + u 1 Y YY

f

vv

y

= "(VXX

f

vyy)

-

-

P,, pY

v = 0. Y

The Prandtl equations (1.1)

a r e f o r m a l l y d e r i v e d from (3.4) by t h e f o l l o w i n g

manner. By t h e t r a n s f o r m a t i o n

rl =

(3.5)

we have

"-1/zy

-u(x,rl)

,

x = x

= u(x,y),

and

-v(x,n)

= "-1'2v(x,y).

i(X,il)

= p(x,y),

Shin'ya MATSUI and Taka SHIROTA

100 N

u

rln

-

uux

=

-

N

(3.6)

-P, N

ux +

-

U

r

vu

+

N

dv,,

$=

n

rl

-

-

p,

= -wuxx,

uiixx) + v m x + vi;G

rl'

0.

Next, n e g l e c t i n g t h e o r d e r v terms o f t h e r i g h t s i d e s i n (3.6),

we o b t a i n

t h e (dimensionless) P r a n d t l equations

u rlrl

where

?,(XI = Fx(x,O).

and denoting

(;(X,V-'/~Y),

-

--

-

uux

--

vu rl = F x ( x ) ,

Then r e t u r n i n g t o t h e o r i g i n a l c o o r d i n a t e s ( x , y ) ~ ~ ' ~ Y ( x , v - ~ / b~yy (U(x,y) )) ,V(x,y))

f o r conven-

ience o f d i s c r i p t i o n s , we o b t a i n t h e Prandtl equations (1.1) f o r (U(x,y), V(X,Y)

1. I n o r d e r t o study t h e approximation r e s u l t s , c o n s i d e r i n g a l l f u n c t i o n s

below t o be continuous, f o r p o s i t i v e constants a ( < l ) , bo ( > a ) and we d e f i n e t h e c a l s s 0 o f the Zaminar Navier-Stokes @ous (u",v",pv) separation points i n an i n t e r v a l [O,A] The f l o w (u",v",p") (i)

E 0

(u",v",p")

M (> 1)

uithout

(see [l]and i t s r e f e r e n c e s ) .

means t h e f o l l o w i n g : i s t h e s o l u t i o n o f (3.4) w i t h t h e kinematic v i s c o s i t y

u (< - 1) i n t h e domain D = [O,A]

x

[0,2],

which s a t i s f i e s t h e boundary c o n d i -

tions uv = v" = O (ii)

o< u"(x,y)

< 1

on

for

y = 0.

D.

(iii) The absolute values o f u",

v" and t h e i r x - d e r i v a t i v e s up t o o r d e r

101

On Prandtl Boundary Layer Problem 2 are each bounded by M on D. (iv)

V

Iv(uxx

uiy)

+

V

1,

Iu(vXx

+

vv YY

1

I v ( v i x + vv ) I YY x

and

a r e each

I on D. bounded by F

(v)

a.min{l , v - 1 / 2 y ~ < uv(x,y)

l,v-1/2y}

on

[O,A]

x

i n (3.7) and t h e i n i t i a l data ij:(,-,)

IZta i n (1.3) as

follows : For a p o s i t i v e constant B

provided t h a t

i s monotonous and

uv(O,y) (3.9)

' Ipi(x,y)

I

[0,1].

c @ we may t a k e t h e s u f f i c i e n t l y smooth pres-

Now f o r each (u",vv,pu) sure g r a d i e n t p;.(x)

2 bo.min I

bl

on

D

f o r some p o s i t i v e c o n s t a n t bl

Finally, putting

we may t a k e t h e e x t e r i o r f l o w U"(x)

as f o l l o w s :

For some p o s i t i v e constant b2 depending o n l y on bl

.

102

Shin'ya MATSUI and Taka SHIROTA 0

(3.10)

< b;

1

5 ( U v ) 2 ( x ) 2 b2 i f 0 'x b21 and B

Then s e t t i n g b = maxIbo, bl,

>

b

LA.

+ 1 , we mention the follow-

i ng Theorem 3.

For the positive constants B , M, a and b l e t t h e flow (u",

and s a t i s f y ( 3 . 9 ) ' . Then, i f v 2 vo f o r a c e r t a i n vo = vo(B,M,a,b), -v -v -v there e x i s t s the Prandtl flow ( u , v , p ) s a t i s f y i n g ( 3 . 8 ) , (3.9) and (3.10)

vv,p")

E

up t o x

=

A such t h a t f o r a constant c = c(B,M,a,b)

For convenience of the proof we consider the transformations ( 2 . 5 ) by v i r t u e o f the streaming functions $(x,n) and $(x,rl) of (3.6) and (3.7) respectively :

x

=

x and

=

$(x,n)

x

=

x and $

= $(x,n)

in

(3.61,

in

(3.7).

-v2 Then we have t h a t u v ( x , F ) = ( u ) ( x , n ) and GV(x,$) = ( i v ) 2 ( x , 1 ~ )s a t i s f y

pu$

-

ox

/Fi;$

-

wx -

where v 1 / 2 f ( x , r l ) = 2vfl =

-

-v

v(vrln

+ V V X X ) + v Nv-v u vx

- 2<(x,O) 2Qx)

= v1j2f

and

= 0,

af2/ax dn', vfl(x,n)

2V1/2/i

= v9 u x x and ~ ' / ~ f ~ ( x , 1 1 )

+ v -vw v v

11'

The o u t l i n e of the proof of Theorem 3 .

From t h e assumptions mentioned

103

On Prandtl Boundary Layer Problem above and by t h e F i f e ' s t h e o r y ([l]) we need o n l y t o prove t h e f o l l o w i n g : Y

For any v 5 vo = uo(B.M,$,b) 2 P ([O,A]) such t h a t ;"(x,$)

there e x i s t s a Prandtl flow N

L $ on [O,A]

x

[0,1],

where

(~",V",~") E 2 2a X = pg .

0 Now i n o r d e r t o o b t a i n t h e c o n t i n u a t i o n GV mentioned above up t o x =

A, we p u t A" = supIa; f o r any v '

5 w the solution lv'(x,$) exists

2+a i n PM ( [ O , a ] ) and s a t i s f i e s

-v

(3.i2)

on [ 0 , a l

-

I

(XYU

L;?

[0,111.

x

Obviously from the. p r o o f o f Lemma 4 ( i n p a r t i c u l a r see t h e one o f Lemna 2 i n [ll])i t i m p l i e s t h a t A1 p o i n t such t h a t a (i),...,(v),

A1 and A1

c

- a

0.

>

<<

1.

Assume t h a t A1 < A. Then from ( 3 . 1 2 ) ,

L e t a be a

a l l assumptions

(3.10) and by the analogous way as i n t h e p r o o f o f

(3.8),...,

Lemma 5.3 i n [l]we o b t a i n t h a t f o r a c e r t a i n

p o s i t i v e constant El=

u

(B,M,$b) Iw"(x,U i f (x,?)

E

[0,113

x

[0,1]

-

."(x,U

I

5

C1V

1/2JI

and v < 1.

< r n i n { (- d )2 ,l}z Therefore from ( v ) i n Q we have t h a t f o r w vO 2E1

Accordingly from ( 3 . 9 ) , (3.9)', (3.10) and by u s i n g C o r o l l a r y 1 and t h e p r o o f o f Lemma 4 we conclude t h a t f o r such u

Gv can be continued and s a t i s -

f i e s (3.12) up t o x = a t 6, where t h e p o s i t i v e c o n s t a n t 6 does n o t depend on II. Thus we see t h a t A

vO = A.

may conclude t h a t A" 0

>

A1.

Furthermore b y t h e same way as above we

Shin'ya MATSUI and Taka SHIRO'I'A

104

F i n a l l y we must mention t h e f o l l o w i n g : vo i n Theorem 3 tends Therefore i n

t o zero r a p i d l y i f a tends t o zero and i f M, B y b are f i x e d . order t o keep v0 n o t t o o small, we must assume t h a t 1 > a

0.

>>

References

[l] F i f e , P.C. :

C o n s i d e r t a t i o n s r e g a r d i n g t h e mathematical b a s i s f o r

P r a n d t l ' s boundary l a y e r theory, Arch. Rat. Mech. Anal.

, 28

(1968)

,

184-21 6. : Corrigendum, Considerations r e g a r d i n g t h e mathematical

b a s i s f o r P r a n d t l ' s boundary l a y e r theory, Arch. Rat. Mech. Anal.,

46

(1972) , 389-393. [2]

Glimm, J . : S i n g u l a r i t i e s i n f l u i d dynamics, Math. Prob. i n T h e o r e t i c a l

Phys., ed. R. Schrader, R. S e i l e r and D.A.

Uhlenbrock, Springer-Verlag

( 1981 ) , 86-97. [31

Hartman, P.: Ordinary D i f f e r e n t i a l Equations, John Wiley and Sons I n c . , New York (1964).

[4]

Hastings, S.P.:

Reversed f l o w s o l u t i o n s o f t h e Falkner-Skan equations,

SIAM. Appl. Math. Vol. 22, No.2 (1972), 329-334.

[5]

I l ' i n , A.M.,

Kalashnikov, A.S.

and O l e i n i k , O.A.:

L i n e a r equations o f

t h e second o r d e r o f p a r a b o l i c type, Russian Math. Survey, 17-3 (1962), 3-1 46. *[6]

Lagerstorm, P.A. : S o l u t i o n s of Navier-Stokes equations a t l a r g e Reynolds number, SIAM. Appl. Math. Vol. 28, No.1 (1975), 202-214.

*[7]

Landau, L.D. and L i f s h i t z , E.M.:

F l u i d Mechanics, Pergamon press,

105

On Prandtl Boundary Layer Problem Oxford (1966). Matsui, S. and S h i r o t a , T.: On separation p o i n t s o f s o l u t i o n s t o Prandtl boundary l a y e r problems, Hokkaido Math. Jour. Vol. 13, No.1 (1984), 92-108. Von Mises, R. and F r i e d r i c h s , K.O.:

F l u i d Dynamics, Appl. Math Sciences

5, Springer-Verlag (1971). N i c k e l , K.:

P a r a b o l i c equations w i t h a p p l i c a t i o n s t o boundary l a y e r

theory, P. D. E . and C o n t i . Mech., ed. R. Langer, The Univ. Wisconsin Press, Madison, Wisconsin (1961), 319-330. O l e i n i k , O.A.:

On a system o f equations i n boundary l a y e r theory,

U. S . S. R. Comp. Math. Phys.,

3 (1963), 650-673.

: Mathematical problems o f boundary l a y e r theory, Uspehi

Mat. Nauk, Vol. 23, No. 3 (1968), 3-65.

: Weak s o l u t i o n s i n the Sobolev sense f o r a system o f boundary l a y e r equations, Amer. Math. SOC. Trans. ( 2 ) ,

105 (1976), 247-

264. and Kruzhkov, S.N.:

Q u a s i - l i n e a r second-order p a r a b o l i c equa-

t i o n s w i t h many independent v a r i a b l e s , Russian Math. Surveys, Vol. 16 n.5 (1961), 106-146. S e r r i n , J.: Asymptotic behavior o f v e l o c i t y p r o f i l e s i n t h e P r a n d t l boundary l a y e r theory, Proc. London Math. SOC. A 299 (1967), 431-507. Williams, 111, J.C.:

Incompressible boundary-layer separation, Ann.

Rev. F l u i d Mech., 9 (1977), 113-144.

An a s t e r i k s * i s used t o mark t h e references developing t h e t h e o r y o f F l u i d Mechanics i n p h y s i c a l and e n g i n e r i n g p o i n t s o f view.

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 8, 107-125 (1985) Recent Topics in Nonlinear PDE I t , Sendai, 1985

On a Free Boundary Problem in Ecology

*

**

Masayasu MIMURA, Yoshio Y A M A D A and Shoji

*Department

*** YOTSUTANI

of Mathematics, Hiroshima University

Hi rosh i ma 730, Japan

**Department

o f Mathematics, Nagoya University

Nagoya 464, Japan

***Department

of

Applied Science, Miyazaki University Miyazaki 880, Japan

We shall be concerned with a free boundary problem for semilinear parabolic equations, which describes the habitat segregation phenomenon in population ecology.

The main purpose

is to show the global existence, uniqueness, regularity and

asymptotic behavior of solutions for the problem.

The stability

or instability o f each stationary solution is completely determined using the comparison principle.

107

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

108

§I. Problem We consider the following one-dimensional free boundary probl em.

= dluxx + f(u)u

in

S-,

(1.2) ut = d2uxx + g(u)u

in

s+,

(1.1)

I I

(')

I

u

t

(1.3) u(O,t) = m l ,

t E (O,-),

(1.4) u(l,t) = -m 2,

t E (O,-),

(1.5)

u(s(t),t) = 0, t E ( O , - ) ,

(1.6)

(1.7) I(1.8) s ( 0 ) = R,

+)

S- (resp. S

where (0,l))

in which

x

(resp. ux(s(t)+O,t))

is the open set of

<

s(t)

(resp.

x

>

Q = I x

s(t)

denotes the limit of

,

(I =

(O,-)

ux(s t ) - 0 ,t )

and

u(x,t)

x = s(t)

at

from the left (resp. right).

This is a model describing regional partition of two species, which are struggling on a boundary to obtain their own habitats.

In our model, the function

u

in

S-

(resp. -u

in

the population density of the species which lives in S').

S + ) denote S-

(resp.

These two spieces are supposed to undergo diffusion and

109

Free Boundary Problem in Ecology growth.

Here the boundaries

x

x = s(t)

intermidiate boundary

= 0, 1

are fixed, while the

is not prescribed a priori.

The latter boundary is determined by the interactions between the t w o species there (see (1.5) and ( 1 . 7 ) ) . x = s(t)

called a free boundary.

is

f

In the absense of

and

In this sense,

For details, see C31.

g, the problem is reduced to a

two phase Stefan problem in the one-dimensional space, for which there are many contributions (see Rubinstein C51, Kamenomostskaja

C21, Yamaguti & Nogi C61, Yotsutani C73 e.t.c., and references therin).

We shall show several interesting results on the global

existence, uniqueness, regularity and asymptotic behavior of solutions of ( P ) .

Especially, a bifurcation phenomenon occurs

in ( P ) , differently from the one in the case

f = g

= 0

(Stefan problem).

82.

Assumption

We summerize the assumDtions:

f

is locally Lipschitz continuous on LO,-),

non-increasing on f(u) g

>

0 on

CO,l),

and satisfies

f(1) = 0 and

f(u) 4 0

is locally Lipschitz continuous on

non-decreasing on g(u)

0

C0,ll

<

>

0

on

ml S 1

C-1,Ol

(-1,Ol, and

0

<

monotone

on

(l,=).

(--,Ol, monotone

and satisfies

g(-I)=O m2 4 1.

and

g(u) 4 0

on

(--,-I).

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

110

<

(A.4) 0

<

Q

1.

Our first result i s concerned with the global solvability of ( P I . Theorem

I. (Global existence and uniqueness).

assumptions (A,l)-(A.5),

Ctb)

Cu, sl

s ( 0 ) = Q,

(ii)

u

s E

satisfying the following properties:

L3 ( 0 , m )

b E (0,l)

and for some constant

satisfies (1.3), (1.4), (1.5)and (1.6) everywhere and

( i i i ) Let u*

there exists a unique pair of functions

X C(C0,m))

(i)

Under the

u

+

= max (u, 0 )

E C(C0,m);H I( 1 ) )

+

sup lu (t)l OSt<m H (I)

<

and

u

-

= - min Cu,O>.

Then

and

0,

sup lu-(t)l OSt<-

Hi(

I)

<

m.

111

Free Boundary Problem in Ecology (u)

ut, u

E C(S-)

xx

I7 C ( S + )

and (1.2) everywhere in (vi)

6

For each

S*

n

Ct28)

>

0, u

and

and

is H6rder continuous in

s>

(9,

set

(a,-).

is a smooth solution

a).

Theorem

I

Let

assures the global existence

(u(x,t;P,L), s(t;p,R)>

so that its asymptotic behavior as

be discussed.

t E C6.m).

denote the smooth solution of ( P ) with

of the smooth solution

@-limit

(u,

t E

(x,t) E

if it has the properties in Theorem I.

C0,m)

initial data

respectively.

is HAlder continuous in

s

Cu(x,t;9,Q), s(t;P,R)>

R>,

S',

S- and

In this report, we say that

(9,

satisfies (1.1)

satisfies (1.7) for every

(uii) Cu, sl

of ( P I on

u

for every

t +

is now to

For this purpose, we conviniently introduce the

defined by

U(9,R)

= ((u

* , s* >

6 H

1 ( I ) X I ; there exists a sequence

We say that the sequence converges to

(u

* , s* 1

in

cu(.,ti:9,t), s(ti;p,Q)>

&topology

if it has the convergence

properties in (2.1) Theorem 11. (structure of @-limit set).

Assume

(A.I)-(A.5).

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

112

Then

(i)

@(@,a)

is non-empty and connected in

* , s* 1

( i i ) If

Cu

i

*

E

+

@ ( @ , a ) , then f(U*)U*

= 0, u

R-topology.

Cu

* , s* 1

satisfies

*

2 0,

x E (0,s * ),

dlUxx

II *

*

+ g(u*)u*

= 0, u* 5 0,

x E

* ,l),

( 5

d2Uxx

(sap)

u (0) = m

I

- ulu*x(s*-o)

* *)

u (s

*

*

= 0,

*

+ u2u x(s + O )

u ( 1 ) = -m

2’

= 0.

Theorem I 1 gives very useful information about the asymptotic behavior of solutions of ( P ) .

For example, if one

can show that solutions of ( S P ) are isolated in &topology, then as t

(u(t;@,!Z), s(t;@,L)1 +

data.)

m,

approaches one of them in 0-topology

(The limiting function w i l l depend an the initial

Therefore, it is very important to determine the

structure of the set of solutions for (SP). We now study the stationary problem ( S P ) with the aid of the following auxiliary problem

where

E E I

is any fixed number.

I13

Free Boundary Problem in Ecology Then we haue

1 1 1 . (Stationary solutions).

Theorem

Assume (A.l)-(A.3).

Then

(i)

For every

(ii)

Cw(x;s

E

I,

E

* ) , s* 1

( 2 . 2 ) has a unique solution

is a solution of ( S P ) ,

w

= w(x;E).

if and only if

s

*

is a zero point o f

( i i i ) If (iv)

El < E,,

then

w(x;E1)

<

w(x;E2)

( S P ) has a (unique) minimal solution

* , s* >

s S s

*

-

S s

and

g S u

*


-

S u

in

I.

Cg, 5) =

Cu, s> =

and a (unique) maximal solution the sense that any solution

in

Cw(.;s), s>

Cw(.,s),

in

of ( S P ) satisfies

I.

-

Here

s_

and

zero point of

s

are the least zero point and the greatest W

on

I,

respectiuely.

By the properties ( i i i ) o f Theorem 111, the set of solutions for ( S P ) has an obvious order relation.

For

< u i , s i >E

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

114

if u1 5 u2

on

-

I,

s1 5

S2'

We are now in a position to study stability or instability of each solution of ( S P ) in connection with the asymptotic behavior of solutions for ( P ) . Theorem (A.l)-(A.S).

(i)

IV. (Stability of stationary solutions).

Assume

Then

The minimal solution

{g, 21

of ( S P ) is globally

aymptotically stable from be ow in the sense that

f

( P , t 1 5 {g, 21, then the smooth so satisfies

and

(2.4) lim Cu(t;P,R), s(t:g),R)1

= (g, 2)

in

R-topology.

t+-

( i i ) The maximal so 1 ut ion

Cu,

of ( S P ) is globally

asymptotica ly satab e from above, that is, the assertion is valid if

of ( i )

Ci,

s>

( i i i ) Let

and

El < E ,

(2, 2)

and

4

are replaced by

2 , respectiuely.

be adjacent zero points of

W(E) defined

115

Free Boundary Problem in Ecology

for every

t E CO,=).

W(E)

Moreouer, if

Cw(*,E2),

then

E2>

>

0

(resp.

<

W(E)

(resp. Cw(.,E1),E1>)

0)

for

E

E

is asymptotically

stable from below (reep. from above) in the sense that, for any

CP, El

satisfying ( 2 . 5 ) with

w ( - , E 2 ) f 9 on Cg, 2)

-I

),

replaced by

Remark 2.1.

h

Cw(.,E2), E21 (resp. C w ( * , E l ) ,

ul

=

u2

= u,

ml

-

m2 = m ,

is a function satisfying ( R . 1 ) .

h(m,d,h) =

(resp.

the convergence property ( 2 . 4 ) holds with

We define

with

I

As a special case, w e take

dl = d2 = d,

where

w(.,E1) # 9 on

(x) 1’2 I

:(

H(m)

-

H(u)

)

-112 du

El>).

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

I16

H(u) =

v h(u) dv.

After some calculations, it is proved that zero point points

E =

E = 1/2

A 2. 112

if

s', 1/2, I-s'

with

(us,

and

Cu,

CK,

A 2 1/2, then ( S P ) has

In the case

s ' 1 , Cus,l/21 and

1-3'1

If we take, say,

is smaller than a critical value

and two bifurcated solutions

h

as a parameter,

then

If

h

{us, 1/21

(g, s ' 1

However, if

<us, 1/21.

and

loses its sability

Cu,

1-s'1

get the

the asymptotic stability, Example 2.1.

112,

ho (recall the definition of

A ) , every solution of ( P f converges to h,,,

<

C u , 1 - s ' 1 , of

these cosiderations exhibit a bifurcation phenomenon.

becomes greater than

A

are asymptotical 1y stable from

below and above by Theorem IV.

h

112.

1/21, which is globally asymptoyically

( S P ) has three solutions Cg, s ' 1

<

A

if

s' E (0,1/2)

stable from below and above by Theorem IV.

which

has only one

and that it has three zero

Therefore Theorem I 1 1 implies that, if a unique solution

W(E)

Suppose that

Then ( S P ) exhibits bifurcation phenomenon when bifurcation parameter (see Figures 1, 2, and 3 ) .

h

>

0

is a

117

Free Boundary Problem in Ecology

1/ 2

0 Figure 1.

The bifurcation diagram of

( S P ) with respect t o

A.

1

*

s

in

118

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

Figure 2.

Profile o f

*

u (x)

with

A = 10

119

Free Boundary Problem in Ecology

1

+ x

-m

Figure 3.

Profile of

*

u (x)

with

X = 40

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

120

f3. Concluding remarks Remark 3.1. of

Even if we drop the monotonicity assumption in ( A . 1 )

f (resp. g)

(resp. ( A . 2 ) ) ,

Theorems

I

and I 1 remain

However, the uniqueness of solutuions for (2.2) is not

true.

assured, so that the analysis of ( S P ) will become more However the stabil ity analysis o f solutions of

compl icated.

( S P ) can be carried out along the line developed in this report

(for details, see C 4 l i . Remark 3.2.

x = s(t)

free boundary x

= 0, 1

In the case where

ml = 0

or

rn2 = 0, the

may hit one of the fixed boundaries

in a finite time.

Therefore, we need more careful

analysis to get complete information about the behavior of solutions for ( P ) (for details, see C41). Similar situation occurs when (1.3) and (1.4) are replaced by Neumann boundary conditions. Remark 3.3.

In this report, we have discussed a one-

dimensional model only.

The analysis of higher dimensional

models can not be done by the same method as used here.

is still open.

Here we display a numerical simulation of a

2-dimensional model where the region C(x,y)

:

x E

I,

This

y E I>

(see Figures U

K

is the unit square

- 8).

The equations are

described by the 2-dimensional version of (1.1)-(1.8),where

The boundary conditions are

Free Boundary Problem in Ecology u(O,y,t) = 0,7,

y

E I,

t

>

0,

u(l,y,t) = 0.7,

y

E I,

t

>

0,

u (x,O,t) = 0,

x E

I,

t

>

0,

u (x,l,t) = 0.

x E

I,

t

>

0,

Y

Y

It seems that 2-dimensional uniformly in

y E

I

steady state solutions

(which are essencially the one-dimensional

solutions obtained in the above)

are also stable in

2-dimensionel problems. We note that the papers C13, C81 end the references therein are worthly o f attention.

121

122

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

I! C1:

m:

0.lBBBE O . m

1 0

0 : 0.1BBBE

m:

0.-

1 0

G1:

0.1BBBE

1

(2:

0.1BBBE

1

3

(2:

0.1BBBE

3

Figure. 4

Cl:

m:

0.1O . m

1 0

0 : 0.1-

1

0 . -

0

m:

Gl:

0.1-

Figure. 5

123

Free Boundary Problem in Ecology Tat 3

51:

mi:

8.10e

1

B.;OBE

B

t

a:

8.t80M 1 B U ~ : 0 . m ~B

1

= 0.2

GI:

B.1mE

3

I Z : B.1000E

3

GZ:

3

Figure. 6 Teat31t=I.I

V C1:

m:

0.1BBBE 8.;138E

1 0

a: 0.1em: 8.78BE

1

G1:

0.1-

B

Figure.,7

3

0.lBBBE

124

Masayasu MIMURA, Yoshio YAMADA and Shoji YOTSUTANI

C1: 801:

0.lBBBE 1 0.768 0

0 : 0.1-

BCN:

1 0.788 E

G1: 0.lBBa

Figure, 8

3

GZ:

0.lBBa 3

125

Free Boundary Problem in Ecology References

C11 R, K, Alexander and B.A.

Fleishman.

Perturbation and

Bifurcation in a Free Boundary Problem.

J. 0. E.,

45

(1982). 34-52, C21 S. L. Kamenomostskaja.

On Stefan’s problem. Mat. Sb., 53

(1961), 489-514, C31 M. Mimura, Y. Yamada and S. Yotsutani. problem in ecology

I.

To appear in

A

free boundary

J.J.A.M.

C41 M. Mimura, Y. Yamada and S, Yotsutani,

A free boundary

problem in ecology 11. In preparation.

157 L. I. Rubinstein.

The Stefan Problem. Translation of

Mathematical Monographs 27. Amer. Math, SOC. prouidience,

R.

I.,

1971.

C61 M. Yamaguti and T. Nogi,

The Stefan problem. Sangyo-Tosho,

1977, ( i n Japanese).

C73 S. Yotsutani.

Stefan problems with the unilateral boundary

condition on the fixed boudary

IV.

Osaka J. Math., 21

(1984), 149-167

C81 D, Aronson, M. G. Crandall and L, A . Peletier. Stabilization of solutions of a degenerate nonlinear problem. Nonlinear Analysis

, 6 (1982). 1001-1022.

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 8, 127-132 (1985) Recent Topics in Nonlinear PDE 11, Sendai. 1985

On the Vanishing Viscousity of the Incompressible Fluid in the Whole Plane Rfiichi Mizumzchi Mathematical Institution, T8hoku University Sendai 980, Japan Many works are done as to the convergence of the solutions of the Navier-Stokes equations (NS) in space domain R2 to those of the Euler equations (E) as the viscousity parameter v goes to 0 ([1]-[4I).

("

1

It=o = u 0

Among others, a recent paper of T. Kato [ 2 ] showed that if u o E H S ( R 2 ) f o r s > 2 , then there are unique solutions u") and u (0) to (NS) and ( E ) , respectively, in C ( [ O , - ) ; C([O,T);

Hs) a s

v * 0, f o r any T>O.

Hs) and u")

127

u(O) in

(The same result has been

obtained by KyQya Masuda in an unpublished paper.) this problem in HElder spaces.

-+

Here we discuss

Ryiichi MIZUMACHI

128

We f i r s t c o n s t r u c t s o l u t i o n s of

Let T be a r b i t r a r i l y f i x e d .

(NS) and ( E ) i n t h e domain ( 0 , T ) X R 2 ,

referring t o

[3].

Let

F1 be

t h e o p e r a t o r d e f i n e d by

where xA=(x2,-x,)

to

an

{ u E ( C ( [ O , T ) X R ~ ) ~u :( t , * ) E ( C1 ( R 2) ) 2 and d i v u ( t , * ) = O ,

o p e r a t o r from

Oct
F o r ~ 2 0 ,l e t u s d e f i n e F;"',

f o r x=(xlyx2).

C([0,T)XR2),

by FLv)u=w, where w i s t h e u n i q u e bounded

s o l u t i o n o f t h e f o l l o w i n g e q u a t i o n (1) w i t h a g i v e n bounded c o n t i nuous i n i t i a l d a t a w o :

Let F(v)=FkV)F1.

nLm((O,T); B,)

Let B,=Ca(R2)AL

w i t h t h e norm

1 ( R 2 ) , O < a < l , equipped w i t h t h e

lgl

w 0= r o t u o

of F(')

O
and w i s a f i x e d p o i n t of ) ' ( F

t i o n of (NS) o r (E) ( s e e

[3]).

, gcXa. Then, i f Bci i n X,, t h e n Flu i s a s o l u -

= sup I g ( t , - ) l

'a

Unique e x i s t e n c e o f t h e f i x e d p o i n t

i s e s t a b l i s h e d by t h e f o l l o w i n g theorem.

Theorem 1.

( ' ) point of F

Suppose w O c B a .

i n Xa t o e a c h "20.

bounded i n X, f o r small

O u t l i n e of p r o o f .

Then t h e r e i s a u n i q u e f i x e d These f i x e d p o i n t s a r e u n i f o r m l y

V.

To c o n s t r u c t t h e s o l u t i o n w of (1) l e t u s

Vanishing Viscousity of the Incompressible Fluid

p r e p a r e some n o t a t i o n s .

For xeR 2 and O & , t < T ,

U

t ,s

129

( x ) i s defined

a s t h e s o l u t i o n of t h e d i f f e r e n t i a l e q u a t i o n

A function Z ,

i n t e g r a l operators W,

@

and y , and a d i f f e r e n t i a l

o p e r a t o r L a r e g i v e n by

Then t h e s o l u t i o n w o f (1) can b e g i v e n by

We n o t e t h e f o l l o w i n g lemma d u e t o McGrath [3].

Lemma 1 (McGrath). FIT and U

Suppose 6eLm((0,T); L 1( R 2)AL"(R 2 ) ) , u=

t , s ( x ) i s t h e s o l u t i o n of ( 2 ) . Then t h e r e e x i s t 6 and M , depending o n l y on T and s u p ( l c ( t , * ) l + l i z ( t , * ) l m ) y such t h a t O
Ryiichi MIZUMACHI

130

The following lemma is essential to prove theorem 1.

Lemma 2.

Suppose ceXkl, k ' > O , and u=F 15 .

Ir;l

15cx)-s(y)I

where

= sup cy(R2) x,yeR 2 ,x+y

Let O < h-k < l . Then

for y>O and c is a

IX-YP

constant depending only on T.

By these lemmas and the expression (3) of w, we can find a closed convex bounded subset S of X g a such that F ( ' ) S C S v.

for small

is a compact operator in X for each h Hence Schauder's theorem assures the existence of fixed

On the other hand, F")

h>0.

points w(')

of F")

in S.

By lemma 2 we can also show that w ( V )

are uniformly bounded in Xa.

The uniqueness follows from the next

lemma.

Lemma 3. some '20.

Suppose w and w ' are fixed points of F")

Then for any OLt
in Xa for

13 1

Vanishing Viscousity of the Incompressible Fluid

where C is a constant depending on wo,

w,

w',

T and v but not on t.

To go further to prove the convergence o f the solutions u

(V)

of (NS) to u(O) o f (E), we make the following assumptions on w o . 2

2

(3)

FluO G L (R )

(4)

as a function of x,

Iwo(x)-wo(Y) sup

Ix-Y 1<1 Let w")

be the fixed points of F"),

Ix-Y l a

v20,

I

is in L2 ( R 2 ) .

and U ( " ) = F ~ W ( ~ ) .We

obtain

Additional to the assumption of theorem 1, assume u(o) in w satisfies (3) and (4). Then f o r any 8, O
~

P r o o f is done by showing L2-convergence and using the inter-

polation of L2 and Ba.

We can also obtain the rate o f convergence

(see E41).

Theorem 3.

Under the assumptions of theorem 2, the following

estimate is valid for any 8, OzB
where C is a constant depending on wo and T.

Detailed proof o f these theorems will be given elsewhere.

132

RyQichi MIZUMACHI References

[11

K.K.

Golovkin,

V a n i s h i n g v i s c o s i t y i n Cauch’s problem for

hydromechanics e q u a t i o n s , P r o c . S t e k l o v I n s t . Math.

92

(1968) 33-53. [2]

T . Kato,

Remarks on t h e E u l e r and N a v i e r - S t o k e s e q u a t i o n s

i n R‘.

[3]

F.J.

McGrath,

i d e a l fluids,

N o n s t s t i o n a r y p l a n e flo w of v i s c o u s a 1 5 Arch. R a t i o n a l Mech. Anal.

3 (1968)

323-

348.

[4]

J.T.Beale and A . Majcla,

R a t e s of c o n v e r g e n c e f o r v i s c o u s

S p l i t t i n g of t h e N a v i e r - S t o k e s e q u a t l o n s , Math. Comp.

(L981) 243-259.

2

Lecture Notes in Num. Appl. Anal., 8, 133-161 (1985) Recent Topics in Nonlinear PDE II, Sendai, 1985

Index Theorems and Bifucations in Duffing's Equations

Fumio Nakajima Department of Mathematics, Iwate University Morioka 020, Japan

1.

Introduction. We shall consider the Duffing's equation with periodic

forcing term ; (0)

+ k; + au + bu3

=

2

B cos 2n

(

'

=

d

)

where k > 0, a 2- 0, b > 0, B > 0 and u > 0 are constants. It is known that equation ( 0 ) has several periodic solutions for appropriate k and B, which consist of harmonic solutions and subharmonic solutions, cf. [3] , [ 4 ] ,[6], [9],[lo]. The initial values at t = 0 of periodic solutions with period nu are called nu-periodic points for integer n > 0 in the following, and they are fixed points ofthe n-th power of Poincarg mapping. Several authors considered the indices of periodic points and obtained index theorems which show among periodic points , cf. [ 21 , [ 51 , [ 71

relationships

. However their definition

is done under the assumption that any periodic point is simple,

133

I34

Fumio NAKAJIMA

that is, the modulus of characteristic multipliers of the periodic point are different from 1, and this assumption is not true in general for equation (0). Recently G.Seifert and the author showed that the number of nu-periodic points of equation (0) is finite for each n > 0, cf.[8]. Moreover K.Shiraiwa at Nagoya University pointed out to the author that the above result enables us to define the indices of periodic points and state index theorems without the simplicity assumption. Theorem 1 and 2 are based on his idea. In Corollary 1, it is proved that if equation (0) has a directly unstable periodic solution, then there exist at least two other periodic solutions. Index theorems seem to be useful to the study of bifucations of periodic solutions. In Theorem 3 , 4 and Corollary 3, we shall show that the bifucation of second order subharmonic

solutions arises from the existence of an inversely unstable harmonic solution and that the bifucation of harmonic solutions arises from the existence of a direstly unstable harmonic solution. These results are illustrated by the physical data of 141 which will be stated later.

We denote by Rn the n-dimensional Euclidean space and set R1

=

R. By setting (

=

v in the equation (0), we have

u = v v

=

27r -kv - au -bu3 t B cos 7

135

Duffing 's Equations

The system (1) is a particular case of ( 2 )

,- u = U(t,u,v,B) where U(t,u,v,B) and V(t,u,v,B) are continuous for (t,u,v,B) R

4

and periodic in t with period

> 0;

The parameter B is considered to be fixed in the section 2 and to be

variable in the section 3 . Let (u(t,x,y),v(t,x,y)) be the solution of ( 2 ) through

(x,y)t R2 at t = 0 for fixed B. (x,y) is an nu-periodic point if (u(t,x,y),v(t,x,y)) is periodic in t with period nu. We assume that (i) the system (2) is dissipative, that is, there is a compact subset D of R 2 such that any solution (u(t,x,y),v(t, x,y)) exists in the future and remains in D for large t _> - 0, (ii) U(t,u,v,B) and V(t,u,v,B) are analytic in u and v for fixed (t,B), and (iii)

3u +ax au av

<

0

for (t,u,v,B) e R

4

.

We can verify that (1) satisfies (i), (ii) and (iii).

2.

Index theorems. We shall consider Poincarh mapping T

: R2-

R2 defined by

136

Fumio NAKAJIMA

it follows from Proposition that C has no a-periodic point on it. Therefore I(T,C) is defined. If

C

is shrinked down to P

continuously without crossing any o-periodic point, the change of I(T,C) is also continuous. Since I(T,C) is an integer, it is constant. Therefore we can define I(T,P)

=

I(T,C)

and call it the index of P by T. Similarly we can define I(Tn,P) for nu-periodic point P. Now we shall state relationships between the indices and the stabilities of w-periodic point P p2

=

(x,y). Letting p1 and

be the characteristic multipliers of P I that is, the eigen-

values of the matrix

we define that (il P is completely stable, if (ii) P is directly unstable if

1 pll < 1 and I p21 < 0

1,

and (iii) P is inversely unstable if

p1

< -1 < p 2 < 0.

The rest cases are the following; (iv)

IP1

(v)

IP1

> 1

and

= 1

or

and

The case (iv) should not happen, because condition (iii) implies

lp1p21

< 1. On the other hand, the case (v) happens

for system (1) where k,a,b and B are appropriate constants,

137

Duffing ' s Equations

Since u(t,x,y) and v(t,x,y) are analytic in T(x,y) is also analytic Tn(xry)

x and y by (ii),

in x and y. Clearly, (u(nu,xry) ,V(nurXrY)

=

and (x,y) is an nu-periodic point if and only if

We have obtained the following result in [ 8 ] Proposition.

;

For each integer n 7 0, the number of nu-periodic

points is finite. Let C be a simply closed curve which has no u-periodic point on it. For Q, C C, let Q1 = TQo and Q 3 1 be the vector from Qo to Q,, which is not a zero vector. The number of revolu-

+

tions made by QOQl as Qo traces out C once is an integer, since Q, returns to its starting position. Denoting the number by

Ior we set I(T,C)

=

I .

+

if the orientations of the revolutionsmade by Po and by QOQl are the same, and I(T,C) = -Io if they are reverse. I(T,C) will be called the index of C by T. Similarly, the index of C by Tn, say I (Tn,C), can be defined if C has no nu-periodic point on it. It should be noted that if t h e interior of C contains no nw-periodic point, then I(T",c) = 0. Now we shall consider w-periodic point P. If C is a circle with center at P and with sufficiently small radius,

Fumio NAKAJIMA

138

and it will be seen in the proof of Theorem 3 that this case i5 concerned with the bifucation of periodic Solutions. In [SI, it is shown that if P is completely stable or inversely unstable, then I(T,P)

= +1

and that if P is directly unstable

then I(T,P) = -1. Since P is also a 2w-periodic point, we may define I(T2 ,PI. It should be noted that if P is inversely unstable , then 2

I(T ,P) = -1, which

will be important in the proof of Theorem 3 .

We shall State index theorems.

Theorem 1. Let n be any positive integer. (i)

If P is an no-periodic point, then (I(TnrP)l 5 1.

(ii) If C is a simply closed curve which has no nw-periodic 0.

point on it and { Pj}j,l

is the set of no-periodic points in

the interior of C, then

(iii) If

{

PjIjEl is the set of all nw-periodic points of (2)r

then B

z I(T",P.)

j-1

3

= 1.

The proof is similar to one in [5] except assuming no simplicity condition.

Duffing' s Equations The proof of

139

( i ) .For t h e s i m p l i f i c a t i o n w e assume

C i s t h e c i r c l e w i t h c e n t e r a t (0,O) and

t h a t P = (0,O) and

w i t h r a d i u s r 2 0 . The expansion of T n ( u , v ) around (0,O) i s that

where a , b , c and d a r e c o n s t a n t s such t h a t

Since O < a d - b c ( 1 by Abelk e q u a l i t y , w e have a t l e a s t f o l l o w i n g case 1 o r c a s e 2 ; casel:

a # l o r b # O

case 2 : c

#

0 or d

f

1.

Here w e s h a l l c o n s i d e r case 1, because t h e c a s e 2 can be t r e a t e d s i m i l a r l y . Denoting C$ be t h e a n g l e between t h e v e c t o r (ul

-

u , v1

-

w e have

v) and u-axis,

For ( u , v ) & C , w e s e t

u = r cos e

,

v = r sin 0 ,

140

Fumio NAKAJIMA

where the domain of 8 is [0,2~). From ( 3 ) we have

=

(4)

tan-l (d-1)sin 8 (a-1)cos 8

+

+

+ +

c cos 8 b sin 8

o(r) o(r)

.

Our purpose is to show that

Therefore, it is sufficient to show that for the denominator of the right hand side of ( 4 ) F(r,8) F(r,e)

=

(a

=

-

,

say

1) cos 8 + b sin 8 + o(r),

0 has exactly two roots in [0,2a) for a small and

fixed r > 0. We have F(O,8)

(a

=

-

1) cos 8

+ b sin

and hence F(O,8) = 0 has exactly two roots

8,

el

and O 2 in

[O,~T)such that = tan

-1 b

and e2=e1+T. Since

we have

sJ( eo . e i )

+o

for i = 1,2.

By the implicit function theorem, F(r,8) roots e,(r)

and e2(r)

=

0 has exactly two

in the neighbourhoods of

and of €I2

,

141

Duffing’s Equations

respectively, for sufficiently small r > 0, such that

e,(o)

=

el

and

e,(O)

= 8,.

Here, we understand that a neighbourhood of 0 is the union of neighbourhoods of 0 and 2n. This completes the proof of (i).

The proof

of (ii). For each nu-periodic point P

1 5 j 5 - a, we enclose it by a small circle C j contained in the interior of C and

I(T”,P.)

= I ( T n ,cj)

for j such that C . is 3

.

3

Let us join C . for 1 5 - j 7

-

a and C together by curves

so that if the curves are regarded as cuts, then the curves and cycles, C . and C form a simple closed curve as illustrated in 7

Fig.1 and denote this curve by .’l

Fig.1

Fumio NAKAJIMA

142

In determing the index of T r each curve segments, or and C is traversed first in one direction and j then the other for 1 f j 5 a. Thus the net effect of these cut, joining C

segments on the index is zero, and we have c1

I(Tn,r)

Since

r

=

I(TnrC) -

1 I(Tn,C.). j=1 7

contains no nu-periodic point in its interior, we have

and hence

This completes the proof of (ii).

The proof of (iii). Since ( 2 ) is dissipative, there is a simply connected set K such that T ~ K C K

as is stated in [51. Letting C be the boundary of K which is a simply closed curve, we have by ( 6 ) that I(Tn,C) Since

{

B Pj Ijs1

= +1

.

is contained in the interior of C, (ii)

implies that B

c I(T",P.)

j=1

3

=

I(T",c) = +1

.

This completes the proof of (iii). If P is an nu-periodic point, then TiP are also nuperiodic points for 1 -5 i - n-1

. Then we

have following result.

143

Duffing's Equations

Theorem 2. (i) If P is an nu-periodic point, then (7)

for 1 5 - i 5 - n-1

I ( T ~ , T ~ P=)I(T",P) a

(ii) If {Qj}j=l

is the set of nu-periodic points

with least period nu, then a

c I(T",Q.)

(8)

j=1

3

o

(

mod. n

)

Proof of (i). It is sufficient to show that for any nu-periodic point P

,

I Let

Co

be a circle with center at P and with a sufficiently

small rad us such that it contains 1' as a unique nu-periodic point in its interior, and hence

Setting

ct

= {

(u(t,Q),v(t,Q)) E R~ : Q e:

c0 I

for t 1 0,

by the uniqueness of solutions of initial value problems we can see that Ct is a simply closed curve and continuous for t. Since

Cw

contains TP

as a unique nu-periodic point in its

interior, we have

Now we shall consider the following mapping St on

R 2 to R2

:

Fumio NAKAJIMA

144

where

(

u(t,s;x,y),v(t,s;x,y)

(x,y) at t For Q

t

= s.

)

is the solution of (2) through

Clearly St is continuous for t and

S

-

0 - su

= Tn.

Ctl the solution through it at t is not nu-periodic,

because it is on Co at t = 0, and hence StQ # Q. Therefore we can define the index of Ct by St as similar to I(T,C). In fact, when Q traces out Ct once, the number

We set I(St,Ct)

=

I.

I(St,Ct) = -Io

and

+ and

if the orientation of the revolutions Q, Q,

same and if they are reverse, respectively.

, Q are the From the

continuity of St and Ct for t, we can see that I(St,Ct) is also continuous for t. Since I(St,Ct) is an integer for each t, it is constant, and hence

.

I(So'Cu) = I(StlCt) = I(S0~CO) By

Su

= So =

Tn, we have

I(Tn,Cs) = I(Tn,CO)

.

Therefore ( 9 ) and (10) implies I(T",TQ) = I(T",Q)

.

This completes the proof of (i).

Duffing's Equations

145

The proof of (ii). Since Q is a periodic point with j n-1 least period nu, {TiQj) is a set o f distinct nu-periodic i=O points. Therefore {Q,}j:l

can be rearranged as a disjoint union;

We have

6

n-1

k=l

i=O

a

c

I(T",Q.) j=l 3 Since I(Tn,TiQj ) k (i) , we have

=

c t

=

=

I(T",T~Q~1 1.

k

-

for 0 5 i 2 n-1

I(Tn,Q. ) lk

crc I(T",Q.) i=l 3

c

c' nI(Tn ,Q. k=l ' k

=

by the above

)

6 n I(T~,Q~ k= 1 k

which shows ( 8 ) .

We have following corollaries. Corollary 1. If ( 2 ) has a directly unstable w-periodic point, then there exist at least two other w-periodic points. Proof. Let P1 be the directly unstable w-periodic point. Since I ( T , P

1

) =

-1, (iii) of Theorem 1 implies

6

Therefore, (i) of Theorem 1 implies the existence of two wperiodic points with index +l.

146

Fumio NAKAJIMA

The above result is illustrated in Fig.2 by the data given in [ 4 1 which shows the location of 2n-periodic points of the equation ,#

u

+

0.2;

+ u3

=

0.3~0s t

.

Figure

2.

Here point 1 is a directly unstable 2 ~ p e r i o d i cpoint, and points 2 and 3 are 27r-periodic points with index +l.

Corollary 2.

Let

c1

be the number of w-periodic

points of ( 2 ) . If n is a prime number such that a + l ,

n

then for the set of periodic points with least periodsay {Qk},=,B

nu

, we have

(11)

' k=l

n

C I ( T ,Qk) = 0.

Consequently, all the periodic points of least period of nu cannot be completely stable.

,

Duffing 's Equations

Proof. c1

{Pj}j=l

.

147

We denote the set of w-periodic points by

Since n is a prime number, the set of nu-periodic

points consists of only {Pj}jzl and {Qk}k=l B

. By

(iii) of

Theorem 1, we have c1

'

c I(T",P.) j=l

+ cB

n I(T , Q ~ )= +i k=l

,

and hence

Since

II(Tn ,Pj) I 2 1, we have

By (ii) of Theorem 2 , there is an integer q such that

B

c I(T",Q~)

=

nq

k= 1

.

Therefore we have lnql Since n

>

c1

2

1

+ a

*

+ 1, we obtain q

Remark.

In [ 7 1 ,

under the assumption

= 0

which shows (11).

(11) is proved for all odd number n

that every periodic point is simple.

Fumio NAKAJIMA

148 3 . Bifucations.

A second order subharmonic point is the initial value

of second order subharmonic solution at t = 0, that is, it is a 2w-periodic point but not a w-periodic point. In the following, the parameter B of (2) is variable and the Poincar’e mapping and W-periodic point are denoted by T(B) and P(B) , respectively. Since U(t,x,y,B) and V(t,x,y,B) are continuous

.

(x,y,B) t R 3

for (t,x,y,B).$ R 4 , T(B) (x,y) is continuous in

Theorem 3 . We assume that there is a Bo & R and

E~

> 0

such that (2) has an w-periodic point P(B) which is continuous in Bo

-

e O < B < Bo

+

E

~

completely , stable for Bo

< Bo and inversely unstable for Bo < B < Bo

Then

,

for a sufficiently small

following conclusions (i), (ii) or (i) for any B,BO < B < Bo +

E,

both

E

+

E

~

-

E

0

< B

.

> 0, we have

;

(2) has at least two second

order subharmonic points Q1(B) and Q2(B) such that Qi(B)

__*

P(BO)

as B

--f

Bo

and 2

1 (T (B)rQi(B)

for i = 1 and i

=

= -1

2 (T (B),P(B) )

2,

(ii) for any B, Bo

-

E

C

B < B o r the same as in (i) holds.

The above result may be illustrated by the following figures.

149

Duffing’s Equations

(i)

(ii)

7

Fig.3 For example, in (i), the arrows

represent the manners

how Q1(B) and Q2(B) bifucate from P(B) as B is increasing. The number +1 or -1 attached to Q1(B) ,Q2(B) ,P(B) denotes the index of the point by T 2 ( B ) . We shall prove Theorem 3 by three steps. Step 1. Letting Po = P(Bo), we can see that Po is a fixed point of T(BO) and T 2 (Bo). By Proposition, there is a circle C with center at Po and with a small radius such that C has no fixed point of T2 (Bo) on it and Po is

a

unique

fixed point of T2 (Bo) >in the interior of C. Therefore we have (12)

If

E

is

2

I ( T ~ ( B ,pol ~ ) = I ( T ( B ~,c) )

sufficiently Small

continuity of P(B) and T(B)

, then

.

it follows from the

that if IB

-

Bol <

E,

Fumio NAKAJIMA

150 (*)

P(B) is contained in the interior of C

and (**)

2

C has no fixed point of T (B) on it.

Now we shall show that if

IB - B 0 I <

and

E,

E

is

sufficiently small

then

( * * * IP(B) is

a

unique fixed point of T(B) which

is contained

in the interior of C.

In fact, fixed points of T(B) are the zero points of the following function;

We have

where

p1

and p q are the characteristic multipliers of P

0'

Since P(Y) is inversely unstable for Bo < B < Bo+

E

,

we have that P1

2 -1 5

P* < 0

which implies

By the implicit function theorem, the zero point of S(x,y,B) is unique in the neighbourhood of Po for IB E

is

of C

-

sufficiently small. Therefore, taking

BOI <

E

if

the radius

so small that its interior is contained in the above

neighbourhood of Po, we have (***I

.

15 1

Duffing’s Equations Step 2. In the following, we take (*)

,( * * )

and

(***)

so small that

E

hold, and let IB - B , l

E

. By

2

2

I (T (B),C) is defined and continuous for B. Since I(T ( B )

,C)

is an integer, it is constant, and hence

2

Letting {Pj}jEl be the set of fixed points of T (B) contained in the interior of C, we have from (ii) of Theorem 1

and it follows from (12),(13) and (14) that c 1 2 C I(T (B),Pj)

(15)

j=1

By

(*)

,

,

P(B) is contained in the interior of C

P(B) = P . 3

inversely unstable for Bo < B < Bo

+

-

E

We shall show that if Bo or B # Bo

,

and hence

for some j.

Since P(B) is completely stable f o r B o -

E

.

E

E

< B < Bo

and

, we have

B < Bo

,

Bo < B < Bo +

2

then T (B) has at least two fixed points

the interior of C whose indices are +1 and -1. For Bo

-

E

in <

B < Bo, suppose that TL(B) has no fixed point with index -1. Then we have by (i) of Theorem 1 that

Fumio NAKAJIMA

152

which shows by ( i )of Theorem 1 t h a t 2

I ( T ( B o ) I P o ) = +1

(16)

.

S i m i l a r l y , s u p p o s i n g t h a t T 2 ( B ) h a s no f i x e d p o i n t w i t h i n d e x +1 i n t h e i n t e r i o r of C f o r some B, Bo < B < Bo

+

E,

we can

show by t h e same argument a s above t h a t 2 I ( T (Bo), P o ) = -1

which c o n t r a d i c t s t o ( 1 6 ) . T h e r e f o r e w e o b t a i n t h e above assertion. Since t h e f i x e d p o i n t of T ( B ) i s unique i n t h e i n t e r i o r

,

o f C by (***I

one o f t h e above two f i x e d p i n t s o f T 2 ( B ) i s

n o t a f i x e d p o i n t of T ( B )

I

and t h e r e f o r e i t i s a second o r d e r

subharmonic p o i n t . S t e p 3 . L e t t i n g t h e second o r d e r subharmonic p o i n t be Q1(B)

and Q 2 ( B )

= T(B)Q1(B)

I

w e have by ( i ) o f

Theorem 2 t h a t Q 2 ( B ) i s a second o r d e r subharmonic p o i n t and

I n t h e above argument, w e have s e e n t h a t

and t h e r e f o r e

153

Duffing’s Equations I(T2(B) ,Qi(B)) NOW, l e t t i n g B

= -I(T

f o r i = 1 and i = 2 ,

(B) , P ( B ) )

-

r Bo,

P(B)

2

w e have

*

I n t h e above a r g u m e n t s , t h e r a d i u s o f C i s o n l y r e q u i r e d t o

is

s a t i s f y t h a t C c o n t a i n s P ( B ) i n i t s i n t e r i o r . S i n c e Q1(B)

also c o n t a i n e d i n t h e i n t e r i o r of C , w e have Po

Q1(B)

as B

-’ Bo

r

and h e n c e

l i m Q2(B) = B+Bo

l i m T(B)Q1(B) B i B o

= T ( B o ) P o = Po

.

This completes t h e proof.

2 R e p l a c i n g T ( B ) by T ( B ) i n t h e above p r o o f

, we

can

o b t a i n t h e following r e s u l t . C o r o l l a r y 3 . F o r some Bo 6 R and

, we

> 0

E~

assume

t h a t ( 2 ) h a s a n w - p e r i o d i c p o i n t P ( B ) which i s c o n t i n u o u s

€or B

0

-

c O < B < Bo

+

so, c o m p l e t l y s t a b l e f o r Bo

Bo and d i r e c t l y u n s t a b l e f o r Bo

B < Bo

Then f o r s u f f i c i e n t l y s m a l l c o n c l u s i o n s (i), (ii)o r b o t h ;

E

+

E

~

-

E~

< B <

.

> 0 , w e have f o l l o w i n g

and > Q(B) -

-

(ii) for B

as B >-

B0 ' < B < Bol the same as in (i) holds.

E

P(Bo)

We shall return to the Duffing's equation

;

u = v

v

=

-kv

-

au -bu3

+

.

2n B cos t0

Theorem 4 . If there is a parameter B* such that (1 has inversely unstable w-periodic point for B = B* there is a Bo, 0 < Bo < B*, and a small number

E~

,

then

> 0 such

that (1) has a periodic point P(B) which is analytic for Bo

-

E~

< B < B

0

+

E~~

completely stable for Bo

and inversely unstable for Bo < B < Bo +

E

~

-

E~

< B <

. Consequently

the conclusion of Theorem 3 holds.

Remark. In the above we considered the case where B is increasing for a fixed k. The same conclusion can be obtained by considering that k is increasing for a-fixed B under the same assumptions as of Theorem 4 .

155

Duffing 's Equations

We shall prove the theorem

by three steps.

Step 1. Let (u(t,x,y,B),v(t,x,y,B)) be the solution of (1) through (x,y) at t

= 0

and set

f(X,y,B)

=

U(W,X,Y,B)

-

X

g(x,y,B)

=

V(w,X,y,B)

-

y

-

Since u(t,x,y,B) and v(t,x,y,B) are anlytic for (x,y,B) C R

3

,

.

f (x,y,B) and g(x,y,B) are analytic for (x,y,B) Clearly (x,y)

is an w-periodic point if and only if f(X,y,B)

=

g(Xry,B)

= 0.

Let (x*,y*) be the given inversely unstable w-periodic point for B

=

B* and pl, p 2 be the characteristic multipliers

such that P1 < -1 < p 2 < 0 .

We have f(x*,y*,B*)

=

g(x*,y*,B*) = 0

and

Therefore, by the implicit function theorem, there exists analytic functions x(B) and y(B) defined in the neighbourhood of B* such that

Fumio NAKAJIMA

156

and

,

X(B*) = x*

y(B*) = y*

.

Considering the analytic continuation of (x(B),Y (B) for B < B*, we have the maximal interval (B1,B*) such that

(x(B),y(B)) is analytic for B1 < B < B*. Since ( 1 7 ) i s satisfied in the neighbourhood of B*, it follows from the unicity theorem that (17) holds on (B1,B*), and hence (x(B),y(B)) is an

w-

periodic point on (B1,B*).

Step 2 . Let multipliers of

(

p1

(B) and p2 (B) be the characteristic

x(B) ,y(B) 1 such that they are ordered to

be continuous for B and p1

(B*) < -1 < p2(B*) < 0 .

By Abel’s eqality, we have

*

We shall show that there is a B2 f (B1’B ) such that

To the contrary we suppose that (20)

If B1 >

-m

points as B

pl(B)

-

, P(B)

2 -1

for all B

6

(BIrB*).

is bounded on (B1,B*) and has accumulation B1. Letting (xl,yl) be one of them, we

157

Duffing's Equations

have

and

Since the characteristic multipliers

p1

and

p2

of (xl,yl)

satisfy by (18) and (20) P1

2 -1 2

P2

< o l

it follows from the same argument as of Step 1 that P(B) can be defined to be analytic for B1

-

E~

< B < B1

and for a

small number cl > 0. This contradicts to the definition of Bl. Therefore we have that

On the other hand, for B

= 0,

(1) is reduced to the system

;

u = v v

=

-kv

-

au

-

bu 3

.

Clearly,this system has the unique periodic point (0,O). Therefore p1 (0) and

p 2 (0)

is the characteristic multipliers

of (0,O) which is completely stable, and hence

158

which contradicts to (21). Thus (19) is proved.

Step 3. Since pl(B) is continuous for B,by (19) there is a B t- (B2,B*) such that (22)

pl(B)

= -1

which implies with (18)

.

p2(~)= -eVwk Since pl(B) # p 2 ( B )

,

it is known that pl(B) and p2(B) are

analytic at B. Therefore there is a Bo t (B2,B*) and a small number

E

> 0 such that

-1 < pl(B) < 0

pl(BO)

=

for Bo -

E

< B < Bo

,

-1

and pl(B)

< -1

for Bo < B < B O + € .

Therefore P(B) is completely stable for Bo and inversely unstable for Bo < B < Bo

+

E

-

E

< B < Bo

. This complete

the proof.

Thoerem 4 is illustrated by the following data of [ 4 1 which shows the location of 2Tiperiodic points and second

order subharmonic point of the equation

159

Duffing's Equations u

+

0.26

+

u3 = B cos t,

where B is increasing. w

U

When B = 0.3, there is a completely stable 2n-periodic point

B6. When B = 3, there is a 2n-periodic point B7 from which two second order subharmonic points bifucate. When B = 5.5

,

there are an inversely unstable 2n-periodic point B8 and two completely satble

second order subharmonic points D 8 and E8.

Fumio NAKAJMA

160

Acknowledgement. The author wishes his invaluable thanks to Professor K.Shiraiwa at Nagoya University for his comments and suggestions. Moreover the author wishes his invaluable thanks to Professors C.Hayashi, Y.Ueda and H.Kawakami

for thier permitting that thier interesting data of [ 4 ] may be used here.

References

[l]

K.T.Alligood, J.Mallet-Parct and J.A.York, An index for the global continuation of relatively isolated sets of periodic orbits, Geometric Dynamics, Springer Lecture Note in Math. 1007(1983)

[21

G.D.Birkhoff, Dynamical systems with two degree of freedom, Trans.Am.Math.Soc. 18(1917)

[31

Funat0 and Maekawa, On the existence of subharmonics for Duffin&

[4]

equation, Math.Japonica, 5f1958 v 59) , pp.27-32.

C.Hayashi, Y.Ueda and H.Kawakami, Transformation theory as applied to the solutions of nonlinear differential equations of the second order, Int. J.Non-linear Mechanics, vo1.4(1969) ,pp.235-255.

161

Duffing's Equations [5]

N.Levinson, Transformation theory of nonlinear differtial equations of the second order, Ann. Math., 45(1944), pp.723

[6]

- 737.

W.S.Loud, Periodic solutions of

+

c;

+

g ( x ) = Ef (t),

Amer. Math. SOC. Mem., No. 31(1958).

[7]

J.L.Massera, The number of subharmonic solutions of nonlinear differential equations of the second order, Ann. Math., SO(19491 , pp. 118 - 126.

[8]

F.Nakajima and G.Seifert, On the number of periodic solutions of 2-dimensional periodic sytems, J. Diff. Equations, vol. 49, No.3(1983), pp.430 - 440.

[91

Y.Shinohara, Numerical investigation of

1

- subharmonic

solutions to Duffing's equation, Memoirs of Numerical Mathematics, No. l(1974).

[lo]

M.Urabe, Numerical investigation of subharmonic solutions to Duffing's equation, Publ. RIMS, Kyoto Univ., vo1.5 (19691, pp.79

-

112.

This Page Intentionally Left Blank

Lecture Notes in Num. Appl. Anal., 8 , 163-174 (1985) Recent Topics in Nonlinear PDE I I , Sendai, 1985

Global Solutions for Some Nonlinear Parabolic Equations with Non-monotonic Perturbation Mitsuhiro NAKAO Department of Mathematics,College of General Education,Kyushu University,Fukuoka 810,Japan

0.

Introduction

In this article we are concerned with the &stenoe,

uniqueness

and decay property of the solutions of the following two problems

and

where R is a bounded domain in R” with smooth boundaryas and f(XrU) iS a (locally) Halder continuous function on a X R + such that

with some d 3 0 and ko 7 0.

163

Mitsuhiro NAKAO

164

The existence and the nonexistence of global solutions for (P1) with 8 (x,u)=-I uId u were first investigated by Tsutsumi [ I 1 ] using the concept of the ' potential well' introduced by Sattinqer [ 9 I . The essential assumption in [ I l l for global existence is the growth condition on (xlu),i.e.,

P

where

m; =

{

(mN+2m+4)/ (N-m-2) arbitrarily large

if N ) m+2 if 1 < N ( m+2.

The result of [ f I I was generalized by 6tani [ 8 I ,Ishii [ 3 1 and Nakao & Narazaki [ ? ] etc.. Concerning the uniqueness very little is known,i.e., the solution is known to be unique only for the case 1 s N<m+2.( This case is rather trivial by the inclusion Wo1 ,m+2 LW ) .

c

A parallel result to [ I l l was proved also for the problem (Pz) by Galaktinov [ 2 ] and Nakao [ 4 ] independently. For (P2) the restriction on d is

where

mf =

{

(mN+4)/ (N-2)

if N)2

arbitrarily large

if N-1,2.

The decay properties of solutions of (P1) and (P2) were discussed in [ 7 1 and [ 4 1 ,respectively (see a l s o [ 3 1 ) .

(7)

Nonlinear Parabolic Equations

165

Recently in [ s ] we have derived L o o estimate of the solutions both for ( P ) and (P2) using the so-called Moser’s 1 technique, and consequently proved the existence and uniqueness theorem for (P1) and (P2) under the assumption ( 4 ) and ( 6 ) , respectively. Saks [ l o ] also has proved closely related result for (P2) with the use of quitebdifferent method. In [lo] the assumption ( 6 ) is made implicitely. The object of this article is to show the existence,uniqueness and some decay properties of the solutions of (P1) and (P,) without any restriction on d,

.

1. Some lemmas and the statement of result. First we recall the Sobolev‘s Lemma and(a variant of) Gagliardo-Nirenberg inequality. Lemma 1. WAfP(R ) is continuously embedded into Lq(R ) provided that (i) N 7 p 3 1 and I s g Np/(N-p) (ii) N=p > 1 and 1 s q < O D or(iii) 1 (,N< p and 1 < q go

<

.

d ut W1 Lemma 2. For all u with \ul

have

with C 7 0 and a = ( d + l ) (r-’-q-’){ N-’-p-l+ (3’ +1)r-’] provided that q 4 dil and (if 1 s r S q 5 (d+l)Np/ (N-p) if N > p (ii) 15 r,c q < f l if N=p=l or(iii) 1 5 r s q 5 w if 1 s Nc p. (The case &‘=O is the origina G-N inequal ty.) For the L v - boundedness of solutions the following lemma will play an essential role.

166

Mitsuhiro NAKAO

Lemma 3. Let w(t)rw(x,t) be an appropriately smooth function defined on R x R+, satisfying

for any A

> xo) max(0,r-m-1,(m-r+l)/(r-1) )

C o O O ) , C 1 ( ~ O ) , 8 , ( 3 01,

C

>0

and d

>0

with some constants

Gl(40) and r a l . Suppose that

such that

Lemma 3 is a generalization of Alikakos 11;Lemma 3.21 and can be provrd by Moser’s technique (cf. [ 5 : Appendix]). Our result reads as follows. Theorem 1. Let d 7 m and let f satisfy (3). Assume that uo for some p, with poz, 0 and po7 m+2 N (a -m)- 2 .

e W tfm+2n LpO+2

Then, there exists do 7 0 such that

if 11 u0\\, +2

<

do the problem

0

(P1) admits a global solution u(t) which satisfies

and

167

Nonlinear Parabolic Equations

with some

C depending on \\qu o m+2' Moreover , under the additional assumption uo E L

(R

,

we have U E L w ( ~ + ; W2'm+2/\

L~

Such solution is unique if

f(x,u) is locally LiFshitzian in u .

)

and

Theorem 2. Let d ) m and

iln L(p1+2) (m+l)

with p

1

-

satisfy ( 3 ) . P-ssumethat[ u ~ ~ u , ~ such that p1 2 0 and p1 7 N(~i-m)-~,

Then, there exists

dl> 0 such that if (luO(lp +2( 1 (P2) admits a solution u(t) satisfying;

dl the problem

and

Moreover , under the additional assumption uo t L@the solution u belongs to Lw(R+; L D " ) and satisfies

Such solution is unique if

/I(XIU) is locally Lipshitzian in u .

168

Mitsuhiro NAKAO

2. Outline of the proof of Theorem 1. The solution will be given as a limit of smooth approximate solutions. For this we must derive the estimates (8) and (9) for the approximate solutions. Here we shall give an outline of the proof of such estimates for (assumed) smooth solution (P1). We write uq for iu\q-'u rq b 1. Multiplying the equation (1) by uP+',p

3 0, we have

Here we utilize Lemmas 1 and 2 to get

for some C o ) 0 under the assumption on p0 ' From (12) and (13) we obtain

where we set

169

Nonlinear Parabolic Equations

Now, making the assumption IIu

11

<

d;

, we conclude that

is monotone decreasing and

(p0+m+2) / (m+2) m+2 11 m+2 L O

d

for a certain

-

~

~

7 This 0 . inequality implies

II u (t)llp0+2 5

C(l+t)

Next, multiplying the equation

(15)

, in particular,

-1/m

(16)

(1) by ut we have

where we set

Here, note that if N) m+2 and dJm; ( and consequently p0 + d ) we see 11 u(t)11,(+2 Cd; ( P O When po
<

.

for a certain C1) 0. Hence, under the additional assumption

d6 we have

d +2

(c-)

1f (d-m)

(19)

Mitsuhiro NAKAO

170

for a certain E l > 0. Thus, setting do=d; if pozo( and do=min(dA,dG) if poco( we obtain the boundedness of ] \ T j u ( t ) ) 1 m + 2 . Since

we can take To) 0 such that (20) is valid for t ?To and a l s o Next, multiplying the equation by u and using (21) for po7d we may assume

.

with some E 2 7 0.

Combinning (17),(20) and (22) we can obtain

or

which implies

Integrating (17) and using (24) we also have

Nonlinear Parabolic Equations

171

Now,the desired estimate (8) has been established. We proceed to the proof of ( 9 ) . The case N < m+2 is trivial and we assume in the sequal that N m+2. Let us return to the inequality (12). By Holder’s inequality

where q=(p+m+Z)N/(N-2-m) if N 7 m+2 and q=V(p+m+2) if N=m+2 ( U being fixed sufficiently large number), and Qi(i=l12,3)are given by

with

r=m(q-po-2)+(po+2) (q-p-2). Since 11 u(t)il

where

C

P C P

is bounded we have from (25) that

is a constant satisfying

<

e

const. (l+p)

4

with 0

=

o(q/(r-q).

Thus we arrive at the inequality

d

(p+m+2)/ (m+2) m+2 11m+2

Mitsuhiro NAKAO

172

for any p 2 0 and some C2'C3(> 0) independent of p . Applying Lemma 3 to (26) we see thatI{u(t)(/,$const.c@ for any t >, 0 . under the additional assumption u0e LP Finally, setting u(t) (l+t)l'mzw(t) and t=log(l+t) ,we have

Applying the argument obtaining the boundedness of Ilu(t)ll,to (27) we can prove similarly the boundedness of ((W(t)IIcs,which shows (9). 3. Outline of the proof of Theorem 2.

It suffices again to derive the a priori estimates (10) and (11) for assumed smooth solutions. and um+ 1 Multiplying the equation (2) by up+' ,

we have

Nonlinear Parabolic Equations

173

and I \ m+l ~ U (t)\\, 2

-

ko JQum+d+2 dx

~i,t~m'4 dx

where we set

With the use of (28)-(30) we can prove the desired estimates in a parallel way to that of the previous section. The details o f the p r o o f s w i l l b e s t a t e d in [ 6 ] . Re ierences 1. N.D. Alikakos, LP-bounds of solutions of reaction diffusion equations, Comm.Partia1 Differential Equations 4(8)1979) p.827-868. 2. V.A. Galaktinov, A boundary value problem for the nonlinear parabolic equation ut= A u3(+l + upI Differential Equations 17 (1981),p.551-555 (in Russian)

.

3. H.Ishii, Asymptotic stability and blowing up of solutions of some nonlinear evolution equations, J.Differentia1 Equations ,2 6 (1977) ,p .291-319. 4 . M.Nakao, On solutions of perturbed porous medium eauations, Proc. 7th Conf.Ordin.& Part.Diff.Equat. at Dundee Univ.,

Lect.Notes of Math.,Springer,N0.964(1982),p.539-547. 5. M.Nakao, LP-estimates of solutions of some nonlinear degenerate diffusion equations, J.Math.Soc.Japan,to appear. 6. M.Nakao, Global solutions for some nonlinear parabolic equations with non-monotonic perturbations,in preparation.

174

Mitsuhiro NAKAO

7. M.Nakao & T.Narazaki, Existence and decay of solutions of some nonlinear parabolic variational inequalities, International J.Math.& Math.Sciences,Vol.2,No.l 19801, p . 79-102.

+vl

Existence of strong solutions for @ dt u) - 3 $ (u) 3 f l J.Fac.Sci.Univ.Tokyo,Sect. IA,Math.24(1977) p.575-605.

8. M.&ani,

9. D.H. Sattinger, On global solutions of nonlinear hyperbolic equations, Arch.Rationa1 Mech.Anal. 30(1968),p.148-172. 10. P.E. Sacks, Global behaviour for a class of nonlinear evolution equations, to appear. 11. M. Tsutsumi, Existence and nonexistence of global solutions €or nonlinear parabolic equations,Pub.R.I.M.S.,Kyoto Univ. 8 (1972-73),~.211-229.

Lecture Notes in Num. Appl. Anal., 8, 175-228 (1985) Recent Topics in Nonlinear PDE I I , Sendai, 1985

On a G l o b a l E x i s t e n c e T h e o r e m of Neumann P r o b l e m f o r Some Q u a s i - L i n e a r H y p e r b o l i c E q u a t i o n s Yoshihiro SHIBATA* and Yoshio TSUTSUMI**

*

Department of Mathematics, University of Tsukuba Ibaraki 305, Japan

**

Department of Pure and Applied Sciences, College of General Education, University o f Tokyo, Tokyo 113, Japan

1. P r o b l e m , N o t a t i o n s and R e s u l t s

Let a be an unbounded domain in an n dimensional Euclidean space Rn with compact and Cm boundary

an. Let us denote a time variable by t o r xo

and space variables by x = (xl we write a t = a.

slat,

,. . . , x n ) , respectively.

aj = a / a x j ,

j = 1 ,..., n .

For d i f f e r e n t i a t i o n

In t h i s note, we

consider the following problem:

@(nu) = n u

-

cij=, n ai(aij(Dxu)aju) 1

t

1 ohij Dxu)aju

B ( D 1u )

=

f

+ ~ ( u )= g

i n [O,-)xn,

on

[O,m)xaQ,

in a.

0,x) = u1 ( X I

1 1 % 1 Here, Dxu = ( a i u i = 1 ,..., n ) , D'U = (a,u, D X u ) , AU = ( D ~ U . ~ ) , AU = 2 2 1 ( D u , a . a . u , i , j = O ~ I , . . . , ~o)=, a t - A = a: - c:=, a j ( A : Laplacian, 1 J

0: d'Alembertian , -awa_ - zjj=l n v j ( x ) a j , and w i ( x ) , i

=

valued functions i n Cm(Rn) such t h a t v ( x ) = ( v l ( x )

,. . . , v n ( x ) )

the u n i t outer normal o f aa a t x

E

an.

175

l,...,n,

are realrepresents

Yoshihiro SHIBATA and Yoshio TSUTSUMI

176

The t y p i c a l examples o f 9 and Q~ a r e t h e f o l l o w i n g : 1

I n f a c t , s i n c e by T a y l o r expansion we can w r i t e 1 = aju-2aju(zk=i(aku) 1 n

a.u(l+ck=l(aku)z)-T n

2

)I

1

--3

[l+a(zF=,(aku)2)l

'da

0

J

we have

where 6ij

= 1 i f i = j and

= 0 if i

# j.

I t i s t h e purpose of t h i s note t h a t we prove t h a t t h e r e e x i s t s one

and o n l y one c l a s s i c a l s o l u t i o n o f (1.1) f o r s u f f i c i e n t l y small and smooth

f and g.

data u o y ul,

Notations.

F i r s t , we summarize n o t a t i o n s .

n and N, N ' , N " , II always r e f e r t o t h e dimension and

t h e i n t e g e r s d e f i n e d i n Theorem 1.3 below.

K,

L, M and p, q always r e f e r

t o non-negative i n t e g e r s and extended r e a l numbers w i t h 1 5 p, q respectively. integer s

Functions considered below a r e a l l r e a l - v a l u e d .

2 1, a f u n c t i o n u, a v e c t o r valued f u n c t i o n f

and a m u l t i - i n d e x

a;

(a;u;

$D U ;

qD:f Do,X

= (DxDxfl 4 M

0

= D u = u

lal M 5

=

F o r an

= ( f l ,...,fS)

set

a = (al,...,an),

= afll**-a;n,

m,

1

n'

If1 = lfll +...+ IfS!,

5 M+L), -DL DM u - ( a i a i u ; FF -

,... ,DxDxfs), -LM

-DL DM f = ( D- L DM fl

-LM -LM (DxDxu, D x D x f ,

...

2 jtlal

M+L),

,...,-DL DM fS),

a r e a l l v e c t o r valued f u n c t i o n s ) .

I77

Some quasi-Linear Hyperbolic Equations I n p a r t i c u l a r , A U = ~ ’ O ’ U and xu = n i u .

IRs, r e s p e c t i v e l y . space over G and

L e t G and I be domains i n Rn and

L e t X be a Banach space.

II-lL,P

L H ~ ( G =) i u

i t s usual norm.

L ~ ( G ) ;( I U ( ( G , ~ , ~ =

Put

Lp(G) denotes t h e usual Lp

Ilflb,p

=

S cjZ1

llfjlb,p.

Set

( -L( D ~ u
L C (1;X) denotes t h e space o f f u n c t i o n s which a r e L-times c o n t i n u o u s l y differentiable for t

E

L I w i t h range i n X. CL(T;X) and Co(I;X) denote t h e

L spaces o f f u n c t i o n s i n C (1;X) a l l o f whose d e r i v a t i v e s o f o r d e r 5 L have continuous extensions t o -f and w i t h compact support i n I,r e s p e c t i v e l y . Put

S = S ( IRn ) denotes t h e Schwartz c l a s s .

I f g . E A, j = 1 ,..., s, we w r i t e g J L EL(G) and u E E (G) and a r e a l number k, p u t P P

a vector.

E

2 R-1, p u t

Q~ = { x

E

R;

1x1 <

rl.

For f = ( f l

A.

R always r e f e r s t o a r e a l number such t h a t Rn - R For any r

,. . . ,gs) ,...,f s ) 6

L e t A be a s e t and g = (gl

c

{x

E

IRn ; 1x1 5 R-13.

For s i m p l i c i t y , we w r i t e

Yoshihiro SHIBATA and Yoshio TSUTSUMI

178

For any r e a l number k , [k] i s t h e " i n t e g r a l - p a r t " f u n c t i o n used i n number theory. 6 . denotes Kronecker's d e l t a ( c f . ( 1 . 3 ) ) . 1j (u,v) =

1

u(x)v(x)dx,

<

u,v > =

n

\

u(y)v(y)dSy, < u

Put 2

>

= < u,u >

an

,

where dS denotes t h e surface element o f an. L e t us d e f i n e t h e s p e c i a l Y constants depending on dimension n as f o l l o w s :

4/3,

n 26,

615,

3

1,

n 26,

2,

3 'n

n = 5,

q(n) =

n = 4,

2 n 2 5,

4/7

n = 3.

Y

r(n) = (5,

For a n o n - l i n e a r f u n c t i o n H(t,x,y19...,Ys)y

Put

L e t Xj, j = 0 ,..., n,

.. ,n

and xi j , i,j = 0,.

..

be independent v a r i a b l e s

..

corresponding t o 3 .u, j = 0,. ,n, u and a . a .u, i, j = 0,. ,n, respecJ 1 J t i v e l y . Put A = (xi, i = 0 ,...,n, A . ., i , j = 0, ..., n ) , A ' = (A,, ,... ,A,), 1.I

x"

= (?,,,...,An),

, I ,

x

= (x",A

n+l ) .

I n t h e course o f c a l c u l a t i o n s below

v a r i o u s constants w i l l be s i m p l y denoted by C.

Especially

C(***.*)

denotes a constant depending on q u a n t i t i e s appearing i n parentheses.

I79

Some quasi-Linear Hyperbolic Equations Special constants w i l l be denoted by co, c1 ,.

.. e t c .

NOW, we i n t r o d u c e our assumptions.

Assumption 1.1. 1" 2'

a

'3

fi y

B

aij,

ij

E

The s p a c i a l dimension n

2 3.

a r e a l l r e a l - v a l u e d f u n c t i o n s such t h a t

Brn({h"

E

Rn;

nj s - non-trapping i n t h e f o l l o w i n g sense:

Let G(t,x,y)

be t h e Green

f u n c t i o n o f t h e f o l l o w i n g problem:

where y i s an a r b i t r a r y p o i n t i n R a

fi b

supp v

6 i s the Dirac d e l t a function.

p o s i t i v e numbers such t h a t R - 1 c Ra,

a 2 b.

For any v

E

Let

2 L (n) & iy-

put

(Gv)(t,x)

=

J R G(t,x,y)v(y)dy

Then, __ the r e e x i s t s & p o s i t i v e constant T depending o n l y on a, b,

such _ _t h a t ( G v ) ( t , x )

c C"([T,-)xT).

n

fi fi

Yoshihiro SHIBATA and Yoshio TSUTSUMl

180

Remark 1.2.

1"

The above definition o f "non-trapping" is due to

Vainberg [15]. 2"

If the complement of n is convex, then n is non-trapping (see e.g.

Me1 rose [4]). 3"

It follows from (1.3) that the operator defined in (1.2) satisfies

the assumtion 2 " . The purpose of this note is to prove the following. T h e o r e m 1.3. Assume that Assumption 1.1

5 satisfied.

II

be 5

non-negative integer and put

N"

=

max(et1, 2n+[n/p(n)]+8),

N'

Then, _ there _ -exist positive numbers

do

=

2N"+1, N

=

N'+Zn+[n/p(n)]t8.

and 6l depending 0" N , aij, 6,

Y, n

such that if data uo, u l , f and and n ~ ~ -g satisfy the - following conditions: ~

u1

8N

E

(3n

g

HqN(,,)(a),

E

EN-2,d(n)r(n)

(1.4) N-4,d(n)r(n) q(n)

n E

and

N'+1 ,d(n)r(n) 3

E2

th _ order_ compatibility condition g satisfy the N-1 -

uo, ul,

f

section

v below),

Ig12,N-2,d(n)r(n)

with 0 <

for some

d

of (1.1)

a

L -

"

=

(6.

61 '6

6 5 60, then there exists a solution

U E C"'([O,m)x$

Some quasi-Linear Hyperbolic Equations

Furthermore, t h e r e e x i s t s a small number 62 > 0 such t h a t i f u are solutions in C

2

([O,m)xE)

of (1.1)

and v

f o r t h e same data such t h a t

then u = v.

Remark 1.4.

I f u i s a solution o f (I.]),

i t i s necessary t h a t u

s a t i s f i e s the conditions:

I(nu)(t,x)I 5- 1 f o r a l l ( t , x )

lu(t,x)I 5 1 f o r a l l (t,x)

c [O,m)xan

(cf.

c

[O,m)xE and

Assumption 1.1- 2").

Our p r o o f o f t h e existence theorem f o r (1.1) i s a s t r a i g h t f o r w a r d adaptation of a q u a d r a t i c i t e r a t i o n scheme w i t h a process o f "smoothing" which i s well-known as t h e Nash-Moser technique. why we use t h e Nash-Moser technique. where

aij

= B = 0 i n (1.1).

We e x p l a i n t h e reason

For t h i s , l e t us consider t h e case

I n t h e usual sense,

t h e l i n e a r i z e d problem

i s the following: Ou = f i n [O,m)xn,

-=

g on [o,mjxan,

(1.6) U(O,X) = u o ( x ) , (atu)(o,x)

= u,(x)

For s i m p l i c i t y , we assume t h a t uo = u,

i n R. 0.

According t o a r e s u l t due t o

Miyatake 151 we have

where u i s a s o l u t i o n o f (1.6) and C and u a r e some p o s i t i v e constants

Yoshihiro SHIBATA and Yoshio TSUTSUMI

182

depending on n and a.

To o u r knowledge, we can n o t prove t h e g l o b a l

existence theorem f o r (1.1) by u s i n g (1.7)

because o f t h e term: eut.

111

On t h e o t h e r hand, by t h e well-known energy method ( c f . s e c t i o n below) we have

T h i s estimate i s good f o r t h e constant ( t h e term eut disappears). encounter t h e d e r i v a t i v e l o s s because o f t h e term:

But, we

To

IlD2g(t,.)IIR.

overcome t h e d i f f i c u l t i e s caused by t h e d e r i v a t i v e l o s s o f t h i s type, t h e Nash-Moser technique i s very u s e f u l and the p r o o f i s easy and s h o r t as compared w i t h o t h e r methods.

From t h i s p o i n t o f view, we adopt t h e Nash-

Moser technique t o prove t h e g l o b a l existence theorem f o r (1.1).

2.

Preliminaries 2.1.-

SMOOTHING OPERATOR..

Choose @ ( t ) E S(R1) and $ ( x )

t

S ( R n ) so

that m

(2.1.1) 0 L J ,2 1 , supp JI

c

m

$ ( t ) d t = 1,

[O,m),I

-a

(2.1.2)

0 0'

zl,

$(x)dx = I ,

I,

$ ( t ) t j d t = 0, j

1,

J

~ ( X ) X " dx = 0, la1 2 1. Rn I n f a c t , t h e e x i s t e n c e o f f u n c t i o n s s a t i s f y i n g t h e p r o p e r t i e s (2.1 . l )

Rn

f o l l o w s immediately from Boas' theorem ( c f . Widder [17]). i n v e r s e F o u r i e r t r a n s f o r m a t i o n o f a f u n c t i o n i n C;(Rn) t h e o r i g i n , $ s a t i s f i e s (2.1.2). = 0 i f t < 0.

52

+

i s the

which i s 1 near

E L , p u t uo = u i f t 2 0 and P i s Cm and compact, v i a l o c a l map,

For any u

Since t h e boundary o f

If

E

by u s i n g well-known Seeley's extension method, we have t h a t t h e r e e x i s t s a function u'(t,x)

d e f i n e d on R'x Rn s a t i s f y i n g t h e f o l l o w i n g p r o p e r t i e s :

183

Some quasi-Linear Hyperbolic Equations

(2.1.4)

n+l

S(T)U =

T

UJ( (t-S)T)$((X-Y)T)U' (syY)dsdY.

I n the same way as i n Shibata [ 9 , Lemma 11.11, by u s i n g (2.1.1), and (2.1.3)

L~~~ u

E

(2.1.2)

we have k

2.1.1.

EL.'Kn CL([O,w)xn), &t P

where I ~denotes t h e -

&a

non-negative r e a l number.

Then, f o r any

following assertions are v a l i d .

i d e n t i t y operator.

2.2- INTERPOLATION INEQUALITIES.

I n t h e paragraphs 2.2 and 2.3,

D and k, m always r e f e r t o a domain w i t h Coo and compact boundary and nonnegative r e a l numbers, r e s p e c t i v e l y .

Lemma 2.2.1

(see Shibata [8, Lemma 2.2.71).

polation inequalities

=.

The f o l l o w i n g i n t e r -

Yoshihiro SHIBATA and Yoshio TSUTSUMI

184 By Lemma 2.2.1

Lemma 2.2.2 inequalities

and Young's i n e q u a l i t y , we have (see Shibata [ 8 , Lemma 2.2.91).

The f o l l o w i n g

hold.

2.3- ESTIMATES OF COMPOSED FUNCTIONS AND PRODUCTS

~ e m a2.3.l(see

__ be a function =

0.

Let u(x)

*

Klainerman [3, Lemma 5.11).

Bm(DxIy = (y,

= (ul(x),

Lemma 2.3.3. assume t h a t --

three

H(t,x,O)

K

,. . . ,ys)

...,u s ( x ) )

E

E

!& G(x,yl ,...,y,)

lRs ; IyI 5 11) such t h a t G(x,O)

L Hp(D) such t h a t l l u l l m =< 1.

H be t h e same as i n Lemma 2.3.2. = 0.

Let u(t,x) -

OF FUNCTIONS.

=

(ul(t,x)

Then,

I n addition, -

,...,u S ( t , x ) )

E

EL'k(D) n P

185

Some quasi-Linear Hyperbolic Equations L <~ 1 . Em(D) such t h a t I U I D , ~ , ~ , =

I H(

3

Then,

) ID,p ,L ,k 5 c(P,L, k, [ I' I D,p,L ,k

' Yu( '9

+

I I D,p

,O ,k

IuI DYm,L,Ol'

Proof. By T a y l o r expansion, we have (2.3.1

H ( t ,x,u(t,x)

=

c;=~

% t ,x,au( ayj

where G .( t ,x ,u( t ,x) ) = J t o (2.3.1), we have (2.3.2)

)uj( t ,x),

G j ( t ,x,u(t,x)

K

( D H ( * Y ' Y U ( ' , * I ) I~,p,o,k 2

s

K

I G( j'

( L, C j = 1 'M=O

Y

9

u(

I 0 y m ,M I0 IUjlD,p,K-M,k

3

Applying Lemmas 2.2.2 and 2.3.2 t o (2.3.2),

Lemma 2.3.4. r e a l numbers -____--such t h a t 1 L,k. E J(D) pj

Applying L e i b n i z ' s formula

t ,x))da.

Let s

we o b t a i n t h e

be an i n t e g e r L 2, k, kl,.

such t h a t k = k,+-**+ks, p, p1 ,. p, pl,

... ,ps

z=m

p-

1

=

.. ,p,

s

-1

zjZ1 p j

., ,kS

O (-K ( L-. emma. non-negative

extended r e a l numbers

.

Then, f o r any u

j '

the i n e q u a l i t y : (U1"'U

s I D,p,L,k

= <

--holds w i t h some constant C Proof. IU1"'U

CE?i=l[IUiID,pi.L,ki

= C(p,L,k,s,D)

n

+

IUjIO,pj,O,kj

> 0.

By L e i b n i z ' s formula and H o l d e r ' s i n e q u a l i t y , we have

s I D,p,L,k

(2.3.3)

c =L IU1 D,P, C(L) j,+...+j

J, ,kl

...'UslD,pS.js,kS

S

Applying Lemma 2.2.2 t o (2.3 3 1 , we o b t a i n t h e lemma.

1

Yoshihiro SHIBATA and Yoshio TSUTSUMI

186

2.4- SOME INEQUALITIES FOR THE TRACE OPERATOR,

Lemma 2.4.l.(see

f o r an y-

r 2 R-1

e.g.

Mizohata [6, Chapter 3]),

the i n e q u a l i t y :

If n

Lemma 2.4.2.

the inequality:

If u

E

1 H2(n),

then

I I U ~ ~ ~ h, o~l d,s,.

u

>

2 C(n,n,r)

23 @u

E

1 H2(n), t h e n f o r any r 2 R-1 we have

<

1 IIDxuII

llulhrY2 5 C(n,n,r)

.

Lemma 2.4.2 can be proved by u s i n g R e l l i c h ' s compactness theorem (see e.g. Mizohata [6, Theorem 3.33) and t h e i d e n t i t y :

r-n+2

[ai(r(n-2)/2u)12

=

+

Clu2,-2

2 n zj=l(a.u) J

n-2

For f u r t h e r d e t a i l s , Heywood [l]can be r e f e r r e d . and 2.4.2,

a

-1 2

u ) (r = 1x1, n ' 3 ) .

Combining Lemmas 2.4.1

we have

Lemma 2.4.3.

Assume t h a t n 2 3.

Then, t h e r e e x i s t s a c o n s t a n t co

1 depending o n l y on n @ R such t h a t f o r any u E H2(n) 1 < u > 5 co IIDxuII

2.5"

+

2

i=l

LOME

the i n e q u a l i t y :

holds.

ELLIPTIC ESTIMATES.

I n t h i s paragraph, we g i v e some

a p r i o r i " estimates o f s o l u t i o n s o f t h e f o l l o w i n g equation:

(2.5.1)

AU

+ cyj=, a.(a..a.u)

Lemma 2.5.1.

1

1J

J

= f in

Assume t h a t

n, av+ au z?j=l viaijaju

+ bu = g on an.

187

Some quasi-Linear Hyperbolic Equations

Then,

there ~ exists constant c1 ,L

n -that i f CijZl

llaijlL

the following

estimates

0 depending o n l y on n, n

clYLy then f o r any s o l u t i o n u

E

H;+'(n)

and L such of (2.5.1)

E.

I n o r d e r t o prove Theorem 2.5.1,

we need t h e f o l l o w i n g two w e l l -

known r e s u l t s

Recall t h a t Rn

-

n

c

Ix

E

n

R ; 1x1

< R-11.

Choose $ ( X I

t h a t $ ( x ) = 1 if 1x1 2 R-(1/2) and = 0 if 1x1 5 R-1. s a t i s f i e s t h e equation (2.5.11, (2.5.2) where R-1

F

A($u)

= f

-

c

n

J

2 1x1 2 R-(1/2)1,

1J

J

E

+ n$.u

Since supp aj$,

a p p l y i n g Lemma 2.5.2

i n rtn SUPP

t o (2.5.2),

~g

Cm(lRn) so

H2L+2 (n)

we have

n gF + 2zjZl ajg-a.u

~ ai(a. ~ .a.u). = ~

If u

E

{x

we have

Yoshihiro SHIBATA and Yoshio TSUTSUMI

188 G =

-2’ ij=1

(2.5.4)

v a

AU =

i ij

-

a.u J

bu

F i n n, =:

+

g, u s a t i s f i e s

G on an.

Applying Lemma 2.5.2 t o (2.5.4), we have

By Lemmas 2.2.2 and 2.4.2 and L e i b n i z ’ s formula, we have

Choose a p o s i t i v e constant c < 1/2, C(L,n,n)

(2.5.8)

so small t h a t c llvjlL)C(Lynya) 1 YL 1 YL (1 + I:=, being the c o n s t a n t i n (2.5.6). S u b s t i t u t i n g (2.5.7) and

i n t o (2.5.6),

we have t h e theorem.

3 . L 2-Estimates f o r Linearized Equations. I n t h i s s e c t i o n , we g i v e L 2 -estimates (energy estimates) o f s o l u t i o n s o f t h e f o l l o w i n g h y p e r b o l i c equation:

a 2t u - Cij=l n

ai((dij

+ a i j) aJ. u ) + cnj = O bj1a j u = f

in

[O,m)xQ,

(3.1)

zYjz1

vi(dij

+ a

)a.u + bou

ij J

g

on [O,-)xan,

189

Some quasi-Linear Hyperbolic Equations U(0,X) = (atu)(0,x) Theorem

Put A

3.1.

in

= 0

1 0 bj, b ) ,

(aij,

=

Let L

a.

be an i n t e g e r ?- 4.

Assume _ _ _ -t h a t

and clYL

where co 3”

6ij

c

_ f o_ r -a l l

4”

+

f

aij(tyx)

= dji

= (cl,...,~,)

c

E

respectively,

1 2 ~ + a=i j ( t ,~x l ) c ( ic j6 -L2-~ 151~

~

-

(t,x)

R”

6

[o,m)xn,

L+l ,d(n)r(n)

gEE2

9

= (aig)(o,x)

Then, there exists __

n ~

aji(t,x).

+

L L-1 , d ( n ) r ( n ) E2 n E2

(aif)(o,x)

and 2.5.1,

a r e t h e same as i n Lemmas 2.4.3

IxEaP

Y

= 0,

a unique s o l u t i o n u

E

0 Ij 2L-1.

IE2L+1

of (3.1)

which has t h e

f o l l o w i n g estimates:

1

I D U12,M,0

= <

C ( M ’ n y n ) [ t f IE,M,d(n)r( n)

A Im ,M+1 ,d ( n ) r ( n )

f o r any i n t e g e r M

R e m a r k 3.2.

with 0 2 M IfA

E

I

4

R lg12,M+2,d( n ) r ( n )

’

R

I 2 ,O ,d ( n) r ( n) ’ I 2 ,2 ,d (n ) r( n )

’

2 L-1.

s”([O,m)xn),

according t o Miyatake [5],

t h a t t h e r e e x i s t s a unique s o l u t i o n o f (3.1). l e a s t B ([O,m)xE),

+

we have

But, i f A belongs t o a t

estimates of Miyatake‘s t y p e f o r (3.1) can be obtained.

So, by t h e l i m i t i n g process we o b t a i n t h e unique e x i s t e n c e theorem f o r

(3.1) under t h e assumptions 1”-4”

(see Shibata [ l o ] ) .

Yoshihiro SHIBATA and Yoshio TSUTSUMI

190

Proof o f Theorem 3,1,

Noting t h a t u

E

E i t l and t h a t 0 'M

(L-1

and

d i f f e r e n t i a t i n g (3.1) M times w i t h r e s p e c t t o t, we have

a2t ( aMt u)

-

-

A(atu) M

~ y a,[~!=~(:)a:a~~a:-~a~u~ ~ = ~

+

'

M M k M-ka.u = atfM 'J=OEk=o(k) atbj at J

(3.3) da~a t u ) + zyj=l

vi[z:=o(:)a:aija:-kaju~

=

-

ga:

i n [O,-)xn,

M M k 0 M-kU Ck=O(k)atbat

on [O,-)Xan,

by p a r t s , we have t h e i d e n t i t y :

dd< a t Mg , atu> M 7zij=l((&ij I n

- F1

bO atu, M

-


'k=l fM) k

a!$>

+ &atb O atu, M

k+l O M-ku, aMu> at

-

'k=l

(M)
-

t

2 ij=1

aMa t j u)

M k+la..aM-kaiu, Eij=l'k=l(k)(at ij t

ataju). M

4-1 llD'a:u(t,*)lI

2

M-k uyajatu)l M

( a a..a Ma.u, a Ma u) t

ij

t

1

t j

+

-

-

M u l t i p l y i n g (3.3) b y a r l u and i n t e g r a t i n g over [O,t]xn,

(3.5)

+

+ E ~ = 2 ( ~ ) < a : b 0 a ~ 1 - k u ,au:> atu> M

' i nj = l E kM= 2 ( kM) ( a tka..aM+'-ka.u, ij t i

assumption 3" we have

a'u> t

Ma u) + 'ij=l'k=l(kf(ataijaiat n M M k t j

+ a..)ataiu,a M 1J

(3.4)

atu> M

by (3.4) and t h e

Some quasi-Linear Hyperbolic Equations

We are going to evaluate each term.

For simplicity, let us put

By the assumption 2", Lemnas 2.4.1, 2.4.3 and 2.2.2, we have

191

192

Yoshihiro SHIBATA and Yoshio TSUTSUMI

( t h i s term disappears when aiay”(t,.),

3.2 Mu(t,.))J

M

=

O),

2

J t ( t h i s term disappears when M = 0),

193 193

Some quasi-Linear Hyperbolic Equations ( t h i s term disappears when M = 0 ) ,

C(M)I

+

J(t)

( t h i s t e r n disappears when 0 5 M 5 l ) ,

C(M)I

+

J(t)

( t h i s term disappears when 0

M 5 l),

where f o r n o t a t i o n a l convenience t h e same l e t t e r C(M) i s used t o denote constants depending e s s e n t i a l l y on M,n and n. Combining (3.5) and (3.6), by G r o n w a l l ' s i n e q u a l i t y we have

Yoshihiro SHIBATA and Yoshio TSUTSUMI

194

for a l l M with 0

2 M 5 L-1 , where n(M) i s a c o n s t a n t such t h a t n(M)

i f M = 0 and = 1 i f M

2 1.

I n p a r t i c u l a r , we have (3.2) when M = 0.

Now, we s h a l l prove (3.2) by i n d u c t i o n on M. 1 'M

= 0

We may assume t h a t

5 L-1 and t h a t (3.2) a r e a l r e a d y proved f o r s m a l l e r values o f M.

We s h a l l prove

We prove (3.8) a l s o by i n d u c t i o n on K.

For s i m p l i c i t y , l e t us p u t

Applying t h e i n d u c t i o n hypotheses t o (3.7), we have t h a t (3.8) a r e v a l i d f o r K = 0 and 1.

We may assume t h a t 2 5

a l r e a d y proved f o r s m a l l e r values o f K.

K 5 M+l and t h a t (3.8) a r e D i f f e r e n t i a t i n g (3.1) M+1-K

times w i t h r e s p e c t t o t, we have M+l-Ku)

-dat

n

-

' i j = l ai(a. IJ.a.aM+l-Ku) J t

= FK

i n n,

(3.9) QI-I

where FK = a

M+1-Kf t

-

ap3-Ku

+

(3.10)

k aiataijajat

M+l -K-ku)

an,

195

Some quasi-Linear Hyperbolic Equations G K

= aM+l-KS t

-

n M+1-K M t l - K k Mtl-K-ka.u cij=l " i E k = l f k )ataijat J

M+l-K M t l - K k 0 Mtl-K-kU 'k=O ( k ) a t b at

n By t h e assumption 2", we have zij=l we can apply Theorem 2.5.1

By Lemnas 2.3.6,

laijlm,O,O

t o (3.9).

2.2.2 and 2.4.2,

= <

c ,Lly

which i m p l i e s t h a t

Thus, we have

L e i b n i z ' s formula and t h e assumption

we have

k R atai j I-,K-l

M+1-K-k

,01 at k

R

a J.uI R2,090 M+1-K-k

I ata ij I m ,O ,O I a t R

R

aj

R

a taij I m ,K+k-2,0 I a j I 2 ,M+1-

K- k ,O

+

I2,K-l,(3]

5

Z0,

Yoshihiro SHIBATA and Yoshio TSUTSUMI

I96

I D1x u 12,M+l-K,0

lm,K-2,0

= <

C(M)B,

where f o r n o t a t i o n a l convenience t h e same l e t t e r C(M) i s used t o denote constants depending e s s e n t i a l l y on M, n and R. Combining (3.10),

(3.11) and (3.12), we have

f12,M-l,0

+

IDxK-2,M+3-K t u12,0,0

+

1912,M,0 R

+

Applying t h e i n d u c t i o n hypotheses t o (3.13), we have (3.8), which completes t h e p r o o f .

4. Uniform Decay Estimates.

In t h i s s e c t i o n , we g i v e t h e u n i f o r m decay estimates o f s o l u t i o n s o f t h e equation (3.1).

Let

Theorem 4.1.

be an i n t e g e r 2 2n+5

I n a d d i t i o n t o a l l assumptions o f Theorem 3.1, -

1"

n L - 3

'"

A

3"

f c E

and ci

and u

assume t h a t

i s non-trapping,

L-2 , d ( n ) r (

' Ep(n)/r(n)

n)

L-S,d(n)r(n) q(n)

'

a s o l u t i o n o f (3.1).

IAtp(n)/r(n),lYd(n)r(n)2 l Y

197

Some quasi-Linear Hyperbolic Equations

0 =i; M 2 L-2n-5 we have

Then, for any integer M y J t i

IF( I

(4* )

n) ,O,d( n)

fI[‘

Iq(n) ,2n+2,d(n)r(n)

R I2,2n+4 ,d( n) r( n)

’ ID1 U12y2n+4,0

+

+

1

{IAlp(n)/r(n) ,2n+3,d( n)r( n) lD1ulp(n),M,d(n) = < C(M)CI

IAlm,2n+4 ,d( n) r( n)

Iq( n) ,M+2n+2 ,d( n) r( n)

R Ig12,M+2n+4,d( n)r( n)

(4*2)

+

ID

I2 ,O ,01 ’

+

1

+

ID ’12 ,M+Zn+4,0

{ I A l p ( n)/r(n) ,M+2n+3,d( n)r( n)

+

IAlm,M+Zn+4,d(n)r(n)

+

’ID

1

‘12 ,0,03*

Before proving the theorem, we discuss the decay rate of solutions of the Neumann problem for the wave equation:

nu

=

f in

[o,m)xn,

3 = g on [O,-)xan, av

(4.3) u(0,x)

=

uo(x),

(atu)(O,x)

=

u,(x) in n.

For simplicity, assume that u is a Cm solution of (4.3) and that all norms for uoy u l , f and g appearing below are finite. If g = 0, then we have

n),L+2n+3 llUl I‘q(n) ,L+2n+2

+

I

’

Iq( n) ,L+2n+2 ,d( n)r( n)’.

The proofs of (4.4) and (4.5) are given in Appendix. When g f 0, we can represent u as follows: Let v7 be a solution of the equation:

Yoshihiro SHIBATA and Yoshio TSUTSUMI

198

+ ,,v,

-Av,

=

o

in

nR,

av $=

an,

g ( t , * ) on

av

o

$=

w h e r e r = 1x1, t i s r e g a r d e d a s a p a r a m e t e r a n d

i~ i s

c

IR'; 1x1 = RI,

a sufficiently large

~R-(l/2) Choose $ ( x ) c C i ( R n ) so t h a t $ ( X I = 1 if 1x1 -

p o s i t i v e number. a n d = 0 i f 1x1

on {x

R and p u t

v ( t , x ) = $ ( x ) v l ( t , x ) if x

E

QR

and = 0 i f x

QR.

Let w be a s o l u t i o n o f t h e Neumann problem: Ow = f

- Ov

aw

i n [O,m)xn,

w(0,x) = uo(x)

-

v(O,x),

= 0

on [o,m)xan,

(atw)(o,x) = u,(x)

-

(atw)(o,x) i n R.

Then, by t h e u n i q u e n e s s t h e o r e m for ( 4 . 3 ) we h a v e u = v + w.

S i n c e by t h e

u s u a l e l l i p t i c estimate i n a n i n t e r i o r domain we h a v e

R

R

1'1 12,Kyd(n ) r ( n ) 2 C ( K y n y n ) l g2 ,/ K , d ( n ) r ( n ) and s i n c e 1 2 q ( n ) 5 2, by S o b o l e v ' s i n e q u a l i t y , ( 4 . 4 ) and ( 4 . 5 ) we h a v e

(4'6)

R

-1

ID ' l p ( n ) , O , d ( n )

= <

'[ 11UOlb(n),2n+3

+

I I q ( n ) ,2n+2 ,d( n ) r ( n ) (4*7)

1'

IIU11b(n),2n+2

+

Ig l

+

R

2,2n+4 ,d ( n ) r ( n ) "

1

'[ IIUOlk((n),L+2n+3

'lp(n),L,d(n)

I

I q ( n ) ,L+2n+2,d( n ) r ( n )

Proof o f Theorem 4 . 1 .

+

+

11U11b(n),L+2n+2

'

i R I gl 2,L+2n+4,d( n ) r ( n)"

Let u b e a s o l u t i o n of ( 3 . 1 ) .

Then, we c a n

regard u a s a s o l u t i o n o f t h e equation: (4.8) Ou = f +

n cijZl ai(a..a.u) 1J J

- c jn= O

b .1a . u J J

in

[O,m)x~,

199

Some quasi-Linear Hyperbolic Equations

g-zyj=lviaijaju-b0u on

-= av

2-1 + (p(n)/r(n))-',

=

Since q(n)

[O,m)xaR,

~ l ~atu(t=O = ~ 0 in R. =

=

it follows from Lenmas 2.3.6, 2.2.2

and 2.4.2 with some constant C = C(K,n,fi) that O R U12,K+2n+4,d(n)r(n) 2

lb

R

O R

I

(4'9'a) '[

lm,K+2n+4,d( n)r( n) I u I2,O ,O

Iu

R

12,K+2n+4,01 1

1

'[ID

+

'I2 ,K+2n+3,0

+

I A I-,K+Zn+4 ,d( n) r( n) ID

12,O ,01 '

lai(a.i j.a.E)\ J q(n),K+2n+Z,d(n)r(n) jIp( n)/r( n) ,K+2n+2 ,d( n)r( n) ai aju12,O ,O

*

laijlp(n)/r(n) ,O,d(n)r(n) l a ia jU I 2,K+2n+2,0

+

'[Iai

(4.9.b )

I aiaijIp( n)/r(

n) ,K+2n+2,d( n)r( n) I aju12,0,0

+

Iaiaijlp(n)/r(n) ,O,d(n)r(n) laju12,K+2n+2,01 1

'[

1 (4.9.c)

ID

'I2 ,K+2n+3 ,O

bla

'[

+

IAl

p( n)/r( n) ,K+Zn+3 ,d( n) r( n) I

1

2 ,O ,01 '

1

ju q( n) ,K+2n+2 ,d( n) r( n) 5

1

I2

K+2n+2 ,0

+

IAJ

1

p( n)/r( n) ,K+2n+2 ,d(n)r( n)

I D "I 2 ,O ,01'

la. . a . u l R ij J 2,K+Zn+4,d(n)r(n) (4.9.d)

ID 1

2,K+Zn+4 ,O

+

IA Im ,K+2n+4 ,d ( n

r ( n ) Dl

I 2 ,0,0'

Thus, applying (4.6) and (4.7) to (4.8)and using (4.9), we have the

theorem .

Yoshihiro SHIBATA and Yoshio TSUTSUMI

200

5. Compatibility Condition and Reduction of Problem. I n t h i s s e c t i o n , we s h a l l i n t r o d u c e t h e d e f i n i t i o n o f c o m p a t i b i l i t y c o n d i t i o n and reduce t h e problem (1.1) t o t h e case where uo = u1 = 0.

For t h e f i r s t purpose, we i n t r o d u c e some n o t a t i o n s :

q j ( x l ,...

n1 =

y x

@ 1J .-(A,,

. . . y

An) +

n

c k = l ( a h ."fk) (hl . * * Yhn)XkS Y

J

(5.1)

~ ( A U )=

-

CijZl n

ai(u..(Dxu)aju) 1

+ B(D 1u ) ,

1J

$mb(Dxu) 1 = cijZ nl

.(O,u)a.u. 1 J

w.".

1 1J

I t i s obvious t h a t

(5.2)

ak(cj"=l a i j (D'u)a.u) J

L e t us d e f i n e f u n c t i o n s G (5.3)

k s at(m(nu)) =

k

=

,

~ y a/j(D:u)ajaku, = ~

k = O,l,...,n.

k 2 0, by t h e formula:

-2 G k (Dxu ,...,-2~ , ak ~ u k+l , a ~u ) .

L e t us define successively u ~ + k~ , 0, by ( 5 ~ 4 ) Uk+2 = Auk + By Assumption 1.1-2'

(5.5)

G k ( -2D x U ~ ~ - - . -2y D x U k y U k + +l ) akf)(o,X). we have t h a t t h e r e e x i s t f u n c t i o n s IT^, k20, such t h a t

+kf0) = 0, Uk+2 = "k(Dx -k+2 Uo(x)y y l u l ( x ) y

($f)(oyx)).

Next, l e t us d e f i n e v e c t o r valued f u n c t i o n s o kb ( x Y * ) by at(mb(Au)) k a

= CijZl n

(5.6)

wi(x)(Gij

+ a;j(D;u))a.a J kt u + y l ( u ) a tku + ob(x,lu,.. k

By Assumption 1.1-

2 O ,

k we have mb(x,O)

= 0

f o r a l l k 2 0.

. ,Latk-1 u).

20 1

Some quasi-Linear Hyperbolic Equations NOW, we i n t r o d u c e t h e d e f i n i t i o n o f t h e c o m p a t i b i l i t y c o n d i t i o n .

Now, we reduce t h e problem (1 . l ) t o t h e case where uo = u1 = 0. t h e same way as i n Hormander 12, Lemma 5.6.11,

Let p1 ,.. . p,, be extended

Lemma 5.2.

L e t Land M be i n t e q e r s

i = l,...,m. v

j

E

m n H ~ - ~ ( Q j) , k = l pk

there exists function --

L e t u o y u, j = 2, and u jy

,f

V(t,x)

O,I

with 0

In

we have r e a l numbers w i t h 1 2 pi 5 'M

5 L.

my

Then, f o r a n 1

y . . . , ~ ,

such t h a t

and g s a t i s f y t h e N-1 t h o r d e r c o m p a t i b i l i t y c o n d i t i o n

...,N+1,

then we have by (5.5)

be f u n c t i o n s d e f i n e d by ( 5 . 4 ) .

and Lemma 2.3.1

If

Yoshihiro SHIBATA and Yoshio TSUTSUMI

202

f o r any i n t e g e r s k and L w i t h k+L 2 N t l and k

uo, u1 , f

Lemma 5.3.

(5.7).

b i l i t y condition

and g

2 and f o r r = q ( n ) and

m.

s a t i s f y t h e N-1 t h o r d e r compati-

Then, t h e r e e x i s t s a f u n c t i o n v ( t , x )

such

that

for k

= q(n)

n, a, N,

aij,

Proof.

and-. Here, 6 &Y.

L e t V be a f u n c t i o n s t a t e d i n Lemma 5.2 w i t h L = M = N+1,

m = 2, p1 = q ( n ) , p2 =

so t h a t

p(t)

= 1 if

-

It[

By (5.81,

p(t)V(t,x).

c2 i s some p o s i t i v e c o n s t a n t depending o n l y on

...,

= u , j = 0, N+1. Choose p ( t ) E C;(R1) j j & 1/4 and = 0 i f It1 2 1/2 and p u t v(t,x) =

and v

Lemma 5.2,

(5.31,

(5.4) and t h e d e f i n i t i o n o f t h e

c o m p a t i b i l i t y c o n d i t i o n , we have t h a t v ( t , x ) properties

s a t i s f i e s the required

. Q.E.D.

From now, v ( t , x )

always r e f e r s t o t h e f u n c t i o n s t a t e d i n Lemma 5.3.

I f u i s a s o l u t i o n o f (1.11,

p u t t i n g u = v + w, w s a t i s f i e s t h e equation:

203

Some quasi-Linear Hyperbolic Equations

i n [O,=J)xn,

OW + "(Aw) = F ( t , x ) (5.9)

on

= G(t,x)

t yb(?w)

w(0,x) = (atw)(O,x) =

o

[O,m)xaR,

i n R,

where

(5.10)

Yb(?w)

(cf. (5.1)).

=

'd;,(DXv 1 + Dxw) 1

- %b(Div)

+ y ( v + w)

-

y(v),

Conversely, i f w i s a s o l u t i o n of ( 5 . 9 ) , then u = v

Thus, we solve (5.9) below.

a s o l u t i o n of ( 1 . 1 ) .

t

w is

In p a r t i c u l a r , i t

follows from Lemna 5.3 and (5.10) t h a t (5.11)

(aiF)(O,x) =

6. Iteration Scheme I n t h i s s e c t i o n , we d e s c r i b e our i t e r a t i o n scheme f o r solving ( 5 . 9 ) .

The scheme constructed below i s c a l l ed the Nash-Moser-Hormander scheme

Let

= ej. j Let S ( T ) be a smoothing o p e r a t o r defined i n paragraph 2.1 of s e c t i o n

(see Klainerman [3]).

and p u t S.u J

s(eJ.)u.

smoothing operator S

8

be a fixed c o n s t a n t

1 and put 8

11

By Lemna 2.1.1 we have t h e following p r o p e r t i e s of

-

j'

if u

E

E L S kn P

CL([O,m)xc)

(k

0 ) , then

Yoshihiro SHIBATA and Yoshio TSUTSUMI

204

where a l l constants a r e independent o f j and u. L e t us f i x wo by r e q u i r i n g t h a t

F

Owo =

- G av

i n [O,-)xn,

i n [O,m)xan,

(6.1) wo(O,x)

(atwo)(O,x)

i n n,

= 0

where F and G a r e t h e same as i n (5.10).

We s h a l l d e f i n e wj,

j 2 1,

successively

For t h i s , we prepare some n o t a t i o n s . r e q u i r i n g t h a t $ ( x ) = 1 i f 1x1 section

I).

L e t us f i x $ ( x ) c Ci(Rn) below by

2 R-1 and

= 0 i f 1x1 2 - R

Put

L.u = Ou + (d$)(SjAwj)Auy J B.u = J

e+

1 1 (d% ) ( S . D w.)D u + y ' ( S . ( $ w . ) ) u , b J X J x J J

= (d%b)(D:wj)DA1j e' b,j

e'

J

(d8)

-

(d%b)(S.D 1w . ) D 11. t J X J XJ

( c f . Notation i n

205

Some quasi-Linear Hyperbolic Equations el! = 'L Q ( A W ~ + ~- ) 'S(Awj) L J e

j

= el

J

+

j,

el'

(d%)(Awj)Mj,

ebyj = e i y j + e"b,j

j-1 EJ. = Zk=O ek'

EO = EbyO = 0,

fo

-

= -S 0 ~(AWO),

' j -1

Eb,j

C ~ = O eb,k,

j '1,

Sjej-l,

j 21,

go = -sO[$rb(lwO)l,

f J. = - ( S .J- S . J-1 ) Y ( A W 0)

-

-

(Sj-Sj-l)Ej-l

g j = -(Sj-Sj-l)[$Yb(X~o)]

-

(S.-S. J J - 1 ) E b,j-1

-

j '1.

'jej-l9

We define s u c c e s s i v e l y fij, j 2 0, as s o l u t i o n s of t h e f o l l o w i n g equations: L.I. = f J J j

in

B.1. = g J J j

[O,m)xn,

on [O,m)xan,

(6.4) a.(O,x) J

= ( a h.)(o,x) t J

= 0

i n R.

Noting t h a t

by (6.2),

(6.3) and (6.4) we have

OW^,^ +

wjtl(O,x)

Y(Aw~+,) =

F +

= (atwj+,)(o,x)

(I-S.)Y(AW~) J

=

o

+

(I-S.)E. J J

+ ej i n

[O,m)xn,

i n n.

Yoshihiro SHIBATA and Yoshio TSUTSUMI

206

7. Estimates o f Non-Linear Terms. I n o r d e r t o evaluate e

jy

ebYj and c o e f f i c i e n t s o f l i n e a r o p e r a t o r s

we g i v e some c a l c u l u s lemmas i n t h i s s e c t i o n . L e t D be a J j' y = (yl ,. . . ,yS) , a domain w i t h compact and Cm boundary and H(t,x,y)

L. and B

function i n Brn([O,m)xbIy

E

Rs ; IyI 513). Assume t h a t

Put e'(H;U,V,W)

= (dH)(U)W

-

(dH)(V)W,

(7.2) = H(U+V)

e"(H;U,V) where U = (U,(t,x), (W,(t,x),..

-

H(U)

...,Us(t,x)),

, ,WS(t,x))

- (dH)(U)V,

V = (V,(t,x),

E

L EJD)

and W =

a r e vector-va ued f u n c t i o n s .

2 6.

f o r any U --

...,V,(t,x))

such t h a t

IU~D,rn,o,o

Then, -we have -

1.

Assume t h a t 3 2 n 2 5. Then, we have (b) ---

I ( d 2H)(U)ID,p,L,d(n) f o r any U --

E

Proof.

E,(LD)

I\

= <

C(H,L~D,p)[l

E L y d ( n ) ( D ) such that P

ulD,p,L

,d( n)

' I ulD,myL ,OI

\ ' l D , ~ y O , O ~ ~ U ~ ~ , p , O , d ( n= <) 1.

By (7.1) we have t h a t (d2H)(y) = O(ly[ r ( n ) - l )near y = 0.

Since r ( n ) = 1 i f n 2 6 and = 2 i f 3

n 5 5, t h e lemma f o l l o w s

i m n e d i a t e l y from Lenunas 2.3.2 and 2.3.3. Q.E.D.

207

Some quasi-Linear Hyperbolic Equations

By Taylor expansion, we can write

I

1

e' (H;U,V,W)

=

0

(7.3)

(d2H) (V+a( U-V))da(U-V ,W) ,

1

el'( H;U,V)

=

i, (1 -a) d2H) (U+aV)da(V (

,V)

.

Applying Lemmas 2.3.4 and 7.1 to (7.3), we have the following immediately. Lemma 7.2.

q-l

=

Let p and q be extended real numbers such that

r(n)p-' + 2-l.

x, the following estimates E:

' IU-' ID , 2 ,O ,O IwI D ,p ,L ,d( n)

+

IvlO,p/( r( n)-1) ,I-,(r(n)-1 )d(n)

('+ +

I D ,p/ (r( n) -1 ) ,L ,(r(n) -1 )d( n)

IUID,mYL ,O

+

Iu-'1D,2

~V~D,,,L ,O) x ,O ,01 10 ,p ,O , d ( n)]

+

Yoshihiro SHIBATA and Yoshio TSUTSUMI

208

for an^ U -

and V

such that

< 1/2. ~u~D,p/(r(n)-l)yOy(r(n)-l)d(n)y ~ " ~ D , p / ~ r ~ n ~ - l ~ , O ,=~ r ~ n ~ - ~ ~ ~ ~ n

Here for notational convenience we have put --

-1

0

and 0- 1

=

m.

Next, we give a lemma to evaluate the coefficients of Lj and €3j' For this, we begin with

Lemma 7.3.

The following

estimates hold.

Proof. By Taylor expansion, we can write

(7.4) (dH)(U)

=

i:,

(d2H)(aU)da*U.

Applying Lemnas 2.3.4 and 7.1 to (7.4), we have the l e m a immediately. Q.E.D. Put

209

Some quasi-Linear Hyperbolic Equations

( c f . (6.3)).

By Lemma 7.3 we have

Lemma 7.4. c

Assume t h a t Assumption 1.1- 2"

~ be, t h~e same as i n Theorem 3.1 IV1m,N+l,d(n)'

Then,

[I] c,,"

the f o l l o w i n g

with L

IVlp(n),N+l ,d(n)

= N.

= <

satisfied.

co

and

Assume t h a t

1/2.

assertions are v a l i d .

There e x i s t s a p o s i t i v e c o n s t a n t c3 depending e s s e n t i a l l y o n l y on c o y aij,

6 and

IAvl-,2,d(n) then _f o r

+

y

such t h a t i f

R IVI-,2,d(n)

%function

the f o l l o w i n g -

'3'

I"Ip(n),l,d(n)

f such t h a t

four assertions are v a l i d :

+

R Ivlp(n),lyd(n)

'3

F i n a l l y , we g i v e estimates of terms: u(Aw0) and @Y,,(AW~).

Since

a p p l y i n g Lemmas 2 . 3 . 4 and 7.3, we have

Lemma 7.5.

Assume t h a t Assumption 1.1- 2”

wo be t h e same as i n s e c t i o n s

negative i n t e g e r 5

N-1.

v

~

satisfied.

V I , r e s p e c t i v e l y . Let L

Let v a non-

21 1

some quasi-Linear Hyperbolic Equations

then,

the following

(a)

Jy(hwO)I2 ,L ,d( n)r(n )

fi:

estimates

c(L)(IvIm,N+l, d ( n )

(b) I y ( A w O ) l q ( n ) , L , d ( n ) r ( n )

+

= <

)r(n)

I AW0I 2,L ,o

IAWOlm,L,d(n)

C(L)l"WOl2,L,O

,N+1 ,d(n)

+

IAWOl-,L,O

+

l"Olp(n),L,d(n)

y(n)'

< -

c ( L ) ( IVlm,N+l,d( n )

+

R )r(n) R IXWOlm,L,d(n) I l w o I 2 ,L ,O'

8. Convergence o f Iteration Scheme and Proof o f Theorem 1.3 In what follows, we use notations defined i n sections F i r s t , by induction we shall prove t h a t f o r a l l j

V

and

VI.

0 the following

statements a r e valid:

E!'

n B " ( [ O , - ) ~ ~ L ) , BJ. c

(St.1)

AB.

(st.2)

[najlL + [aj]:

J

c

-N"+L

bej

E~~ 2 8

n BN'([O,m)XnR),

, o 5 L 5 NO.

Here 6 i s a positive constant determined l a t e r , B

jy

same as i n section ['"'L

=

VI

j = 0,1,2,...

a r e the

and we have p u t

IAU12,L,0 I A u U l p ( n ) , L , d ( n )

R = Iu12,L,0

+

R +

Iulp(n),L,d(n)

+

IhulmyL,d(n)' R

+

IUl-,L,d(n)'

Let a be a non-negative integer and assume t h a t

(A.1)

ho, ...,ha a r e already given and t h e statements ( S t . 1 )

and (St.2)

Yoshihiro SHIBATA and Yoshio TSUTSUMI

212

a r e valid f o r G o , . . . , G a ,

wo, being a solution of the equation ( 6 . 1 ) , s a t i s f i e s

(A.2)

the

conditions: AMo c

' ! E

n B"([O,-)~C)

/vlm,N+l ,d(n)r n )

(A*3)

I n p a r t i c u l a r , from (A.3 (A-3)'

' /viq(n),N+l, d ( n ) r ( n )

<

=

6.

i t f o l l ows t h a t

lVlpyN+l,d(n)r(n)

'

f o r a l l p w i t h q(n) 2 p 2".

To prove ( S t . 1 ) and ( S t . 2 ) , we need several lemmas.

I n the same way

as in Klainerman [3, 971 and Shibata [8, 553, these lemmas can be proved by using Lemna 2.2.1, the properties o f smoothing operators (Sm.l)-(Sm.4) in section

VI,

Lemmas 7 . 2 , 7.4 and 7.5, and the f a c t t h a t

ej

=

e j , so

we may omit t h e i r proofs.

Lemma 8.1.

Assume t h a t t h e assumptions ( A . l )

& (A.2)

are

satisfied. w j - cJ-l k=O &k

'

WOY

j = 1,2

,..., a + l .

213

Some quasi-Linear Hyperbolic Equations

e and

c6 are some positive constants depending

where c4, c5

VI.

constants appearing & (Sm.2)-(Sm.4) o f section

By Lemma 8.1 and (Sm.4) in section V I , we have CS.Aw.1 J J L

+

fSj(rnWj)l,

-N"+L 2 -1,

cp,

for some positive constant c7. Assume that (A.4) 6 2 114, c76

114, c46

114, c76

c3/2, c46 2 c3,

where c3 and c4 are the same as in Lemmas 7.4 and 8.1, respectively. Assume that Assumption 1.1- 2"

Lemma 8.2.

(A.4)

assumptions (A.1)-

are satisfied. Then, the following assertions are valid

for any

j = 0,l,..., a.

leji2,L,d(n)r(n)

(b)

leb,j I 2,L,d(n)r(n)

' lejlq(n),L,d(n)r(n)

= <

c 86 ej -2N"tL,

< c & 2 -2N"+L e 8 j '

N' ,

0 2L

0 5- L IN'. -

=

Here c8 i s some positive constant independent of j and L Assume that Assumption 1.1- 2"

Lemma 8.3.

(A.4)

and assumptions

(A.1)-

are satisfied. Then, the following assertions are valid. IE"

(a)

lu(Awo)

(b)

Iy("O)12,N',d(n)r(n)

E

q(n)

n

B~'([o,-)G), $Pb(t,)

' I'("O)lq(n),N',d(n)r(n)

E

E!'

n ~"([o,-)xii~).

'

I$yb(liwO)12,N',d(n)r(n)

= <

2

C96 .

214

Yoshihiro SHIBATA and Yoshio TSUTSUMI

Here c9 and cl0 are

some positive constants.

Assume that Assumption 1 .l- 2"

Lemma 8.4.

(A.4)

and assumptions

(A.1)-

are satisfied. Put Eo

Eb,o

= 0,

Ea

a-1

a-1 ej, Eb,a = ' j =O eb,a

= Cjz0

(if a 21).

Then, the following assertions are valid.

EC~;~,

BN' ([o,m)xs~), E ~ ,E~ N'~

(a)

E,

(b)

IEal2,L,d(n)r(n)

n

n

B~'([o,~)G,).

' lEa1q(n),L,d(n)r(n) '

lEb,alZ,L,d(n)r(n) h '116 z '

('I

IEa12,L,d(n)r(n)

-2N"+L 5 -1.

' lEalq(n) ,L,d(n)r(n)

+

2 -2N"+L

lEb,alZ,L,d(n)r(n)

(dl

I('-Sa)Ea12,L

,d(n)r(n)

'1Z6 +

ea

-2N"+L 2 1 , 0 z L 5 N' .

9

l('-sa)Ealq(n)

,L,d(n)r(n)

2 -2N"tL

I(1-Sa)Eb,a12,L,d(n)r(n)

'13& 'a

I

' O2LzN'.

215

Some quasi-Linear Hyperbolic Equations I('a+l- S a ) E a I E,L,d(n)r(n)

I(Sa+l-Sa)Ealq(n) ,L,d(n)r(n)

1 ( S a t l - S a ) E b , a 1 2 9 L,d(n)r(n)

Here cll , c12 and ~

-

< C(L)6

2 -2N"+L

+

vL 20.

ea

Y

1 are 3 some p o s i t i v e constants.

Combining Lemnas 8.3 and 8.4, we have

L~~~ (A.4)

8.5.

Assume t h a t Assumption 1.1- 2"

are s a t i s f i e d . Let fo, g o y f a+l

and ga+

assumptions (A.1)be t h e same as i n ( 6 . 3 ) .

Then, t h e f o l l o w i n g a s s e r t i o n s a r e v a l i d .

- _ .

(b)

Ifa+l12,L,d(n)r(n)

If012,L,d(n)r(n)

+

+

R Ig012,L,d(n)r(n)

I f a + l I q(n),L,d(n)r(n)

I f O l q ( n ) ,L,d(n)r(n) < C(L)6

=

+

'

2 -2N"+L

eo

vL

Y

I n o r d e r t o use t h e r e s u l t s obtained i n s e c t i o n s

111 and I V Y we

LO. have

t o evaluate t h e c o e f f i c i e n t s o f t h e operators t . and B J j'

Lemma 8.6. (A.4)

Assume t h a t Assumption 1 .l-2"

are s a t i s f i e d . Put

Then, - f o l l o w i n g

assertions

are v a l i d .

and assumptions

(A.1)-

Yoshihiro SHIBATA and Yoshio TSUTSUMI

216

f o r a l l 5 = (~ly...y$,)

E

_.-

('1

IR",

a;j being t h e same as i n (5.1).

I A j Im,O,O = < min( ( 8 ~ : ) ~ , ~c1 ,N)

and c1 ,N

where co

a r e t h e same as i n Theorem 3.1

(')

IAjlm,2,d(n)r(n)

(dl

I A j l p ( n ) / r ( n ) , l ,d(n)r(n)

(e)

I A jlm,L,d(n)r-(n)

(f)

IAjIp(n)/r(n),L,d(n)r(n)

(g)

IAjlm,L,d(n)r(n)

(h)

l*jjp(n)/r(n),L,d(n)r(n)

and c15

= N.

51*

'. -N"+L 5 - -1.

2 '14'

= <

-1. -N"+L 2 -

2 '15'

-N"+L > 1, 0 5 L 5 N-1. - -

C ( L )( 1+6eiN'lfL) -N"+L

Here c14

with L

< C(L)(l+6ej

) , -N"+L 51, 0 i L IN-1.

=

a r e some p o s i t i v e constants.

NOW, we g i v e estimates o f fi and Qo. a+l

Note t h a t

L.u = U U + (d%)(S.Aw.)Au = J J J atu 2 B.u = J

-

z ~ , , , = ~ak((6krn + a;m(SjDxwj))amu) 1

+

+

~nm , o ( ~ ) ( s j D 1 w j ) a m u y

rn

1 1 (d% )(S.D w.)D u + ~ ' ( S . ( $ W . ) ) U= b J X J X J J

+ a ' (S.D1w.))a,u 'L,rn=l

'k('krn

km

J

x

+

~'(S~(+w~))u,

J

By v i r t u r e o f Lemna 8.6, we can a p p l y being t h e same as i n (5.1). ij Theorems 3.1 and 4.1 t o ( 6 . 4 ) . F i r s t , by Theorem 3.1 w i t h L = N, Lemnas

a'

8.5 and 8.6 we have (8'1)

ID

1 'a+lIz,L,O

2 -2N"+L+2

5 c166 ea+l

0 2L

N-1.

217

Some quasi-Linear Hyperbolic Equations By Lemma 2.4.2 and (8.1) we have 0

L 5 N-2,

0 ( L IN-1. I n particular, i f C(L) being t h e same as i n (8.2), by t h e f a c t t h a t N"

2 3 and (8.2) we have

Next, we a p p l y Theorem 4.1.

Since N'+[n/p(n)]+Z

= N-l-(2n+5),

a p p l y i n g Theorem 4.1 w i t h L = N-1, by Sobolev's i n e q u a l i t y , Lemmas 8.5 and 8.6 and ( 8 . 2 ) we have

ID (8'5)

1

1 'a+l Im,L,d(n)

2 'ID1'a+l 5 '17*

+

I D 'a+l

I p ( n ) ,L,d(n)

Ip(rt) ,L+[n/pfn)]+l

,d(n)

2 -2N"+L+[n/p( n)]+2n+7

R I'a+lI-,O,d(n)

+

I'a+l

IR p(n),O,d(n)

(8.6) = <

If we choose

0 2 L (N'+l,

Y

c p l l sa + l I p(n),[n/p(n)],d(n) R

-2N"+2n+6 5 c186 2 ea+l

6 so small t h a t

then, by t h e f a c t t h a t N" 2 2n+[n/p(n)]+8,

(8.5) and (8.6) we have

Yoshihiro SHIBATA and Yoshio TSUTSUMI

218

Combining (8.4) and (8.8), we have (8.9)

[Awa+lIL

[hatlIL

+

i f 6 s a t i s f i e s (A.4),

R

-N"+L 2 6eatl

0 'L

9

(8.3) and (8.7).

zN',

I n t h e same way, we have

Thus, by i n d u c t i o n we o b t a i n t h a t t h e statements (St.1) and (St.2) a r e v a l i d f o r a l l j 2 0 under t h e assumptions (A.2) and (A.3). Now, we g i v e a c o n d i t i o n f o r d a t a i n o r d e r t h a t (A.2) and (A.3) a r e valid.

Since wo i s a s o l u t i o n o f (6.1),

by (4.6) and (4.7),

Theorem 3.1

w i t h A = 0 and Lemma 2.4.2 we have

1"012,N',0

+

R IW012,N',0

C19[1F12,N'+1, d ( n ) r ( n )

I"Ol-,N'

R ' IG12,N'+3,d(n)r(n)"

R

,d(n)

(8.11)

+

IWOI-,N' ,d(n)

+

IAW0lp(n),N',d(n)

'[

1

'OI

+

+

p( n ) ,N ' +[ n/p ( n ) I t 2 ,d ( n )

< c 2 0 [ 1 F 1 q (n ) ,N'+[n/p(n)]+2n+4,d(

I d wO 1Rp ( n ) ,N '+[n/p(n) 1,d ( n )

n ) r ( n)

IGI By (5.10), t h e f a c t t h a t N = N'+[n/p(n)]+8

R IWOlp(n),N' ,d(n)

+

R 2 ,N'+[n/p( n)]+2n+6,d( n ) r ( n ) ] .

and Lemma 2.3.3 we have

219

Some quasi-Linear Hyperbolic Equations (8' 12'a)

IF12,N'+1 ,d(n)r( n) 5

(8.12'b)

If

+

12,N'tl ,d(n)r(n)

IG +

R

12 ,N It3,d (n)r( n)

1gl!,N1+3,d(n)r(n)

IF lq( n) ,N '+[n/p( n)]+2n+4 5

If

lq(n),N-4,d(n)r(n)

,d (n)r(n) +

R (8.12'c) 1G12,N'+[n/p(n)]+2n+6,d

c~v12yN'+4,d(n)r(n)'

+

Clvlq(,,),N'+3,d(n)r(n),

n)r(n)

R

5 1g(2,N-2,d(n)r( n)

provided that IvIm,N+l ,o

= <

1.

12,N ,d( n)r(n) Noting (A.3)' and combining (8.10), (8.11)

and (8.12), we have that there exists a constant cZ1

0 such that i f

>

R

lf1q(n) ,N-4,d(n)r(n)

+

If12,N'+1 ,d(n)r(n)

+

1g12,N-2,d(n)r(n)

+

(8.13)

IVl-,N+l ,d(n)r(n)

+

Ivlq(n) ,N+1 ,d(n)r(n)

'216'

then (A.2) is valid. Therefore, by Lemma 5.3 we have that there exists a positive constant

61

such that if (1.5) holds then (A.2) and (A.3) are

valid. If we put

w

=

m

cj=o flj +

W0'

noting that N" 2 a + l , by (St.2) we have

[MI,t ;1.

2 (l+(e-l)- 1 )6.

By (6.5), Lemmas 8.2,8 . 3 and 8.4 we have that w satisfies the equation (5.9), which proves the existence theorem for (1.1).

Finally, we prove the uniqueness theorem for (1.1). begin with

For this, we

Yoshihiro SHIBATA and Yoshio TSUTSUMI

220

Let

Lemma 8.7. a large -

number w i t h b 2 2 ( n + l ) . r(r,T)

I(t,x);

=

x

a, 1x1 5 r + b ( T - t ) , 0 2 t

E

j = 0 ,... ,n,

Let a..(t,x), i,j = l,...,n, bj(t,x), 1J 1 f u n c t i o n s j~ C ([O,-)xT) such t h a t

If u

E

la..(t,x)l

21/2,

T}.

c(t,x) &real-valued

= aji.

aij

lJ

C 2 ([o,m)xsL)

satisfies

Pu = a 2 u t

ij=1

- cn

the equations: + a

ai((hij

+ cn j=o

)a.u) ij J

+ a i j1a.u J + cu =

E ! ~ = ~ui(dij u(0,x)

b.a.u = 0 i ~ r ( r , T ) , J J

o

on [O,TlxaQ, -

(atu (0,x) = 0

then u = 0

and b

Put

_ .

sup (t,x)cr(r,T)

2R

T be any p o s i t i v e numbers w i t h r

r

ar+bT'

~

r(r,T)

I n t e g r a t i n g Pu-atu over r(r,T) and u s i n g t h e divergence theorem, o b t a i n Lemma 8.7 i n t h e usual way.

So, we may o m i t t h e p r o o f .

Using Lemma 8.7, we s h a l l prove t h e uniqueness theorem. be s o l u t i o n s i n C (8,141 I " ~ 1 ~ , 0 , 0

2

+

([O,m)xm

lul,

R Y

=

zlj,l

R

0 3 0 5 1, lAvlm,oyO

1 ai(aij(Dxu)a.u)

J

L e t u and v

o f (1.1) f o r t h e same data such t h a t

where h 2 i s a c o n s t a n t determined l a t e r . (8.15) zyj=l

we

-

c;j=l

+

IvI-,O,o

I 1,

1 IDx~Iw,O,O

By T a y l o r expansion,

ai(aij(Dxv)ajv) 1

ai(a. . ( D1~ u , ~ 1, v ) a . ( u - v ) ) , 1J J

< 62 '

=

Some quasi-Linear Hyperbolic Equations

1

n

vi(x)a. .(D,u,1

= cij=l n

B(D

-

vi(x)a..(D u)a.u 1J X J

EijZl

221

vi(x)a. .(Dxv)a.v 1 1J J

Cij=l

Dxv)a.(u-v), 1 J

1J

1U) - B(D1V) = Cnj = o bj(D 1U, D 1v)a.(u-v), J

Here we have p u t aij

= aij(Dxu,1

Dxv) 1 I

1

= a i j ( D x l O + zk=l

(8.16)

1 1 1 (a, .aik)(Dxv+s(Dxu-Dxv))ds.akv,

'0 J 1 1 b . = b.(D u, D'v) = (a, B)(D1v+s(D'u-D 1v ) ) d s , J J J'o j 1 c = c(u,v) = y'(v+s(u-v))ds.

1,

P u t t i n g w = u-v, we have by (8.15) ai( (8.17)

z?j,l

+ cw

+ aij)ajw

vi(Aij

w(0,x)

+ aij)ajw)

6ij

= (atw)(o,x)

=

+ c jn= O b.a.w J J

= 0

i n [O,=)xa, on [O,m)xan,

= 0

o

i n Q.

t

Since v ( t , x )

= u(0,x)

+

(atv)(s,x)ds,

by (8.14) we have

0

(8.18)

1 ID,v(t,x)l

2 2S2,

(t,x)

E

[0,621~E.

Since by (8.16) and Assumption 1.1- 2" we have

f o r some p o s i t i v e constant c Z 2 depending o n l y on

aij,

i f we choose g2 so

Yoshihiro SHIBATA and Yoshio TSUTSUMI

222 small that (8.20

362c22

we have by (8.14), (8.18), (8.19) and (8.20)

zcj=l l a i j ( t , x ) I 2 1/2,

(8.21) Since u , v

E

C

2 ([O,m)x;),

(t,x)

[0,621xE.

E

we have t h a t a i j y b j , c

Lemma 8.7 t o (8.17), we have that w = u which implies t h a t u = v in

-

v

= 0

Since

[O,ti,]x~.

ti2

E

1 C ([O,m)xE).

Applying

in r ( r , s 2 ) for any r L R , depends only on a i j ( c f .

(8.19) and ( 8 . 2 0 ) ) , replacing 0 by s2 and s2 by 2s2 and repeating the argument, we have that u = v in [ s 2 , 2 s 2 ] x ~ , which implies t h a t u = v in [Oy262]x~,

By repeated use of the argument, we have t h a t u = v in

[O,T]xE for any T

>

0 , which implies t h a t u = v in [O,-)xn.

This com-

pletes the proof of the uniqueness theorem for (1.1). Appendix. Proofs of ( 4 . 4 )

and

(4.5)

In order t o prove (4.4) and ( 4 . 5 ) , we need the following two lemmas.

Let a , b and d be real numbers -Assume that n 2 3 and a & non-trapping.

Lemma Ap.1 (local energy decay).

with

0

< d

2 n-1 & a , b,

Let u be a Cm solution

of

R-I.

(4.3) for data uoy u1

norms appearing below are f i n i t e . supp f

c [O,-)xRa,

Il++’u(t,

* )llaby2

,f

g = 0, supp u i

g.

Assume that a l l

c slay i = 0,1,

@

5 C(Lya, b y Q ) (l+t)’d[

IIUOI~,L+~+

11 u l l l 2 , ~+ I f 1 2 , ,dl ~ -

If n 2 3 and n i s odd, Lemma Ap.1 follows from Morawetz [71.

n

zZ 4

and

and n i s even, Lemma Ap.1 can be proved in the same way as in

If

223

Some quasi-Linear Hyperbolic Equations

S h i b a t a and Tsutsumi [12]. referred.

The f o l l o w i n g result i s well-known

k a solution

13= 0

Then,

~

[0,-)

( s e e Wahl [16]).

p be an extended real number and q = p/(p-1 )

Lemma Ap. 2.

Let v

For f u r t h e r d e t a i l s , Tsutsumi [14] can be

.

o f Cauchy problem:

xRn, v(0,x) = v,(x),

( a t v ) ( 0 , x ) = v l ( x ) j t ~Rn.

the f o l l o w i n g twc e s t i m a t e s @.

and ( 4 . 5 ) .

I n t h e course o f t h e p r o o f , by S(D) =

we d e n o t e the s o l u t i o n o f Neumann problem:

+ av = o on And a l s o , by So(Do) = So(t,x;Do)

[o,-)xan,

, Do

=

ultzO

=

uo, atult=O = u1 i n n.

(voyv, $9), we d e n o t e t h e s o l u t i o n

o f Cauchy problem: UV =

g in [o,-)xR~,

~ l = vo,~ =atvlt=O~

=

v1 i n R ~ .

The proof i s d i v i d e d i n t o t h r e e s t e p s . 1 s t step.

We c o n s i d e r t h e case where f

=

0 , supp u i

c

Rn

- nR+2,

Yoshihiro SHIBATA and Yoshio TSUTSUMI

224 i = 0,l. 4.71,

I n t h e same way as i n Shibata and Tsutsumi [ l l , Proof o f Lemma

by u s i n g Lemmas Ap.1 and Ap.2,

we have

f o r any t > 0, where M = 0 o r L, D = (uo,ulYO) (Ap.2)

a =

n-l 2

and we have p u t

i f n 2 4 and 1-E i f n = 3

f o r any small p o s i t i v e number

E.

On t h e o t h e r hand, by t h e usual energy method ( c f . Theorem 3.1 w i t h A = g = 0 ) and Lemma 2.4.2

we have

I n t e r p o l a t i n g (Ap.1) w i t h M = L and (Ap.3) and i n t e r p o l a t i n g (Ap.1) w i t h M = 0 and (Ap.3)',

we have

f o r any p w i t h 2 5 p 5 - and q = p/(p-1). 2nd step.

We consider t h e case where uo = u1 = 0 and supp f

I n t h i s case, we can use Duhamel's p r i n c i p l e f o r t h e

[O,-)X(R"-Q~+~). mixed problem.

Thus, we can w r i t e

1, t

(Ap.5)

S(t,x;D)

where D = (0,O.f)

c

=

S(t-s,x,D'(s))ds

and D ' ( s ) = (O,f(s,*),O).

Note t h a t

225

Some quasi-Linear Hyperbolic Equations -a( )1:-

2

ds z . C ( l + t ) where (Ap.6)

2

6 = ci(1--)

P

2 if a(1--) P

> 1 and =

2

1

1 + i~f a(1--) P

f o r any s u f f i c i e n t l y small p o s i t i v e number

Applying (Ap.4) t o (Ap.51,

K.

we have

f o r any t > 0.

I n p a r t i c u l a r , combining (Ap.4) and (Ap.7) and t a k i n g

2

p = p(n), q = q(n), a(l-im) = d ( n ) , B = d(n )r(n ), we o b t a i n (4.4) and (4.5) when supp ui

C Rn

-

aR+2, i

0,1,

and supp f

c

[O,m)x(R n

-

aR+2).

To complete t h e p r o o f , we prove (4.4) and (4.5) under

3rd step. t h e assumptions:

From Lemna Ap.1 i t f o l l o w s t h a t

f o r any t z 0.

i f 1x1 5 R+3. p(x)S(t,x;D) w(t,x) we have

Choose

p E

Camn) so t h a t

p(x)

= 1 i f 1x1 2 R+4 and = 0

By t h e uniqueness theorem f o r Cauchy problem, we have = So(t,x;Do)

= S(t,x;D)

if x

E

where Do = (0,0,-2~j,l(aj~)(ajw)-(Ap)w) n and

n and

= 0 i f x E Rn

- a.

By Duhamel’s p r i n c i p l e

Yoshihiro SHIBATA and Yoshio TSUTSUMI

226 (Ap. 10)

p(x)S( t ,X;D)

where D;(s)

=

=

J,‘

So( t - s ,x;OA( s))ds,

( O , - ~ C J , ~ ( a j p ( x ) ) a,w( s , x ) - ( h p ) (x)w(s ,x) ,O).

Since p ( n ) >

n o t i n g t h a t t h e supports o f a . p and A P a r e contained i n J and a p p l y i n g Lemma Ap.2 t o (Ap.lO), we have by (Ap.9)

Z(n+l)/(n-l), ‘R+4

By Sobolev’s i n e q u a l i t y , (Ap.9) and ( A p . l l ) , we have

(Ap. 12)

(1’1 ((2,M+n+l

-dI-m) L < -

C(M,a)(l+t)

Taking B = d ( n ) r ( n ) and d(n) = a(l-&), (Ap.121, which completes t h e proof.

,

[ ’0 Ilq ( n ) ,M+2n+3 ‘1

’ If12,M+n+l ,B 1

Ib( n) ,M+2n+2

+

+

I I q( n) ,M+2n+2 ,B’.

we have (4.4) and (4.5) by

221

Some quasi-Linear Hyperbolic Equations

Reference 8 A uniqueness theorem f o r n o n s t a t i o n a r y Navier-Stokes

3.6. Heywood,

flow p a s t an obstacle.

Ann. Scuola Norm. Sup. Pisa, 11 (1), (1979),

427-445. linean P a h t i a e Ui,56ment;d OpehcLto~, B e r l i n ,

L. Hormander, S p r i nger , 1963.

S, Klainerman,

Global e x i s t e n c e f o r n o n l i n e a r wave equation,

Comm. Pure Appl

. Math.,

R.B. Melrose,

S i n g u l a r i t i e s and energy decay i n a c o u s t i c a l

33, (1980)

, 43-101.

Duke Math. J . , 46, (1979), 43-59.

scattering, S. Miyatake,

Mixed problem f o r h y p e r b o l i c equation o f second o r d e r ,

J . Math. Kyoto Univ., 13 ( 3 ) , (1973), 435-487.

Jheoky oh PatLt.id U-id6eh/rentiae Equationb, Cambridge

S. Mizohata,

UP.,

1973.

C.S.

Morawetz,

Exponential decay o f s o l u t i o n s o f t h e wave equation,

Corn. Pure Appl. Math., 19 (4), (1966), 439-444. On t h e g l o b a l e x i s t e n c e o f c l a s s i c a l s o l u t i o n s o f mixed

Y. Shibata,

problem f o r some second o r d e r non-1 i n e a r hyperbol i c o p e r a t o r s w i t h d i s s i p a t i v e term i n t h e i n t e r i o r domain,

Funk. Ekva.,

25, (1980),

303-345. ___

,

On t h e g l o b a l e x i s t e n c e theorem o f c l a s s i c a l s o l u t i o n s

o f second o r d e r f u l l y n o n l i n e a r h y p e r b o l i c equations w i t h f i r s t order d i s s i p a t i o n i n t h e e x t e r i o r domain,

Tsukuba J . Math., 7,

(1983), 1-68.

,

On t h e e x i s t e n c e and r e g u l a r i t y theorem o f Neumann

problem f o r some hyperbol i c - e l l i p t i c system,

i n preparation.

Yoshihiro SHIBATA and Yoshio TSUTSUMI

228

[ll] Y. Shibata and Y. Tsutsumi,

On a g l o b a l e x i s t e n c e theorem o f small

amplitude s o l u t i o n s f o r n o n l i n e a r wave equations i n an e x t e r i o r domain, 1121

t o appear. and

~

,

~

Global existence theorem o f n o n l i n e a r wave

equations i n t h e e x t e r i o r domain, Applied Analysis, [13]

Y. Tsutsumi,

Vol. 6,

L e c t u r e Notes i n Numerical and

155-195, K i n o k u n i y a l North-Holland.

Local energy decay o f s o l u t i o n s t o t h e f r e e

Schrodinger equation i n e x t e r i o r domains, Sec. I A , 31 , (1984) ~141

___ ,

[15]

B.R.

, 97-108

Doctor Thesis, Tokyo Univ.,

Vainberg,

J . Fac. Sci. Univ. Tokyo

i n preparation.

On t h e s h o r t wave asymptotic behaviour o f s o l u t i o n s

o f s t a t i o n a r y problems and t h e asymptotic behaviour as t s o l u t i o n s o f n o n s t a t i o n a r y problems,

+

-

of

Russian Math. Surveys, 30 ( 2 ) ,

(1975), 1-58. [16]

W.V. Wahl,

Z., [17]

LP-decay r a t e s f o r homogeneous wave equations,

Math.

120, (1971), 93-106.

D.V. 1941.

Widder,

The Laplace T h a n n ~ o t r n , P r i n c e t o n U n i v e r s i t y Press,