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

RECENT TOPICS IN NONLINEAR PDE I1

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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)

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@ 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.

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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

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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 .........................

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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/




,

< 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
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)


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)
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*


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


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*
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

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
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




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.

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


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)
> U2 ( x ) = U2 (X,)


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


>

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

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




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


,

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)




s S s

*

-

S s

and

g S u

*

=

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.


)

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,,,


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. 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


-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.

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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


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


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


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


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

=
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,

=
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) =


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

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