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RECENT TOPICS IN NONLINEAR PDE

This Page Intentionally Left Blank

NORTH-HOLLAND

MATHEMATICS STUDIES

98

Lecture Notes in Numerical and Applied Analysis Vol. 6 General Editors: H. Fujita (University of Tokyo) and M. Yamaguti (Kyoto University)

Recent Topics in Nonlinear PDE

Edited by

MASAYASU MIMURA (Hiroshima University) TAKAAKI NlSHlDA (Kyoto University)

1984

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK. OXFORD

KINOKUNIYA COMPANY LTD. TOKYO JAPAN

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM'NEW YORK'OXFORD KINOKUNIYA COMPANY -TOKYO

@ 1984 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 b y any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87544 1

Publishers NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM * OXFORD. NEW YORK

*

*

*

KINOKUNIY A COMPANY LTD. TOKYO JAPAN

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

Distributed in Japan by KINOKUNIYA COMPANY LTD.

Lecture Notes in Numerical and Applied Analysis Vol. 6 General Editors H. Fujita University of Tokyo

M. Yamaguti Kyoto Universtiy

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

PREFACE The meeting on the subject of nonlinear partial differential equations was held at Hiroshima University in February, 1983. Leading and active mathematicians were invited to talk on their current research interests in nonlinear pdes occuring in the areas of fluid dynamics, free boundary problems, population dynamics and mathematical physics. This volume contains the theory of nonlinear pdes and the related topics which have been recently developed in Japan. Thanks are due to all participants for making the meeting so successful. Finally, we would like to thank the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan for the financial support. M. MIMURA T. NISHIDA


CONTENTS

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kiyoshi ASANO and Seiji UKAI: On the Fluid Dynamical Limit of the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . Hiroshi FUJI1 and Yuzo HOSONO: Neumann Layer Phenomena in Nonlinear Diffusion Systems . . . . . . . . . . . . . . . . . . . . . . . . Tadayoshi KANO and Takaaki NISHIDA: Water Waves and Friedrichs Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuichi KAWASHIMA : Global Existence and Stability of Solutions for Discrete Velocity Models of the Boltzmann Equation . . . Kyiiya MASUDA: Blow-up of Solutions for Quasi-Linear Wave Equations in Two Space Dimensions . . . . . . . . . . . . . . . . . . . Tetsuro MIYAKAWA: A Kinetic Approximation of Entropy Solutions of First Order Quasilinear Equations . . . . . . . . . . . . . . Yoshihisa MORITA: Instability of Spatially Homogeneous Periodic Solutions to Delay-Diffusion Equations . . . . . . . . . . . . . . . . . Shinnosuke OHARU and Tadayasu TAKAHASHI: On Some Nonlinear Dispersive Systems and the Associated Nonlinear Evolution Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hisashi OKAMOTO : Nonstationary or Stationary Free Boundary Problems for Perfect Fluid with Surface Tension . . . . . . . . . . Yoshihiro SHIBATA and Yoshio TSUTSUMI : Global Existence Theorem for Nonlinear Wave Equation in Exterior Domain Kazuaki TAIRA: Diffusion Processes and Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atusi TANI: Free Boundary Problems for the Equations of Motion of General Fluids .............................. Masayoshi TSUTSUMI and Nakao HAYASHI: Scattering of Solutions of Nonlinear Klein-Gordon Equations in Higher Space Dimensions .......................................

v 1 21

39 59 87

93 107

125

143 155 197 21 1

221


Lecture Notes in Num. Appl. Anal., 6, 1-19 (1983) Recent Topics in Nonlinear PDE,Hiroshima, 1983

On the Fluid Dynamical Limit of the Boltzmann Equation

K i y o s h i A S A N O * and Seiji UKAI** *Institute of Mathematics, Yoshida College, Kyoto University Kyoto 606, Japan **Department of Applied Physics, Osaka City University Osaka 558, Japan

1.

Problem and Results This paper i s a continuation o f our paper C161 concerned w i t h the Euler

l i m i t o f the Boltzmann equation. density d i s t r i b u t i o n path E(>O)

f (t,x,S)

tends t o zero.

I n C161 we studied the behavior o f the

o f r a r e f i e d gas p a r t i c l e s , when t h e mean f r e e

More precisely, i f the i n t i a l density d i s t r i b u t i o n

i s s u f f i c i e n t l y close t o an absolute Maxwellian and ' s a t i s f i e s some r a t h e r

f,(x,S)

r e s t r i c t i v e conditions, then t h e s o l u t i o n f"(t,x,E)

o f the Boltzmann equation

w i t h i n i t i a l data fo e x i s t s i n a time i n t e r v a l [O,T1 independent o f and

when

F

E

e(t.x)}

E E

(O,m),

0

converges t o a l o c a l Maxwellian f (t,x,E):

tends t o zero.

Moreover, the f l u i d dynamic q u a n t i t i e s {p(t,x),v(t,x),

( i . e , mass density, f l o w v e l o c i t y and temparature) s a t i s f y the com-

p r e s s i b l e Euler equation w i t h i n i t i a l data s p e c i f i e d by fo(x,S).

This l i m i t -

i n g process i s the f i r s t approximation t o the H i l b e r t expansion o f the s o l u t i o n o f the Bol tzmann equation. I n t h i s paper we make a more d e t a i l e d treatment o f the H i l b e r t expansion

and e s t a b l i s h an asymptotic formula such as

1

2

Kiyoshi ASANO and Seiji UKAI

.

and behaves l i k e exp(-oT) w i t h u > 0 (j=O,l....)

-.

However, t h e general f o r -

mula t o c a l c u l a t e fJ and fJ i s so complicated t h a t we prove o n l y t h e s p e c i a l case (1.2)

fE(t,x,c)

0

= f (E,t,Xsc)

-0

f (E,t/EsX,E)

Ef'**(E,t,X,c),

and suggest t h e method t o prove t h e n e x t s t e p of t h e expansion. The 1 i m i t i n g process from t h e Boltzmann equation t o t h e compressible E u l e r e q u a t i o n was described i n d e t a i l i n 1101 and C161, and we s t a t e o n l y t h e conclusion.

The Cauchy problem o f t h e Boltzmann e q u a t i o n i s described as

a f t c*v,f at

Here f = f(Est,X.c)

1 QCf,fl,

=

t>O, (x.6)

E

Rn

x

Rn

(nr3),

i s t h e d e n s i t y d i s t r i b u t i o n o f gas p a r t i c l e s w i t h t h e

p o s i t i o n x and t h e v e l o c i t y 5 a t time t, E-V, = E,a/axl

+.a*+

cna/axn

and QCf,hl i s t h e s y m e t r i z e d c o l l i s i o n i n t e g r a l which i s a q u a d r a t i c o p e r a t o r The s c a t t e r i n g p o t e n t i a l i s assumed t o be t h e c u t -

a c t i n g on t h e v a r i a b l e 5. o f f hard t y p e o f Grad C51.

E>O i s t h e mean f r e e path.

Since we c o n s i d e r (1.3) near an a b s o l u t e Maxwellian , we p u t 2 g(E) = p ( 2 r e ) - n / 2 e -161 / ( 2 e ) , p > 0, e > 0, (1.4)

f(E,t,X,c) fo(x,e)

= = g

+ +

g1/2~(c,t,X,~)

g1l2 u 0 ~ x . c )

,

.

Then we o b t a i n t h e e q u a t i o n f o r t h e unknown

-au- - -s.vxu at

t

1

LU t

1 r[u,ui

,

u :

Fluid Dynamical Limit of the Boltzmann Equation

3

(1.5) where

Denoting by Q(k,S) = Fxu(.,S)

the Fourier transform o f u,

O(k,S) = (2a)-"' we convert (1.5)

u(x,E)dx,

t o the f o l towing

3 at = -

iS.kQ

filt,o

= Oo(k,S),

+ 1 LO + 1 rCO,01, ^

i = fl ,

(1.6)

where (1.7)

F[u,vl(k,S)

= (21r)-"'

The equation (1.6) i s a c t u a l l y solved i n t h i s paper (see also

According t o C 3 1 ~ the c o l l i s i o n i n t e g r a l

151 2/21,

{h.(E) ; O<jsn+ll = {l,El,***,Sny J (1.8)

.

rCu(k-k',*),v(k',*)I(E)dk'

QCf,fl(5)hj(S)dS = 0

,

I: 61).

Q t f , f l has (n+2 i n v a r i a n t s

i.e,

j = 0,1,***.

n+l,

By the f o l l o w i n g formula we define f l u i d dynamic q u a n t i t i e s associated w i t h

t h e density f ( t , x , S )

o f gas p a r t i c l e s , i.e, the mass density p(E,t,x),

flow v e l o c i t y v(c,t,x), tensor p(E,t,x)

i n t e r n a l energy e(E.t,x).

P ( ~ , t , x ) = (P. . ( ~ , t , x ) ) ,

temparature

heat f l o w vector q(E,t,x)

1J

fluid

e ( ,t,x), ~ stress

and pressure

(see C101): P(E,t,x) P ( E ,t ,x)v

hO(S) dS

f'(t,x,S)

=

R" (E

,t ,x)

P(E,t,x){e(E,t,x)

=

sf"

( t,x,5) h

, (5Id5

+ 2 / v ( E . ~ , x ) / ~ =)

(1"jsn

5 fE(t,x,E)h,+l

, (E)dS,

4


(1.9)

The l a s t i s t h e i d e a l gas c o n d i t i o n . Combining (1.3) and (1.81, we o b t a i n

na P+

Since

VX'(PV) = 0,

P = PI ( I = t h e i d e n t i t y m a t r i x ) and q =O

f o r t h e l o c a l Maxwellian f,

t h e equation (1.10) reduces t o t h e compressible E u l e r e q u a t i o n f o r E = 0 (and t > 0). According t o P r o p o s i t i o n 3.1 o f C121, we have P.

-

.(E,*)

1J

q.(E,') J

=

P(E,*)Gij

-

EK(e)

a ?&-e

+

o(E

2 axi )

Vi)

j

- -n1v

x* v I +

O(E

I

j

Thus t h e f l u i d dynamic q u a t i t i t e s {p,v,el f(E,t,x,5)

2

a v. + a J

i

= -2E!.l(R){-(

o b t a i n e d from t h e d e n s i t y

g i v e n i n Theorem 1.2 w i l l s a t i s f y t h e compressible Navier-Stokes

e q u a t i o n w i t h t h e e r r o r o f O ( E ~ ) . A more d e l i c a t e t r e a t m e n t w i l l be g i v e n elsewhere.

To s o l v e t h e e q u a t i o n (1.6) we use several f u n c t i o n spaces and norms ( c f . C161). We i n t r o d u c e these spaces. (1.11)

3 u(k,O

I'Ia,~,~=

A l l f u n c t i o n s a r e measurable o r continuous. +===c.

sup

k,c Rn

e'("l

k'

(1+1k l f ( l + [ 5 [ )B\u(k,E)I
R)

Here

i s the c h a r a c t e r i s t i c f u n c t i o n o f the s e t {(kyS)cR$Rn;

( k ( + l S ( > R l . With a Banach space X,

0 B (D;X)

denotes the space o f X-valued,

bounded and continuous functions defined on D. RT* = RT \ i ( O , O ) >

m2

2

0

(1.14)

(resp. m

Z!$;i'*

2

and RT =

{(E,T);

We p u t RT = CO,lIxCO,Tl,

m = (mlym2)y ml ( E , E T ) E R ~ ~ For .

0 ),

:Bmyy(R*;X" T R,B )

Theorem 1.l. Let

g

a > 0, 9. > n+l, B

i s defined s i m i l a r l y .

be an absolute M m e l l i a n and l e t 5

1.

2

0,

6


Then there e x i s t positive numbers a1,b0,b0 data fo = g + g1/2uo

and bQ such t h a t for each i n i t i a l

satisfying

the following statements hold with constants Y > 0 , T > 0 (a-yTzO) and u > 0 .

For each

lil

E E

(0,1],(1.3)

(resp. Q(E,t,k,S))

f(E,t,X,c)

f = g

For

t

on the time interval [O,T],

and there hold

+ g1/2u, +

= u0 ( E , t )

U(E,t)

liil

(resp.(l.6)) has a unique solution

E

(O,T],

+

$(E,t/E)

f(O,t,x,S)

E“l’*(E&)

= g(5) +g(E)

1 / 2 u0 (O,t,x,S)

Mamellian whose f l u i d dynamical quantities Cp,v,Ol

i s a ZocaZ

are the soZution of

the compressible Euler equation (1.10) w i t h P = PI and q = 0. liiil

Moreover, there hold

“

A0

Theorem 1.2.

lP,Y ‘ ‘k,B,T

’ 1

A0

IIl,a,y,L,B,T

Let g be an absolite Maxuellian and l e t

a > 0, II > n+3,

B

2

2.

’bj (j=O,l),

Then there e x i s t positive numbers a2,bj,

(j=O,l, a. = a, a , -yOT 2 a,, a l data

fo 3 g

+

’ bb I Q O l a , L , B 1

g1/2uo

satisfying

- ylT

2

bt, yj, a j

0) a n d o such t h a t for each i n i t i a l

7


the solution

f(E,t.x,S)

the following formula 0 U(E,t) = u (E,t) + “U(E.t/E)

a €-a aE

2,*

= {y

E

.t E U

1 (E,t)

1

.t E i j (E,

+

t/E)

E2U2’*(E,t),

(€,t

We note t h a t i f 0

Ba

of (1.6)) is described i n

of (1.3) ( r e s p . O(E,t,k,C)

c

X;,B

, then

u(x,E) i s a n a l y t i c i n x

E

Rn ; IyI < a 3 , and u n i f o r m l y bounded on Rn + iE6,

Rn + Bi ,

.

0 < 6 < CL

According t o t h e r e s u l t s o f Theorem 1.l,we p u t

(1.16)

= g(5) + g(E)1’2Uo(E,tsX,5) f0(€,t,X,5) 0 1/2 -0 P (E,t,/E,x,S) = g(5) u (E,t/&.x,S)

fl’*(E.t,X,S)

= g ( 5 ) 1/2u19*

(E

,t ,x ,5 )

9

,

.

Then we have t h e d e s i r e d formula (1.2). S i m i l a r expansion formula can be estableshed using t h e r e s u l t s o f Theorem 1.2. Considering t h a t

fo,

i0 and fl’*

are analytic i n x

E

Rn

.t

iBa,yt

for

0 < t < T, our existence theorem i s o f Cauchy-Kowalewski type ([8],[91). hope t o f i n d more n a t u r a l existence theorems.

2.

Some estimates Denoting t h e unknown by u(k,S)

i n s t e a d o f ^u(k,S), we w r i t e (1.6) as

We


8

We d e f i n e t h e l i n e a r i z e d Bol tzmann o p e r a t o r

-

B(k) =

(2.2)

i 5 - k + L.

Then t h e e q u a t i o n ( 2 . 1 ) reduces t o

au -

at

(2.3)

B(Ek)u t 1 rLu,Ul, A .

E

= uo(k,S).

U I t.0

5 w i t h t h e parameter k

The o p e r a t o r B(k) a c t s on t h e v a r i a b l e

Rn.

E

B ( k ) generates a s t r o n g l y continuous semi-group e t B ( k ) i n v a r i o u s f u n c t i o n spaces on Rn5,

(2.4)

.m

f o r example i n Lg,

where

Li

B

=

i"8 =

is measurable and bounded 1 ,

; ( l t \ E ] ) f(E)

{f(E)

If(E)l

; (1t151)'

{ f c L;

+

o

I E I "1,

u n i f o r m a l y as

+

w i t h t h e norm

lflg=

(2.5)

(l+lE1)B

sup

5

.

If(S)I

Thus t h e e q u a t i o n ( 2 . 3 ) can be r e w r i t t e n as t h e i n t e g r a l e q u a t i o n

Now we quote some fundamental p r o p e r t i e s o f L and by c ( A ) , d ( B ) , * * *

t h e constants 2 0

r

([51,[61).

We denote

depending on t h e parameters A, 8,

.*..*.

Lemma 2.1 f i )The operator L has the decomposition L = -A

+

K,

A i s a multipZication operator,

A

and K %s an integral operator i n 5.

= v(E)x,

Moreover

v ( 5 ) i s contiouous and v o

(2.8)

v i t h p o si t i v e constants vo and v 1 (2.9)

lKulB

(iil

, and o i t h a constant B

c

c(B)

2

0

R. m

am

,B

E

R, and

L has 0 as an i s o l a t i e d eigenvalue of m u l t i p l i c i t y

Denoting th e corresponding eigenprojection b y P(0) ( = CP.(O),

Lema 2.2.

(2.10)

v 1 (1+151)

The s p e a t m a ( L ) of L i s inoariant i n Lg and LB

contained i n (-m,01. nt2.

c(B)lu18-ll

5 U(5) 5

J

f i ) ( c i l , ue have P(O)~CU,VI = 0

,

m

U,V

L~ ( B

2

o),

see

Fluid Dynamical Limit of the Roltzmann Equation

(2.11)

IP(o)ulB

c ( ~ , 8 ' ) l u l g , f o r any B , B '

2

R.

E

m

(iii) f i e operator A - l r L , J i s a continuous mapping from L0 (resp.

i;

x

ii

.m

t o ;L

Ih- 1rCu,vllg

(2.12)

.0

d(B)lulglvlg

5

;L

f o r B > 0, i . e ,

(resp. L~

2

0.

The f o l l o w i n g Lemna i s concerned w i t h the spectral p r o p e r t i e s o f B(k), e s s e n t i a l l y due t o E l l i s - P i n s k y C41, and c r u c i a l i n the study o f the

1.

Boltzrnann equation (e.g, C111, C141, C151 and C161

Lemma 2.2.

( i l There i s a p o s i t i v e nwnber

KO

such t h a t f o r

Ikl

s

K~

, n+l ) and euresponding eigen-

B( k ) has (nt2) eigenualues A . ( k ) (j=O,...

J

projections P . ( k ) of rank 1 s a t i s f y i n g the foZZowing f a ) , Ib) and ( 0 ) . J B(k)P.(k) = A.(k)P.(k) , j = O , l , * * * , n+l, I k l I K ~ . (a)

J

A

Cm(nK) ,

E

J

J

Re A.(k)

J

A(!)J

with the c o e f f i c i e n t s Pj(k)

(b)

E

R

E

lk12 + 0 ( [ k l 3 )

- A ( ?J)

J

0 and

5

J

j c h . ( k ) = +ih('.)lkl

A(?) J >

and

Cm(EK) , and there

(lkl

5

e x i s t s a constant C.(B ,B ' ) such that

J

(By@'

Cj(B,B')lulO,

f i t P(k) = CPj(k).

(el

0)

0.

0

1Pj(k)ul0

+

E

R).

Then u ( B ( k ) ( l - P ( k ) ) )

; Re A
0 . P(0) = C P j ( 0 ) i s the eigenprojection i n (2.10). (0)

If I k l

2 K

~

a, ( B ( k ) )

(ii) Let u = u ( E )

Let

x(k)

E

E

ii

c

{A

; Re A
0, y

0 ) , and m ' = m or m +(0,1).

With appropriate constants bm(L,B',B),

2

5 j 5

m2

0,

> n+lml,

B

2

\mi,

T > O

Then a l l the claims i n L e m a 2.6 hoZd bm(l,B) and hm(k,BP)

To prove Lemma 2 . 6 , we use t h e f o l l o w i n g f o r n u l a

.

2

0.

Kigoshi ASANO and Seiji UKAI

14

lo

,(t-s)B(Ek)

=

{-iS.k}

For f u r t h e r d e t a i l s o f t h e p r o o f o f (2.23),

,sB(Ek)dSm

see C161.

The p r o o f o f (2.24)

-

(2.26) i s n o t d i f f i c u l t .

3.

Proof o f Theorem 1.1 With t h e n o t a t i o n d e f i n e d by (2.21) and (2.28), t h e equation (2.7) i s

r e w r i t t e n as (3.1)

+ G(t/E,k.Ek)uo

u(E,t) = F(t,k,Ek)uo

+ FCu,ul(E,t)

+ G[u,u~(E,~/E).

We p u t (3.2)

u ( E , t ) = u0 (E,t) +

0 (E,t/E)

+ EU”*(E,t)

.

By i n t e g r a t i o n by p a r t s we have (3.3)

GCuo,uo1(E,~) = =

-

B(Ek)-lQ(Ek);

i,

e ( T - S ) B ( E k ) Q ( E k )~ C U ~ ( E , E S ) , U ~ ( E , E S ) ~ ~ S

[Uo(E,ET),Uo(E&T)l

+ eTB(Ek)B(Ek)-’Q(Ek) ~ l u o ( ~ , O ) , u o ( ~ , O ) l

+

z c c e (T-s)B(Ek)B(Ek)-lQ(Ek)

A 0 rC$c ,r s ) ,u’(E

,~s)lds

.

Then t h e e q u a t i o n (3.1) i s decomposed i n t o t h e f o l l o w i n g t h r e e equations (3.4)

uo(c,t)

= E(t,k,Ek)uo

(3.5)

~ ( E , T )=

+ F Cuo9u01(E,t)

B(Ek)-’Q(Ek) G(t,k,Ek)uo

rCu a

.

( ~ , t ) ~ u ~ ( ~ I, t ) l A

.

+ eTB(Ek)B(Ek)-’Q(Eklr Cu

(€,a),

+ Z G [ U ~ , ~(E~,TI f + G i t 0,u-0.1 (E ,T )

0 u (c,0)1

,


ul’*(E,t)

(3.6)

= FLHG

0

,th0l ( ~ , t ) +

15

0 0 2FCu ,Hi i ( E , t )

0

+ 2 8 ( k)-lGC ~

2,

uOI(E , t / E )

+ B ~ F l ~ ~ , u ~ ~ * +l (FCHiO,ul’*](~,t)I ~ , t )

+

F[ul’*,u’’*I(~,t)

E‘

+ 2 G t u o , u ~ ~ * l ( ,Et / E ) +2GCto,U1’*,(E,t/E)

.

ul’*l(c,t/E)

+EGCul,*,

Put t h e r i g h t hand s i d e s o f (3.4),(3.5)

@(u 0 ), ” 0) and

and (3.6) as

Q’(U’’*).

A p p l y i n g Lemma 2.5 and 2.6 t o @ ( u o ) , we have

11

(3’7)

o(‘

0

li0,a,y,g,5,T

a > 0,

e0 ( B P B )

7

{ 1 bO(I1,BsB-l)

+

with

< -

II > n, B a 0, y > 0, a-yT

2

IUOla,I1,B

+

d(e,B)lII

uo I120,a,y,g,B,T

9

0.

The q u a d r a t i c e q u a t i o n Y = eoiB,B)

(3.8)

I

I

uo a,ll,B + C

1 7 bo(L,B,B-l)

has two d i f f e r e n t p o s i t i v e r o o t s p r o v i d e d there holds D 0 -= 1 - 4{ 1 bo(L,B,O-l) + d ( L , B ) I e0 ( 1 3 8 ) (3.9)

+

d(L,B)l Y

I UOla,L,B

2

’ O.

0 Denoting t h e s m a l l e r r o o t o f ( 3 . 8 ) by Y , we have

yo = -e ( 0 8 ) I U O I a , L , B 1 4 0 O

(3.10) For

uo

E

0 s a t i s f y i n g ( 3 . 9 ) , t h e succesive approximation (u.} J

ki,B

0 uo ( E , t ) = 0, i s bounded i n

Oa

Ze:B::

,

1

0 0 U ~ + ~ ( E =, ~@)( u . ) J

i.e,

we can show e a s i l y f o r j = 1,2,.-. 0 0 uj+l u j 110,a ,y,e B , ,T

A p p l y i n g L e m a 2.6, (3.11)

< 2 e(J(O*B) I U O l a , L , B ‘

-

( j = O,l,..-)

:

Kiyoshi A S A N O and Seiji UKAI

16

T h i s i m p l i e s t h e convergence o f

,

.

i :n: ; : :Z

{u?l J

The l i m i t

uo

is in

and t h e s o l u t i o n o f (3.4). An easy c a l c u l a t i o n shows t h a t

Hence we have from (3.12)

a o

(3.14)

1

5 -

" R uj+l

,B,T

Ilo,a,y,a-l

1-1 0

0 e(o,l)(B,b)

Iuo Ia,E,B

= y1

Another simple c a l c u l a t i o n shows

Since

Ccuo

J

-

uy-;}

i s m a j o r i z e d by a convergent s e r i e s , (3.13) and (3.15) {z a uj) o

i m p l y t h e convergence o f Noting t h a t

Zosayy 9-,B,T

*

Eauo/at

E(aF/at)uO

Hence

6

,$Z;

d u 0/ a t E;:;:Z;

Z&,B,T ( 0 7 1 ) y a y y , i f 9- > n t l . S i m i l a r l y we have

i n Ze;l'B,T Oav

.

Thus

$ uo

6

ZOsa*y 9--1 ,B.T

Eau?/at i s proved t o be convergent i n J

.

S i m i l a r argument as above shows T h i s f a c t is used i n t h e s t u d y o f (3.6).

'

17


I n t h i s case we have t o p u t two c o n d i t i o n s :

-

(3.17)

1

(3.18)

Do

> 0,

260(II,B,o)Y0

11

E

-4

-

260(II,B,a)Y 0 1 2

bo(".B9o)go(.B)

{

1 ~ Ja,k,B 0

+

Yo12

d(L,B)

>

These c o n d i t i o n s a r e s a t i s f i e d , if Y 0 i s s u f f i c i e n t l y small, i . e , s u f f i c e n t l y small.

Under t h e c o n d i t i o n s (3.17) and (3.18),

approximation f o r Go = ;(Go)

converges i n

z,:B:'* ocr

u

.

0

.

/uola,II,B i s

t h e succesive

Denoting by Y-n t h e

s m a l l e r p o s i t i v e r o o t o f t h e corresponding q u a d r a t i c equation, we have (3.19)

I'

"O,a,y,o,L,B,T

11

We n o t e t h a t t h e c o n d i t i o n imp1 i e s (3.20)

11

G ~~C,a,y,o,e,B,T

Z

E

t h e s t u d y o f (3.4), we can show

2'

0

/aT

~

60

E

~

/~o,cr,y,u,II,.B,T2 -0 Y

.~

'- a

llO,cr,y,~,~,B,T

By~ t h e~ s iym i l a' r argument as i n

Z ~ ~ ~ ~ ~ y a, 'i fy yIIa > n + l .

and 8 i j o / a E i s proved i n a s i m i l a r way.

B u t we may have t o t a k e a s m a l l e r uo i n {aG./aE:l 0

,/I c

' ( l - Jbo)i

llO,a,y,o,k,B,T

Thus (3.5) has a s o l u t i o n Go

I a i j0. / a T l and

.

-

The e x i s t e n c e o f

- 70
n + l , B

2

I), for

t o be convergent.

J

J

The t h i r d e q u a t i o n (3.6) i s r a t h e r complicated, b u t i t can be t r e a t e d similarly. omitted.

Thus we have almost proved Theorem 1.1.

The r e s t o f t h e p r o o f i s

*


18

4. Remarks

To prove Theorem 1.2 we have t o p u t 0 u ( E , t ) = u ( E , t ) + GO(E,t/E) t

(4.1)

EU

1

( c , t ) t EG1(E,t/E)

2 2,*

t E u

(E,t). 0

S u b s t i t u t i n g (4.1) i n t o (3.1), we o b t a i n t h e same equations f o r uo and ii

.

Making o t h e r i n t e g r a t i o n s by p a r t s F [ u O , i O 1 ( ~ , t ) = B ( ~ k ) E ( t , k , ~ k ) ( 2,I "-uo (E,o),u~(E,O)I + T 1 ( ~ , O ) j

-

B(Ek){2FCuo(c,t),

-

B(Ek)(2Fho,171

GCu0 ,u 1 I ( E , T ) =

5'0

G:(E,~/E)~ +

1;

+

r, (E,t/E))

F(t-s,k,Ek)r^,

( ~ , s / ~ f d s,j

e ~ ~ ~ s ~ B ~ ~ k ~ Q ( ~ k ) ~ C u o ( ~ , ~ s ) , u ' ( ~ , ~ ~ ) l d s 1

O O

= -B(Ek)- Q(Ek)rCu (E,ET),u~ ( E , E T ) ]

+ eT B ( E k ) B ( E k ) - l Q ( E k ) ~ C u o ( E ,0) ,ul (E,O) 1

1

0

+ EB(Ek)-l {G C$,

u ' l + GCuO,

I 1,

..... ,

& $1l' =, at where

0 and

i1 are

t h e i n d e f i n i t e i n t e g r a l s o f Go and ;CGO,GO1

respectively,

w i t h some n i c e p r o p e r t i e s . We can s o l v e t h e e q u a t i o n f o r u1 and then t h e e q u a t i o n f o r G we can s o l v e t h e equation f o r u2'*,

1

. Finally

by u s i n g o n l y t h e successive approximations.

The r e q u i r e d p r o p e r t i e s o f these s o l u t i o n s a r e proved by t h e s i m i l a r method as i n t h e above and by u s i n g Lemma 2.8.

References Local s o l u t i o n s t o t h e i n i t i a l and i n i t i a l boundary v a l u e problem f o r t h e Boltzmann e q u a t i o n w i t h an e x t e r n a l f o r c e I , I I , (p r e p r i n t )

C11 Asano, K.:

.

C21

Caflisch,

R.:

The f l u i d dynamic l i m i t o f t h e n o n l i n e a r Bolttmann e q u a t i o n . Comm. Pure Appl. Math. 651-666 (1980).

s,


19

131 Carleman, T.: "Probleme Mathematiques dans l a Theorie Cinetique des Gaz" Almqvist-Wiksel I s , Uppsala (1957). C41

E l l i s , R and Pinsky, M.: The f i r s t and second f l u i d approximation t o t h e l i n e a r i z e d Boltzmann equation, J . Math. Pures Appl. 3, 125-1 56 (1 975).

15:

Grad, H.:

C 61

Asymptotic theory o f t h e Boltzmann equation, Rarefied Gas Dynamics I , 25-59 (1963). Asymptotic equivalence o f the Navier-Stokes and nonlinear Boltzmann equation, Proc. Symp. Appl. Math., Amer. Math. Sot., 154-183 (1965).

n,

C71 Kaniel, S. and Shinbrot, M.:

The Boltzmann equation, Corn. Math. Phys., 58, 65-84 (1978).

181 Nirenberg, L.: An a b s t r a c t form o f t h e n o n l i n e a r Cauchy-Kowalewski theorem, J . D i f f . Geometry., 6, 561-576 (1972). [91

Nishida, T.:

1101

A note on a theorem o f Nirenberg. J . D i f f . Geometry, 629-633 (1 977).

12,

F l u i d dynamical l i m i t o f the nonlinear Boltzmann Equation t o the l e v e l o f the compressible Euler equation. C o n . Math. Phys., 61,119-148 (1978).

C l l l Nishida, T. and Imai, K.: Global s o l u t i o n s t o the i n i t i a l value problem f o r t h e n o n l i n e a r Boltzmann equaiton, Publ. Res. I n s t . Math. Sci., Kyoto Univ., 12, 229-239 (1976). On the f l u i d dynamical C121 Kawashima, S., Matsumura, A. and Nishida, T.: approximation t o t h e Boltzmann equation a t the l e v e l o f t h e Navier-Stokes equation, Commun. Math. Phys., 70, 97-124 (1979). C13l Ukai, S.:

On t h e existence o f global s o l u t i o n s o f mixed problem f o r t h e nonlinear Bol tzmann equation, Proc. Acad. Japan, 50, 179-188 (1974).

C 141

Les s o l u t i o n s globales de 1 'equation n o n l i n e a i r e de Boltzmann dans l'espace t o u t e n t i e r e t dans l e demi-espace, Compte Rendu Acad. Sci. Paris, 3,317-320 (1976).

151 Ukai, S. and Asano, k . : S t a t i o n a r y s o l u t i o n s o f t h e Boltzmann equation f o r a gas f l o w p a s t an obstacle, I Existence ( t o appear i n Arch. Rat. Mech. Anal.), I1 S t a b i l i t y ( p r e p r i n t ) . [

18

The Euler L i m i t and i n i t i a l l a y e r o f the nonl i n e a r Boltzmann equation, Hokkaido Math. J . , 12, 303-324 (1 983).


L e c t u r e Notes in Num. Appl. Anal., 6 , 21-38 (1983) Recent Topics in Nonlinear PDE, Hi?mhinza, 1983

Neumann Layer Phenomena in Nonlinear Diffusion Systems

Hiroshi FUJI1 and Yuzo HOSONO Department of Computer Sciences, Kyoto Sangyo University Kyoto 608, Japan

1.

Introduction T h i s paper concerns t h e c o n s t r u c t i o n o f a new c l a s s o f s t a t i o n a r y

s o l u t i o n s t o a couple o f n o n l i n e a r r e a c t i o n - d i f f u s i o n equations :

29 n

t

f(u,v)

0,

=

dx

O < X < l ,

w i t h t h e no f l u x boundary c o n d i t i o n s :

x = 0 where t h e n o n l i n e a r i t i e s .tajr

f

and

and

1,

a r e assumed t o be o f ncLiuatoh=ivikibi-

g

t y p e , which appears t y p i c a l l y i n mathematical b i o l o g y . Roughly speak-

i n g , we assume t h a t t h e zero l e v e l c u r v e o f

f

i s sigmoidal throughout

t h i s paper. By n new d a s h we mean here

solutions

(u(x;c),v(x;~)),

(E

E-families o f large amplitude layer-type

> 0, where u > 0 i s kept f i x e d ) ,

c h a r a c t e r i z e d by t h e f a c t t h a t i n t h e l i m i t

E

which a r e

4 0, U(X;E) becomes a

continuous f u n c t i o n which have b o t h

hutcivfA7~7hyandlo&

i n t e h i o f i &uzv~s.iLiond i n c o n t i m i t i e s

-

intdufi

hLi&

disand

t h e d i s c o n t i n u i t i e s o f t h e former 21

Hiroshi FUdIJ and Yuzo HOSONO

22

t y p e we c a l l here Neumann b when

E

> 0.

following.

U

(N-sl i t s ) , and Neumann Layehn ( N - l a y e r s )

We s h a l l r e f e r t o such s o l u t i o n s as N-hot~Lioion6 i n

the

The e x i s t e n c e o f such N - s o l u t i o n s has been announced by t h e

authors a t t h u U.S.-Japm Seminah on N o d i n m Pahtiae Uiddehentiae €quation4 [ 7

1.

I t i s noted here t h a t l a y e r - t y p e s o l u t i o n s which possess o d y

i n t e r i o r t r a n s i t i o n s have been c o n s t r u c t e d f o r t h e same system by Mimura,

1.

Tabata and Hosono i n [ 8

The s i g n i f i c a n c e o f t h i s c l a s s of N - s o l u t i o n s may l i e n o t o n l y i n t h e f a c t t h a t t h e y a r e new, b u t r a t h e r i t l i e s i n t h a t t h e y p l a y a key r o l e i n understanding

t h e g l o b a l b i f u r c a t i o n s t r u c t u r e o f t h e system (1 . l ) i n t h e

parameter space e.,

E

.L 0 )

(E,u)

E

R,.2

Roughly speaking, t h e y r e p r e s e n t s g h b d (i.

dentir.iation4 o f secondary b i f u r c a t e d branches, b i f u r c a t e d from

p r i m a r y branches o f s o l u t i o n s w i t h c e r t a i n s p a t i a l group symmetry.

The

l a t t e r ones have been born as p r i m a r y b i f u r c a t e d branches from t h e t r i v i a l

( = constant s t a t e ) solutions. t h e phenomenon

06

Thus, t h e N - s o l u t i o n s a r e r e s p o n s i b l e t o

a e c o v u ~ y06 b h b & L t y

o f primary branches.

do n o t discuss such p o i n t s here, and would l i k e t o ask r e f e r t o our paper

C71.

However, we

t h e reader t o

[ 51, [ 61.

See, also,

We s h a l l i n s t e a d d i s c u s s about how N-layers a r e c h a r a c t e r i z e d .

As

mentioned above, Mimura e t a1 [ 8 1 have shown t h e e x i s t e n c e o f c - f a m i l i e s o f s i n g u l a r l y p e r t u r b e d s o l u t i o n s which e x h i b i t i n t e r i o r t r a n s i t i o n l a y e r s . T h e i r s o l u t i o n s , which we r e f e r t o as M-boLutio~d, have jump d i ~ c o n t i n u U e i n the l i m i t

E

c 0, as i n F i g . l . 1 .

(Note:

t i o n s o f p r i m a r y b i f u r c a t e d branches.

M-solutions a r e g l o b a l d e s t i n a -

See, [ 9

1,

[lo].)

On t h e o t h e r

hand, N - s o l u t i o n s , o f which we have proposed t h e e x i s t e n c e i n [ 71, have, i n a d d i t i o n t o i n t e r i o r jumps, N - 4 L i L l h ) a t one o r b o t h o f t h e boundaries and/or a t t h e point

06

t h e symm&g.

The depth3 o f these s l i t s a r e d e t e r -

mined by t h e o t h e r d i f f u s i o n c o e f f i c i e n t a-1.

See, Fig.1.2.

23

Nonlinear Diffuciim Sv-stems

E = o

E > O

E > O

E = O

Note : All profiles in the present paper correspond t o the May-Mimura model, i.e., Eqs.(l.l), (1.7).

m.-m E = O

E

>o

24

Hiroshi FLI.111 and Yuzo HOSONO

Fig.l.2

The f o l l o w i n g arguments may j u s t i f y why we c a l l them Nmrcnn L a y m . F i r s t l y , we s h o u l d n o t e t h a t f o r D i r i c h l e t b o u n d a r y - v a l u e problems,

the

appearance o f b o u v i h y k y m i s w e l l - k n o w n f o r s m a l l enough

The

0.

E

e s s e n t i a l r e a s o n o f t h i s L q e h phenomenon i s t h a t boundary c o n d i t i o n s a r e o f D i r i c h l e t type.

See, e.g.,

as Du~,hiceet l a y m .

such

[

11.

Thus, i n t h i s c o n t e x t , we may c a l l

On t h e c o n t r a r y , as w i l l become c l e a r f r o m o u r

c o n s t r u c t i o n , t h e l a y e r s w h i c h we c o n s i d e r h e r e appear e i t h e r a t Neumann b o u n d a r i e s o r a t p o i n t s o f g r o u p symmetry o f s p a t i a l p a t t e r n s o f s o l u t i o n s . T h i s means t h a t t h e appearance o f N - l a y e r s depends e s s e n t i a l l y on "boundary" c o n d i t i o n s o f Neumann t y p e . However, i t i s w o r t h n o t i n g t h a t t h e N - l a y e r s do appear n o t o n l y i n Neumann b o u n d a r y - v a l u e problems, b u t even i n D i r i c h l e t problems

-

a t the

m i d p o i n t o f t h e i n t e r v a l , s i n c e t h e y can appear a t iL+'ii!iig p o ~ t ~ Lu4 i 5 p -

rnuky.

:

B e f o r e p r o c e e d i n g , we need t o s t a t e o u r h d l u n p - t c o ~ n on t h e system (A.l)

The z e r o l e v e l c u r v e o f

f(u,v) = O

i s S-shaped, and

t h e u p p e r r e g i o n o f t h e sigrnoidal c u r v e ( F i g . l . 3 ) real roots

u-(v)

5

uo(v)

5

u,(v),

r e s p e c t t o u, i t has t h r e e branches

for v

E

; f = O

f

0

in

has t h r e e

A. When i t i s s o l v e d w i t h

h - ( v ) 5 h,(v)

5 h,(v).

Nonlinear Diffusion Systems

G, ( v 1

(1.2) Then,

dG+ ( v

(1.3) dV

We d e f i n e :

1

=

g (h,(v),v < 0,

for

1 any

E

v

C'(h).

E

A+.

25

Hiroshi FUJI1 and Yuzo HOSONO

26

There a r e a number o f examples w i t h i n t h e s e t t i n g (A.1)-(A.3).

[6

3.

The

May-Uimwra model

See,

f o r d i f f u s i v e prey-predator system p r o v i d e s

an example, i n which

where

2 f o ( u ) = (35+16u-u ) / 9 ,

and

g o ( v ) = 1+(2/5)v.

Now, b e f o r e t h e d i s c u s s i o n o f N-solutions, i t seems convenient t o r e c a l l t h e c o n s t r u c t i o n o f M-solutions which e x h i b i t i n t e r i o r t r a n s i t i o n l a y e r s [ 1 3 , [ 81. The key concept i s “reduced s o l u t i o n s ” , d e f i n e d as s o l u t i o n s o f (1.1) with

E

= 0 , and which a r e candidates f o r M-solutions w i t h

suppose we f i x 17

h

E

;

arbitrarily.

u=h(v;q) satisfies

f(u,v)

=

= 0, f o r

dV dx

I

VEA

{

-

E

> 0. I n f a c t ,

Then, h-(v)

,

v

n.

(ri:.

So, i f

0,

x = 0, 1,

where G(v;n)

=

g(h(v;n),v),

I

has a s o l u t i o n

V‘(x;n),

(assumed t o be monotone decreasing, f o r d e f i n i t e ‘a

-u

ness), then, t h e p a i r (U ,V

) , where

U‘(x;q)

o - f a m i l y o f reduced s o l u t i o n s f o r each

c A.

= h(?(x;l,),q),

-

Obviously, Ua

gives

a

has a jump

d i s c o n t i n u i t y by c o n s t r u c t i o n . Now, t h e fundamental q u e s t i o n i s iuhe2theh t h e dincontinuotln d a U o v t J -0

-o

(U ,V )

can be C.X&nded .to a tayeh xype doeLLti0ylb huh

i s p o s i t i v e i f t h e VaoZ’evc,-Fi,je-M.imwla

E

> 0.

et d. c o n d i t i o n

The answer

21


(see, [ 8 1 ) .

i s satisfied

family o f M-solutions such t h a t as E

+

I n o t h e r words, i f

(u'(x;E),

0 t h e p a i r (u',

v'(x;E)), v')

n

= v:

, we have an

E

-

> 0 ) , f o r each small o > 0,

(E

.~

converges t o (Ua,

V')

i n an a p p r o p r i -

a t e sense.

-

I n Fig.1.3,

V'(x;v:))

t

we p l o t w i t h a boCddaced h a k d f i n e t h e s e t

2

R, ; 0 -~ 5 x 5 1 1 i n t h e (u,v)

E

2

R, plane.

{(U'(x;v~),

:Je may thus summa-

r i z e the above arguments as : t h e A!-oo.&LLovm i o d h intehioh .t/rarb&!%on h y m

WLL

c o v m ~ c t e di n ouch a my that in t h e

E

4

0, t h e y

"Ube"

t h e botd6aced ooe-id f i n e i n F i g . I . 3 . For a l a t e r use, we d e f i n e f o r each small

0

> 0, t h e q u a n t i t i e s :

and

x = t * ( o ) i s defined

I

by t h e r e l a t i o n

r,

( 1 . 1 ' ) v'(t*r-),v!) See, Fig.1.4

.

= 0

(left).

-

"The g m p h a6 Vo = V'(x;v:)" F i g . 1 .4

We propose now, whenever t h e M - s o l u t i o n s e x i s t , t o c o n s t r u c t an f a m i l y o f new l a y e r - t y p e s o l u t i o n s , which "use" i n t h e l i m i t

E

E-

4 0 one o r

b o t h o f t h e b o l d f a c e d broken l i n e s as w e l l as t h e b o l d f a c e d s o l i d l i n e i n Fig.l.3.

Let

(V'(x;v;),

U'(x;v,*))

denote t h e corresponding reduced

Hiroshi FUJI1 and YUZOHOSONO

28

s o l u t i o n s , where

Va z Va,

and

U'

has 6 U ( b ) a t e i t h e r o r both o f x =

0 and 1, as w e l l as the i n t e r i o r t r a n s i t i o n jump a t x = t * ( n ) .

See, F i g .

1.5.

F i g . 1.5 We emphasize t h a t t h e two f u n c t i o n s values a t a l l

x

i.e.,

and/or

at

x=O

i n the i n t e r v a l

T

U"

= [0,1],

and

-

-

the sdme

except a t one o r two p o i n t ( s )

1,

t h e d e p t h o f N - s l i t s a r e determined by t h e

genttalized V a ~ ~ ' e v a - F i 6 e - M h w rel. a o l . colzdLtion

k, = k+( q )

LdKe

x=l.

As i s suggested i n [ 7

where

U"

a r e f u n c t i o n s o f rl c A,,

:

determined by

See, F i g . l . 3 . We s h a l l show i n t h e n e x t s e c t i o n t h a t such s o l u t i o n s a c t u a l l y e x i s t , and can be c o n s t r u c t e d u s i n g t h e s i n g u l a r p e r t u r b a t i o n technique.

In

the

l a s t s e c t i o n , we show our r e s u l t s o f numerical computations o f those l a y e r typed s o l u t i o n s .

2. 2.1.

Construction o f solutions Strategy L e t us b e g i n our c o n s t r u c t i o n .

small

0

>

Since we f i x

0 E

(O,G), f o r some

0 i n t h e f o l l o w i n g , we o m i t t h e a-dependency from t h e symbols


we s h a l l use, whenever no c o n f u s i o n a r i s e s .

I

= (0,l)

i n t o three subintervals

0 < s < t < 1.

with

Here, x = s

I-

=

and

F i r s t , we s p l i t t h e i n t e r v a l

I. = ( s , t ) and I+ = ( t , l ) ,

(O,s), x = t

29

prescribe the locations o f

a Neumann l a y e r and an i n t e r i o r t r a n s i t i o n l a y e r , r e s p e c t i v e l y .

s

and

t

w i l l be determined as f u n c t i o n s o f

l i m S(E) = 0

and

€SO

E

> 0

O f course,

satisfying

l i m t ( & ) = t*. CO

&

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

x = t

can be p e r -

fornied e x a c t l y as i n Mimura e t a1 [ 8 1 , t h e e s s e n t i a l p o i n t i n o u r arguHence, we f i x f o r a moment t h e values

ment i s t h a t o f an N - l a y e r a t x = s. of

(u,t)

i n some neighborhood o f

the i n t e r v a l

I- U I*.

I.

< ho(u)

and

and c o n s i d e r t h e problem i n

We o m i t a l s o t h e (\J,t)-dependency from t h e symbols

u t i t i l i t becomes necessary. h-(v)
0.

F E

C

0,

~(€1.

Construction o f solutions f o r ( P I - ) L e t us c o n s i d e r t h e problem ( P I - ) and i n t r o d u c e t h e new independent

variable

I,

E, = x / s

and s e t

(6,u-,v-)

6 =

t o s t r e t c h the i n t e r v a l

I-

onto t h e f i x e d i n t e r v a l

Then, o u r problem becomes t o f i n d t h e t r i p l e t s

E/S.

satisfying

6

2 d2 u t f ( u , v ) 2

= 0,

dx

6

2 dL 2 v dx

t E

2

ocj(u,v) = 0,

d d dE u ( 0 ) = dE v ( 0 ) = 0,

Setting

E

= 0

in

(2.11, we have

reduced t o t h e s c a l a r problem:

v(E) 5

u , and ( 2 . 1 )

z

(2.3)

is

32

Hiioshi FIJ,JII and Yuzo HOSONO

u ( 0 ) = a. BY t h e phase p l a n e analysys, we can prove t h a t f o r each f i x e d and

a

E

(h-(u),ho(u)), of

U-(f,;a,p)

(2.4)

t h e r e e x i s t s a unique monotone i n c r e a s i n g s o l u t i o n only f o r

We look f o r a s o l u t i o n become

(U-,u)

and

Theorem 1. (h-(u),ho(ll)) t-

Let x

A+.

and

be a neighborhood o f

For each

such t h a t f o r any

the solution

6*(a,p).

6

whose f i r s t approximations

respectively.

6*

N*

6=

(u-, v - )

and a p o s i t i v e f u n c t i o n

(O,E-)

(vt,v:)

p E

(a,p) 6(E;a,p)

c (O,E-)

E

E

(a:,vT)

such t h a t

@

E

N 3 , t h e r e e x i s t a p o s i t i v e constant ( =

E/s(E;~,~)

t h e problem

( u - ( ~ , ~ ; a , p ) , v - ( f , , ~ ; a , l ~ ) ) and

6 =

) , defined i n

(2.1)

2.

~(E;~,LI),

(2.3)

has

satisfying

that

and

l i m ~ ( c ; ~ , L= I6)* ( a , p ) ,

u n i f o r m l y i n a and

p.

t $0

Futhermore, i t holds t h a t

uniformly i n 2.3.

and

iy

p.

Construction o f solutions f o r (PIo) The D i r i c h l e t problem

11, ours.

(PIo)

was a l r e a d y i n v e s t i g a t e d by P.C.

Fife

b u t t h e s i t u a t i o n i n t h e reduced problem i s a l i t t l e d i f f e r e n t from Hence, we f i r s t examine t h e reduced problem: d2 7 V + dx

G(V;V)

= 0,

s < x < t,


33

(2.7) V(s) =

u,

V ( t ) = v,

which i s o b t a i n e d by s e t t i n g Lemma 2. hood

= 0

E

i n (PIo)

Assume ( A . l ) and ( A . 2 ) .

N:

of

(v:,O)

has a unique s o l u t i o n

Then, t h e r e e x i s t s a small neighbor-

such t h a t f o r any Vo(x;u,s),

N:,

E

t h e problem (2.7)

satisfying

Uo

Since t h e reduced s o l u t i o n

s)

(11,

does n o t s a t i s f y t h e

= h+(Vo)

D i r i c h l e t boundary c o n d i t i o n , we i n t r o d u c e t h e boundary l a y e r c o r r e c t i o n s at

x = s

and

x = t.

Let

be t h e unique s t r i c t l y monotone

z(c;p)

solution of

2 z + f(z+h,(u),p) ~ ( 0 =) h o ( u )

-

o
0, t h e r e e x i s t s a unique s o l u t i o n

satisfying

1) We r e f e r readers t o [5] f o r n o t a t i o n s .

171

Water Waves and Friedrichs Expansion

(2.4)

llv(t)-v-,u(t)-u-llp

uniformly w i t h respect t o

It1

for

< R,

43

< a(pO-p), p < po

6 E [0,1], p r o v i d e d

(v,u)(O) E X

with PO

Here we s h a l l show t h a t i f then the s o l u t i o n w i t h respect t o

(v,u)(t,S;6) 6 E [0,1]

(v,u)(O) E X

i s independent o f 6, PO i s i n f i n i t e l y many times d i f f e r e n t i a b l e

w i t h values i n

u

S =

To do t h a t ,

Xp.

O'P d i f f e r e n t i a t i n g (2.1) m-times w i t h r e s p e c t t o for

(v,,u,)(t)

where

and M a r e b i l i n e a r o p e r a t o r s on

L

L = F ~ y u ( v m , u m ) ,M = G;

YU

(vm,um)

and t h e inhomogeneous terms (v,u) = (vo,uo),(vkyuk), F,

and

II = 0,1,2,...;

E+k

respect t o

Fm and

aeA6/a6',

a(;'

= 1,2,3,*..:

v,

and

,u,

i. e.,

a r e Fr6chet d e r i v a t i v e s o f G,,

k = 1,2,**.,m-1,

F and

G,

contain and t h e i r d e r i v a t i v e s w i t h

A6)/aS",

agC6/a6',

5 m u s i n g L e i b n i z ' formula.

I n o r d e r t o s o l v e (2.5) theorem.

m

= (amv/a6m,amu/a6m)(t,5;6),

6, we would have t h e system

-

(2.6) we use t h e a b s t r a c t Cauchy-Kowalevski

By v i r t u e o f t h e p r o p e r t i e s o f o p e r a t o r s ( 2 . 3 ) analyzed i n [ 5 ]

and by t h e u n i f o r m estimates o f (2.4), we see t h a t t h e l i n e a r o p e r a t o r s L

and M s a t i s f y t h e f o l l o w i n g e s t i m a t e :

Lemma 2 . 1 .

For any

P

P I < Po

and f o r any

,.-

v, u E X

P'

we have

Tadayoshi KANO and Takaqki NISHIDX

44

(ti < a(Po-P'),

for

of

6

E

where

[O,ll.

Concerning t h e inhomogeneous terms

where C

i s a p o s i t i v e constant independent

C = CtRl

=

Since

C(R,ml

Fm and

G,

aLC6/a6',

v, u E Xp,

we have

a r e E - d e r i v a t i v e s (except f o r terms o f vk, uk. k = 0,1,2,...,m-l,

.9 = 0 , 1 , 2 , . - - , r n ;

consequence o f t h e f o l l o w i n g estimates: For any

G,

i s a p o s i t i v e constant independent of' 6 E [ O , l l .

o f sums o f terms c o n s i s t i n g o f ak(+6)/XL,

Fm and

1 $i6u)

akA6/aGL,

ktk = m, Lemma 2.2 i s an easy

45

Water Waves and Friedrichs Expansion

C, C,,

where

of

a r e constants independent o f

2 = 1,2;.-,

.

6 E [O,l]

and

P ' < P.

If we apply an a b s t r a c t l i n e a r Cauchy-Kowalevski theorem ( c f . [5] appendix) t o (2.5) Theorem 2.3.

(2.6) using Lemnas 2.1 ( v , u ) ( t ,*;6)

The s o l u t i o n

-

2.2,

we have

o f (2.1) (2.2) satisfying (2. 4)

i s i n f i n i t e l y many times d i f f e r e n t f a b l e w i t h r e s p e c t t o values i n

Xp,

I:I

a unique s o l u t i o n

< alP -P),

0

i . e . , the Cauchy problem

(vm,umi(t,~;6) i n

xP

for

It1

6 E LO, 1 I 12.5)

-

with

(2.6) has

a(Po-P), P
0 be t h e i n i t i a l d a t a and the a b s o l u t e Maxwellian s t a t e , r e s p e c t i v e l y . I t i s proved t h a t i f

Fo

-

M i s small i n Hs(Rn)

(S

-

?[n/2] + l ) , a global s o l u t i o n

o f (1.1 ) e x i s t s and tends t o

M ( i n t h e maximum norm) as t (Theorem i s small i n Hs(Rn) n Lp(Rn) ( s ? [ n / 2 ] + 1 ; p = l f o r n = l , p ~ [ 1 , 2 ) f o r n ? 2 ) , the s o l u t i o n converges t o M ( i n H s ( R n ) ) 5.2).

Furthermore i f

a t the r a t e

Fo

-

-f

M

t-Y ( w i t h y = n ( 1 / 2 p

- 1/4)

) as

t

+ m

(Theorem 5 . 3 ) .

The l a t -

It t e r r e s u l t i s analogous t o t h a t f o r t h e Boltzmann e q u a t i o n ( c f . [14]). should be n o t i c e d t h a t i n o u r r e s u l t s no assumptions a r e made on t h e s i z e m

o f t h e system o r t h e space dimension

n.

The p l a n o f t h i s paper i s as f o l l o w s .

I n s e c t i o n 2 we s h a l l r e v i e w t h e The formu-

b a s i c p r o p e r t i e s o f t h e system (1.1) which a r e developed i n [6].

l a t i o n o f t h e problem and t h e l o c a l e x i s t e n c e theorem a r e g i v e n i n s e c t i o n 3. I n s e c t i o n 4 we o b t a i n energy i n e q u a l i t i e s and decay estimates f o r l i n e a r i z e d equations a t an a b s o l u t e Maxwellian s t a t e .

These estimates a r e

used i n s e c t i o n 5 t o prove t h e g l o b a l e x i s t e n c e and asymptotic s t a b i l i t y o f solutions f o r (1.1). i n i t i a l data

S e c t i o n 6 contains some g l o b a l e x i s t e n c e r e s u l t s f o r Fo - F E Hs(Rn) w i t h > 0, n o t an a b s o l u t e

Fo s a t i s f y i n g

Maxwellian s t a t e .

As a p p l i c a t i o n s o f o u r r e s u l t s , we s h a l l deal w i t h t h e

one-dimensional Broadwell model and t h e two-dimensional 8 - v e l o c i t y model i n s e c t i o n s 7 and 8, r e s p e c t i v e l y . F i n a l l y we remark t h a t o u r c o n d i t i o n (11) i s n o t s a t i s f i e d f o r t h e plane r e g u l a r model w i t h 4 v e l o c i t i e s and t h e three-dimensional

Broadwell model.

T h i s may i m p l y t h a t t h e c o l l i s i o n mechanism f o r these models i s t o o s i m p l e t o guarantee t h e asymptotic s t a b i l i t y o f t h e Maxwellian s t a t e s . hopes t h a t the c o n d i t i o n

(II) w i l l

The a u t h o r

cover many p h y s i c a l l y reasonable models.

2. B A S I C PROPERTIES F o l l o w i n g [6] o r [4] we s h a l l i n t r o d u c e t h e b a s i c concepts concerning (1.1) and s u n a r i z e t h e i r p r o p e r t i e s which w i l l be used l a t e r . D e f i n i t i o n 2.1

A vector

$ =

t

($,,.--,$m)

E

IRm i s c a l l e d a s m a t i o n a Z in-

variant i f A:i(+i/ai

+

$j/aj

-

+k/ak

-

$,/a,)

= 0

for all

i ,j,k,a

= l,..-,m.

Discrete Velocity Models of the Boltzmann Equation

We denote by cause

t(al,

61

-

L e t Q(F

Let ( I ) be assumed and l e t t i o n s are equivalent. Lemma 2 . 1

IRm . The following three condi-

E

n.

(i) (ii)

=

(in) Here

F = t (Fl,---,Fm)

>

0

o

if

F. > 1

o

i=

for all

i s c a l l e d a Zocal Maxwellian

if AiJ(F.F. kn. i J

-

FkF,)

= 0

for a l l

i,j,k,n.

= l,...,m.

I n particular,

F > 0 i s c a l l e d an absoZute m m e l l i a n i f i t i s a l o c a l l y Maxw e l l i a n s t a t e and i s independent o f t and x. Lemma 2.2

Let ( I ) be assumed and l e t

F

=

t

(F1,*--,Fm)

>

0.

m e following

four conditions are equivalent. (i) F . .i s a locally M m e l l i a n s t a t e . (ii)

Aiilog(FiFj/FkFR)

= 0

t(allog F1 amlog Fm) (in) Q(F,F) = 0. (iv)

Proof.

1 ailogFiQi(F,F) See [6] o r [4].

E

for a l l

i,j,k,e

= l,..-,m,

that i s ,

m.

= 0.

I t i s easy t o see t h a t ( i )

The i d e n t i t y (2.1) i s a l s o used i n t h e p r o o f o f ( i v ) d e t a i 1s.

++ (ii) +

+

(iii)

=+

(iv).

(i).We o m i t t h e

62

Shuichi KAWASHTMA

A v e c t o r M > 0 i s c a l l e d t h e locally &zueZZian s t a t e asso0 i f M i s a l o c a l l y Maxwellian s t a t e and s a t i s f y M = F

D e f i n i t i o n 2.3 F

ciatedwith on

>

m. Let ( I ) be asswned and l e t

Lemna 2.3

F > 0 be a given vector. f i e n there F. (We denote

e x i s t s uniqueZy the ZocalZy Mamellian s t a t 2 associated with i t by M = M(F).I *

The p r o o f i s o m i t t e d .

See [6].

Next we c o n s i d e r t h e Bol tzmann H - f u n c t i o n : m

H

=

1

oriFilog

Fi

i=1 M u l t i p l y (1.1) b y ( w i t h $i = 1 + l o g Fi

(2.3)

and add f o r

( n l o g n ) " = l / n > 0. &(ll,s) =

nlog n

-

clog c

By use o f (2.1)

i = l,---,m.

and G = F) we have t h e e q u a l i t y f o r

n l o g n i s s t r i c t l y convex f o r

The f u n c t i o n l o g 0 and

ni(l + l o g F i )

n

>

0

H:

because

(qlog

,,)I

= 1 t

Therefore

-

(1 + l o g

r)(n

-

5)

,

0,

>

0

,

i s p o s i t i v e d e f i n i t e ( &(n,s) = O i f and o n l y i f n . 5 ) . Thus we a r r i v e a t t h e q u a d r a t i c f u n c t i o n associated w i t h t h e Bol tzmann H - f u n c t i o n :

Let ( I ) be asswned. Let M = t (M,,--*,Mm) > 0 be a constant vect o r and l e t ko > 1 be an arbitrary constant. I f F = t (F1,***,Fm) s a t i s fies k i ' s Fi/M. I ko , then Lemna 2.4

1

(2.4)

ClF

- MI 2

5

1 ai&(Fi,Mi) i

holds f o r some p o s i t i v e constants c Remark

s

CIF

-

and C

MI

2

(C

< C ) independent of F.

Compare t h i s q u a d r a t i c f u n c t i o n w i t h t h e ones used i n [12] and [ll].

If M

i s an a b s o l u t e Maxwellian s t a t e ,

(2.2) and (1.1) t o g e t h e r w i t h


(2.1) ( w i t h Oi = 1 + l o g M i

I 1

(2.5)

i

=

-

and G = F ) y i e l d t h e e q u a l i t y f o r

1V ~ - V ~ { ~ ~ & ( F ~ , M ~ ) }

It+

ai&(FisMi)

i

-

ifktii(FiFj

1 ai&(Fi,Mi):

.

FkF,)lOg(FiFj/FkF,)

T h i s e q u a l i t y w i l l be used i n s e c t i o n 5 t o d e r i v e a p r i o r i e s t i m a t e s f o r 2 n L ( R )-norm o f s o l u t i o n s .

3. FORMULATION OF THE PROBLEM AN0 LOCAL EXISTENCE Consider t h e i n i t i a l v a l u e problem f o r (1.1): n

(3.1)

Ft +

.

1 VJFx j=l

= Q(F,F)

,

t r o ,

X E

Rn,

j

,

(3.2)

F(0,x) = Fo(x)

where

m VJ = d i a g ( v j1, . - - , v j ) ,

X E

JJ?,

j = l,..-,n,

and

Q(F,G) = t(Ql(F,G),...

-.,Q,(F,G)). L e t M > 0 be an a b s o l u t e Maxwellian s t a t e . We s h a l l c o n s i d e r t h e case t h a t Fo - M E Hs(IRn) ( s 2 [n/2] + l ) . Here Hs( Rn) denotes t h e 2 n L ( W )-Sobolev space o f o r d e r s, w i t h t h e norm ll.]ls (we w r i t e 11.II i n stead of l l - l / o ) . P u t t i n g

(3.3)

A =

diag(Ml/al,-..,M

m/ am )

,

we s h a l l seek t h e s o l u t i o n i n t h e f o r m

(3.4)

F(t,x) = M + A’/2f(t,x).

Then t h e problem (3.1),(3.2)

i s transformed i n t o

where (3.7)1

L f = -2A-’/2Q(M,A’/2f)

,

63

Shuichi KAWASHIMA

64

L and

The operators

r

have the f o l l o w i n g p r o p e r t i e s .

Let (I) be assumed.

Lema 3.1

Then we have:

L i s r e a l symmetric and positive semi-definite; i t s null space i s given

(i) by n ( L ) = A’’2RZ,

( i i ) r i s bi-linear and s a t i s f i e s r ( f , g ) c ~ ( L I ’ f o r any where ?L(L)’ denotes the orthogonal compZement of Iz(L) i n Proof. uct

(cf.

[6])

L e t f, g

< f, Lg >

E

.Rm be a r b i t r a r y .

by using (2.1)

f, g

IR~,

6

d.

We c a l c u l a t e t h e i n n e r prod-

F = M and 6=A’/2g)

( w i t h $i

as

follows:

= 0 (i.e,,

where we have used Aii(MiMj-i/;tMe)

M i s an absolute Maxwellian

ii

s t a t e ) ; we s e t 7, = (aiMi) fi and = (aiMi)-1/2gi Since the expression (3.8) i s symmetric with respect t o <
= f, Lg > = Taking

= 0 (i-e.,

.

This and the property

.

Therefore

g = f

f

that is,

and

g, we have

f, h > = < h, f >

imply

i s proved t o be r e a l symmetric.

L

i n (3.8), we see

f, L f >

2

0.

Furthermore

n(L))holds i f and o n l y i f

E

ij

Akl(fi

O ) g i v e s t h e e s t i m a t e

(4.4)

x

a (with

(II) be assumed. Let n

Let ( I ) and

P r o p o s i t i o n 4.2 (decay e s t i m a t e )

+

2

1

and L t 0 be i n t e g e r s , and l e t p, q E [I ,2] and T > 0 be constants. Asswne t h a t h E C 0 (0,T;H'(Rn) n Lq(Rn) ) s a t i s f i e s (4.2). I f f ( 0 ) E H L ( l R n ) n f

Lp(LRn), then the solution satisfies

Ip.

]If(+-) 2

(4.6) for

t

E

llf(0)ll~,p +

-

cl

1/4) and

)

t 0

=

Let =

g

Ilfll,

+

IlfllLP

for

f

2

y' =

E

be t h e F o u r i e r t r a n s f o r m o f

(zn)-"'

/ e - i X . c g(x) dx

.

of (4.1)

(1 +t-.)'2Y'lIh(T)/le,qd?

n(1/2q

-

1/4), and

Here ue use the notation

Ilfll,,p

L e t us d e f i n e

C(1 + t ) - "

Co(O,T;HE(Rn)) n C 1 (0,T;H'-'(lRn)

where y = n(1/2p

[O,T],

is a constant.

Remark 4.2

2

E

H'(IR") g:

n LP(R")

.

C > 1

Discrete Velocity Models of the Boltzrnann Equation

67

where

Then (4.1) i s transformed t o t h e i n t e g r a l e q u a t i o n

t f ( t ) = e-tS f ( 0 ) +

(4.8)

e - ( t - T ) s h ( r ) dT

.

0

Therefore, t a k i n g

h = 0, we have by v i r t u e of (4.6)

(4.9

This decay e s t i m a t e was proved in [18] f o r more general systems. Proof of P r o p o s i t i o n 4.1

where

0 ( a w i l l be determined l a t e r ) . with a constant

Proof o f P r o p o s i t i o n 4.2

(1 +

1 ~ 1 +~ (4.14) ) lLfl

Since

xa

t^

2

C I P f l , we have

w i t h some c o n s t a n t

Ca

,

where

I t i s easy t o see t h a t t h e r e e x i s t s a c o n s t a n t

~ 1 ~ 51 E"2

(O,aO],

where

If(t,c)12 C

21fI2

holds f o r a l l

a. E

>

0 such t h a t i f a E a = min

R n . Now choose

Then i t f o l l o w s f r o m (4.15) t h a t

( a o , c/C}.

(4.16)

5

5

4e-t@(6)

li(o,c)~t

c

t

e-(t-T)@(')

l h ( ~ , c )I 2 dr ,

i s a constant and

The d e s i r e d e s t i m a t e (4.6) i s a consequence o f (4.16) and t h e r i n e q u a l i t y (see

[81, [ I 4 1 o r D 8 1 ) (4.17)

/(l

where

y = n(1/2p

For any (4.9).

f

g

1512)ee-t'(5)

in

1/4).

Ii(c)12dc

5

Cte-6t

We o m i t t h e d e t a i l s .

I[gll:

+ (1 +t)-"

( ( g l (2 L

,

This completes the p r o o f .

He(Rn) n Lp(Rn)), we have proved t h e decay e s t i m a t e

Here we s h a l l show t h a t i n some case t h e decay r a t e

t-Y i s improved

to t-(Y + 1/2)

P r o p o s i t i o n 4.3 (decay e s t i m a t e ) and k z 0 fan integer), and l e t Lp( R1) and

Let ( I ) and (II) be a s s m e d . Let n = 1 p E [1,2]. Assme that g E H'(R 1 ) n


69

Then the decay estimate (4.9) i s improved to (4.19) Proof.

IIe-tS gll,

C ( l + t ) - ( v + 1 / 2 ) l(gl(,,p

5

n = 1,

When

S(g) =

f a m i l y o f matrices.

-

1/4.

L + igV ( < = c l c I R ' and V = V ' ) i s a one-parameter

Therefore we can apply t o

of matrices (see [9]).

y = 1/2p

%

the p e r t u r b a t i o n theory

S( 0

where

R(F)

2

dTdx

C

5

)Ifoll2

..

- FkF,)log(FiFj/FkFe)

1 ALi(FiFj

R(F) =

(5.5)

I0/ R(F)(T,x)

0.

2

Since

A::

Fi/Mi

5

a r e non-nega

has t h e e s t i m a t e

C ] Q ( F , F ) ] ~ f o r any

ki’

5

S u b s t i t u t i o n o f (5.5)

i s a constant,

because Q ( Mtn’/‘f,M

F with

= n’/‘r-Lf

thl’*f)

ko

,

i n t o (5.4) y i e l d s (5.3)

t r(f,f)j.

Next, a p p l y i n g (4.4) ( w i t h a = l and h = r ( f , f ) we have

E

n(L)*) t o t h e s o l u t i o n

of (3.5),

f

1‘

-

lILf(T)

r(f,f)(T)l12dT}

5

c

) I f o ) )21

.

0 Moreover a p p l y (4.3) ( w i t h

e = O and h =D,r(f,f))

Combine (5.3),(5.6) and (5.7) so as t o make a =

1/2C.

Then we o b t a i n f o r

5

where

Nr(T)

Crl]fol]f 5

6o

(5.3) t (5.6) x a

t

(5.7) w i t h

s = 1,

t

+

I0 I I D x r ( f , f ) ( T ) I (25 _ 1 d ~ l

for

t

E

[O,Tl

i s assumed. s 2 2, a p p l y i n g (4.5) ( w i t h e = s - 1 and h = D x r ( f , f ) )

I n t h e case the derivative

t o the d e r i v a t i v e D x f :

D,f,

Combine (5.8)(s = 1 ) and ( 5 . 9 ) t o conclude t h a t (5.8) is a l s o v a l i d f o r h

2.

Since

to

we o b t a i n

//Dxr(f,f)/)s-,

5

s

C ~ ~ f ~ ~ s ~ ~ D ,x the f ~ ~d es s-i r,e d e s t i m a t e (5.1)

Discrete Velocity Models of the Roltzmniin Equation

f o l l o w s from (5.8), p r o v i d e d t h a t

Ns(T)

5

61

f o r some

61

71 This

(0,60].

E

completes t h e p r o o f o f P r o p o s i t i o n 5.1.

If n

Remark

2

2, we can s i m p l i f y t h e above p r o o f as f o l l o w s .

(4.5) ( w i t h i l = s and h = r ( f , f ) )

t o t h e s o l u t i o n o f (3.5),

On t h e o t h e r hand t h e N i r e n b e r g ' s i n e q u a l i t y (see [13])

Therefore t h e d e s i r e d e s t i m a t e f o l l o w s f r o m (5.10) i f small.

Applying

we have

gives

i s suitably

Ns(T

Combinig Theorem 3.2 and P r o p o s i t i o n 5.1, we can prove the e x i s t e n c e o f g l o b a l s o l u t i o n t o (3.5),(3.6). L e t ( I ) and (11) be asswned.

Theorem 5.2 ( g l o b a l e x i s t e n c e ) s

2

[n/2.]

+

1 be inte ge rs.

Let

n

2

1 and

fo E Hs(IRn).

Suppose that t h e i n i t i a l data

Then

there e x i s t s a positive constant 62 f < 6 1 1 such that if l\folls5 6 2 , then the i n i t i a l value problem (3.5),(3.6) has a lmique global solution f E C 0 (0,m; Hs(Rn) ) n C 1 (O,m;Hs-'(Rn) ) s a t i s f y i n g (5.1) for t E [0,m). Furthermore the s ol ut i on decays t o zero (uniformly in x E IRn I a s t + m

.

Proof.

Choose

6 2 = S1/2C1

.

Then t h e s o l u t i o n o f (3.5),(3.6)

ued g l o b a l l y i n t i m e p r o v i d e d t h e c o n d i t i o n f a c t we have s t a n t To =

IlfolIs

t

Ns(TO) s 2 ~ 55 ~61

e s t i m a t e (5.1) f o r by t a k i n g

t = To

t

E

61

.

E

e s t i m a t e (5.1) 2c1 I / f O I l s (5.1) f o r

( f o r t E [O,T,])

.

5

[O,TO]

and s a t i s f i e s

be as i n P r o p o s i t i o n 5.1.

Ns(T)

P r o p o s i t i o n 5.1 g i v e s t h e

62,

Noting Ilf(T0)l(,

as t h e new i n i t i a l time.

2562 = t E [0,2T0]. 5

Let

by t h e d e f i n i t i o n o f

2T01 w i t h t h e e s t i m a t e I l f ( t ) l l s

In

Therefore, by Theorem 3.2, t h e r e i s a con-

[O,TO]).

[O,To].

can be c o n t i n -

i s satisfied.

62

4

such t h a t a s o l u t i o n e x i s t s on

Ilf(t)lls 2 2 Ilfo(ls ( f o r Since

s

5 62

0

>

I(folIs

5 A1

,

we a p p l y Theorem 3.2

Then we have a s o l u t i o n on

2 Ilf(To)l(s ( f o r

and t h e d e f i n i t i o n o f

t c

62

[T0,2T01).

,

we have

[To,

By t h e Ns(2TO)

5

Therefore P r o p o s i t i o n 5.1 a g a i n g i v e s t h e e s t i m a t e I n t h e same way we can extend t h e s o l u t i o n t o t h e

Shuichi KAWASHIMA

72

interval [O,nTo] successively n = 1 , 2 , . . - , and get a global solution. Finally we prove the asymptotic behavior of the solution. Set @,(t)= k 2 IIDxf(t)II (1 s k s s ) . Then i t follows from ( 5 . 1 ) and (3.5) t h a t m

0

I @ k ( t ) l dt

’ 0 l a t @ k ( t ) l dt ‘

IlfOll: k

with some constant C . From t h i s we can deduce t h a t @ k ( t )= I I D , f ( t ) l 1 2 as t + - . This and the Nirenberg’s inequality (see [13])

llgllL,

6

+

o

+ 1 and a = n/2r C ~ ~ g ~ ~ l - a ~ ~with D ~ gr ~= ~[n/2] a

give the decay law stated i n Theorem 5.2.

This completes the proof.

Finally we shall show the asymptotic decay of solutions f o r i n i t i a l data fo

E

H’(R”) n L P ( R ~ ).

Theorem 5 . 3 (asymptotic decay) Let ( I ) and (II) be assumed. Let n 2 1 and s t [n/2] + 1 be inte ge rs, and l e t p = 1 for n = 1 and p E [I ,2) f o r n t 2 . Suppose t h a t f o E H’(IR”) n L P ( R ” ) . Then there eccists a p o s i t i v e constant 63 ( ~ 6 such ~ ) t h a t if IlfolIs,p : l ] f o l l s + IIfoIILP 5 63, the solution of Theorem 5.2 s a t i s f i e s

where

y =

n(l/2p

-

1 / 4 ) , and

Proof. Let n 2 2 and p k = s and q = l ) gives

Set

E

C > 1 i s a constant.

[1,2).

IIlf(t)I[ls,y = sup (1 +,)’ 0s.rst

2 Since ~ ~ ~ ( f , f ) 5~ C~ Isl f l,I sl

~ ~ f ( ~ Noting ) ~ ~ the s . inequality

t

( l + t ) 2 y / (1+t-,)-n’2(1+,)-4vd~ 0 we can deduce from (5.13) t h a t

6

C,

, (4.6) (with

Discrete Velocity Models of the Boltxmann Equation

73

The d e s i r e d e s t i m a t e (5.12) i s an immediate consequence o f (5.14). n 2 2

proof f o r

Thus t h e

i s completed.

I n t h e case n = 1, we apply t o ( 4 . 8 ) ( w i t h h = r ( f , f ) ) t h e e s t i m a t e (4.9) and (4.19) ( w i t h

n. = s and p = 1 ) t o o b t a i n

Therefore, by t h e same arguments we can prove t h e a s s e r t i o n o f Theorem 5.3 n = 1.

also f o r

The d e t a i l s a r e o m i t t e d .

T h i s completes t h e p r o o f .

6 . SOME FURTHER REMARKS Let E

be a c o n s t a n t v e c t o r which may be o t h e r t h a n a b s o l u t e

> 0

Maxwellians. Hs(Rn).

(6.2)

Fo

We c o n s i d e r (3.1),(3.2) f o r t h e i n i t i a l d a t a F i r s t o f a l l we s t u d y t h e a u x i l i a r y problem:

G(0) =

with

Fo

- F

T.

Let ( I ) be assumed and l e t M = M(F) > 0 be the M m e Z l i a n s t a t e associated with a given vector F > 0 ( s e e Lema 2.31. Then there e x i s t s a Lemma 6.1

al

p o s i t i v e constant (6.1),(6.2)

IF - MI

such t h a t i f

has a unique global s o h i o n

(6.3)

- MI

IG(t)

5

Ce-"IT

-

MI

G

(3.7)2,

For

M

= M(F), l e t us d e f i n e

respectively.

Set

t

for

u

where C = C(a,) > 1 i s a constant, and t i v e eigenvalues of L. Proof.

E

A, L

G(t) = M

+

problem fC l([O,=)) ' the i ns ai tt ii safly ivalue ng

>

[0,-),

E

0 i s the m i n i m of the posi-

and

/1'/2g(t).

r

by (3.3),

( 3 . 7 ) 1 and

Then t h e problem (6.1),

(6.2) i s transformed i n t o t h e i n t e g r a l e q u a t i o n

where

g(0) =

A-1/2(F- M).

n i t i o n o f M=M(T) ) i m p l i e s (6.4) t h a t

g(t)

E

n(L)'

n(L) =

Since g(0) for all

E

n(L)'. t.

A1'2@l,

F- M

E

a '

(cf. the defi-

Therefore we can deduce f r o m

Hence (6.4) has t h e e s t i m a t e

Shuichi KAWASHIMA

74

This i n e q u a l i t y gives the a p r i o r i estimate 1 g ( t ) l .s Ce-pt I g ( 0 ) l f o r s u i t ably small Ig(O)l, from which we can conclude the existence o f a global solution. Thus the proof i s completed. Now we s h a l l seek the s o l u t i o n o f (3.1),(3,2) i n the form

,

F(t,x) = G(t) + A’/2f(t,x)

(6.5)

where G(t) i s the s o l u t i o n o f (6.1),(6.2) problem ( 3 . 1 ) , ( 3 . 2 ) i s reduced t o

n

given i n L m a 6.1.

.

1 VJfx

+ Lf = A(t)f

(6.6)

ft +

(6.7)

f(0,x) = f o ( x )

j=1

t

r(f,f)

,

j 5

n-’/*(F0(x)

- T)

.

Here A , L and r are given, respectively, by (3.3), M = M(F), and A ( t ) i s defined by

Compare (6.5)-(6.7)

w i t h (3.4)-(3.6).

n(L)’

(6.9)1

A(t)f

(6.9)2

IIA(t)flls

E

5

Then the

for

Ce-ut

(3.7)1 and (3.7)2 w i t h

Note t h a t [OP)

and

f E

lRma

I$ - MIIlfll,

for

t

[O,-)

t

E

E

and f

E

Hs(lRn).

By the estimate (6.9)2 the i n i t i a l value problem f o r (6.6) can be solved

l o c a l l y i n time as follows: Theorem 6.2 ( l o c a l existence) L e t ( I ) be assumed. Let n 2 1 and s [n/2] + 1 be integers. We prescribe the i n i t i a l data a t t = T 2 0 : (6.10) If

f(T,x)

fT c H’(IR”),

on llfTlls and Zem (6.6),(6.10) H~”(IR“)

= fT(X)

,

-X E

2

IRn.

then there e&sts a p o s i t i v e constant T ~ , depending only Mi findependent of Ti, such that the i n i t i a t value probhas a unique 8olution f E Co(T,TfT1;HS(lRn) ) n C 1(T,T +TI;

-

satisfying

76


Next we prove a p r i o r i estimates o f solutions f o r (6.6),(6.7). Let ( I ) and

Proposition 6.3 (a p r i o r i estimate)

(n)

be assumed.

Let

n

2

1

and s 2 [n/Z] + 1 be integers and l e t T be a positive constant. Suppose that fo E H ~ ( I R ~ )and , that f E c 0 (o,T;H’(IR~) n c 1( O , T ; H ~ - ~ ( R ” ) i s a Then we have: s o h t i o n of (6.6),(6.7). (i.) In the case n 2 2 there e x i s t positive aonatants a2 ( 5 al ), 64 and C2 = C2(a2,ci4) > 1 such t F a t i f IF MI s a2 and NS(T) SUP I l f ( t ) l l S 5 OstsT ?i4, then

-

s 1 1 1 ( i i ) In the case n = 1 we assume that fo E H ( R ) n L ( R 1. Then there e x i s t positive constants a3 ( 1 such that i f 65’ then IF MI s a3 and IlfOlls,l l l f O l l s + llfOllLl

-

(6.13)

I I f ( t ) lls

5

C3(1 + t)-1/4 11 fo 11 s ,l

for

t

E

[O,T]

.

Proof. Applying (4.5) ( w i t h L = S and h = A ( t ) f + r ( f , f ) ) t o the s o l u t i o n o f (6.6), we obtain

where we have used (6.9)2 and (5.11). The desired estimate (6.12) follows e a s i l y from (6.14). Next apply t o (4.8) ( w i t h h = A ( t ) f + r ( f , f ) ) the estimate (4.9) ( w i t h L = S and p = l ) and (4.19) ( w i t h L = S ; p = 2 f o r g = A ( t ) f , p = l f o r g = r ( f , f ) ).

Then we have

From t h i s i n e q u a l i t y we can deduce (6.13) i n the same way as i n the proof o f Theorem 5.3.

This completes the proof o f Proposition 6.3.

Shuichi KAWASHIMA

76

Combining Theorem 6.2 and Proposition 6.3, we have: Theorem 6.4 ( g l o b a l existence)

Let ( I ) and

s 2 [n/2] + 1 be integers. ( i ) In the case n 2 2 oe U88Mne that

(n) be aeswned.

fo c H

s

(IR n ).

Let

n z 1 and

Then there e x i s t s a

-

p o s i t i v e constant 66 ( 5 64) such that i f MI 5 a2 and llfOlls 5 66 , then the i n i t i a Z vaZue problem (6.6),(6.7) has a unique global solution f E Co(O,m;Hs(IRn) ) n C1(O,-;HS-’(IRn) ) s a t i s f y i n g (6.12) for t E [ O , m ) . Furt h e n o r e the soZution decays to zero (uniformly i n X E ]Rn) as t + 1 1 1 ( i i ) rn the case n = 1 we a s s m e that f o E H’(IR J n L (R 1. I f (?-MI

-.

i a3 and Ilf0lls,, 5 65, then the probZem (6.6),(6.7) has a unique gtobat sotution f i n the same space. The solution s a t i s f i e s the decay e s t i m t e (6.13) f o r t E COY-)

.

Remark I f the estimate (4.19) remains t r u e f o r n z 2, we can conclude t h a t t h e s o l u t i o n o f ( i ) decays a t the r a t e t-B( B = min{y, 1/21) as t -+ m f o r small i n i t i a l data i n Hs(JRn) n Lp(Rn)), where y = n(1/2p - 1/4), Proof o f Theorem 6.4

Taking

global s o l u t i o n t o (6.6),(6.7) The d e t a i l s are omitted.

66 = 64/2C2

, we

can show t h e existence o f a

i n the same way as i n the p r o o f o f Theorem 5.2.

7. EXAMPLE, I (ONE-DIMENSIONAL BROADWELL MODEL) Here we s h a l l discuss t h e one-dimensional Broadwell model ( c f . [ l ] ) , t h e simplest example o f (1.1): (7.1) where

F = t (F1,F2,F3),

,

t s o ,

V = diag(v,O,-v)

x a I R

1y

and

and a are p o s i t i v e constants. We s h a l l v e r i f y the conditions ( I ) (II) f o r t h i s one-dimensional model. By (7.2) we have

Here v and

Ft t VFx = Q(F,F)

Discrete Velocity Models of the Boltzmann Eauation = : A;

:A:

=

A13 22 = u and

.. A:;

=

0

77

otherwise.

22 Therefore ( I ) i s checked. To v e r i f y

(II)

we need some preparations.

The space

o f sumnational

i n v a r i a n t s c o n s i s t s o f vectors + = t (+1,+2,03) s a t i s f y i n g F1 + 2 ( + 1 + a,) = 0 . Therefore nZ. and t R.' a r e spanned by {$(1),+(2)} and {$(3)}, respec-

-

t i v e l y , where

F

t

(F1;F2,F3) > 0 Therefore i t has t h e expression F = F1 (1 ,a, a2)

On t h e o t h e r hand a l o c a l l y Maxwellian s t a t e i s a v e c t o r

=

s a t i s f y i n g F; - F1F3 = 0 . w i t h F1 > 0 and a = F2/F1 > 0 . Let

for

be an absolute Maxwellian s t a t e :

t M = M~ (1, a, a2)

(7.3) where

M > 0

,

M > 0 and a = M /M > 0 are constants. Set F(t,x) = M 1 2; A = M,diag(l, aI4, a and s u b s t i t u t e i t i n t o (7.1):

+

A1I2f(t,x)

where

(7.5)1

L =

-

Since n ( L ) = A1I281, spanned by

,

aM1

a simple c a l c u l a t i o n shows t h a t n ( L )

{e(1),e(2)l

and

{e(3)),

r e s p e c t i v e l y where

and

n(L)'

are

Shuichi KAWASHIMA

78

.

2 1/2 bl = ( 1 + 4 a + a 2 ) l 1 2 and b2 = (1 + a + a ) Now we r e p r e s e n t t h e m a t r i c e s L and V w i t h r e s p e c t t o t h e orthonormal b a s i s {e(i)}i:l o f lR3 :

with

N

L

(7.6)1

(

5

Le(i),

2a112b23 0

Let

a

and

B

) l s i , j s 3 = oMlb2

3a(l

- a*)

2

diag(0, 0 , l )

,

a112b

-(1

al/*b:

- a2 )b, 2

be p o s i t i v e constants, and l e t

N

(7.7)

K = a

-B

\ o

-1

0

A d i r e c t c a l c u l a t i o n shows t h a t t h e r e e x i s t s a p o s i t i v e c o n s t a n t t h a t i f B E ( O , B ~ ] and a > 0, then (7.8)

2

a ( B c / f l j 2 + c ( f 2 12

-

B~

such

C(f3I2)

f = t ( f ,f ,f ) E R3 , where c and C ( c < C ) a r e p o s i t i v e constants 1 2 3 independent o f a and B ; [fi]’ denotes t h e symmetric p a r t o f From

f o r any

E.

(7.6)1 and (7.8) we can conclude t h a t t h e r e i s a p o s i t i v e c o n s t a n t a. such t h a t f o r a E (O,ao] arid B E ( O , B ~ ] , [El‘ + L i s p o s i t i v e d e f i n i t e . Thus t h e c o n d i t i o n (11) has been checked, S u n a r i z i n g t h e above c o n s i d e r a t i o n s , we have: Leima 7.1

The one-dimensional Broaddell model (7.1) s a t i s f i e s the conditions

( I ) and (11) for a general absolute Maxuellian s t a t e (7.3). In particular, the a n t i - s y m e t r i c matrix K can be taken as i n (7.7) (with respect t o the b a s i s k(i)li21, f o r suitably small constants a > 0 and B > 0 . Remark T h i s lemma enables us t o e s t a b l i s h t h e g l o b a l e x i s t e n c e and asymptoSee Theorems 5.2, 5.3 and 6.4 ( i i ) . t i c s t a b i l i t y o f s o l u t i o n s f o r (7.1).

Discrete Velocity Mcdels of the Eoltzmann Equation

79

8. EXAMPLE, I1 (TWO-DIMENSIONAL 8-VELOCITY MODEL) I n t h i s s e c t i o n we s h a l l p r e s e n t a two dimensional model w i t h 8 v e l o c i t i e s f o r which t h e c o n d i t i o n s ( 1 ) and (11) a r e s a t i s f i e d . i The v e l o c i t i e s v ( i = 1 ,8) o f t h e model considered a r e

,.-.

v

1

v 2 = (O,v),

= ( v , 01,

v5 = (v, v ) , where

v

v3 = - v

v6 = (-v, v ) ,

v7

=

-

1

,

v

4

v8 =

“5,

2

= - v ,

-

6 v ,

Note t h a t [vi[ = v ( i = I , . - - , There are s i x n o n - t r i v i a l c o l l i s i o n s :

i s a p o s i t i v e constant.

= 6 v (j=5,.**,8).

We assume t h a t f o r each o f t h e above types t h e values o f :A: p e c t i v e l y by

where

a1

,

u2

and

a3

are p o s i t i v e constants.

d i t i o n ( I ) from a p h y s i c a l p o i n t o f view.

-.,1),

lvjl

are given res-

Moreover we assume t h e con-

Then, l e t t i n g

(ctl,-*-,a8)= (1,s.

we o b t a i n t h e f o l l o w i n g equations.

(8.1) where

) and

(Fi)t Qi(F,F)

+ vi*OxFi

a r e g i v e n e x p l i c i t l y by

Q5(F,F) = Uz(FgF8 and SO on.

Let

V1 (8.2)

V

2

i = 1,..-,8,

= Qi(F,F),

F

-

F5F7) + u31(F1F6

= t(F1,-**,F8),

= v d i a g ( 1, 0, -1, = vdiag(0,

Q(F,F)

=

t

-

F3F5) + (F2F8

(Ql(F,F) ,...,Q,(F.F))

0, 1, -1, -1, 1 ) ,

1, 0, -1, 1, 1, -1, -1 ) .

-

F4F5)} and

,

Shuichi KAWASHIMA

80

Then (8.1) can be w r i t t e n i n t h e form

Now we w i l l show t h a t f o r t h i s two-dimensional model t h e c o n d i t i o n (11)

i s satisfied. $ = t ($l,..-,$8)

I t i s easy t o see t h a t satisfying

dimn= 4

Therefore t h e orthonormal b a s i s f o r 112 (resp.

=

1

0

6

J4)= 12 =

=

+

,

?l,-1, 1, -1, 0, 0, 0, 0) 0, 0, 0, 1, -1, 1, -1)

,

1?2,

0, -2, 0, -1, 1, 1, -1)

,

-& ?o, 2 43

2, 0, -2, -1, -1, 1, 1 )

.

On t h e o t h e r hand a l o c a l l y Maxwellian s t a t e i s a v e c t o r .-,F8)

>

0

{$ ( i ) ,i=l 4

,

7 1 , 1, 1 , 1, -1, -1, -1, - 1 ) a

2 6

J8) =

is g i v e n by

, 1, 0, -1, 1, 1, -1, - 1 1 ,

= +t(O,

$(7)

a')

bn c o n s i s t s o f v e c t o r s

1, 1 , 1, 1, 1, 1, 1 1 ,

2 a

$ (3) = q

and

F = t (F1,--.

satisfying F2F4

-

F1F3 = 0

F3Fg

-

F1F6 = 0 ,

,

F F - F5F7 = 0 , 6 8 F F - F2F7 = 0. 4 6

By Lemma 2.2 t h i s i s e q u i v a l e n t t o

t ( l o g F, ,..-,log

F8)

E

m ; so

we have f o r

81

Discrote Velocity Models of the Boltzmann Equation

Putting

Fo = e x p ( ( c l + c 4 ) / 2 a

exp(c3/&)

and

F

(8.4)

+

(c2 + c 3 ) / 6 } ,

, we

c = exp(c2/&)

a = exp(c4/2&)

,

b =

a r r i v e a t t h e expression

= Fot (b, c, bc2, b2c, a2, a2c2, a2b2c2, a2b2)

.

c2 = c3 = 0 (i.e., b = c = l M > 0 be an a b s o l u t e Maxwellian s t a t e of t h e s i m p l e form:

For s i m p l i c i t y we t r e a t here t h e case where Let

and Fo = Fl), (8.5) where

t M = M~ (1, 1, 1, 1, a',

M1 > 0

and

A = M1 d i a g ( l , l ,

1,1,

a2, a2, a 2 ) ,

a = (M /M ) l l 2 > 0 52 2 a2, a , a', a ).

l a t i o n g i v e s t h e orthonormal b a s i s

a r e constants.

I n t h i s case we have

Since E ( L ) = ~ ~ / ~ l al Lsimple , calcu( r e s p . { e ( J ) lj = 5 ) f o r a ( L )

{e(i)}i:l

1:

(resp. R(L)'

e ( 2 ) = L t ( l , 0, -1, O, a, -a, -a, a )

fib,

e(3) =

1t ( ~ , 1,

0, -1, a, a, -a, - a )

fib2

e(4)

-

,(5)

= J5)

,(7)

-

e(8) =

1 Zbl

, ,

t ( a , a, a, a, -1, -1, -1, -1)

,

= $(6)

1 t (2a, 0, -2a, 0, -1, Zb2

1 t ( ~ , 2a,

,

, 1, 1, - 1 )

,

I , 1) ,

0, -2a, -1, -1,

Zb2

2 112 bl = ( 1 + a ) 'I2 and b2 = ( 1 +2a ) 1 V J w .J w i t h r e s p e c t t o t h e o r We r e p r e s e n t t h e m a t r i c e s L and V ( W ) thonormal b a s i s { e ( j ) l i Z l o f lR8. By L e m a 3.1 ( i ) we have

.

where

0 *

(8.6),

L

f

( < Le(i),

e ( J ) > )lsi,jr8

Shuichi KAWASHIMA

82

c2,

where inite.

,

t h e square m a t r i x o f o r d e r 4, i s r e a l symmetric and p o s i t i v e d e f -

Also, we o b t a i n by a d i r e c t c a l c u l a t i o n

, where 0

’

-aal 9

-aa2

w1

( 0

and

g , 2 ( ~ )= Let a

Y

i s r e a l symmetric. be p o s i t i v e constants, and l e t t h e anti-symmetric m a t r i x

t g 2 1 ( w ) ; V2*(w) and

“Ku) = 1 X J w j (8.7)

-w2

Z(w1

B

t o be

[

= a

Y

BKll(w) “Kl(w)

K12(w)

N

K21(w) =

-

tK12(w)

where aw2 @b,

“Kl(W)

=

-awl

0

0

- b Oi w l

-aw2

0

0

-b2 w2

Z q

l

I’

Discrete Velocity Mcdels of the Boltzmann Equation

I o

0.

83

0

Then a simple c a l c u l a t i o n shows t h a t there i s a p o s i t i v e constant that i f

B

( O , B ~ ] and

E

holds f o r any

w

E

S1

-K ( L I ) ~ ( u ) .

and

f = t(fl,**-,f8)

p o s i t i v e constants (independent o f a and 5); r i c part o f

+

c

B~

such

a > 0,

R 8 , where

E

c

and

a > 0

and

are

denotes the symnet-

[F(w)i(u)]'

From (8.6)1 and (8.8)we can deduce t h a t

i s p o s i t i v e d e f i n i t e f o r s u i t a b l y small

C

B

>

[K(u)~(u)]'

0.

Thus we have proved: Lemma 8.1

The two-dimensional 8-velocity model (8.1) s a t i s f i e s t h e condi-

t i o n s ( I ) and

(8.5).

(n)

(Gt

l e a s t ) .for an absolute M m e l Z i a n s t a t e of t h e form

In p a r t i c u l a r , the a n t i - s y m e t r i c matrices

as i n (8.7) ( w i t h respect t o t h e b a s i s {e(i))i!,) a >

0 and

B

KJ (j = 1 ,2) can be taken f o r s u i t a b l y small c o n st a n t s

0.

This lemna enables us t o apply Theorems 5 . 2 and 5.3 (reSP. Theorem Remark 6.4 ( i ) ) t o the model (8.1) if M i s an absolute Maxwellian s t a t e o f t h e form (8.5) (resp.

M(r)

-

F

i s a constant s t a t e such t h a t t h e corresponding M =

i s o f the form (8.5)).

REFERENCES

[l] J.E. Broadwell, Shock s t r u c t u r e i n a simple d i s c r e t e v e l o c i t y gas, Phys. o f Fluids, 7 (1964), 1243-1247. [2]

H. Cabannes, S o l u t i o n g l o b a l e du problPme de Cauchy en t h e o r i e c i n g t i q u e d i s c r s t e , J . de Mcanique, 17 (1978), 1-22.

Shuichi KAWASHIMA

H. Cabannes, S o l u t i o n g l o b a l e d'un probleme de Cauchy en t h e o r i e c i n e t i -

que d i s c r e t e . ModSle p l a n , C, R. Acad. Sc. P a r i s , 284 (1977), 269-272. H. Cabannes, The d i s c r e t e Boltzmann equation (Theory and a p p l i c a t i o n s ) ,

Lecture Notes, Univ. o f C a l i f o r n i a , Berkeley, 1980. R.S. E l l i s and M.A. Pinsky, Limit theorems f o r model Boltzmann e q u a t i o n s with s e v e r a l conserved q u a n t i t i e s , Indiana U n i v . Math. J . , 23 (1973), 287-307. R. Gatignol, Theorie c i n e t i q u e de gaz 'a r g p a r t i t i o n d i s c r e t e de v i t e s s e s , Lecture Notes i n Phys. 36, Springer-Verlag, New York, 1975.

R. I l l n e r , Global e x i s t e n c e results f o r d i s c r e t e v e l o c i t y models of t h e Boltzmann e q u a t i o n i n s e v e r a l dimensions, J . de Mgcan. Theor. Appl. , 1 (1982), 611-622. K. Inoue and T . Nishida, On t h e Broadwell model o f the Boltzmann e q u a t i o n f o r a simple d i s c r e t e v e l o c i t y g a s , Appl. Math. O p t . , 3 (1976), 27-49.

T. Kato, P e r t u r b a t i o n theory f o r l i n e a r o p e r a t o r s , (second e d . ) S p r i n g e r Verlag, New York, 1976. [ l o ] S. Kawashima, Global s o l u t i o n of the i n i t i a l value problem f o r a d i s c r e t e v e l o c i t y model o f the Boltzmann e q u a t i o n , Proc. Japan Acad., 57 ( 1 9 8 1 ) , 19-24. [ l l ] S. Kawashima, Smooth global s o l u t i o n s f o r two-dimensional e q u a t i o n s of electro-magneto-fluid dynamics, t o appear. [12] S. Kawashima and M. Okada, Smooth global s o l u t i o n s f o r the one-dimensiona1 e q u a t i o n s i n magnetohydrodynamics, Proc. Japan Acad., 58 (1982), 384387. [13] L . Nirenberg, On e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s , A n n . Scuola Norm. Sup. P i s a , 1 3 ( 1 9 5 9 ) , 115-162. [14] T. Nishida and K. Imai, Global s o l u t i o n s t o t h e i n i t i a l value problem f o r t h e n o n l i n e a r Boltzmann e q u a t i o n , Publ. RIMS, Kyoto Univ., 12 (1976), 229-239. [I51 T. Nishida and M. Mimura, On the Broadwell's model f o r a simple d i s c r e t e v e l o c i t y g a s , Proc. Japan Acad., 50 (1974), 812-817. C161 T. Nishida and M. Mimura, Global s o l u t i o n s t o the Broadwell's model o f Boltzmann e q u a t i o n for a simple d i s c r e t e v e l o c i t y g a s , i n "Mathematical Problems i n t h e o r e t i c a l physics", Lecture Notes i n Phys. 39, SpringerVerlag, New York, 1975.


[17]

L. Tartar, Existence globale pour un systeme hyperbolique s e m i - l i n g a i r e de l a t h e o r i e c i n 6 t i q u e des gaz, Ecole Polytechnique, Seminaire Goulaouic-Schwartz, 28 octobre 1975.

[18]

T. Umeda, S. Kawashima and Y . Shizuta, On the decay o f s o l u t i o n s t o the

l i n e a r i z e d equations o f electro-magneto-fluid dynamics, p r e p r i n t .


L e c t u r e N o t e s in Num. Appl. Anal., 6, 87-91 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1983

Blow-up of Solutions for Quasi-Linear Wave Equations in Two Space Dimensions

KyCiya MASUDA Mathematical Institute, Tohoku University Sendai 980, Japan

Abstract I t i s shown t h a t a s o l u t i o n of q u a s i - l i n e a r wave e q u a t i o n

azu t

-

au = (atu)'

i n two space dimensions, w i t h t h e i n i t i a l f u n c t i o n s o f

compact support, blows up i n f i n i t e time.

- Au = ( a t u ) z i n t h r e e dimensions.

r e s u l t on blow-up o f s o l u t i o n s f o r

1.

Introduction.

T h i s i s a complement t o John's

Consider t h e Cauchy problem f o r q u a s i - l i n e a r wave equa-

t i o n s o f t h e form

(1)

nu =

$(U',U"),

XEP,

t>O

w i t h the i n i t i a l condition: (2) (0

U(Xl0)

= f(x),

ut(xlO)

denotes t h e D'Alembertian

0

= g(x), =

a2/at2

XERn

-

A).

Here u ' , u " r e p r e s e n t t h e

v e c t o r s o f f i r s t and second d e r i v a t i v e s o f u w i t h r e s p e c t t o xk ( x = ( x l ,

..., x,))

and t ; and $ i s a smooth f u n c t i o n o f u ' , u " w i t h $ and i t s

f i r s t derivatives vanishing f o r u '

=

u " = 0.

There i s e x t e n s i v e l i t e r a t u r e on e x i s t e n c e o r non-existence o f q l o b a l s o l u t i o n s o f s o l u t i o n s of t h e form:

mu = $ ( u ) . (See [ l ] , [5] and

t h e r e f e r e n c e s g i v e n i n those papers). S. Klainerman [3] showed t h a t a g l o b a l smooth s o l u t i o n s o f ( I ) ,

( 2 ) e x i s t s f o r a l l " s u f f i c i e n t l y s m a l l " i n i t i a l data f , g i f n t 6 and 87

KyCiya MASUDA

88

$(u',u") =

O(I u ' l2 + 1 u"I

near u '

2 ,

=

u " = 0.

We a r e concerned with

t h e problem whether o r n o t Nlainerman's r e s u l t h o l d s f o r the case excluded t h e cases n = 1 , 2, 3 a r e o f s p e c i a l importance f o r a p p l i c a t i o n s .

i n [3];

c3-

F . John [ l ] showed t h a t any n o n - t r i v i a l

s o l u t i o n o f e.g.,

nu = ( a t u ) 2 i n t h r e e space dimensions f o r which u ( x , 0 ) , atu(x,O), a r e o f compact support, blows up i n f i n i t e time.

a2u(x,0) t

H i s method can n o t be

a p p l i e d a t l e a s t d i r e c t l y t o t h e case o f two space dimensions, s i n c e he considered t h e incoming and outgoing waves, and used t h e r e f l e c t i o n o f t h e incoming wave a t t h e o r i g i n ; t h e p r o p e r t y of t h e r e f l e c t i o n i s p e c u l i a r t o t h e t h r e e space dimensions. We s h a l l show: Theorem. Rn,

L e t n = 1, 2 , 3 .

o f compact support.

L e t g be a smooth non-negative f u n c t i o n on

I f u i s a c2 s o l u t i o n of the equation

(3)

ou = ( a t u ) 2 , xsRn, t > O w i t h t h e i n i t i a l c o n d i t i o n :

(4)

u ( x , 0 ) = 0; atu(x, 0 ) = g ( x )

then,

2.

u = u ( x , t ) vanishes i d e n t i c a l l y i n xcRn, t > O .

Representation o f s o l u t i o n For a s o l u t i o n u o f ( 3 ) , we s e t

1 ~h ( x , t ) = fi ( u ( x , t + h ) Then the uh s a t i s f i e s atuh

-

-

u (x,t)),

A U ~= $h

h>O.

(u)

where $ h ( u ) (x, t ) =

((atu(x, t + hl2)-((atu(X,

t))2).

We a s s o c i a t e a f u n c t i o n f c C (Rn) w i t h i t s s p h e r i c a l means on t h e u n i t sphere

(

1 Sm-d

Sm-l

about t h e o r i g i n :

: t h e surface area o f t h e u n i t sphere).

89

Quasi-Linear Wave Equations

Hence by D'Ambert's formula, 1 uh(r, t ) = ( i h ( r + t, 0 ) + i i h ( r

where Tr

-

t, 0 ) ) +

{r

r+t atiih(s,

0 ) ds

r-t

i s the c h a r a c t e r i s t i c t r i a n g l e w i t h vertex ( r , t ) : Tr,t

= { (p,~);

T

+

p 5 t

+ r,

-

T

L e t t i n g h 4 i n (51, and s e t t i n g v ( r , t )

=

p 5

t

-

r, T 2 0

ati(r,t),

1

we get, by p a r t i a l

in t e g r a t ion, r+t

(6)

v(r,t)

(O) so small t h a t

(8)

A

-*1

(n-1)(3-n)

P

1 dp dT 5 7

Tr,t f o r a l l Ost

JJsgn(u-t)[.l 1=1

holds f o r every sgn(y) = 1

-t

L ~ ( R " ~ ( O , T ) )f o r every

kE R1 and e v e r y nonneaative

(y > 0); = 0

O

1

( y = 0); = -1

@ €Ci(Rnx(O.m)),

( y < 0).

93

where

which

Tetsuro MIYAKAWA

94

Hereafter the solution

u

above w i l l be c a l l e d entropy s o l u t i o n o f (M)

s i n c e ( E ) g e n e r a l i z e s t h e e n t r o p y c o n d i t i o n o f O l e i n i k [8] t o t h e case of several space v a r i a b l e s .

I n t h i s paper we p r e s e n t a new approach t o t h e problem ( M ) which i s based

on an analoay w i t h t h e k i n e t i c t h e o r y o f gases.

Namely, we regard t h e problem

(M) as a model o f macroscopic conservation laws i n f l u i d mechanics, and then

i n t r o d u c e as i t s microscopic model t h e f o l l owing 1 inear problem:

c(x,t,S)

= F(C(x,t),S),

n

C(x,t) =

-1

Ai.(x,t,O) i=l 1

if

1

-

B(x,t,O),

O < e -~ w ,

if w 5 5 < 0,

-1

0

otherwise.

The f o l l o w i n g a r e e a s i l y checked.

w =

F(w,S)dg

f o r any

weR 1 .

-m

I

m

Ai(x,t,w)-Ai(x,t,O)

=

ai(x,t,[)F(w,c)dc,

-CO

(C1

From (C) and (0) we e a s i l y see t h a t i f with

fo = F(uo(x),s),

f = f(x,t,c)

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

l e a s t f o r m a l l y ) t h e problem ( M ) a t

t = 0.

i s t h e s o l u t i o n o f (m)

v(x,t) = /f(x,t,c)dg

T h i s suggests t h a t f o r small

approximate s o l u t i o n may be c o n s t r u c t e d so t h a t i t s a t i s f i e s j = O,l,

... ;

satisfies (at

see Section 1 f o r p r e c i s e statement.

(M) a t t

h > 0

= jh,

The p r e s e n t work c o n t i n u e s

t h e previous ones [ Z ] , [3] which a r e w r i t t e n j o i n t l y with Y . Giga and

First Order Quasilinear Equations

I n [ 2 ] we considered t h e case A i = Ai(u),

S. Oharu.

96

B =

I)

and a p p l i e d t h e

method i l l u s t r a t e d above t o c o n s t r u c t a g l o b a l weak s o l u t i o n . [3] discusses t h e i i B = B(x,u) and proves t h a t our s o l u t i o n s a r e e n t r o p y case A = A (x,u), s o l u t i o n s , w i t h t h e a i d o f t h e t h e o r y o f n o n l i n e a r semigroups.

I n t h i s note

we extend t h e r e s u l t i n [ 3 ] t o general time-dependent case and g i v e a p r o o f which does n o t use t h e t h e o r y o f n o n l i n e a r e v o l u t i o n o p e r a t o r s .

I n the f i n a l

s e c t i o n we d i s c u s s another approximation, due t o Y . Kobayashi [5],

B = 0, which uses t h e l i n e a r Bolttmann e q u a t i o n i n s t e a d o f t h e

A' = A ' ( u ) ,

l i n e a r equation

1.

i n t h e case

(m).

Main r e s u l t

We c o n s i d e r t h e Cauchy problem ( M ) under t h e f o l l o w i n g assumptions: (A.l)

For each

r > 0

and each

T

>

0

a i, a i x , , axi

the functions

j k

J and

b, b x ,

a r e a l l bounded and continuous on

J (A.2)

T > 0, C ( x , t ) =

F o r each

bounded and continuous on (A.3) a

2

-

' 1

' i Ax (x,t,O)-B(x,t,O) i=l i

1

and

Cx

are j

Rnx[O,T].

T > 0

For each

-

Rnx[O,T]x[-r,r].

t h e r e a r e constants

aXi(x,t,6)-b(x,t,6), i

6 2 -b(x,t,t)

CI

'> 0 and

for a l l

6

2 0 so t h a t

(x,t,6)€Rnxx[0,TlxR 1

i=1 Let

IU5(t,s);

problem (m) w i t h

0

=
0

so t h a t

Then we have

C(u) = C(.,a).

IK(t,s)v-K(t,s)wll,,

5 e ( B+w)( t - S ) IV-Wl1 ,r

We n e x t c o n s i d e r e s t i m a t e s f o r d e r i v a t i v e s the s e t o f functions

(x,t,E)ERnx[O,T]x[-r,r]~

one can show t h e f o l l o w i n g lemma

and

(ii)

1

v e Lm(Rn)

such t h a t

for

aK(t,s)v/axi.

O i s 5 t ~ T .

Let

h(R")

be

Tetsuro MIYAKAWA

98

r

is finite for all

0.

>

IDx~ll,r

v t A(Rn),

Notice t h a t i f

F o r each

IDxvll,r

r > 0

and each

t h e r e i s a sequence

{vm}

such t h a t

If v t A ( R n ) ) , t h e n

LEMMA 2.4.

; see [4].

a r e Radon measures w i t h vx i The following two lemmas a r e shown i n [3].

v€A(Rn)

Rn

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

jprlDxvI

then t h e derivatives

locally f i n i t e total variation.

LEMMA 2.3.

i s o f t e n denoted b y

F(v(-),E)€A(R~)

IDxF(~(.),E)ll,rd~

=

f o r a.e. ~ E R ' ; and r > 0.

for a l l

-m

U s i n g t h e s e lemmas, we c a n e s t i m a t e

LEMMA 2.5.

Let

Y

1

2

i ,j

If

vCA(Rn)

for

O

~

and

s

Now l e t z ( x , t ) where

fo(x,c)

on

suPIlai,(x,t,c)l: J

lvl,

~

2 r

IC(x,t)l

IDxK(t,s)vll,,

Rnx[O,T],

for

and choose

V€A(Rn).

y

2 0 so t h a t

(x,t,E)ERnxx[O,Tlx[-r,rl},

2 r, t h e n we have

t

~

T

= jb(x,t,S)VS(~,ilfod:;

= F(v(x),E)

and

vCLm(Rn).

yi(x,tf

=

Ii

a (x,t,cfVe(t,sffodE,

Then i t i s c l e a r t h a t

F i r s t Order Quasilinear Equations

n

1 ayi/axi i=1

aK(t,s)v/at +

(2.6)

i n t h e sense o f d i s t r i b u t i o n s .

LEMMA 2.6. IC(x,t)l 2 r T

>

Let

on

on

R"(~,T)

r > 0

and choose

so t h a t

Then t h e r e i s a c o n s t a n t

IDxvll,r

K

1v1, 0

>

and

r

depending on

so t h a t

-

lK(t,s)v

c

From (2.6) and Lemma 2.5 we o b t a i n

v€A(Rn)

Rnx[O,T].

0, r > 0, and

+ z =

99

K(T,s

A p p l y i n g t h e foregoing r e s u t s r e p e a t e d l y , we can now show t h e e s t i m a t e s f o r t h e approximate s o l u t i o n s

PROPOSITION 2.7.

Let

( v ~ ( r,~ and assume

uh:

uo

and

vo

be i n

(C(x,t)( 5 r

on

Rnx[O,T].

aoproximate s o l u t i o n s w i t h i n i t i a l d a t a R

2 reaT(l+T),

for

t ((0,T)

(iii)

uo

luOlrn

Lm(Rn) w i t h

and

Let

uh

and

r,

vh

vo, r e s p e c t i v e l y .

be If

t h e n we have t h e f o l l o w i n q e s t i m a t e s :

h > 0.

and

h h ( @ + w ) t l u -v I Iu (t)-v (t)ll,R 2 e 0 0 l,R

PROPOSITION 2.8. as i n Lemma 2.5 w i t h

Let r

T,

r

and

r e p l a c e d by

Let

u0€A(Rn)

Then: h IDXU ( t ) 11 ,R

2

( B+O+Y 1t

( l D x U Q l l,R

t ( (0,T)

and

be as i n P r o p o s i t i o n 2.7.

R R.

for

+

Y t l u g l l ,R)

with

h

>

Define

luOlrn 5 r .

0.

y

Tetsuro MIYAKAWA

100

for

tt(0.T)

and

h

L e t T, r

PROPOSITION 2.9.

IDxU"ll,R

R

be as above and

0 depending on

>

R

and

- uh(

S ) I ~ 2, ~K l t

-

for

s1

t, s€[O,T]

and

h

>

0.

Convergence t o t h e entropy s o l u t i o n s

3. Let

uo

be i n h(Rn).

Then, P r o p o s i t i o n s 2.7, 2.8 and 2.9 t o g e t h e r show

1

that

K

u o € ~ ( R n ) be such

so that h Iu ( t )

L -norms and t h e t o t a l v a r i a t i o n s o f

compact subset o f >

and

( u o l m 5 r . Then t h e r e e x i s t s a c o n s t a n t

that

h

0.

>

uh

a r e u n i f o r m l y bounded on each

F u r t h e r , P r o p o s i t i o n 2.9 i m p l i e s t h a t , f o r any

Rnx(O,T).

0,

luh(t)

-

[t/hl

n

5 K(t-h[t/h])

K(,jh,(j-l)h)uoll,R

for

tE[O,T].

j=l

Thus, a we1 1-known compactness theorem ( [ 4 , Theorem 1.191) y i e l d s

PROPOSITION 3.1. hm+ 0

Let

uo

u on Rnx(O,-)

and a f u n c t i o n

w i t h the following properties:

h (i)

[t/hml

u m(. , t )

-+

u(. , t ) ,

(. , t ) z 'hm

in

L1 (Rn) 9. oc (ii) (iii)

u

uniformly i n is in

t

L"(R'~(O,T))

The map: t

-+

o f P r o p o s i t i o n 2.9.

n

K(jhm,(j-l)hm)uo

+

u(*,t)

j=1

2 0 on every compact s u b i n t e r v a l . T > 0.

f o r every

u(. ,t) i s continuous from [ O p )

Notice t h a t the u n i f o t m i t y i n

solution of

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

be i n n(Rn)).

t

1 LLoc(RF).

o f t h e convergence i n ( i ) i s a consequence

We now show t h a t t h e f u n c t i o n

(M) w i t h t h e i n i t i a l f u n c t i o n uo.

known t o be unique, i t t u r n s o u t t h a t

into

h {u 1

u above i s t h e e n t r o p y

Since t h e e n t r o p y s o l u t i o n i s

i t s e l f converges t o

u

as

h

-+

0.


101

I n v i e w o f (ii)and ( i i i ) above, i t s u f f i c e s t o show t h a t i n e q u a l i t y (E).

I n d o i n g t h i s t h e f o l l o w i n g Lemma 3.2,

C r a n d a l l and Majda [l],p l a y s a fundamental r o l e ,

s

R',

satisfies

u

w h i c h i s suggested by

F o r s i m p l i c i t y we w r i t e

then

t 2 0.

for

PROOF.

Since

s =

F(s,c)dg,

+

Since

t

t

](U:(h)-l

the definition o f

oives

)F(s,S)dC. t

I K ( h ) v - s l = (K ( h ) v - s ) s g n ( K ( h ) v - s )

oreservino,

t K (h)

(3.1) f o l l o w s from (3.2).

and s i n c e

t UE(h)

i s order-

Tetsuro MIYAKAWA

102

where

U (t,s)*

5

a r e s o l u t i o n o n e r a t o r s o f t h e (backward) Cauchy problem:

so t h a t

I"k o y ( s - k ) q ( s ) d s

lim j-

1

= q(k)ssn(w-k)

1

f o r g t C (R ) ,

J

and m u l t i o l y b o t h sides o f (3.4) by

q . ( s ) I o'!(s-k) 3 J

1 (k€R ).

I f we n o t e t h e

i d e n t i t i e s (see [ 9 1 ) : (Ui(h)-1

)J, =

(Ui(h)*-l)w

=

h L(h[t/h]+u)U,(h[t/~l+o,

n

h joL(h[t/h~+u)*U:(h,~)**du

h[t/hl)$do;

for

$I€ Ci(Rn)),


where

103

L(t)$ =

i t i s e a s i l y seen t h a t T

(3.5) l i m l i m h-'

(3.6)

q.(s)ds

h+O

.j-

J

lirn l i m h-l j-m h+O m J:

J, J

d t ($(x,t-h)-@(x,t))lu,(x,t)-sldx

h+O

j-

(3.8

. (A'(x,t,u)-Ai(x,t,k))$,

J

(3.7) l i m l i m h-'

n

q.(s)J1(s.h)ds = 2

1:-

!: I

2

d t sgn(u-k)@(B(x,t,u)-B(x,t,k))dx

:I I

q.(s)J2(s,h)ds = 2 J

d t spn(u-k)C(x,t)$(x,t)dx

;

l i m l i m h - l r qj(s)J3(s,h)ds h-+O --

j-Ko

sgn (u- k I@ I A:, 1

From (3.4)-(3.8)we see t h a t

u

( x ,t ,k )+B ( x ,t ,k )+C ( x ,t ) j d x . 1

satisfies inequality (E).

4. An aoproxirnation u s i n g t h e l i n e a r Boltzmann e q u a t i o n T h i s s e c t i o n d e a l s w i t h t h e Cauchy problem: n

ut +

(MI'

1

i=1

. A ' ( U ) ~ , = 0,

u(x,O) = uo.

1

The argument g i v e n below i s due t o Y . Kobayashi [ 5 ] . nonnegative f u n c t i o n i n 6 ( ~ =) 6(l~l);

Using such a f u n c t i o n

6

Rn

J

6(n)

be a smooth

w i t h supp 6 c o n t a i n e d i n the u n i t b a l l such t h a t 6 ( v ) d n = 1;

J

qi6(n)dn = 0

i = 1,

..., n.

we d e f i n e

W

(4.1)

Let

F(w,n) = j o d ( n - a ( s ) ) d s ,

Then i t i s e a s i l y seen t h a t

a ( s ) = (a

1

(s), .... a n ( s ) ) ,

ai(s)

=

Ab(s).

dx i ;

Tetsuro MIYAKAWA

104 F(w,n)dn;

'

and

Ai(w)-Ai(0)

= JniF(w,n)dn

for all

wcR

1

.

be t h e s o l u t i o n o f t h e l i n e a r Boltzmann equation:

f = f(x,t,n)

Let (m)

J

w =

(4.2)

n i!l "ifxi

ft +

= 0 ;

f(x,O,n)

= F(u~(x),~)~

and p u t

I

( S t ~ O ) ( ~=) f ( x , t , n ) d n .

(4.3) Note t h a t

StuO f o r m a l l y s a t i s f i e s ( M ) ' a t

t = 0.

Kobayashi [5] proved t h e f o l l o w i n g r e s u l t :

THEOREM 4.1 ([5]). o f t h e problem (M)' w i t h

uniformly i n

t 2 0

Let

uo

u(.,O)

be i n

Lm(Rn)

= uo.

u

and

t h e entropy s o l u t i o n

Then

on every compact s u b i n t e r v a l .

T h i s may be shown i n t h e same way as described i n t h i s paper; so t h e d e t a i l s are omitted.

REMARK.

Kobayashi's approximation described here does n o t always g i v e

so sharp r e s u l t s as ours. function

For example, i f

n

= 1

and

A(u)

i s convex, t h e

d e f i n e d i n S e c t i o n 1 g i v e s t h e exact s o l u t i o n o f

K(t,O)uo

i n t h e t i m e i n t e r v a l [O,tO) where

to

(M)'

i s t h e t i m e when shock begins t o develop.

Furthermore, f o r t h e Riemann i n i t i a l value problem f o r t h e nonviscous Burgers

2 t h e case: A ( u ) = u / 2 ) , we can show t h a t

equation (i.e.,

h

u (-,t)

uniformly i n

t

-

u(*,t) = O(h)

i n LiOc(R1)

2 0 on every compact s u b i n t e r v a l .

On t h e o t h e r hand, t h e


105

scheme of Kobayashi seems t o be useful i n some o t h e r problems.

For i n s t a n c e ,

i t may be a p p l i e d ( [ l o ] ) t o o b t a i n approximate s o l u t i o n s o f t h e equations w i t h v i s c o s i t y term: '

Ut

i

A (ti),, + i1 =1

= vAU, 1

u(x,O) = u,(x).

References

[l]M. G. Crandall and A. Majda, s c a l a r c o n s e r v a t i o n laws, [2] Y . Gipa and T. Miyakawa,

Monotone d i f f e r e n c e approximations f o r

Math. Comp. 34 (1980), 1-21. A k i n e t i c construction o f global solutions o f

f i r s t o r d e r q u a s i l i n e a r equations,

Duke Math. J . 50 (1983), t o appear.

[3] Y . Giga, T. Miyakawa and S. Oharu,

A k i n e t i c approach t o general f i r s t

o r d e r q u a s i l i n e a r equations,

[4] E. G i u s t i ,

Preprint.

Minimal surfaces and f u n c t i o n s o f bounded v a r i a t i o n ,

Notes

on Pure Mathematics no. 10, A u s t r a l i a n N a t i o n a l U n i v e r s i t y , Canberra, 1977. [5] Y . Kobayashi, [6] S . N. Kruzkov,

variables, [7] W . Mazja,

P r i v a t e communication. F i r s t o r d e r q u a s i l i n e a r equations i n s e v e r a l independent

Math. USSR-Sb. 10 (1970), 217-243. Einbettungssatze f u r Sobolewsche Raume,

Teubner, L e i b z i g ,

1980. [8] 0. A. O l e i n i k ,

Amer. Math. SOC. T r a n s l . ( 2 ) 26 (1963), 95-172.

equations, [9] H. Tanabe,

Equations o f e v o l u t i o n ,

[lo] T . Miyakawa, equation

Discontinuous s o l u t i o n s o f n o n - l i n e a r d i f f e r e n t i a l

Pitman, London, 1979.

Construction o f solutions o f a semilinear parabolic

by u s i n g t h e l i n e a r Boltzmann equation,

Preprint.



Instability of Spatially Homogeneous Periodic Solutions to Delay-Diffusion Equations

Yoshihisa MORITA Research Institute for Mathematical Sciences, Kyoto University Kyoto 606, Japan

§1

Introduction

There a r e v a r i e t y o f o s c i l l a t o r y phenomena i n e l e c t r o n i c s , b i o l o g y , b i o c h e m i s t r y etc.,

which a r e described by d i f f e r e n t i a l equations w i t h t i m e

f o r i n s t a n c e , proposed t h e f o l l o w i n g d e l a y e q u a t i o n Hutchinson [l],

delay.

as a s i n g l e species b i o l o g i c a l model e x p r e s s i n g an o s c i l l a t o r y phenomenon: d -y(t) dt where

a, r, K

1

= a(

-

a r e p o s i t i v e constants.

The e q u a t i o n (1.1) i s transformed

into d

(1.2) where +;

v(t) =

-

(

;+ u ) ( 1 + v ( t ) ) v ( t - 1 ) ,

p = a r , and t h e steady s t a t e

o f (1.2).

y

I

K

o f (1.1 ) corresponds t o

I t i s Known t h a t (1.2) has a p e r i o d i c s o l u t i o n f o r

[3]) and t h a t t h e r e occurs a Hopf b i f u r c a t i o n a t

p=O ( [ 5 ] ) .

vE0

p > O ([2],

Furthermore

t h i s b i f u r c a t i n g p e r i o d i c s o l u t i o n i s s t a b l e near t h e b i f u r c a t i o n p o i n t ~ 4 1 ,[ g i ) . Here we s h a l l c o u p l e t h e e q u a t i o n (1.2) w i t h a d i f f u s i o n term. p r e c i s e l y , we c o n s i d e r t h e f o l l o w i n g i n i t i a l - b o u n d a r y v a l u e problem:

107

More

108

Yoshihisa MORITA

1i g=O, aV(t,x)

= dAv(t,x)

-

(;

+p)(l+v(t,x))v(t-l,x),

(t,x)t(O,m)xn,

at

(1.3)

where a/an A

Q i s a bounded domain i n Rn

w i t h a smooth boundary

denotes t h e o u t e r normal d e r i v a t i v e t o

stand f o r

1

a2

i=l

aR

20,

and

.

It i s clear t h a t f o r

p> 0

t h e e q u a t i o n (1.3) has a p e r i o d i c

s o l u t i o n corresponding t o t h a t o f (1.2).

This periodic s o l u t i o n i s a

s p a t i a l l y homogeneous p e r i o d i c one (independent o f s p a t i a l v a r i a b l e s ) .

I n t h i s paper we s h a l l d i s c u s s t h e s t a b i l i t y o f t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n t o such a e q u a t i o n (1.3).

As f o r s t a b i l i t y o f t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n t o (1.3),

Yoshida [ 7 1 has proved t h a t t h e b i f u r c a t i n g p e r i o d i c s o l u t i o n

near t h e b i f u r c a t i o n p o i n t

p=O

i s stable.

However, i t has n o t

been made c l e a r how t h e s t a b i l i t y r e g i o n o f t h e b i f u r c a t i o n parameter

u

d

depends on t h e o t h e r f a c t o r s such as t h e d i f f u s i o n c o n s t a n t

and t h e shape of t h e domain n = l , L i n and Kahn

[a]

0. In t h e case where t h e space dimension

have suggested by a p e r t u r b a t i o n method t h a t t h e

b i f u r c a t i n g s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n l o s e s i t s s t a b i l i t y f o r some

u

f a i r l y near

\ 1 = 0 when

d

i s s u f f i c i e n t l y small.

In t h i s paper we s h a l l study t h i s problem and discuss t h e d e s t a b i l i z a t i o n of t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n i n q u i t e a general framework. Applying t h e r e s u l t s i n

55 t o (1,3), we see t h a t f o r any RCRn

t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n becomes u n s t a b l e near p=0

i f the d i f f u s i o n c o e f f i c i e n t

d

i s t a k e n s u f f i c i e n t l y small;

and, moreover, i n t h e case o f several space dimensions ( i . e . ,

n12),

Delay-Diffusion Equations

f o r any f i x e d

d, such d e s t a b i l i z a t i o n a l s o occurs when t h e shape o f

R i s varied.

t h e domain

109

More p r e c i s e l y , t h i s occurs when t h e second

eigenvalue o f t h e L a p l a c i a n on

R w i t h homogeneous Neumann boundary

c o n d i t i o n becomes s u f f i c i e n t l y s m a l l . In

12 we f o r m u l a t e t h e d i f f e r e n t i a l e q u a t i o n w i t h t i m e d e l a y ( 1 . 2 )

i n a f a i r l y general form o f f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n and i n we g i v e t h e Hopf b i f u r c a t i o n theorem f o r t h i s equation.

93

I n 14 we s h a l l

d i s c u s s t h e l i n e a r i z e d s t a b i l i t y around t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n by u s i n g t h e i m f o r m a t i o n o b t a i n e d i n t h e Hopf b i f u r c a t i o n thorem in

93.

Main theorems i n 55

f o l l o w from the r e s u l t i n

I n t h e l a s t s e c t i o n we s h a l l a p p l y t h e theorems i n

55

14

immediatly.

t o the equation

(1.3) and examine t h e c o n d i t i o n f o r t h e occurrence o f d e s t a b i l i z a t i o n i n t h e above sense.

12

Some r e s u l t s f o r f u n c t i o n a l d i f f e r e n t i a l equations

Let continuous

X

be a Banach space.

e u c l i d e a n space.

t ([r,T+a],

w i l l denote a s e t o f a l l

X-valued f u n c t i o n s d e f i n e d on [ a , b l w i t h supremum norm

For s i m p l i c i t y , C[a,b]

Let

C([a,bl;X)

Cm

~t R ' , O,r t h e symbol

by t h e r e l a t i o n

denotes

C([a,b];Rm),

where

R"

i s t h e m-dimensional

and vt

a > 0.

For any

vCCCr-r,r+al

w i l l denote t h e element i n

-rcecO.

and

C[-r,O]

It i s clear that

defined vt(0) = v ( t ) .

L e t us c o n s i d e r t h e f o l l o w i n g f u n c t i o n a l d i f e r e n t i a l e q u a t i o n

where

a l l .

stands f o r t h e m-dimensional complex Space.

vt(e) = v ( t t e ) ,

(without diffusion) :

11

Yoshihisa MORITA

110

F : I ~ c[-~,oI X i s of c l a s s C 4 ,

L(p)

and

o r d e r ( n o n l i n e a r ) p a r t of

+

G(p,-)

F(p,-).

R"' a r e t h e l i n e a r p a r t and t h e h i g h e r Furthermore we assume for P C I ~ ,

F(P, 0) = 0 where

I.

0

i s an i n t e r v a l c o n t a i n i n g

For example, t h e e q u a t i o n (1.2) i n L(p1 and G(p,-)

respectively

6R

1

I1

. s a t i s f i e s above c o n d i t i o n s ;

a r e g i v e n by

.

We c o n s i d e r t h e l i n e a r equation associated w i t h (2.11,

The r e s u l t s i n t h e r e s t o f t h i s s e c t i o n w i l l be found i n AS

~ ( p ) i s a continuous l i n e a r mapping o f

t h e r e i s an

mxm

matrix function

e

have bounded v a r i a t i o n i n

Moreover, t h e domain o f (2.3)

a l s o denotes When

u = 0,

holds f o r

L(p)

R'",

whose elements

0C

CC-r,Ol.

i s n a t u r a l l y extended i n t o

0 cC([-r,Ol;Cm).

C([-r,Ol;Cm).

-rcecO,

into

[-r,O], such t h a t

L ( v ) @ = f,.[dde;~~)l@(eI,

(2.3)

and

on

n(e;p),

c[-r,ol

[51.

C([-r,Ol;Cm)

Hereafter the notation

C[-r,O]

The readers w i l l n o t confuse t h e n o t a t i o n .

we simply w r i t e

L e t us d e f i n e t h e c h a r a c t e r i s t i c e q u a t i o n a s s o c i a t e d w i t h (2.1);


where

I

i s the

r o o t s o f (2.41,

mxm

i d e n t i t y matrix.

111

There a r e c o u n t a b l y many

each o f them being a t most f i n i t e l y degenerated.

It i s known t h a t t h e s e t o f t h e r o o t s o f (2.4) c o i n c i d e s w i t h t h e s e t o f t h e eigenvalues o f t h e l i n e a r system (2.2). A(p)

be t h e i n f i n i t e s i m a l generator o f t h e semigroup o f a s s o c i a t e d

w i t h (2.2);

where

More p r e c i s e l y , l e t

namely

&(A(p))

spectrum o f

A(p)

A(p)

i s d e f i n e d as

denotes t h e domain o f t h e o p e r a t o r

A(p).

Then t h e

c o n s i s t s o n l y c f eigenvalues, each o f which i s a

r o o t o f (2.3) w i t h t h e corresponding m u l t i p l i c i t y . g e n e r a l i z e d eigenspace i n

C[-r,O]

I n particular, the

s u b j e c t t o each eigenvalue o f

A(p)

i s f i n i t e dimensional. We s h a l l i n t r o d u c e t h e formal p r o d u c t d e f i n e d by

where ( a , . )

'J,

denotes t h e transpose o f t h e

m-vector

stands f o r t h e h e r m i t e i n n e r p r o d u c t i n

The a d j o i n t o p e r a t o r

A*(O)

of

A(0)

@

and t h e n o t a t i o n

Cm, t h a t i s ,

with r e s p e c t t o (2.6) i s g i v e n by


respectively, where

co and c;

-I-,

(2.13a

( iuoI

(2.13b)

( -iwoI

113

satisfy

e

iw e

Cdn(e)l ) c o

=

o ,

' tCdn(e)l 1 c i

-iw 0

I t i s shown i n

e

C5;Chap 7l

t o the range of the operator

t h a t a function

(iwo

-

= 0

.

Q e C[-r,Ol

i f and only i f

A(0))

< Q , < ? > = 0.

Thus the space C[-r,O]

i s deco posed as

(2.14)

C[-r,Ol = n / ( i w , - A('))

BR(iuO

/J(iuo

-

A(0)) = { 4

From (2.14) we see

and we may normalize

< e l , q >=

(2.16)

and

e l , cf

t2

i s given by

as

1

.

I

(iwo

-

A(O

belongs

4 satisfies

Yoshihisa MORITA

114

53 The Hopf bifurcation of functional differential equations

Theorem HZ Consider the equation (2.1). hold.

Assume that (Al) and (A2) in 52

Then (2.1) has a family of periodic solutions: More precisely,

there are a positive constant

such that for each solution p(t;EJ

E

E~

and C’-functions

e ( 0 , ~ ~ and ) u = U(E)

with period 2n/w(~).

has Floquet exponents 0 and periodic solutions p(t;Ef,

a = B(E).

E C(O.E~)

P ( E ) , w(E),

B(E),

there exists a periodic

This periodic solution p(t;E) Except for the family of there is no non-trivial periodic

solution in a sufficiently small neighborhood of (0,O) t I,,

x

Rm.

Delay-Diff usion Equations

If

f o r each

B2 < 0,

then t h e r e i s a c o n s t a n t the periodic s o l u t i o n

E. C ( O , E ~ )

E,

115

0 < cO< cH, such t h a t

P(-;E)

i s asymptotically

s t a b l e ( w i t h asymptotic phase).

H.

Corollary

Assume t h e hypotheses i n Theorem The c o e f f i c i e n t s

iw

(3.3) where

B,

where

cl, 68

and

p2

- u2

w2

dX G(0)

H.

i n (3.1) a r e determined by t h e equation,

=

B, ,

i s g i v e n by

and

c2,

i2

a r e d e f i n e d i n (2.121, (2.13) and ( 2 . 1 7 ) ,

(2.18).

F o r t h e p r o o f o f Theorem H, see [91.

(3.4) i n C o r o l l a r y H

The equations (3.3) and

a r e found i n [13; 521.

H e r e a f t e r we assume t h a t

14 L i n e a r i z e d s t a b i l i t y o f t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n

L e t us i n t r o d u c e some f u n c t i o n spaces. Sobolev space o f a l l r e a l valued up t o o r d e r

2

W2’P(n)

the

f u n c t i o n s whose d e r i v a t i v e s

belong t o LP(n), where Q is a bounded domain i n Rn

with a smooth boundary

au/an = 0 on

LP(n)

We denote by

aR.

L e t us p u t

an I y where a/an

L4~yp(Q) =

u t W2’p(Q),

denotes t h e o u t e r normal d e r i v a t i v e

Yoshihisa MORITA

116

to

an.

I n what f o l l o w s we s h a l l understand t h a t

s u f f i c i e n t l y l a r g e , f o r instance, p > n/2

( P , 0) w

p

i s taken

so t h a t t h e correspondence

F(u $1 I

d e f i n e s a mapping F : I 0 x (W2yp(f?))m

of

C4

c l a s s , where

F(u,*)

-t

(M2yp(Q))m

i s as i n (1.1)

(satisfying (Al),

(A2) and

(A3)).

Yow we s h a l l c o n s i d e r t h e f o l l o w i n g equation:

where

To a v o i d l e n g t h y argument on t h e well-posedness o f (4.1), which i s n o t t h e s u b j e c t o f t h e p r e s e n t paper, we assume t h a t f o r any

C([-r,O] ; (Wcyp(Q))m) t h e r e e x i s t s a unique s o l u t i o n V ( t , * ) c([-r,-) ; ( W ~ ~ P ( Q ) ) " ' )t o (4.1) sucn t h a t See, f o r instance, Let

A N(s2)

[lo] for

Qoe

6

a / a t v ( t , - ) c c ( c o , ~ ); ( ~ P ( n ) ) m ) .

such e x i s t e n c e theorems.

be a c l o s e a o p e r a t o r i n LP(Q), w i t h dense domain


a A N ( 2 ) )=

wiyp,

d e f i n e d by denotes

s i m p l i c i t y , AN

AN(Li)v = A v

hereafter.

AN(”)

v E

for

B(AN(R)).

tor

Thus (4.1) i s w r i t t e n as

(4.2)

t>O,

D

For any m a t r i x t h a t f o r each

E

.~(O,E,,)

$ 2 , i t i s c l e a r from Theorem

t h e e q u a t i o n (4.2)

H

has a s p a t i a l l y homogeneous

U f t ) = p(t;&)

w i t h period

And by t h e assumption

(A3), p ( t ; E )

periodic solution IJ=IJ(E).

and any donlain

t o s p a t i a l l y homogeneous p e r t u w a t i o n f o r

E

2n/w(c)

occurring f o r

i s stable w i t h respect t(O,ro).

Note t h a t t h e

s t a b i l i t y i n t h e above sense does n o t n e c e s s a r i l y i m p l y t h e s t a b i l i t y w i t h r e s p e c t t o a l l p o s s i b l e p e r t u r b a t i o n s ( e i t h e r s p a t i a l l y homogeneous o r inhomogeneous).

As mentioned i n

51, Yoshida C7] has shown f o r some s p e c i f i c

e q u a t i o n t h a t t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n

P(-;E) i s

s t a b l e i n t h e r i g h t above sense near t h e b i f u r c a t i o n p o i n t . precisely, the s t a b i l i t y region f o r E

f o r which

P(-;E)

p(-;Ej

More

( t h a t i s , the set o f a l l

i s s t a b l e ) i s n o t empty f o r any d i f f u s i o n

c o e f f i c i e n t s and any domain

0.

I t i s c l e a r t h a t t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n p(.;~)

t o ( 4 . 2 ) i s v i r t u a l l y independent of t h e m a t r i x

domain

R; hence i t i s d e f i n e d on some f i x e d

depend on

D

and

c o n t i n u e s t o be not-empty.

0

and

and t n e

c - i n t e r v a l t h a t does n o t

R. However, t h e s t a b i l i t y r e g i o n f o r

mentioned above may v a r y according as

D

p(t;i)

as

R vary, even if i t

T h i s f a c t suggests t h e p o s s i b i l i t y of t h e

occurrence o f d e s t a b i l i z a t i o n t h a t m i g h t be observed when we v a r y

or

a.

More p r e c i s e l y , i t w i l l be shown t h a t t h e s t a b i l i t y r e g i o n

s h r i n k s when t h e d i f f u s i o n c o e f f i c i e n t s

d . l i = l , - - - - ,n) become v e r y 1

D

Yoshihisu MORITA

118

small o r t h e shape o f

R

becomes f a r from being convex; hence,

a c c o r d i n g l y , t h e b i f u r c a t i n g p e r i o d i c s o l u t i o n loses i t s s t a b i l i t y We s h a l l discuss t h i s i n t h e p r e s e n t

very near t h e b i f u r c a t i o n p o i n t . and n e x t s e c t i o n s .

To see how t h e d e s t a b i l i z a t i o n o f t h e s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n occurs, l e t us c o n s i d e r t h e f o l l o w i n g l i n e a r i z e d equation o f (3.2) around t h e p e r i o d i c s o l u t i o n

For any where

E C (O,E,,),

(4.3) i s a p e r i o d i c system w i t h p e r i o d

i s g i v e n i n Theorem

E~

y

I f f o r some

T ( E ) = PTT/w(E),

H.

We s h a l l seek f o r t h e s o l u t i o n

We c a l l

p(t;E):

z(t)

t a k i n g t h e form,

a Floquet exponent o f (4.3) i f such a s o l u t i o n e x i s t s . y

with

Rey > O

then t h e p e r i o d i c s o l u t i o n

z ( t ) o f (4.4) i s a s o l u t i o n t o ( 4 . 3 ) ,

p(t;E)

Now we adopt t h e new v a r i a b l e s

i s unstable. s = u ( ~ ) t ,y ( s 1 = z ( s / w ( E ) ) .

Then

(4.3) i s trnasformed i n t o

where

Let

be t h e j - t h eigenvalue o f t h e o p e r a t o r -AN and j e i g e n f u n c t i o n corresponding t o h j , i .e., X

JI

j

be t h e

1)elay-Diffusion Equatioiis

Considering t h a t y

WGyp(i2)

i s spanned by

i s a F l o q u e t exponent o f t h e l i n e a r

and o n l y i f t h e r e e x i s t a f u n c t i o n

119

{$jlj=1,2,...

, we

see t h a t

2 n - p e r i o d i c system (4.5) if

q(s)

and a p o s i t i v e i n t e g e r

q(s)

i s a continuous

j

such t h a t

s a t i s f i e s t h e e q u a t i o n (4.5), f u n c t i o n and

where

2n-periodic

q f s ) f 0.

S u b s t i t u t i n g (4.7) i n t o (4.51, and compairing t h e c o e f f i c i e n t s o f $.

J

on t h e b o t h s i d e s o f (4.5), we g e t o ( E )dx q ( s ) =

(4.8)

- (Y +

XjD)q(s)

The e q u a t i o n (4.8) i s independent o f t h e s p a t i a l v a r i a b l e

x.

When

j = 1 , t h e e q u a t i o n (4.8) c o i n c i d e s w i t h t h e one induced from t h e l i n e a r i z e d e q u a t i o n of (2.1) i n t h e absence o f d i f f u s i o n , t h a t i s ,

The e q u a t i o n (4.9) has F l o q u e t exponents where

B(E)

i s as i n (3.2).

0

and

B(E) < 0

for

E

~ ( O , E ~ ) ,

Moreover, we see from t h e s t a b i l i t y

assumption t h a t a l l t h e remaining F l o q u e t exponents have s t r i c t l y negative r e a l parts. Next c o n s i d e r t h e case

j # 1

i n (4.8).

i t ; and p u t

E

=

X.D J

Then t h e e q u a t i o n (4.8) i s w r i t t e n as

.

Take any

j >1

and f i x

Yoshihisa MORITA

120

After Scaling exponents as

E2

-+

E=

we s h a l l seek f o r t h e p a i r of F l o q u e t

$E‘,

y + ( c ) o f ( 4 . 1 0 ) such t h a t

Y-(E),

y-(E)

+

B(E), y+(~)

+

0

0.

Then t h e f o l l o w i n g lemma i s o b t a i n e d ( t h e p r o o f i s f o u n d i n [13 ; 531):

Leiiima A. C o n s i d e r t h e l i n e a r i z e d e q u a t i o n ( 4 . 5 ) o f (4.1) s p a t i a l l y homogeneous p e r i o d i c s o l u t i o n

Let

E2

be an

p(t;e)

around t h e

i n Theorem

H.

diagonal matrix withnon-negativeelements.

inxm

Define the equatim,

where

co,

ct

and

El

a r e d e f i n e d i n (2.13) and ( 3 . 4 ) r e s p e c t i v e l y .

Assume t h a t ( 4 . 1 1 ) has two d i s t i n c t r e a l r o o t s f o r

U
18

t

1

and i s s u f f i c i e n t l y c l o s e t o

Our main t o o l i s a g e n e r a l i z e d i m p l i c i t f u n c t i o n theorem. what s o p h i s t i c a t e d estimates a r e r e q u i r e d . and t h e ideas.

yo

ro.

Hence some-

We o n l y o u t l i n e t h e methods

Complete p r o o f w i l l be presented elsewhere.

Although

our r e s u l t s a r e f a r from t h e assurance o f t h e s t a b i l i t y o f t h e s o l u t i o n , we t h i n k t h a t our i n v e s t i g a t i o n g i v e s an i n s i g h t f o r t h e s t a b i l i t y and instability. Acknowledgment.

The w r i t e r i s g r a t e f u l t o Professor

H. F u j i i who gave

him i m p o r t a n t comments on t h e b i f u r c a t i o n equations w i t h 2.

O(2)-symmetry.

Formulation by t h e p e r t u r b a t i o n method.

I n t h i s s e c t i o n we s o l v e Problem (NS). We f i r s t n o t e t h a t (1.8) i s s a t i s f i e d i f (1.8)* below i s s a t i s f i e d f o r some f u n c t i o n

We n e x t d e f i n e f u n c t i o n spaces.

Let

f(t):

T > 0 and 0 < 6 < a < 1 be f i x e d :

Free Boundary Problems for Perfect Fluid

We f i r s t g i v e a f u n c -

Our p l a n t o c o n s t r u c t a s o l u t i o n i s as f o l l o w s . tion

u

E

X.

Then we c o n s t r u c t a time-dependent c l o s e d Jordan curve

( see t h e c o n d i t i o n (2.1),(2.2)

s a t i s f y i n g (1.13)

{ y u ( t ) )Ol o f A pass through t h e o r i g i n ( see F i g . II ). putting

a**

152

Hisashi OKAMOTO

O i a < a *

a = a* Fig.

REMARK 3.1.

a** < a

II

One can see f r o m t h e c o n s i d e r a t i o n above t h a t t h e t r i v i a l

solution loses the s t a b i l i t y a t l e a s t i n Now we t u r n t o (3.5). claim that tunately

f

E

Co([O,T]

a = a**.

Using a Sobolev imbedding theorem, we can

; H3+(’/2)+a(S’))

0 C ([O,T] ; H3+(1/2)+a(S1))

i m p l i c i t f u n c t i o n theorem.

Y.

implies t h a t

w

E

X.

Unfor-

Hence we use here a g e n e r a l i z e d

The w r i t e r , however, can n o t say a n y t h i n g

f o r t h e n e c e s s i t y ( as i n Hanzawa [ 8 ] ) o f a g e n e r a l i z e d i m p l i c i t funct i o n theorem.

Anyway t h e v e r i f i c a t i o n o f t h e assumptions needed f o r t h e

i m p l i c i t f u n c t i o n theorem i s t o o l o n g t o g i v e here.

Hence we s t o p here.

4. Remarks on t h e s t a t i o n a r y problem. I n [5] we show t h a t b i f u r c a t i o n s from t h e t r i v i a l s o l u t i o n do occur i f

an

i s simple, i . e . ,

an

4

tamlmfn.

I f an = am f o r some

n f m, then we have t o deal w i t h an O ( 2 ) - e q u i v a r i a n t b i f u r c a t i o n problem. The eigenspace i s spanned by s i n n e , con ne

, sinme

t h e Lyaponov-Schmidt r e d u c t i o n we o b t a i n a mapping

F : R4xR+ W

4

and

cosme

.

By

Free Boundary Problems f o r Ferfect Fluid

F(x,y,z,w;a-an)

such t h a t t h e s o l u t i o n o f

mapping

F

i ) For a l l

is a

Q= Q o ( r )

for

t i o n s o f Problem ( S )

= 0

153

correspond t o t h e s o l u -

i n a one-to-one manner.

This

O ( 2 ) - e q u i v a r i a n t i n t h e f o l l o w i n g sense:

E

[0,2rr)

the equality

F(xcos na + y s i n na , - x s i n na +ycos na ,zcos ma +wsin ma ,-zsin ma +wcos ma ;a-a )

n

1 F1 (x,y.z,w;a-an)cos

-

F (x,y,z,w;a-a 1

n ) s i n na+ F2 (x,y,z,w;a-a

F3( x,y,z,w;a-an)cos

-

F3(x,y,z,w;a-a

na+ F2(x,y,z,w;a-an)sin

n

na

)cos na

ma + F4( x,y,z,w;a-an)sin

ma

n ) s i n m + F4(x,y,z,w;a-an)cos

[TKX J

holds true. ii)

n ) : ( F1 , -F2 , F 3 , -F4 )(x,y,z,w;a-an).

F(x,-y,z,-w;a-a

Complete a n a l y s i s o f t h e s o l u t i o n s e t of However, Problem ( S )

F = 0

i s n o t made so f a r .

has t h e f o l l o w i n g p r o p e r t y : I f

t o t h e space spanned by

cos n6 and

F

i s restricted

cosme , then t h e image o f

F

is

T h i s r e s t r i c t e d mapping can be analyzed d i r e c t l y .

a l s o i n t h a t space.

Indeed, o u r circumstance i s t h e same as t h a t i n F u j i i , Mimura and N i s h i u r a [l]. The two b i f u r c a t i o r i parameter respond t o t h e parameter o u r problem ( S )

a

and

u

d,

i n o u r problem.

R4x

R

Hence, i n general,

F(x,y,z,w;a-a

n) = 0

But we have r e c e n t l y succeeded t o g i v e a

normal f o r m f o r t h e 4-dimensional case where n = 2m.

i n [l] c o r -

has more complicated s o l u t i o n s e t , which i s n o t

so f a r s t u d i e d completely.

and

d2

has a complicated s o l u t i o n s e t ( see [l] ) . F u r t h e r -

more, as n o t e d by P r o f e s s o r F u j i i , t h e e q u a t i o n i n t h e whole

and

an = am

( 0 < m < n )

T h i s is c a r r i e d o u t by t h e technique in [2,3].

Hisashi OKAMOTO

164

REFERENCES. H. F u j i i , M. Mimura and Y . N i s h i u r a : A p i c t u r e o f t h e general

b i f u r c a t i o n diagram i n e c o l o g i c a l i n t e r a c t i n g and d i f f u s i n g system, Physica D,

5

(1982) 1-42.

M. G o l u b i t s k y and 0. Schaeffer: A t h e o r y f o r i m p e r f e c t b i f u r c a t i o n

v i a t h e s i n g u l a l i t y theory,

Comm. Pure Appl. Math.,

32

(1979121-98.

M. G o l u b i t s k y and D. Schaeffer: I m p e r f e c t b i f u r c a t i o n i n t h e presence o f symmetry,

Comm. Math. Phys.,

67 (1979)

205-232.

I. Imai: Conformal Mappings and T h e i r A p p l i c a t i o n s , ( i n Japanese ) , Iwanami Shoten, Tokyo (1 979). H. Okamoto: B i f u r c a t i o n phenomena i n a f r e e boundary problem f o r a

c i r c u l a t i n g f l o w w i t h s u r f a c e t e n s i o n , ( submitted t o

Math. Meth.

Appl. S c i . ) . H. Okamoto: A s t a t i o n a r y f r e e boundary problem f o r a c i r c u l a r f l o w

w i t h o r w i t h o u t s u r f a c e tension, Proc. Japan Acad.,

58

(1982) 422-

424. [71

H. Okamoto: S t a t i o n a r y f r e e boundary problems f o r c i r c u l a r f l o w s

w i t h o r without surface tension,

t o appear i n t h e Proc. o f U.S.-

Japan Seminar on N o n l i n e a r P.D.E.

i n A p p l i e d Sciences, e d i t e d by

H. F u j i t a , P.D.

Lax and G. Strang. h

E . I . Hanzawa: C l a s s i c a l s o l u t i o n s o f t h e S t e f a n problem,

J . Math.,

2

Tohoku

(1981) 297-335.

E. Zehnder: Generalized i m p l i c i t f u n c t i o n theorems w i t h a p p l i c a t i o n s t o some small d i v i s o r problems, I , (1975) 91-140.

Comm. Pure Appl. Math.,

28


Global Existence Theorem for Nonlinear Wave Equation in Exterior Domain Yoshihiro SHIBATA* and Yoshio TSUTSUMI**

*

*Department of Mathematics, University of Tsukuba Ibaraki 306, Japan **Department of Pure and Applied Sciences, College of General Education, University of Tokyo Tokyo 113, Japan Supported in part by the Sakkokai Foundation.

81.

Introduction.

The g l o b a l e x i s t e n c e o f s o l u t i o n s f o r t h e n o n l i n e a r wave e q u a t i o n has been extensively studied. improvement r e c e n t l y .

F o r t h e Cauchy problem Klainerman [ Z ] has made a remarkable That i s , he showed t h a t i f t h e s p a t i a l dimension i s n o t

s m a l l e r than 6 and i n i t i a l d a t a a r e s m a l l and smooth, then t h e Cauchy problem f o r t h e f u l l y n o n l i n e a r wave e q u a t i o n has a unique c l a s s i c a l g l o b a l s o l u t i o n .

On t h e o t h e r hand i t i s i m p o r t a n t t o c o n s i d e r t h e i n i t i a l boundary v a l u e problem f o r t h e n o n l i n e a r wave e q u a t i o n i n an e x t e r i o r domain i n o r d e r t o s t u d y s c a t t e r i n g o f a r e f l e c t i n g o b j e c t f o r t h e n o n l i n e a r wave equation.

I n t h e p r e s e n t paper we

s h a l l prove t h a t i f t h e s p a t i a l dimension i s n o t s m a l l e r than 3 and i n i t i a l data a r e small and smooth, then we have t h e g l o b a l unique e x i s t e n c e theorem o f c l a s s i c a l solutions f o r

a

l a r g e c l a s s o f n o n l i n e a r wave equations i n e x t e r i o r domains

w i t h t h e homogeneous O i r i c h l e t boundary c o n d i t i o n , which i n c l u d e s t h e n o n l i n e a r v i b r a t i o n equation.

n

Let

be an unbounded domain i n lRn

,n2

3, w i t h Cm and compact boundary an.

We denote a t i m e v a r i a b l e by t o r xo and a space v a r i a b l e by x = (x1,.-.,xn), respectively. o r ,a,

(*)

a . and ,;a J

We s h a l l a b b r e v i a t e a / a t , a/ax. and (a/axl)al...(a/axn)an

J

r e s p e c t i v e l y , where

a

i s a multi-index w i t h

Supported i n p a r t by t h e Sakkokai Foundation. 155

101

to a t

= al+***+a

n

Yoshihiro SHIBATA and Yashio TSUTSUMI

166

.

and j = l,...,n

We s h a l l consider t h e f o l l o w i n g problem:

u = o

on [OP)

~ ( 0 ~ =x 1$o(X), where

2

t11= at

-

A =

a t2

x

an,

(atu)(o,x)

- j.11n a Jz. and

= $,(XI

hu = (aiu,

i n n,

a J. a ku, j,k=O,.-.,n).

i=O,.-.,n;

Before we s t a t e assumptions and t h e main theorem, we s h a l l g i v e n o t a t i o n s .

For any i n t e g e r N 2 0 we w r i t e

L e t 6 be an a r b i t r a r y open s e t i n Rn

.

F o r any p w i t h 1 5 p 5

standard Lp space d e f i n e d on p and i t s norm by Lp(S) and

ml

II.llB,py

we denote t h e respectively.

For a v e c t o r v a l u e d f u n c t i o n h = ( h l y . - * , h s ~ we p u t 2 ~ y~ p~ h j ~ ~ ~ , p l h I 2 = l h 1 I 2 + - . * + l h s l , ~ ~ h ~= / 1 j=l We a l s o w r i t e

We s e t HP~ ( c - ) = t f

E

L’’(F)

;

IIfll,,p,N

ro we denote t h e subset I x

E

n ; 1x1

< r 1 by

nr.

E

P R n ; 1x1 < ro I .

For

F o r any r > ro and any

i n t e g e r k 2 1 we p u t 2 ~,(n) = I u

E

~ ~ ( ;n supp ) uct x

E

R” ; 1x1 5 r I 1,

Ok Hr(n)

E

k Hz(n) ; supp u

E

R n ; 1x1 5 r 1,

I u

c( x

“0 2 kle s h a l l sometimes use Hr(n) = lr(n). (u,v),

and t h e D i r i c h l e t norm

llullD by

a:ulas2 = 0 (la1 L k-111.

We d e f i n e t h e D i r i c h l e t i n n e r p r o d u c t

Nonlinear Wave Equation in Exterior 1)wnain

157

we denote t h e completion o f $(n) i n t h e D i r i c h l e t norm. By ,uN(&) we N denote t h e s e t o f C ( b ) - f u n c t i o n s having a l l d e r i v a t i v e s o f o r d e r 5 N bounded i n E . F o r two Banach spaces X and Y we denote t h e Banach space c o n s i s t i n g o f a l l

By H,(Q)

bounded l i n e a r o p e r a t o r s from X t o Y and i t s norm by B(X,Y) and II-II ]B(X,Y) ' 1 r e s p e c t i v e l y . For an i n t e r v a l I(.- R and a Banach space X we denote t h e s e t

o f m-times c o n t i n u o u s l y d i f f e r e n t i a b l e X-valued f u n c t i o n on I by Cm(I;X).

We s e t

For 1 ;< p 5 =, a nonnegative number k and a nonnegative i n t e g e r N we w r i t e

hL =

I

U

c IELr\CL-l([O,-);H,(s2));

L atu(O,x)

= 0 1.

F o r s i m p l i c i t y we a l s o use t h e a b b r e v i a t i o n s :

f(c)

= f(cl,-*-.~n)

where xc = x

=

1

e x p ( - a x c ) f ( x ) dx,

IRn

1 1

v . = (v~,.**,vJ)

+...+xncn.

5,

i, v e c t o r s u = (u, . * . . , u s ) ,

( 1 5 j 2 i) and a s c a l a r f u n c t i o n H(t.x,u)

J .

(dLH)(t,x,u)(vl

For p o s i t i v e integers

,*-*.vi)

by

(V

E

IRs) we d e f i n e

Yoshihiro SHIRATA and Yoshio TSUTSUMI

158

We s h a l l make t h e f o l l o w i n g assumptions.

Assumption 1.1. (2)

(1)

The s p a t i a l dimension n 2 3.

The n o n l i n e a r mapping F i s a r e a l - v a l u e d f u n c t i o n belonging t o

m

([0,-)

x

sl

x

{ A

E

R 2(n+1)

.> I XI

5 - 1 I).

(3) F(t,x,A)

A =

0(1~1~)

x

near

0,

= 0,

if n 1 6 , if 3 5 n

5.

The e x t e r i o r domain n i s "non-trapping" i n t h e f o l l o w i n g sense:

(4)

G(t,x,y)

O ( 1 ~ 1 ~ ) near =

Let

be t h e Green f u n c t i o n f o r t h e f o l l o w i n g problem

2 (at

-

A ~ ) G= 0

i n ( 0 , ~ ) x n,

where y i s an a r b i t r a r y p o i n t i n n and ax i s t h e Laplace o p e r a t o r w i t h r e s p e c t t o x.

L e t a and b be a r b i t r a r y p o s i t i v e constants such t h a t b 2 a 2 ro. F o r

f o r any v

E

L:(n),

Remark 1.1.

where To depends o n l y on n, a, b and n.

I t is w e l l known t h a t i f t h e complement of B i s convex, t h e n

Assumption 1.1(4) i s s a t i s f i e d (see, e.g.,

Melrose [5]).

1.io

Nonlinear Wave Equation i n Exterior Dxnain

Now we s h a l l s t a t e t h e main theorem.

Theorem 1.1.

L e t m be an a r b i t r a r y i n t e g e r w i t h m

(Existence).

G.

Let

Assumption 1.1 be a l l s a t i s f i e d . 1)

P u t m = 2max(4[n/2]+7,

m+l) t 4[n/2]

4-

1. I f n 2 6, then t h e r e e x i s t

p o s i t i v e constants a and 6 o having t h e f o l l o w i n g p r o p e r t i e s : $1 E.6 2h[n/21+2(5)

and

f

€3:

2m+[n/21+1([0,-)

x

i)

If

@o

,

2m+[n/2]+3(;)

€1,.

s a t i s f y f o r some 6 w i t h 0.

6 ~6~

and t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r m , then Problem (M.P) has a s o l u t i o n

u

E

Cm +2

([o,-)

x

IAu12,0,m 2)

+

ii)

satisfying

lAu14,(n-1)/4,m

6’

Put m = 2max(3[n/2]+6,

m+l) + 3[n/2]

+ 7.

If 4

n 5 5, then t h e r e e x i s t

p o s i t i v e constants a and 6 o having t h e f o l l o w i n g p r o p e r t i e s :

@1 E J ~ ‘ ~ ’ ( ; )

and f

Il@OIlm,2iT;+2

+

E F2Fn ([0,-)

1141 Ilm,2iii+l

x

+

5)

s a t i s f y f o r some 6 w i t h 0

Iflm,o,2in‘

If

o0

< 6

E),

2m+ 2

(E),

A0

2 a6

and t h e c o m p a t i b i l i t y c o n d i t i o n of o r d e r ‘m, then Problem (M.P) has a s o l u t i o n

u

E

c ~ + ~ ( [ o x, i) ~ )satisfying IAU12,0,m

3)

$2

Let

+

E

IAUl-,(n-1

)/z.in

=

1 and

1, then f o r each nonnegative i n t e g e r N

( 2 ) i f n 2 3 and p and p' a r e p o s i t i v e numbers ( p may be i n f i n i y ) such t h a t 1-1 1 , = 1, then f o r any s u f f i c i e n t l y small a > 0 and each -12 = 1 and -1 + -(1P P 2 P onnegative i n t e g e r N

Nonlinear Wave Equation in Exterior Domain

163

We s h a l l d i v i d e t h e p r o o f o f Theorem 2.1 i n t o s e v e r a l steps.

The s t r a t e g y

o f t h e p r o o f f o l l o w s Shibata [12] (see a l s o Tsutsumi [13]).

2.1. Local Energy Decay,

Theorem 2.2.

Here we s h a l l show t h e f o l l o w i n g theorem.

L e t n 2 3, Assume t h a t Assumption 1.1(4) holds.

L e t a and

b be a r b i t r a r y p o s i t i v e constants w i t h a, b 2 ro. L e t t h e d a t a J I ~J I, ~and f be smooth f u n c t i o n s s a t i s f y i n g t h e c o m p a t i b i l i t y c o n d i t i o n o f o r d e r i n f i n i t y such . R ( j = 0, 1) and supp f -. R i x Then, f o r any q w i t h J - a q 2 n-1 and each nonnegative i n t e g e r N t h e smooth s o l u t i o n u o f (2.1)

t h a t supp

0

$J.

s a t i s f i e s t h e f o l 1owing e s t i m a t e :

Following

We s h a l l f i r s t s t a t e t h e theorem needed f o r t h e proof o f Theorem 2 . 2 . Lax and P h i l l i p s C41, we s e t t h e H i l b e r t s p a c e 3 = { f = (fl,f2)

f2

E

2 L ( 8 ) 1 w i t h the inner product (f,g)

g = (g1,g2)),

= (fl,gl)D

; fl

+ (f2,g2)L2(n) 2

where ( - , - ) L 2 ( 8 ) i s t h e i n n e r p r o d u c t i n L ( 8 ) .

H,(n),

E

( f = (fl,f2),

F o r f = (fl,f2)

~ j !

we d e f i n e t h e l i n e a r o p e r a t o r A by

Then, i t f o l l o w s t h a t A i s a skew a d j o i n t o p e r a t o r ori

2 L ( 8 ) n H,(n)

& H 2 ( 8 ) n H,(n).

generated by A.

Theorem 2.3.

(1)

w i t h t h e domain D(A) =

L e t I U ( t ) 1 be t h e one parameter u n i t a r y group

F o r U ( t ) we have t h e f o l l o w i n g theorem.

L e t a and b be a r b i t r a r y p o s i t i v e constants w i t h a, b

Assume t h a t Assumption 1.1(4) holds.

I, 2).

;.

L e t f = (fl,f2) E X w i t h supp f . c J

> Qa

ro. (j =

Then,

i f n i s odd and n 2 3, then t h e r e e x i s t two constants C, 6

0 such t h a t

Yoshihiro SHIBATA and Ycshio TSUTSUMI

164 l u ( t ) f \ p , (b) =
0 such

that

Remark 2.1.

Theorem 2.3(1) i s a l r e a d y w e l l known.

When n i s even and

n 2 4, t h e decay r a t e i n Theorem 2.3(2) seems t o be sharper than t h a t of a l r e a d y Melrose [5!).

known r e s u l t s (see, e.g.,

We d e f i n e 0- = I k

Sketch o f t h e p r o o f o f Theorem 2.3. and =

1 (1" {

k

, E

U(k.)f =

Q*'

;

-

3

n < arg k
0.

s h i f t t h e contour o f t h e i n t e g r a l (2.3) by t h e r e l a t i o n ( A + k 2 ) - ' = k - ' -

+ k2)-'A and t h e f o l l o w i n g t h r e e lemmas ( f o r d e t a i l s , see Vainberg [17]

k-'(,

and Tsutsumi [14]):

Lemma 2.4.

(Vainberg [15]).

Let n

3.

w i t h a, b > ro. The r e s o l v e n t ( A + k 2 ) - l ( k to

D

2 2 as a B ( L a ( a ) , H ( n b ) ) - v a l u e d

function,

L e t a and b a r e p o s i t i v e constants L

D-) adntits a meromorphic e x t e n s i o n

Furthermore, t h e s e t o f a l l

p o l e s o f t h e meromorphic e x t e n s i o n has no l i m i t p o i n t i n 0 and does n o t l i e i n

D- i' ( R 1\

{

0

1).

Below we a l s o denote t h e meroniorphic e x t e n s i o n by ( A

Lemma 2.5. and n 2 3.

(Vainberg [17]).

+

kL)-'.

L e t a and b be p o s i t i v e constants w i t h a, b

Assume t h a t Assumption 1.1(4) holds.

ro

>

Then t h e r e e x i s t p o s i t i v e

constants a , B , C and T such t h a t f o r i n t e g e r s 0 5 s

1 and 0 2 j 5 2

i n the region V = I k s D ;

Lemma 2.6.

Vainberg [ 5, 161 and Tsutsumi [14]).

constants w i t h a, b > ro and n 2 3. such t h a t :

Ikl

(1)

(2)

Then t h e r e e x i s t s a p o s i t i v e c o n s t a n t y

i f n i s odd, ( A + k2)-'

Y j;

i f n i s even,

L e t a and b be p o s i t i v e

i s holomorphic i n t h e r e g i o n W = { k

E

0;

Yoshihiro SHIBATA and Yoshio TSUTSUMI

166

+ k 2 ) - l = Bl(k)

(A

i n t h e r e g i o n W' = I k

t k"'(1og

E

k)B2 t kn-2B3(k),

D ; Ikl

< y

2 2 B ( L a ( n ) , H ( 0, ) ) - v a l u e d f u n c t i o n , B2 i n W ' as a lB(Lg(i?),H2(~,))

I , where Bl(k) i s holomorphic i n W' as a 2 2 B(La(a),H (n,))

E

,

and B3(k) i s continuous

-valued function.

Now we s h a l l s t a t e t h e p r o o f o f Theorem 2.2.

Proof o f Theorem 2.2.

'

i where vo

& V = A V

L e t \r be t h e s o l u t i o n o f

tf,

-

m

0,


167

A t l a s t Theorem 2.2 f o l l o w s from an i n d u c t i v e argument, (2.4) and t h e f o l l o w i n g we1 1-known e l 1i p t i c e s t i m a t e :

Lemma 2.7.

L e t a and b be a r b i t r a r y p o s i t i v e constants w i t h a

L e t a f u n c t i o n u s a t i s f y au = g i n na and u = 0 on an.

b > r 0'

>

Then, f o r each i n t e g e r

N 2 0, u s a t i s f i e s

2.2. Space.

Uniform Decay E s t i m a t e f o r S o l u t i o n s t o Wave Equation i n t h e F r e e I n t h i s s e c t i o n we s h a l l summarize t h e r e s u l t s concerning t h e decay o f

t h e s o l u t i o n t o t h e problem (2.5)

1x11=

i n [o,-)

f

U(O,X) = $,(XI,

x

(atu)(o,x)

R", i n R".

= q(x)

F o r g E Y ( R ~ we ) d e f i n e T ( t ) by a l i n e a r o p e r a t o r which naps g i n t o a s o l u t i o n o f t h e problem ( 2 . 5 ) w i t h

I),=,0,

$1 = g and f = 0.

Taking t h e

F o u r i e r t r a n s f o r m o f T ( t ) , we have

By u s i n g t h e above r e p r e s e n t a t i o n and t h e i n t e r p o l a t i o n technique we have t h e f o l l o w i n g w e l l known lemma (see, e.g.,

Lemma 2.8.

(2.6)

von Wahl [18] and Shatah [ l o ] ) :

F o r each i n t e g e r N 2 0 and any p w i t h 2

N+ 1 [ID T(t)gll;; 2 C(p,N,n)

n-1 --(1--) t

p

2 Ilgllp',N+[n/2]+2'

m

we have


168

1 1 f o r a l l t > 0, where p’ i s a r e a l number w i t h - + -,= 1. P P

From Lemma 2.8 we have t h e f o l l o w i n g theorem:

Theorem 2.9.

L e t n 2 3.

L e t u ( t , x ) be t h e smooth s o l u t i o n o f (2.5) w i t h

, c + ( R n ) and f t h e data $o E Y ( R ~ ) $1

Cm([O,-)

R’)

bounded i n a l l norms 1 1 =l. below. L e t p and p’ be p o s i t i v e numbers such t h a t Y ( 1 - i ) ~1 and - + P P F o r s u f f i c i e n t l y small a > 0 we p u t E

x

-.

Then, f o r each i n t e g e r N

2 0,

Proof o f Theorem 2.9.

u s a t i s f i e s t h e f o l l o w i n g estimates:

u ( t , x ) can be represented as

u ( t ) = zd T ( t ) $ o + T ( t ) $ 1 +

:1

T(t

-

5)

f ( S ) ds.

Therefore, we o b t a i n Theorem 2.9 by u s i n g (2.6) o r (2.7) for t > 1 and (2.8) for 0 < t

i f -(I--) n-1 2

2 P

>

1,

n l 2 if Z ( l - - ) 2 P

o),

= 1.

Next l e t y ( x ) be a f u n c t i o n b e l o n g i n g t o C i ( R n ) such t h a t ~ ( x =) 1 f o r 1x1 2 ro +1 and ~ ( x =) 0 f o r 1x1 2 ro + 2. (2.11)

u2(t,x)

= u(t,x)

-

(1

-

y(x))ul(t,x),

where u ( t , x ) i s t h e s o l u t i o n o f ( 2 . 1 ) . (2.12)

u u 2 =

Y f

+

9

Put

Then u2 s a t i s f i e s

i n [O,-)

x 0,


170

n

where g = 2

1

j=1

a .y a .u

J 1

J

t AY u1

.

From (2.10) and (2.11) we have o n l y t o

e v a l u a t e u2 i n o r d e r t o o b t a i n t h e e s t i m a t e o f u. Applying Theorem 2.2 t o (2.12) w i t h b = ro+ 5, we have f o r any i n t e g e r N z 1

By t h e d e f i n i t i o n o f g and (2.10) we have f o r any i n t e g e r

N 20

where b = ro + 5. We s h a l l n e x t e v a l u a t e u2 f o r 1x1 > ro + 5.

Let

U(X)

be a Cm-function

such t h a t ~ ( x =) 1 f o r 1x1 2 ro t 3 and u ( x ) = 0 f o r 1x1 2 ro + 4. (2.14)

u ( ( 1 - u ) u 2 ) = (1-u)(yf t 9) + h

in

Then

[o,-) x R" ,

n

where h = 2

1 j=l

a

u

j

a u

j 2

+

AU

u2.

Applying Theorem 2.9 t o (2.14), we have by

(2.13)'

' If1p',q,N+2[n/2]t3 where

' If12,q,Nt2[n/2]t2


Therefore, we o b t a i n Theorem 2.1 by (2.10),

(2.13),

171

(2.15) and t h e Sobolev

imbedding Theorem.

(Q. E. D.)

Some Estimates f o r S o l u t i o n s of L i n e a r i z e d Problem.

53.

I n t h i s s e c t i o n we s h a l l show an L 2- e s t i m a t e and a u n i f o r m decay e s t i m a t e o f s o l u t i o n s f o r t h e f o l l o w i n g l i n e a r problem: (3.1)

2

0

= (1 + a (t,x))atu

,f,u

-

where 6.

1j

n ’ 1 aJ(t,x)a.a u j=l J t

n

1 (&ij i,j=l

u = o u(0,x)

+

n

+

t aij(t,x))a.a.u

1 J

on [0,m)

. bJ(t,x)a.u = f ( t , x ) J

an,

x

i n R,

= (atu)(oyx) = 0

= 1 i f i = j and 6ij

1 j=o

= 0 if

iC j

We make t h e f o l l o w i n g assumptions:

Assumption 3.1. = (aJ(t,x),

(1)

g=UO,-)

Put j = O,..-,n;

A l l components o f & x

a

ij

(t.x),

i,j = l,-.-,n;

bJ(t,x),

j = O,....n)

are real-valued functions belonging t o

5).

(2)

aij(t,x)

(3)

F o r a l l 6 = (,

P(t,x,dy)

t h e infinitesimal generator

0

=

W

for a l l

of

> 0

E

and

x E

5.

is d e s c r i b e d a n a l y t i c a l l y as

ITt)

f o l l o w s ( c f . [l], [21, [131) : Let

i) u

E

x

be a p o i n t of t h e i n t e r i o r

2

D ( ( T 0 n C (D)

where

Let

and

c(x)

2

For

0.

choose a system of l o c a l c o o r d i n a t e s

Then

5

b e a ( r e g u l a r ) p o i n t of t h e boundary

XI

neighborhood of

of

we have

10

(aiJ(,))

ii)

,

D

x'

such t h a t

u E D(fl)nC2(z)

x E D

x

=

if

, x ~ -,%) ~

(x1,x2,

%

> 0

of

aD

and

x

E

aD

5,

and

in a if

5=

s a t i s f i e s t h e boundary c o n d i t i o n of t h e form :

= o where

(aij(x'))

(n1,n2,

... ,%)

condition

L

2 0 , y(x')

2

0,

~ ( x ' )5 0 ,

& ( X I )2 0

is t h e u n i t i n t e r i o r normal t o

aD

at

and

n =

X I .

The

is c a l l e d a V e n t c e l ' s boundary c o n d i t i o n .

P r o b a b i l i s t i c a l l y , t h e above r e s u l t may be i n t e r p r e t e d as follows.

0 .

Kazuaki TAIRA

204

A particle in the diffusion process (strong Markov process with continuous paths)

x

-

on D

operator A

is governed by a degenerate elliptic differential

of second order in the interior D of the domain, and it

obeys a Ventcel's boundary condition L on the boundary

'

domain. The terms of L

axiaxj

i,j a

,

aD of the

au

yu,

and 6 Au

i

are supposed to correspond to the diffusion along the boundary,

absorption, reflection and viscosity phenomena respectively. Analytically, via the Hille-Yosida theorem in the theory of semigroups,

-

it may be interpreted as follows. A Feller semigroup {TtItL0 on D described by a degenerate elliptic differential operator A

x

of second

if the paths of its correspond-

order and a Ventcel's boundary condition L ing strong Markov process

is

are continuous. We are thus reduced to the

study of non-elliptic boundary value problems for

in the theory

(A,L)

of partial differential equations.

In this chapter we shall consider the following PROBLEM. Conversely, given analytic data a Feller semigroup -In the case N

=

(A, L )

,

can we construct

1 , this problem is completely solved both from

probabilistic and analytic viewpoints by Feller, Dynkin, I&, and Ray.

So

we shall consider the case N

2

Mckean Jr.

2.

12.1 Statement of Results Let D be a bounded domain in IRN with smooth boundary A

aD.

Let

be a second-order differential operator with real coefficients such

that N A~(X) =

z

i,j=1

,. alJ(x)

aZu i j

N

(x) i=l

+

c(x)u(x)

(x

E

D)

T!iffi:sion Proccsws and Partial DifTerential Equations

where the coefficients of

I

(2.1)

A

205

satisfy:

N

X

2

aij(x)cicj

for all

0

x € R N and

5

E

IRN

i.j=1

,

Now consider the function N

b(x')

=

1 ( bi(x') i=l

N

-

aaij - (XI)) J j=1 ax. Z

ni

on

aD,

which is called the Fichera function for the operator A easily seen that the Fichera function b

We divide the boundary

3D

Each

Xi

(i=O,l,

connected hypersurfaces.

is invariantly defined on the set

into four disjoint subsets :

The fundamental hypothesis for

(H)

( [ 4 ] ) . It is

2,3)

A

is the following

consists

of a

finite number of

Kazuaki TAIRA

206

Note that

Z2uZ3

coincides with the set of all regular points of

aD

(cf. (91). Let L be a Ventcel's boundary condition such that 2

N- 1 aiJ(x*)

E

~u(x*)=

i,j=l

+ ~ ( x ' ) an *(XI)

a u axiaxj -

(XI)

+

N-1 E f3 i=l

(XI)

(x'

G(x')Au(x')

E

+

3

(x') axi

y(x')u(x')

2D)

where the coefficients of L satisfy:

1'

aij are the components of a Cm symmetric contravariant tensor field of type (2.0)

on Z 2 u Z 3

and

3 O

y

E

C"(E2uZ3)

and

y(x')

2

0 on

C2"Z3

4 O

p

E

Cm(C2uZ3)

and

u(x')

2

0

on

C2uE3

. .

5'

6

E

Cm(E2UC3)

and

&(XI)

2

0

on

12"13

.

To state hypotheses for L , we introduce some notation and definitions.

As in 81.1, we say that a tangent vector

is subunit for Lo =

N-1

.

2

j=1

For

x'

E

E3

and

X

=

N-1 . E yJ ax j=1 j

a

at

x'

E3

E

N- 1 I aij a2 if ax ax i,j=l i 1 N-1 I aij(x') i,j=l

p > 0,

rl

rl

i j

€ o r all

II =

N- 1 I n. dx. E. TZ,(13) 1=1 J J

we define a "non-Euclidean ball" (of radius

p

Diffusion Proces3e.s and Pfirtial Differential Equations

about x' ) be joined to a

B o(x',p) to be the set of all points y' 6 L 3 which can L x' by a Lipschitz path y : [ O , P ] + L 3 such that {(t) Is

subunit vector for Lo

BE(x',p)

at

The hypothesis for L

0
0

We denote by

t.

pE1

) , x'

E

M

= {

p > 0

x'

E

Z 3 ; p(x')

we have:

=

01

.

Intuitively, hypothesis ( A . l ) means that a Markovian particle with generator Lo goes through the set M , where no reflection phenomenon occurs, in finite time (cf. Theorem 1.1).

In a neighborhood of

we can write the differential operator A

Z2,

uniquely in the form:

where A

j

a

-

A = A

-+

A2

(j =0,1,2) is a differential operator of order j acting

along the surfaces parallel to restriction AtIZt

of At

to

Note that by hypothesis (H) the

It. Z2

is a second-order differential

operator with non-positive principal symbol, and that u on

Z2.

ball"

Thus, for x'

B

E

It

(x', p )

and

p >

0 and L by

in the same way as

L

0

Z2 and L

- f (A21C2)

The hypothesis concerning L (A.2)

on

There exist constants 0

0 we have :

0 and

b < 0

0 , we can define a "non-Euclidean

-b(A2IZ*) Z3

2

Z2 6

B o(x', L

p)

, replacing

respectively. is the following

5 1

C2 > 0 such that for

208

Kazuaki TAIRA

The intuitive meaning of hypothesis ( A . 2 ) with generator Lo

-

(A,

I z2)

is that a Markovian particle

diffuses everywhere in

The Ventcel's boundary condition L

in finite time.

Z2

is said to be transversal on

if

Z2"Z3

u(xl)

+

&(XI) >

n on z 2 ' J z 3 .

Now we can state the main r e s u l t (cf. [ll], [12]): THEOREM 2 . 1 .

satisfy (2.1)

Let the differential operator A

hypothesis (H) and let the boundary condition L

satisfy ( 2 . 2 ) and be

Suppose that hypotheses ( A . I ) , ( A . 2 ) are satisfied. __-Then there exists 5 ___ Feller semigroup { T t I t L o 0" D whose infinitesimal transversal 0"

CZuZ3.

the restriction of A

to the space

u

E

N- 1

Lu(x') =

i

au

5 (x')

C

i=l

N-1

B

=

i:

+

(x') + y(x')u(x')

(x'

E

.

aD:

on

p(x')

%(XI)

an

i

- G(x')Au(x')

Here

of

C 8 ( D ) ; Lu = 0 0" Z Z u Z 3 }

Further consider the case where aij E 0

(2.3)

C(5)

equals the minimal closed extension &

generator a7

aD).

. a

B1 -

is a real Cm-vector field on

aD.

i=1 We introduce the following hypotheses (replacing hypotheses ( A . 1 ) and (A.2)) :

(A.1)'

B

The operator

A

is non-zero on the set M

integral curve of (A.2)'

5

is elliptic near ={

x' c Z 3 ; u ( x ' )

L3 =

and the vector field

0 ) and any maximal

is not ,entirely contained g~ M.

There exist constants 0

0

such that for

Diffusion Prccessev and Partial Differential Equations

sufficiently

p > 0

209

we have:

Hypothesis (A.1)’ (resp. (A.2)’ ) has an intuitive meaning similar to hypothesis ( A . 1 ) (resp. (A.2) )

,

(Cf. Theorem 1.1.)

Then we have the following (cf. THEOREM 2.2. form (2.3). we have the ----

A

and

[lo])

L be as in Theorem 2.1, L beinp of the

Suppose that hypotheses ( A - l ) ’ ,

(A.2)’

are satisfied. Then

same conclusion % & Theorem 2.1.

REFERENCES J.-M. Bony, P. CourrSge et P. Priouret, Semi-groupes de Feller sur une vari6t6 a bord compacte et problemes aux limites int6grodiff6rentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier (Grenoble), 18 (1968), 369-521. E.B. Dynkin, Markov processes, vols I, 11, Springer, BerlinHeidelberg-New York, 1965. Phong, Subelliptic eigenvalue problems, to appear.

[31

C. Fefferman and D.H.

[41

G. Fichera, Sulla equazioni differenziali lineari ellittico-paraboliche del second0 ordine, Atti. Accad. Naz. Lincei Mem., 5 (1956), 1-30. C.D. Hill, A sharp maximum principle for degenerate elliptic-parabolic equations, Indiana Univ. Math. J., 20 (1970), 213-229.

N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, Kodansha, Tokyo and North-Holland, AmsterdamOxford-New York, 1981. R.M. Redheffer, The sharp maximum principle for nonlinear inequalities, Indiana Univ. Math. J., 21 (1971), 227-248.

D.W. Stroock and S.R.S. Varadhan. On the support of diffusion processes with applications to the strong maximum principle, Proc. of 6-th Berkeley Symp. of Prob. and Math. Stat., vol. 111 (1972), 333-359. D.W.

Stroock and S.R.S. Varadh.an, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math., 25 (1972), 651-713.

Kazuaki TAIRA

210

[lo]

K. Taira, Sur l'existence de processus de diffusion, Ann. Inst. Fourier (Grenoble), 29 (1979), 99-126.

[ll] K. Taira, Semigroups and boundary value problems, Duke Math. J., 49 (1982),

[12]

287-320.

K. Taira, Semigroups and boundary value problems 11, Proc. Japan Acad., 58 (1982), 277-280.

I 1 3 1 A.D.

Wentzell (Ventcel'), On boundary conditions for multidimensional diffusion processes, Theor. Prob. and Appl., 4 (1959), 164-177.


Free Boundary Problems for t h e Equations of Motion of General Fluids

A t u s i TAN1 Department of Mathematics, Keio University

Yokohama 223, Japan

1.

Introduction.

problems'is

The o u t s t a n d i n g f e a t u r e o f many famous hydrodynamical t h e somewhat p a r a d o x i c a l f a c t t h a t t h e boundary o f t h e f l o w , on

which c e r t a i n c o n d i t i o n s have t o be s a t i s f i e d ,

i s i t s e l f n o t given.

There

i s a g r e a t v a r i e t y o f problems w i t h f r e e boundaries, some o f which were a l r e a d y i n v e s t i g a t e d i n Newton's time. t i a l l y nonlinear.

And a l l these problems a r e essen-

I n t h e present paper we c o n f i n e o u r s e l v e s t o t h e f r e e

boundary problems f o r t h e system o f d i f f e r e n t i a l equations o f m o t i o n o f compressible viscous i s o t r o p i c Newtonian f l u i d s (say, general f l u i d s ) . N o t a t i o n . For a domain R i n R3, , any non-negative i n t e g e r n and a

€(O,l),

we d e f i n e :

Cn++"(E)={f(x), d e f i n e d on

(oT)=Cg(x,t),

I

Ilfllp+a)fi 1s

d e f i n e d on

=o

TTT-ilx

211

1 ID:flg)p>o,

and

8*, p o s i t i v e constants)

Remark 1.

f o r some

e*>e>o;

p*

T ' e (0,T).

The r e g u l a r i t y o f t h e f r e e boundary

r(t)

follows d i r e c t l y

f r o m t h e method o f c o n s t r u c t i n g t h e s o l u t i o n ; see t h e p r o o f .

Remark 2. A r e s u l t s i m i l a r t o Theorem 1 i n t h e case o f R b e i n g bounded Z = Q has been o b t a i n e d i n Sobolev space by P. Secchi and A. V a l l i [ 2 ] .

and

The f r e e boundary problem f o r incompressible viscous f l u i d m o t i o n i s s o l v e d by V,A,

Solonnikov [ 3 ] and by

Remark 3. r e g u l a r i t y and

T. Beale [ l ] .

The assumption concerning t h e r i g i d boundary d i s ( r , z ) > O , so t h a t we may t a k e

Z=Q.

If

z r

i s only i t s and

z

p o i n t s i n common, t h e problem i s s t i l l open.

Idea of the proof f o r Theorem 1 .

1".

F i r s t o f a l l , we t r a n s f o r m t h e equations ( 1 ) by t h e c h a r a c t e r i s t i c x t which i s d e f i n e d by t h e r e l a t i o n transformation nXy :(x,t)-(xo,tO)

loo

O Y t O

X=Xo+

i n t o t h e form

c(XO,T)dT

X(Xo,tO)

(v

(Xo.tO) = u x y t

XoJO

V(X,t))

have

Atusi TAN1

214

a * -

{

(6)

V

atop

;-O=a

= -pv-'V*

V - ( p ' V * * Q ) +2V;.(pDc(O)) -V-ptbP, at0 v v V b6S*s=vO*(~~O + p$' )( v O - i ) 2t 2pDQ(3):Di(i) + 2 b S e V A * 3 .

eatO

P V

x t Here b ( x o y t o ) =nxs,t p ( x , t ) , 0 0 vA V = ( v ~ ,, v ~ c Y 2 , v ~ , =~ )

98, 8

m a t r i x w i t h elements

q=

o(xo,tO) = n x l t e(x,t), ( g j k ) = (ax/axo) X0Yt0 (a/ax 0 - 1 , a/axo,2, a/ax0,3)9 is a

-1

D~(V)

+ V ~ , ~ F ~j,k=1,2,3. ) ,

I n t e g r a t i n g t h e e q u a t i o n (6)1, we can reduce o u r problem t o t h e i n i t i a l boundary v a l u e problem f o r t h e p a r a b o l i c system (6)2,3 w i t h b(xo,to) = = po(xo) e x p f - j ?

V O * < ( X O , ~ ) d ~ ] and w i t h t h e i n i t i a l - b o u n d a r y c o n d i t i o n s

i ( x 0 . o ) = vo(x0),

(7)

O(xo,tO) = O .

(8) (9) (6)

= ;e(b

,

(xo,tO)e ZT,

6 ( x o y t o ) = el(xOytO)y

i Q n ( x o ) = - i e l j n ( x o ) , (Kvii).qn(xO) %

i ( x 0 , o ) = eo(xo)

-

I%n(xo)II

( x o s t O )E

rT .

(9) can be w r i t t e n i n a s h o r t e r form

a w = ~ ~ x o , t 0 , w ; 8 ~ w + ~ x 0 , t 0 , w ~i n (3t0

Q,,

W l t o = O = 0,

w= ( 0 , e ( x ,t ) 1 0 0

where

w=

(V - v o y 6 - eo),

- eo(xo))

a(xo,tO,w;?)

and

on

zT,

are matrices w i t h

B(xo,tO,w;;)

elements 2nd and 1 s t o r d e r d i f f e r e n t i a l o p e r a t o r s r e s p e c t i v e l y . We c o n s i d e r an a u x i l i a r y i n i t i a l - b o u n d a r y value problem

2".

R = (o,el(xO,tO) B(xo.tO.w;i) Here w

l\w\\f )

number

T

on

= ~ ( x O y t O , w ) on

zT

I

rT,

i s assumed t o belong t o t h e s e t

2+a,l+a/2 G T = I w c C2 x0't0

(

L

- eo(xo))

=

ME

(aT) I w l t o = o = ~ , I I ~ I"1QT I

jDrDS w l f ) ) '0 T determined l a t e r .

2W s \ = O

(a)l s ~ = 2 ' D x'Ix 0 0' QT

f o r any p o s i t i v e number

M1

cM2}

and a p o s i t i v e

,

Equations of Motion of General Fluids

216

We n o t e t h e f o l l o w i n g two f a c t s ([4,5]): (a)

The system o f d i f f e r e n t i a l e q u a t i o n (11) i s u n i f o r m l y p a r a b o l i c i n t h e 6 ) f o r a s u i t a b l y chosen T. 3 , When we c o n s i d e r t h e same problem as (11) i n R+ = { X ~ = ( X ,~x , ~~ , ~xo,3

sense o f Petrowsky (modulo o f p a r a b o l i c i t y (b)

I xo,3

>

01, t h e complementing c o n d i t i o n holds, i . e . ,

constant

such t h a t f o r any

IS'( < 6)

Rew > - 6 ' 5 l 2 ,

) W ~ ~ + E ' ~(6l2> O - tl2

B( xo, tO,w; i c ) a ( xo, tO,w; i5 ,v)

# (c3 - c J ( ~ ) ( E ' , v ) )

where

j=l

&(xO,tO,w;ic;,v)

i s an

are t h e roots i n

satisfying

t h e row v e c t o r s o f t h e m a t r i x

a r e 1 in e a r l y independent modulo (x0,t0)

i s a fixed point i n

,t ,w;i =

By virtue of Lemma 8

+

we see that

u satisfies (IElV in LP2(I

x

$I c

.-vu,$I> Pvu

E

m

Co(lRt x lRd).

Lp2 (I x lRd 1

. Hence

.

nd

We now prove the uniqueness of solutions to the problem (IE)V

.

Let u and v be two solutions of (IE),,with the same

data. Then, we again make use of Lemma 8 to obtain

for any I with

V B I.

If we choose the length of I so small

that

then dt which implies nomous

,

u(t) E v(t)

S O ,

Since (NLKG) is auto-

a.e. in I. u (t)

it is easily seen that

Therefore we have the following

f

v(t)

for

a.e. t e lR.

:

Suppose that all the hypotheses of Theorem 1 1 d (or of Theorem 2 ) on p hold valid. Then, f o r any (f-,g-) 6 H (lR 2 d x L (IR ) there exists a unique solution uv (t) of (IE),,

proposition 1.

satisfying

u

c Lw(x ;

1 ) p, Lpl(I

x

Bd

n

Lp2(I

V

for any bounded interval I : d

u,, f Lm(lR ;L2 (lRd 1 )

x

wd

. Further-

Nonlinear Klein-Gordon Equations

more, if

v is sufficiently near to

235

E

- m

LP1((-m,Ti

x

nd )

nLp2( (--,TI x IRd) for any T 2 v. If \l(f-,g-)jleis sufficiently small, then

Indeed we have

for any interval I containing v

.

Let

11

(f-,g-)lle be so small

and j o be so large that the equation (31)

-

cM(Il(f-,g-)l/e+ n j )'YP-' 0

has a positive root.

Then

Y + cll(f-,g-)lle + n j

= 0

0

we have

for all j 5 j o and any interval I containing v

, where

Yi is

the least positive root of (31). Hence we have (30) for i

=

1.

Then ( 3 0 ) with i

=

Proposition 2.

Under the same assumptions as in Proposition 1,

if ll(f-,g-)lle LP2uRt

x

Remark 3 .

IRd

2 follows from Lemma 8 .

is sufficiently small, then

uv E LP1(IRtx IRd

)

n

).

Theorem 3 is a special case of Proposition 1.

We now prove Theorems 1, 2. R

Thus we have

such that v n 9

--

Let {vn} be a sequence in 6

and

u

be the unique solution of

n' Then uv satisfies a priori estimates (261-123) with n' n replacing u by uv We prove that {uv 1 is a Cauchy sen n (IE)

.

.

Masayoshi TSUTSUMI and Nakao HAYASHI

236

f o r same T e IR

quence i n Lp2 ( (-=,TI x IRd )

.

Making u s e of

Lemma 8 , w e o b t a i n

If w e t a k e n

s u f f i c i e n t l y l a r g e , we c a n assume t h a t

~(1''" -.m

IIu 'n (t)llP1 P1 a t )

(P-1)/P,

1

5 2 '

Then

W e have f o r

v,

Recent Topics in Nonlinear Partial Differential Equations: v. 1

Nonlinear Partial Differential Equations

Nonlinear Partial Differential Equations

Nonlinear partial differential equations

Partial Differential Equations. Nonlinear equations

Partial differential equations. Nonlinear equations

Nonlinear partial differential equations in differential geometry