RECENT TOPICS IN NONLINEAR PDE
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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
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ISBN: 0 444 87544 1
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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
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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
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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
Kiyoshi ASANO and Seiji UKAI
(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 <
.
2
),
Fluid Dynamical Limit of the Boltzmann Equation
x ( l kl+lSI>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
Kiyoshi ASANO and Seiji UKAI
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
Fluid Dynamical Limit of the Boltzmann Equation
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
Kiyoshi ASANO and Seiji UKAI
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 <
do).
with some u 0 > 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 <
( r e s p . u = u(k,E)
(resp.
etB(k)u c B'([.o,~) 5
xfk)
5
$,B
).
Then
; LB )
E
0
Bo(rO,-)xR:
E
.m
et*(k)u
C:(Ri),
do).
1, X(k)
=
;
%,&
0 :T-P
1). \k/2
KO,
= 1 for \ k l
K0/2,
Kiyoshi ASANO and Seiji UKAI
10 and
- x(k)l.
Q ( k ) = { l - P ( k ) I x ( k ) + 11
1 etB(k)P(k)ulE
s e(B 9 8 ’
letB(k)Q(k)ul,
<
with constants e ( 8 , E ’ ) and g ( 8 )
Then there hold
)I U I E
(E,E’
I
g(B)e‘aotlulB 2
(6
E
R)
,
,
R)
0.
The f o l l o w i n g lemmas a r e simple consequences o f t h e above.
Let CL
Lemma 2.3.
, 1 are
B(Ek)-’Q(Ek);r
2
0, f. > n nni! B
t
Then A- 1I’T
0.
*a
continuous mappings from XR,B x
(2.13)
]A-’
(2.14)
I B ( E k ) - l Q ( E k ) ;Tu,vl
F C U , V l , 1 a,f.,E
, 1 Lzzd
ityBt o $,E.
Moreover
5
w i t h a constant d(L,E) 2 0 .
Lemma 2.4. Define the functions
F1 A .(Ek)
(2.15)
Li.(k,Ek) J
(2.16)
Pj 1 (k,Ek) = 1 EP.(Ek) J
=
J
=
EKo). Moreover
Then both o f p . ( k , ~ k ) and P . 1 (k,Ek) are i n Bm ([O,ll
J
J
(2.17)
I (&)iuj(kyck)l
(2.18)
I(z)a i P j 1 (k,Ek)ulg
5
ci 5
I kl
, i=O,l,*-*,
0 s j
C ~ , ~ ( B , E ’ )I k l i t 1 I u I B l ,
5
n+l
,
i=O,l,*-*,
0
5
j
5
f o r B Y E 1 E R.
I n t h e p r o o f o f (2.14), operator A(k) =
ii and k:,6
-
iS*k
we n o t e t h a t i f we d e f i n e t h e m u l t i p l i c a t i o n
- v(F),
w i t h t h e bound 1. B(Ek)’’A
shows (2.14), bounded i n
=
A(Ek)-’A
1
Thus t h e e q u a l i t y
-
B(Ek)-lKA(e.k)-’A
because Q(Ek) and Q(fk)B(sk)-’
iz w i t h
respect t o
i s a bounded o p e r a t o r i n
then A(Ek)- A
E 2
0 and k
E
= B(Ek)-’Q(Ek)
Rn.
are uniformly
n+l,
Fluid Dynarnieal Limit of the Boltzmann Equation
Now we t r e a t t h e terms appearing i n ( 2 . 7 ) . 1 =
P(EJOx(EJ0
11
First, noting the equality
+ Q(Ek), we have
(2.19) n+l Z F.(t,k,Ek) J=o J
+ G(t/E,k,Ek)
F F
+
G
.
h
Next, n o t i n g (2.10) and t h e corresponding e q u a l i t y P(0)r = 0, we have
+
?
+
;Io
d
(t-S)B(Ek)/EQ(Ek) ~ [ u ( s ) , u ( s ) l d ~ F(t-s,k,Ek)
1:
G((t-s)/E,k,Ek)
We p u t
t
FLu,vl(~,t) =
lo F(t-s,k,Ek)
(2.21
~[u(s),u(s)]~s. ~Cu(s),v(s)Jds, h
G
Then we have t h e f o l l o w i n g
~[u(s).u(s)ld~
( T - s , ~ , E ~ )r“A(ES),V(ES)IdS.
.m
q(R) = the s u p r e m of the norm of Q ( k ) in t h e space La.
Moreover F is continuous as a mapping
G is a l s o continuous as a mapping
Fluid Dynamical Limit of the Eoltzmann Equation
-
Similar i n e q u a l i t i e s t o (2.23)
13
(2.26) hoZd w i t h m on t h e l e f t hand s i d e re -
placed by m ' , and there hold the e q u a l i t i e s
a a t FCu,vl(~,t) =
B(~k)FCu,vl(~,t) A
+ P(ck)X(Ek) i Y u ( E , t )
a
(2.27)
a
,V(E
,t)l
n
GL u ,v 1(E .T ) = B( E k)G[ U , VI ( E J ) + Q( E k)I"u ( E ,ET ) . V ( E ,ET ) 1,
/.
= B(E~)GC<,VI(E,T) + Q ( E ~ ) L ' C < ( E ~ T ) , V ( E , E ~ ) I .
GLU,VI(E,T)
a G[~,;](E,T)
h
=
B(tzk)G[G,
i
Q(Ek)TcU(&,r),
<(&,T)].
Here f o r s i m p l i c i t y v e use the n o t a t i o n G[~~,v](E,T)
(2.28)
=
;1
G(T-s,k,Ek)r"i(E,s),
V(E,ES)]~S etc.
T h i s lemma w i l l be a p p l i c a b l e t o such terms as F [ i , v l
and
F[G,V],
combined w i t h t h e f o l l o w i n g
Lemma 2.7.
Let
<(E,T)
6
Let m = (ml,m2),
y
2
0, u > 0
0 w i t h
a-yT
, and p u t
L,B,T
(HIU)(E,t) = G(E,t/E).
Then t h e r e holds f o r 0
Lemma 2.8.
( a - yT
2
5
i
Let
5
ml
and 0
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
,
Fluid Dynamical Limit of the Boltzmann Equation
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
Fluid Dynamical Limit of the Boltzmann Equation
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 <
(II > 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
*
Kiyoshi ASANO and Seiji UKAI
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,
Fluid Dynamical Limit of the Boltzmann Equation
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).
This Page Intentionally Left Blank
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,
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
Nonlinear Diffusion Systems
(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
Nonlinear Diffusion Systems
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) <
(vg,t*)
v:
Next, we suppose i
11 c
i . Let
(x:
(u,u,s)
E
R3
be such t h a t
= k+(v:).
The c o n s t r u c t i o n o f s o l u t i o n s on I - U I.
con i s i t s o f t h r e e s t e p s .
Given
, f
i n a neighborhood o f
(cr,p)
(cl:,v:
nd t h e € - f a m i l y o f
triplets
s
= 5 0 ; i,Ll),
u- = U-(X,f ; c x , l J ) ,
v- = v - ( x , ~ ; ~ b L l ) , such t h a t
s a t i s f i e s t h e Neumann-Oirichlet problem ( P I - ) :
(u-,v-) 2
u- + f ( u - , v - )
c2
= 0,
dx
w i t h an a u x i l i a r y O i r i c h l e t c o n d i t i o n :
Hiroshi FUJI1 and Yuzo HOSONO
30
and t h a t
lim
s ( ~ ; a , ~ = )0.
E $0
( v , s ) with
11.
Given
s
the
E - f a m i l y o f couples
-8
0
i n some neighborhood o f
uo = UoObE;u,S).
vo =
vo(x,E;v,s),
which s a t i s f y the D i r i c h l e t boundary value problem ( P I o ) :
F i g . 2.1
E2
4 dx
uo + f(Uo,Vo) = 0,
(v:,O),
find
Nonlinear Diffusion Systems
31
The t h i r d s t e p i s :
1
111.
C -patching o f
and f i n d t h e v a l u e o f
(u-,v-)
and
a
( u0 .v 0 )
and
at
u such t h a t
x = s.
@ = Y = 0
We d e f i n e
f o r each
We apply t h e i m p l i c i t f u n c t i o n theorem due t o P. C. F i f e [ 1 1 a t
2.2.
u = v:,
a = a:,
and a t
t o have
a = a ( € ) and
u
=
> 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,
Nonlinear Diffusion Systems
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,
dL
z(*)
ht(u),
z'(x,E)
and
z"(x,E)
i s a Cm-cutoff f u n c t i o n s a t i s f y i n g
i= 1
x 2 - 1/2;
= z ( y , p ) c ( s )
0 5 i5 1 for
1/4
+,
I;
i-
= 0,
which decays e x p o n e n t i a l l y w i t h i t s d e r i v a t i v e s as Set
g c
=
-+
t-x
z(-,v)<(=
for
0
t-x
see,[
21).
, where
5 x 5 1/4;
r,
= 0
5 x 5 1/2.
Now, we have a f i r s t approximation t o a s o l u t i o n of ( P I 0 ) as h
U ( X , E ; ~ , S ) = h,(Vo(x;u,s))
+ z'-'(X,c) + zv(x,c),
h
V(X,E;!A,S)
= V0(x;u.s).
Using t h i s approximation, we can s o l v e t h e problem ( P I o ) through t h e i m p l i c i t f u n c t i o n theorem [Theorem 3.4, 1 1 .
Theorem 3.
Assume (A.1)
e x i s t s some p o s i t i v e c o n s t a n t solution
(A.3). co
Let
(11,s)
c
N:.
Then, t h e r e
such t h a t t h e problem (PIo) has a
( u ~ ( x , E ; ~ , s ) , v ~ ( x , E ; ~ , ~ f) o) r any
E E
(O,co),
satisfying
i(x) for
Hironhi FUJI1 and Yuzo HOSONO
34
u
uniformly i n
2.4.
s.
and
C o n s t r u c t i o n o f Neumann l a y e r s We proceec‘ t o c o n s t r u c t an N-layer by choosing t h e parameters ( a , p )
as functons o f
so t h a t
(u_fx,~;a,vf,v_(x,~;u,u))and (uO(x,~;vlS), 1 = min V ~ ( X , E ; ~ , S ) ) are patched a t x = s ( t ; i ~ . u ) i n C -sense. L e t (E-,E~)
0
and
E
and d e f i n e
D = {(E;CX,P)~
(O,:), ( a , u )
NE}.
E
Y were a l r e a d y d e f i n e d i n t h e subsection 2.1.
Theorem 4.
Let
i
a ( ~ )and
l i m a(€) = a:,
lim E 40
Taking t h e l i m i t
=
U(E)
defined f o r
E E
= v.:
E ~ O i n ( 2 . 9 ) , we have
= -2J(a,h,(v)
and t h e d e f i n i t i o n s o f Q(o;a;,v:)
u(c)
E 40
Y(O;a,u) =
by
be s u f f i c i e n t l y small p o s i t i v e constant.
there e x i s t functions
@(O;~,!J)
The f u n c t i o n s
D.
which are u n i f o r m l y continuous i n
(Proof)
E E
;PI,
dVO
~(o;u,o), a:
and
Y(o;a,v:l
v: = 0,
d i r e c t l y lead t o
Then,
(O,P), s a t i s f y i n g
Nonlinear Diffusion Systems
35
Since t h e diagonal elements o f t h e m a t r i x (2.10) a r e n o t zero (see, Lemna
2 ) , we can a p p l y t h e i m p l i c i t f u n c t i o n theorem [Theorem 4.3, 1 ] t o @ = I = 0, and f i n d t h e s o l u t i o n s
a = a ( c ) and
p = p ( ~ ) . T h i s completes
the proof. We
I - U Io.
have
now t h e s o l u t i o n w i t h an N - l a y e r a t
x = s ( E ; c x ( f ) , p ( E ) ) on
I n order t o construct the i n t e r i o r t r a n s i t i o n layer, i t i s
s u f f i c i e n t f o r us t o r e p e a t t h e arguments i n [ 8 1 .
Then, we o b t a i n t h e
s o l u i t o n which we l o o k f o r .
3.
Concluding d i s c u s s i o n s
As may be c l e a r from t h e c o n s t r u c t i o n , w i t h an M - s o l u t i o n possessing OM^
i n t e r i o r t r a n s i t i o n l a y e r , which we c a l l D1, t h e r e a r e a s s o c i a t e d
t h r e e b r o t h e r s of N - s o l u t i o n s , depending on where t h e N-layers e x i s t ; t h e l e f t end
t
D1, a t t h e r i g h t end Dl# o r a t t h e b o t h ends
I n Fig.3.1,
#
D1
#
.
we show t h e p r o f i l e s o f those b r o t h e r s , i . e . ,
Dl# and #01#, as w e l l as t h e i r ( u , v ) - p l o t s , t h e May-Mimura system ( l . l ) ,
a t c2 = 1/10, and
at
D1,
(J=
# D1,
1/200, f o r
(1.7).
F i g .3.1
" P M J ~ L L0~6
I I
0,''
........ ..............
:
,
Nonlinear Diffusion Systems
37
These f u n c t i o n s a r e o b t a i n e d by t h e numerical methods making use o f t h e group r e p r e s e n t a t i o n t h e o r y .
See, f o r d e t a i l s , [ 3 1 , [ 4 1 .
We have suggested i n 5.1 t h a t those Neumann b r o t h e r s r e p r e s e n t E
+
0 d e s t i n a t i o n s o f secondarv b i f u r c a t e d branches.
and p r o o f s , see our forthcoming paper [ 5 A q u e s t i o n may a r i s e nere.
About such v i e w p o i n t s
1.
Do t h e Neumann b r o t h e r s i n c l u d e a l l p o s s i -
b l e d e s t i n a t i o n s o f secondary branches ? present.
the
The answer i s n o t g i v e n a t t h e
However, t h e a u t h o r s f e e l t h a t t h e r e i s another f a m i l y of l a y e r -
t y p e s o l u t i o n s a p a r t from t h e Neumann f a m i l y .
A further investigation i s
neccesary t o c l a r i f y t h e s i t u a t i o n .
A comment on t h e s t a b i l i t y o f Neumann b r o t h e r s .
Every evidence shows
t h e y a r e un&tabLt s o l u t i o n s , w h i l e as N i s h i u r a showed i n [ l o ] , t h e M-layer t y p e s o l u t i o n s a r e crxixbh.
However, we may emphasize again t h a t t h e s i g -
n i f i c a n c e o f such N-layer t y p e s o l u t i o n s should be understood i n t h e cont e x t o f global b i f u r c a t i o n questions.
Acknowledgements We a r e indebted t o o u r c o l l e a g u e P r o f . Y. N i s h i u r a f o r h i s d i s c u s s i o n s .
References
[1]
P.C.F
fe, Boundary and i n t e r i o r t r a n s i t i o n layer phenomena for pairs of second-order d i f f e r e n t i a l equations, J.Math.Anal.Appl.,
54
(1976), pp.497-521.
[2]
P.C.Fife,
Semilinear e l l i p t i c boundary value problems with small
parameters, Arch.Rationa1 Mech.Ana1.
[3]
52(1973), pp.205-232.
H.Fujii, Numerical p a t t e r n formation and group t h e o r y , Computing Methods i n A p p l i e d Science and Engineering (Eds. R.Glowinski & J.L.Lions)
-
Proc. o f t h e 4 t h I n t e r n a t i o n a l Symposium on Com-
Iliroshi FUJI1 and Yuzo HOSONO
38
p u t i n g Methods i n A p p l i e d Science and Engineering, N w t h - H o l l a n d 1980, pp.63-81.
[ 4 1 H . F u j i i , Looking f o r t h e picture of global b i f u r c a t i o n diagram - i t s nwnerical reaZization ( i n Japanese), Numerical A n a l y s i s and Nonl i n e a r Phenomena (Ed. Irl.Yamaguti), Nihon-Hyoronsha, Tokyo, 1931, pp.45-122.
[ 5 1 H.Fujii, ( i n preparation). [ 6 1 H . F u j i i , M.Mimura & Y.Nishiura, A p i c t u r e of the gZoba2 b i f u r c a t i o n diagram in ecological i n t e r a c t i n g and d i f f u s i n g systems, Physica 0, 5 (1982), pp.1-42.
[7
1
H . F u j i i & Y.Nishiura,
GZobal b i f u r c a t i o n diagram i n n o n l i n e m d i f f u sion systems, N o n l i n e a r P a r t i a l D i f f e r e n t i a l Equations i n A p p l i e d Science - Proc. o f U.S.-Japan Seminar 1982, Tokyo (Eds. H . F u j i t a , P.D.Lax & G.Strang), L N N M 5, North-Holland,
1984.
[ 8 1 M.Mimura, M.Tabata & Y.Hosono, MuZtipZe solutions of two-point bounda r y value problems of Newnann type with a small parameter, S I A M J.Math.Anal.,
11(1980), pp.613-631.
[9]
Y.Nishiura,
[lo]
Y . N i s h i u r a , Every multi-modesingularly perturbed s o l u t i o n recovers i t s
Global structure of bifurcating solutions of some reactiond i f f u s i o n systems, S I A M J. Math.Anal., 13(1982), pp.555-593. s t a b i l i t y - from a global b i f u r c a t i o n viewpoint, t o appear i n Proc. o f t h e I n t e r n a t i o n a l Workshop on Modelings o f P a t t e r n s il: Space and Time, Heidelberg 1983.
L e c t u r e N o t e s in Num. Appl. Anal., 6, 39-57 (1983) Recent Topics in Nonlinear P D E , Hiroshinza, 1983
Water Waves and Friedrichs Expansion
Tadayoshi KANO* and T a k a a k i NISHIDA" * Dedicated t o t h e l a t e K u r t - O t t o FRIEDRICHS *Department of Mathematics, Osaka University, Toyonaka 560, Japan **Department of Mathematics, Kyoto University. Kyoto 606, Japan
61.
Introduction We a r e concerned w i t h s u r f a c e waves o f water whose equations i n t h e
dimensionless form has a parameter wave l e n g t h .
6 which corresponds t o
m a n depth /
We w i l l prove t h a t t h e f r e e s u r f a c e and t h e v e l o c i t y poten-
t i a l a r e i n f i n i t e l y many t i m e s d i f f e r e n t i a b l e w i t h r e s p e c t t o scale
S =
u Xr,
o f Banach spaces o f a n a l y t i c f u n c t i o n s ( 5 2 ) .
6 in a By v i r t u e
0'0
o f t h i s f a c t we can g i v e a j u s t i f i c a t i o n o f F r i e d r i c h s expansion ( c f . [ l ] ) f o r water waves as an a s y m p t o t i c one (553-4). Two space-dimensional i r r o t a t i o n a l motions w i t h f r e e s u r f a c e o f an incompressible p e r f e c t f l u i d a r e c a l l e d "water waves" and a r e governed by t h e f o l l o w i n g system o f equations i n t h e dimensionless form f o r t h e veloc-
i t y potential
0 = @(t,x,y)
(1.1)
2 6 Qxx + 0 = 0 YY
(1.2)
0
Y
= 0
on y = 0
and t h e f r e e s u r f a c e in
n(t),
,
39
r
= T(t,x):
Tadayoshi KANO and Takaaki NISHIDA
40
where
n ( t ) = {(x,y)
:
-m
< x <
m,
i s t h e domain occupied
0 < y < I‘(t,x)l
by t h e water. I n 1948 F r i e d r i c h s [ Z ] proposed a procedure t o d e r i v e approximate
-
equations s y s t e m a t i c a l l y f o r the problem ( 1 . l ) t h e f o l l o w i n g expansions f o r
@(t,x,y)
and
( 1 . 3 ) , assuming t h a t
T(t,x)
w i t h respect t o
S
are v a l i d :
-
S u b s t i t u t i n g t h e expansion i n t o (1.1) t h e same degree o f
6‘
(1.3) and e q u a t i n g the terms o f
t o zero, one o b t a i n s a h i e r a r c h y o f approximate
equations :
I rO,t + ( r o ~ o , x ) x = 0,
1
(1.6)
where
fn-l
an , t + 0,x on,x + r n
rn , t
and
5
gn-l
O(”(t,x,O),
= fn-l,
n = 0,1,2;..,
are functions o f
t i v e s of o r d e r l e s s than On
n = 0,
(roQn,x +@O,xrn)x = gnm1,
+
@,(t,x)
for
for
n = 1,2,.*-,
and t h e inhomogeneous terms Tm, 0
Qm,
2m5
n = 0,1,2;*.,
(Zn+l).
n-1, and t h e i r d e r i v a i s o b t a i n e d from
by t h e f o l l o w i n g ( c f . ( 4 . 1 0 ) )
w2m
(1.7)
o(”(t,x,y)
m Zm a @,_,(t,x)
=
m=O
,
n = 0,1,2,*..
.
ax2”’
The f i r s t approximation (1.5) i s t h e w e l l known s h a l l o w water wave equation, which i s a q u a s i l i n e a r h y p e r b o l i c system f o r t h e f u n c t i o n s (Oo,x,ro)
under t h e c o n d i t i o n t h a t
ro
> 0.
The h i g h e r o r d e r approximate
system (1.6) i s 1 i n e a r inhomogeneous h y p e r b o l i c equations f o r t h e f u n c t i o n s
Water Waves and Friedrichs Expansion
(On,x,rn)
under t h e same c o n d i t i o n on
41
ro.
I n [5] i t was proven t h a t t h e i n i t i a l value problem ( 1 . 1 )
-
(1.3)
has a s o l u t i o n i n a s c a l e o f Banach spaces o f a n a l y t i c f u n c t i o n s l o c a l l y
6 E [0,1],
i n t i m e f o r any
{~(t,x,y;6),r(t,x;6)}
i.e., O(')(t,x,y)
6 -+O
as
wave e q u a t i o n ( 1 . 5 ) .
and t h a t t h e l i m i t o f s o l u t i o n e x i s t s and s a t i s f i e s t h e s h a l l o w water
We n o t i c e t h a t
= OO(t,x)
Q(O)(t,x,y)
( c f . (1.7)).
i s independent o f
y
This i s a j u s t i f i c a t i o n o f
t h e l o w e s t o r d e r o f t h e F r i e d r i c h s expansion (see a l s o [3],
[4]).
I n 93 we g i v e a f u l l a s y m p t o t i c expansion o f t h e e q u a t i o n s f o r r(t,x;6)
and
s = p>o U X0'
Q(t,x,r(t,x;6);6)
6 E [0,1]
w i t h respect t o
Then i t i s shown t h a t t h e asymptotic expansion
i s v a l i d and t h a t t h e c o e f f i c i e n t s
,...
{$n,x,rnln=0,1,2
h y p e r b o l i c system o f p a r t i a l d i f f e r e n t i a l equations. n o t d i r e c t l y make use o f t h e system ( 1 . 3 ) .
t h e e q u a t i o n (3.1) f o r
for
x(t,S,l;6)
and
and
$(t,[,l;6)
on t h e upper boundary and we have t h e expansions
6
w i t h respect t o
A f t e r these expansions we o b t a i n those f o r
I n doing so, we do
I n f a c t we f i r s t expand
$(t,S,1;6)
Q1 = { S E R. 0 < rl < 11 w i t h r e s p e c t t o x(t,S,l;6)
s a t i s f y the
A l l o u r c a l c u l a t i o n s a r e done
on t h e upper boundary as done i n [ 5 ] and (3.1).
of
in
6 as P r o p o s i t i o n 3.1.
O(t,x,r(t,x;6);6)
and
r(t,x;6).
I n 54 we g i v e a j u s t i f i c a t i o n f o r t h e expansion (1.4) o r i g i n a l l y proposed by F r i e d r i c h s .
Using an asymptotic expansion o f t h e p o t e n t i a l we can prove t h e asymptotic
Q on t h e bottom w i t h r e s p e c t t o
6 E [O,l],
expansion o f
as i s g i v e n i n ( 1 . 4 ) .
O(t,x,y)
(1.3) on t h e s u r f a c e equations (1.5),
in
Q(t)
y = r(t,x;6)
g i v e s t h e system of p a r t i a l d i f f e r e n t i a l
(1.6) as t h e c o e f f i c i e n t s o f
I n t h e forthcoming paper,the
Then e q u a t i o n
62n, n = 1,2,--*.
Boussinesq e q u a t i o n and t h e Korteweg-
42
Tadayoshi KANO and Taksaki NISHIDA
de V r i e s e q u a t i o n a r e d e r i v e d r i g o r o u s l y [6], and a l s o t h e t h r e e space dimensional problems w i t h Kadomtsev-Petviashvi 1i e q u a t i o n a r e considered
52.
Smoothness o f s o l u t i o n s w i t h r e s p e c t t o Let
(x,y)
mapping from
be o u r mapping i n [5] from t h e domain
= (x,y)(t,E,n)
{(5,r1): 5 E R, 0
R1 =
5 =
5+
6.
< ri < 1 )
t o t h e domain
z = x + i6y.
i6n to
fi(t),which i s a conformal
The problem (1.1)
-
(1.3)
i s transformed by t h i s mapping i n t o t h e f o l l o w i n g i n t e g r o - d i f f e r e n t i a l system f o r t h e unknowns 0 =
1, 5
E
= xax( t , C , l )
v(t,C)
(V,U)(O,S)
where
@(t,<,l) = Q(t,x(t,S,l),y(t,S,l)),
= &t,S,l) a@
on
= (vo,uo)(5),
i n t e g r a l operators
A6
and
The Cauchy problem (2.1) S =
LJ X p>o
E XD
w = ( v 2 + (A,v)')-',
and t h e
Cg a r e d e f i n e d by
-
(2.2) has been s o l v e d i n [ 5 ] i n t h e
o f Banach spaces o f a n a l y t i c f u n c t i o n s i n such a
way t h a t , w i t h a constant (v,u)(t,-;6)
u(t,5)
R:
(2.2)
scale
and
a > 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 <
poJ
uhich has t h e uniform bound
f o r any
p < pIJ
I tl
< alpl
t h e constants Em, m = I, 2,
Also it holds f o r
- pl,
..-
pl
< pO, and f o r any
depend on
pl
6 E 10,11,
but do n o t on
6E
where LO, 1 1.
n = 0,1,2,*.- t h a t
The property (2.12),
f o r each
n
=
0,1,2,...,
i s a consequence o f
the f a c t t h a t they are s o l u t i o n s o f a homogeneous system w i t h zero Cauchy data
.
46
Tadayoshi KANO and Tnkaaki NISHIDA
53.
Expansions on t h e Surface We now g i v e a j u s t i f i c a t i o n o f F r i e d r i c h s expansion f o r t h e s u r f a c e
r
and t h e v e l o c i t y p o t e n t i a l
5 on r , i. e., we have t h e asymptotic
expansion m
r(t,x;6)
2n - n=O X rn6
and t h e c o e f f i c i e n t s o r l i n e a r ( n = 1,2,-.-) n = 1,2,..*,
rn, in, n = 0,1,2.-.*,
h y p e r b o l i c equations (3.11)0 o r (3.11),,,
respectively.
L e t us remember ([5], and @ = 4(t,5,1;6)
p. 343, (3.28) and (2.1)) t h a t
x = x(t,S,1;6)
i s a s o l u t i o n o f t h e system o f equations
2 w = I X + ( A x )21-1. F; 6 5 as seen i n [5] and i n 14.
where
s a t i s f y t h e n o n l i n e a r ( n = 0)
Therefore i t solves o u r problem (1.1)
Here we want t o expand e q u a t i o n ( 3 . 1 ) w i t h r e s p e c t t o F i r s t we n o t i c e t h e expansions f o r t h e o p e r a t o r s
A6
and
-
(1.3)
6 E [0,1].
C6:
Water Waves and Friedrichs Expansion
where
<(*)
i s t h e Riemann zeta f u n c t i o n .
v = x (t,5,1;6)
5
It1
< a(po-p)
and
u = @ (t,5,1;6)
u n i f o r m l y w i t h respect t o
use the expansion (3.2) f o r the operators side o f equations (3.1),
Next we know t h a t the s o l u t i o n
belongs t o
5
47
X
P'
6 E [0,1].
0 < p ' < po, f o r Therefore i f we
*6 and 6C6
i n the r i g h t hand
we o b t a i n the f o l l o w i n g asymptotic expansions:
n
Here we have w r i t t e n the e x p l i c i t form o f terms on t h e r i g h t hand s i d e 2 O f course one can compute e x p l i c i t forms of equations up t o the order o f 6
.
o f higher order terms.
Also the e l e v a t i o n y
o f the surface on
0 = 1,
5 E R has t h e f o l l o w i n g expansions:
T ~ U Swe obtained the expansion o f equation (3.1). Furthermore, since we have proved i n §2 the Cm-dependence on 6 € [0,1]
(2.1) i n
o f the solution
v = x (t,[,l;6)
Xp, 0 < p < po, f o r
an expansion w i t h respect t o
and
5
It1
< a(po-p),
6 E [0,1]
in
u = @ (t,C,l;6)
5
of
t h i s s o l u t i o n i t s e l f has
Xp.
More p r e c i s e l y we can
s t a t e i t as f o l l o w s :
Proposition 3, I.
For any N = I , 2,
..- and for any
one has the foltowing expansions with respect t o
-
t , It I < alpO p),
6 E r0,lI:
48
Tadayoshi KANO and Takaaki NISHIDA
where
0
n = 0,1,2,
i s i n the sense of X . The c o e f f i c i e n t s x
P n and +n, are independent of 6 and s a t i s f y the foIlowing system
of equations i n
xP, o
<
VP <
pOJ It1 < a(Po
- P):
n = 1,2,3,**., where
Tn-l, gn-,,
n = 1,2,.*.,
are functions of
and t h e i r derivatives with respect t o
Proof.
>x ,,,
",
02 m
2
n-1,
5.
The f i r s t p a r t ( 3 . 5 ) of Proposition i s a consequence o f Theorem 2.3.
I f we s u b s t i t u t e the expansion ( 3 . 5 ) i n t o ( 3 . 3 ) and compare the c o e f f i c i e n t s of
62n i n the expansion, we obtain (3.6),,
n = 0,1,2,.*..
Now we proceed t o consider the expansions for 6 = @(t,x,r(t,x;6);6).
They a r e determined by
r
= r(t,x;6)
Q.E.D. and
49
Water Waves and Friedrichs Expansion
Lemna 3.2.
rJ 8 , and qx E
with respect t o
6 E [0,11.
and the radius of convergence 5,
X-,
P
0<
for
5 < Fo,
P
is detemined by
are Cm-functwns w i t h respect t o 6 E 10,ll
o
pp It\
<
Proof.
- p),
unifonnty
i s the same Banach space a8 X
Here X-
which is the inverse function of
= S(t,x;6)
It1 < a ( ; ,
p
P
and t h a t of
x = x(t,[,1;6).
They
w i t h vatues i n XPS
<
Since xE(t,E,1;6)
T3 V - >
0 (cf.
[51), the proof is an easy
consequence of the inverse function theorem and the same f a c t f o r y(t,S,l;6)
and $S(t,E.1;6):
r(t,x;6) =
,1;6)
i(t,x;6) = $(t,E,,l;6). The l a t t e r part of Lemna is a consequence of Theorem 2.3. We can obtain the equation f o r
r
and 6 which comes from (3.1)
and (3.3). Theorem 3.3.
Proof.
r
and
Q.E.D.
8 S a t i s f y the equation:
W e note the following relations by the definition (3.7):
Tadayoshi KANO and Takaaki NISHIDA
60
Let us express the l e f t hand side o f (3.8) by x
5
and Q
5
by using
(3.114 (3.3) and (3.4)
The r i g h t hand side can be expressed o n l y by expanded with, respect t o
6
r
and
by using (3.2) and (3.4):
iX4 and i t can be i. e.,
S i m i l a r l y we have
I f we expand the r i g h t hand side of t h i s expression, the i n t e g r a l terms w i t h respect t o
5 never appear by v i r t u e o f (3.2) and (3.4). Therefore
i t can be expressed by
r
and 8x and t h e i r derivatives i n contrast
w i t h the i n t e g r a l terms o f the expansions (3.3) and (3.6) o f
x and Q.
I n p a r t i c u l a r the leading term has the e x p l i c i t form:
r t + ( 5 x r ) x = -g2 T $ 1~ g
x5
5 55x 555 + x5 @5 x5555 - Q 5X 55x 5 5 E - X 5 Q m ) + o ( s ~ )=
2
=
- 3r35xx)xx t 0(s4). Q.E.D.
Water Waves and Friedrichs Expansion
r
The expansion for Theorem 3.4.
E
<
Po,
i s given in the following
For any N = 0,1,2,***,
6 E C0,lI
and any
t, It(
we have
The coefficients
or linear In
and
rn(t,xl
= 1,2,-*.1
'n, t 'o,x'n,x +
(3.111,
where
and 5
51
m, t + IroTn,z Fnm1, Cn-l
$n(t,xl
s a t i s f y the nonlinear ( n = 0 )
hyperbolic equations as follows:
+
rn = 'n-1
+ rn'O,xlr
=
Gn-I
n = 1,2,.**, J
f, f,,,, o 5 m 2 n-1, and t h e i r
are functions of
derivatives with respect t o x .
The asymptotic expansion (3.10) i s a restatement of the l a t t e r
Proof.
part of Lemma 3.2.
If we substitute the expansion (3.10) into (3.8). we
obtain the hyperbolic equation (3.11In, n = 0,1,...,
rn, n
which governs
= 0,1,2,***.
One can express explicitly
in,rn,
n = 0,1,2,...
n = 0 , 1 , * - - , by u s i n g (3.7), (3.10) and (3.4), ( 3 . 5 ) . i t here except f o r the following: ~~
2) RN+l[*]
in, Q.E.D.
~
~
stands for the remainder term for
by
$,, xn,
We will n o t give
Tadayoshi KANO and Takaaki NISHIDA
62
- iO,x(t.xo)xl(t,S)s
Tl(t.X0(t,O)
= @l(t,S)
r,(t,x,(t,S))
= x1 , E ( t , ~ ) - ro,x(t,Xo)Xl(t.c).
(3.1211
I t i s easy t o v e r i f y t h a t (3.11)0 comes from (3.12)0 i f we use (3.6)0.
S i m i l a r l y we obtain (3.11)1 i f we s u b s t i t u t e (3.6)*,, derivatives of (3.12)0,1,
xn,
i.e.,
@1,t
@,,,
n = 0, 1 i n the expression (3.11)1 by using
(3.11)1 has the e x p l i c i t form as follows:
SO,X*l
,x 4. r l
= 2-
rooo,xx
(3.13)
r l ,t (r051 ,x V 0 , X ) X
9
Y
so-*
.
)2
1 3-
=
-~~0@0,xx)xx'
Lastly we compare the expansions (3.3), (3.11)n=0
i n t o the time
(3.6)",o,l
,2,.
.., (3.8)
and
(3.3) and (3.8) are expansions o f equations and w i l l
be also used for a j u s t i f i c a t i o n o f Boussinesq equation and Korteweg-de Vries equation [6].
(3.6) and (3.11) are expansions o f solutions and
equations which correspond t o Friedrichs expansion on the f r e e surface.
14.
(1.1)
J u s t i f i c a t i o n o f Friedrichs Expansion Recall how the s o l u t i o n x and 4
o f (3.1) solves our problem
-
and the complex v e l o c i t y
(1.3).
potential f o l 1owing :
f
The conformal mapping z for
(6,~) E sEl =
{[ E R, 0 < q < 1)
are given by the
Water Waves and Friedrichs Expansicn
(4.1)
53
\
The f u n c t i o n
I$ and
$
s a t i s f y t h e f o l l o w i n g equations and boundary
conditions:
A6
where
i s the operator i n (2.3).
5 = :+i6n
holomorphic f u n c t i o n o f u = Q, (t,S,1;6)
5
belongs t o
uniformly w i t h respect t o
X
P
in
for
It1
< a(pl - p ) ,
vpl
f = @+i$ is a
(5,~E ) R1. Since t h e s o l u t i o n
0 < P < p0,
J t l<
6 E [0,1]
w i t h values i n
< po, t h e f u n c t i o n
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 r e s p e c t t o values i n a n a l y t i c f u n c t i o n s o f
It1 <
a(pl - p ) .
y = y(t,c,q;d)
x
( ~ , n )E al, o <
S i m i l a r l y the functions
2
iQ
yn = xe
Vp
in
R1,
Xp,
f = @+i$
(4.1)
6 E [0,1]
with
x = x(t,S,n;6)
o f (4.1) s a t i s f y t h e equations
= - 6 yg,
a ( p o - p)
and s i n c e i t i s i n f i n i t e l y many
6 E [0,1]
times d i f f e r e n t i a b l e w i t h r e s p e c t t o 0 < \ I p < pl,
I n o t h e r words
and
Tadayoshi KANO and Talcanki NISHIDA
54
I t f o l l o w s from (4.1), t h a t the function
(4.3) and t h e a n a l y t i c i t y o f
z = x t i6y p < po,
v = xc(t,C,l;6)
i s a holomorphic f u n c t i o n o f
5 =
z
5 + i6n
It1 < a ( p o - p ) ,
6 E [0,1]
and t h a t
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 i n
6 E [0,1]
w i t h values i n
i n R1
f o r any
a n a l y t i c functions o f
(t,n)
for
E fil
p <
z = z(t,t,q;6)
Therefore, s i n c e t h e mapping
yol < from
It1
Po,
R1
to
is
- p).
< a(pl
n(t)
is
i n v e r t i b l e ( c f . [ 5 ] ) , we can o b t a i n t h e complex v e l o c i t y p o t e n t i a l on n(t):
Then the f r e e surface and t h e v e l o c i t y p o t e n t i a l
s o l v e t h e system (1.1) the potential 6 E [0,1]
It/ <
F
-
(1.3) f o r
w i t h values i n a n a l y t i c f u n c t i o n s o f
a(pl - p ) ,
Ft =
- FzZt
6 E [0,1].
of
Furthermore
< po.
zt
(x,y) E R ( t ) , and
ft
can be
z and f ( ( 4 . 1 ) and (3.20)-(3.21)
i n [5]),
+ ft F
for
z, (x,y) E Q ( t ) ,
The v e l o c i t y p o t e n t i a l
complex v e l o c i t y p o t e n t i a l
Lema 4 . 1 .
Also s i n c e
v p < 'd p1 < po.
has t h e same p r o p e r t i e s as for
tf p
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 r e s p e c t t o
expressed by < - d e r i v a t i v e s o f (4.6)
It1 c a ( p O - p ) ,
F
(4.5).
The velocity potential
(x,Sy), (x,y) E Q ( t ) ,It1
@
It1
< a(pl
- p),
and
i s the real p a r t o f the
Hence we have
cPlt,x,y;GI
a[pg-pl,
i s a harmonic function
6 E CO,~],
i . e . , it s a t i s f i e s
equation f l . l ) , and it i s i n f i n i t e Z y many times d i f f e r e n t i a b t e s with respect t o
6 E CO,11
w i t h values i n analytic functions of
(x,y)
in
Water Waves and Friedrichs Expansion
The d e r i v a t i v e s
0
Ox,
Y
and
s i m i l a r expansions t o ( 4 . 7 1 ,
Proof.
Qt
65
have t h e same p r o p e r t i e s as
0
and t h e
in particular
We have a T a y l o r expansion f o r
N = Oy1,Zy...,
(x,y) E n ( t ) ,
and
v 6 E [0,1]:
0 < 0 < 1.
where
It1
< a(pl
- p),
On t h e o t h e r hand, f o r
v p < ypl
(x,y) E n ( t )
< pol t h e d e r i v a t i v e
and f o r
a2N+20(tyx,y;6)/ax2N+2
i s u n i f o r m l y bounded as an a n a l y t i c f u n c t i o n w i t h r e s p e c t t o
6 E [0,1].
O.E.D.
Thus t h e expansion ( 4 . 7 ) h o l d s .
I n o r d e r t o j u s t i f y t h e F r i e d r i c h s expansion a s an asymptotic one w i t h respect t o respect t o
6, we have t o expand
6 E [0,1].
T(t,x;d)
The f r e e s u r f a c e
r
and
Q(t,x,y;b)
with
has a l r e a d y t h e expansion
(3.10).
On t h e o t h e r hand, s i n c e t h e v e l o c i t y p b t e n t i a l
!(t,x;6)
E O(t,x,0;6)
on t h e bottom i s o f course a i n f i n i t e l y many times
differentiable function o f
6 E [0,1]
by v i r t u e o f Lemma 4.1,
t h e asymptotic expansion w i t h r e s p e c t t o
6 E [OJ]:
we have
Tadayoshi KANO and Takaaki NISHIDA
66
where a l l terms o f odd o r d e r i n remainder term
vanish by v i r t u e o f (2.12).
6
i s u n i f o r m l y founded i n
RNtl[!x]
X=
The
C 6 2Nt2
by
with
P
a constant
C
for
I t [ < a(;,-;),
x-derivatives o f functions
@(t,x,0;6)
@(t,x,y;d),
and
Ot(t,x,y;6)
I n p a r t i c u l a r we have
6 E [0,11,
and
(x,y) E n ( t )
The remainder term
0(62Nt2)
Lo.
r(t,x;6)
The
t- and
have t h e s i m i l a r
and t h e i r space d e r i v a t i v e s have t h e
s i m i l a r expansions.
for
<
Therefore by u s i n g (4.7) and (4.8) t h e
expansions t o (4.9) and (3.10). potential
Z1
<
It1 < a(pl
-PI, t r <~ v ~
has t h e form:
Therefore we can expand each term o f equation (1.3) f o r our
r
s o l u t i o n 0 and (3.10).
w i t h respect t o
6 E [0,1]
Their coefficients consist o f
rpn(t,x),
p a r t i a l d e r i v a t i v e s and a r e independent of t h e c o e f f i c i e n t s o f t h e same o r d e r o f
6
6.
by u s i n g (4.10) and rn(t,x)
and t h e i r
Thus, s i n c e we can equate
t o zero, we o b t a i n t h e equations
(1.5) and (1.6)n=1,2
,.,., f o r
equation ( 1 . 6 ) f o r
n = 1 has t h e f o l l o w i n g e x p l i c i t form: 1
01 ,t
@O,XQl
,x
rl
rn successively.
Qn,
=
2
For example t h e
2
S r O(@O,txx @o,x@o,xxx - @O,xx)'
(4.11)
1
r1 ,t (@O,XTl+ rOQ0,x)x
3
= i?(~O@O,xxx)x'
Water Waves and Friedrichs Expansion
where
ro
= x (t,C,l;O)
5
2
3 4v
-
> 0
57
g i v e s t h e h y p e r b o l i c i t y o f t h e system.
And we have completed t h e p r o o f o f j u s t i f i c a t i o n o f F r i e d r i c h s procedure as an a s y m p t o t i c expansion.
References
[l] J . J . Stoker, Water waves, t h e mathematical t h e o r y w i t h a p p l i c a t i o n s , I n t e r s c i e n c e , New York, 1957.
[2]
K. -0. F r i e d r i c h s , On t h e d e r i v a t i o n of t h e s h a l l o w w a t e r theory, J. J . Stoker, The f o r m a t i o n o f breakers and bores,
Appendix t o :
Comm. Pure Appl. Math., 1 (1948), 1-87. [3]
[4]
n.
8.
OECflHHHKOB,
Bcecoes.
KOH@.
75-fleTHK,
M.
r.
no
0 6 o c ~ o s a ~ nTee O P M H YpaEHeHHflM C
fleTpOBCKOr0.
M e J l K O i BOfibl.
-
WaCTHblMH nPOM3EO&!HblMM,
M3fi-80
MrY,
1978,
C.
B
KH.:
Tp.
nOCEWeHHOk
185-188.
T. Kano - T. Nishida, Sur l e s ondes de s u r f a c e de l ' e a u .
Une
j u s t i f i c a t i o n mathcmatique des equations des ondes en eau peu profonde, C . R. Acad. Sc. P a r i s , t. 287 (17 j u i l l e t 1978), S&ie A, 137-140. [5]
T. Kano
- T.
Nishida, Sur l e s ondes de s u r f a c e de 1 'eau avec une
j u s t i f i c a t i o n mathematique des equations des ondes en eau peu profonde, J . Math. Kyoto Univ., [6]
T. Kano
-
19 (1979), 335-370.
T. Nishida, A mathematical j u s t i f i c a t i o n f o r Korteweg-
de V r i e s e q u a t i o n and Boussinesq e q u a t i o n o f w a t e r s u r f a c e waves, preprint. [7]
T. Kano, Une t h e o r i e t r o i s d i m e n s i o n e l l e des ondes de s u r f a c e de l ' e a u e t l e dfheloppement de F r i e d r i c h s , i n p r e p a r a t i o n .
This Page Intentionally Left Blank
L e c t u r e N o t e s in Num. Appl. Anal., 6 , 59-85 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1.983
Global Existence and Stability of Solutions for Discrete Velocity Models of the Boltzmann Equation
Shuichi KAWASHIMA Department of Mathematics, Nara Women’s University Nara 630, Japan
1. INTRODUCTION The general form o f a d i s c r e t e v e l o c i t y model o f t h e Boltzmann e q u a t i o n i s g i v e n by t h e f o l l o w i n g s e m i - l i n e a r h y p e r b o l i c system o f equations ( c f . [6], [41):
represents t h e mass d e n s i t y o f p a r t i c l e s w i t h t h e v e l o E Rn ( c o n s t a n t v e c t o r ) a t time t and p o s i t i o n x = ( x l ,
where Fi = Fi(t,x) c i t y v i = ( v li, - - - , v A ) -..,xn)
c
where
ai
IRn ; Qi
i s a quadratic operator r e l a t e d t o the binary c o l l i s i o n s :
..
z 0
a r e p o s i t i v e constants, and :A;
f o r some i,j,k,a)
.. (I)
satisfying
..
AiJ = A’J = AJ’ ak ka kn. I n t h e case
(A;.
a r e non-negative constants
and
A i j = Aka ka ij
for a l l
i,j,k,a
= l,...,m
.
n = 1, 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 (1.1) w i t h general
i n i t i a l d a t a (bounded continuous and non-negative) have been o b t a i n e d f o r some p h y s i c a l l y i n t e r e s t i n g models ( c f . [15],[17],[2],[3]).
Moreover i t i s
proved i n [S] t h a t if t h e i n i t i a l d a t a a r e c l o s e t o an a b s o l u t e Maxwellian 1 1 1 s t a t e (see D e f i n i t i o n 2.2) i n HS(IR ) n L ( R ) ( s ? l ) , t h e s o l u t i o n tends t o t h e Maxwellian a t t h e r a t e When
n
t
t-’I4 as t
+-.
2 , g l o b a l e x i s t e n c e r e s u l t s were o n l y known f o r t h e case where 59
Shuichi KAWASHIMA
60
[lo]
t h e i n i t i a l d a t a are small and non-negative (see [16],
and [ 7 ] ) .
I n the p r e s e n t paper we s h a l l study t h e i n i t i a l value problem f o r (1.1) under t h e c o n d i t i o n s ( I ) and (11) (see s e c t i o n 4). L e t F o ( x ) and M > 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)
<
9, Q(F,G) > =
(in) Here
<
9. Q(F,F)
<
E
$
o o
for a l l
F, G
6
IR"'.
> = f o r a l l F E IR"'. , > denotes the standard inner product i n
See [6] o r [4].
Proof.
f o r any
$, F, G
E
IRm.
( i ) 3 ( i i ) and ( i n ) Let l,.--,m.
F =
t
+
Under t h e c o n d i t i o n ( I ) we have
This i d e n t i t y p l a y s a c r u c i a l r o l e i n t h e p r o o f o f ( i ) . We o m i t t h e d e t a i l s .
(F~,-.-,F,)
D e f i n i t i o n 2.2
JRm.
IR"'.
c
A vector
We w r i t e
F >
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
Discrete Velocity Models of the Boltzmann Equation
(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 > = f, Lg > = Taking
= 0 (i-e.,
<
g. L f >
.
This and the property
<
Lf, g >
.
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
<
f, L f
>
n(L))holds i f and o n l y i f
E
ij
Akl(fi
<
, i = l,--..m. f
-
A-ll2f
t
Fj - 7, - 7,)
for a l l
= 0
= l,...,m,
i,j,k,L
= t((M 1/a 1 )~’/2fl,...,(Mm/am)-’/2fm)
E
iVL.
Thus
A-’/2n(L)
= 77L i s proved.
F i n a l l y we show ( i i ) .
Since
from Lemma 2.1 ( i i ) t h a t f o r any
A-’/’JI
JI
E
E &
for
n ( L ) and any
q~
E
& (L), i t f o l l o w s
f, g
E
d,
This completes t h e p r o o f o f Lemma 3.1 Now we s h a l l s t a t e t h e l o c a l existence r e s u l t s f o r t h e i n i t i a l value problem (3.5),(3.6).
Since t h e p r i n c i p a l p a r t o f (3.5) can be regarded as a
Discrete Velocity Models of the Boltzmann Equation
66
f i r s t order symmetric hyperbolic system w i t h constant c o e f f i c i e n t s ,
t h e stan-
dard method allows us t o conclude the existence and uniqueness o f a s o l u t i o n t o (3.5),(3.6) i n the Sobolev spaces: Theorem 3.2 ( l o c a l existence) [n/2] + 1 be integers.
stant
Let (I) be assumed.
n
Let
2
1 and
s
2
If fo E HS(lRn), then there e&sts a p o s i t i v e conTo (depending onZy on 11 ) such that the i n i t i a l value problem
(3.5),(3.6)
foils
has a unique solution
f
E
Co(O,To;Hs(lRn)
) n C 1(O.TO;HS-'(lRn) )
satisfying
4. ESTIMATES FOR LINEARIZED EQUATIONS We s h a l l consider the l i n e a r i z e d equation o f the form
where
L
i s the l i n e a r c o l l i s i o n operator defined by (3.7)1 ( i t should be
L i s r e a l symmetric and p o s i t i v e s e m i - d e f i n i t e ) ; h = h ( t , x ) i s a given f u n c t i o n . We assume t h a t (4.1) i s " d i s s i p a t i v e " i n the f o l l o w i n g sense (see [ l S ] ) .
noticed that
(11)
There e x i s t r e a l anti-symmetric matrices the symmetric p a r t o f = (u1,-.,un)
E
sn-l
.
1 K j V k w .Jw k +
L
KJ ( j= l,--.,n) such 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 any w
Under the conditions ( I ) and (11) we can g e t energy estimates and decay estimates f o r (4.1). Proposition 4.1 (energy estimate)
and 2 0 be i n t e g e r s and Zet E Co(O,T;Ha( Rn) ) and (4.2)
h(t.x)
Then the soZution
E
Q, = [O,T]
x
Co(O,T;Ha(Rn) ) n C (0,T;Ha-'(lRn)
)
n(L)' f
E
Let ( I ) and ( n ) be assumed. Let n 2 1 T be a p o s i t i v e constant. Suppose t h a t h
f o r any
E (t,~
lRn.
o f (4.1) s a t i s f i e s
Shuichi KAWASHTMA
66
for
t
[O,T]. Moreover, i f II
E
+
t
0
C
Here
1
>
1, we have
3
IILf(T)
-
h(T)llE-l dT} t
P
i s a constant and
5
c
Ilf(o)ll,
2
for
t
E
[O,T]. n(L)'.
i s the orthogonal projection onto
Remark 4.1
In t h e case E 2 1, t h e combination (4.3) s u f f i c i e n t l y small c o n s t a n t a > 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
w =
?.
(4.10) w i t h (4.11) where
V(W) =
Since
(
,)
f)
= Re
L
Take t h e i n n e r p r o d u c t ( i n tm) o f a r e r e a l symmetric, i t s r e a l p a r t i s
(h, f ) ,
denotes t h e standard i n n e r p r o d u c t i n
- i ( c I K ( w ) (K(w)
iK(w)
1 V J W J. .
V(W) and
( i l f 1 2 ) t + (Lf,
(4.10) by Since
and
Taking t h e F o u r i e r t r a n s f o r m o f ( 4 . 1 ) , we have
=I KJw.) J
Next m u l t i p l y
Cm.
and then take t h e i n n e r p r o d u c t w i t h
f.
i s hermitian, the real p a r t o f the r e s u l t i n g e q u a l i t y i s
where [K(b)V(w)]' denotes t h e symmetric p a r t o f K ( w ) V ( W ) . N o t i n g t h a t i s p o s i t i v e s e m i - d e f i n i t e , [ K ( w ) V ( W ) ] ' t L i s p o s i t i v e d e f i n i t e and h E n(L)',
we have from (4.11) and (4.12)
i[flP)t +
ClhI2,
(4.13)
(
(4.14)
l - T1[ c l ( i K ( w ) f , ? ) I t +
c(P+fj2
w i t h some p o s i t i v e constants
5
c
c l c l 2 If1 - 2 and
CIS/2 [ Pf Af l 2
C ( c
5
ClLf
- hi2
M u l t i p l y (4.13) and (4.14)
L
Shuichi KAWASHIMA
68
by
( 1 + 1c12)e
IR;.
2 e-1 (1 + (51 )
and
, respectively,
and i n t e g r a t e over
[O,t]
x
Then t h e P l a n c h e r e l ' s theorem gives t h e desired, estimates (4.3) and This completes t h e p r o o f of P r o p o s i t i o n 4.1.
(4.4).
Combine (4.13) and (4.14) so as t o make (4.13) x > 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
Discrete Velocity Models of the Boltzmann Equation
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(<)
This enables us t o represent
e-tS(g) g ( 5 ) e x p l i c i t l y .
Estimating t h i s expression c a r e f u l l y , we o b t a i n the d e s i r e d estimate (4.19) (cf. [El,
[5] o r [14]).
The d e t a i l s a r e omitted.
5. GLOBAL EXISTENCE To prove the existence o f global s o l u t i o n s t o (3.5),(3.6),
we s h a l l g e t
an a p r i o r i estimate o f s o l u t i o n s .
Let ( I ) and (11) be assumed. L e t n 2 1 P r o p o s i t i o n 5.1 (a p r i o r i estimate) and s 2 [n/2] + 1 be i n t e g e r s and l e t T be a p o s i t i v e constant. Suppose t h a t fo E HS(IRn), and t h a t f E Co(O,T;Hs(IRn) ) n C 1 (0,T;Hs-'(Rn) ) i s a soZution of (3.5),(3.6). Then there e x i s t constants 61 > 0 and C, =C1(61) >
1 such that i f
Proof.
N,(T)
5
sup I l f ( t ) J J s 5 6,, then O
The estimate (5.1) can be proved by a method s i m i l a r t o the one em-
ployed i n [12] and [ll].L e t
r = [n/2] + 1, and l e t 6 o
be a p o s i t i v e con-
s t a n t such t h a t (5.2)
Nr(T)
where
F(t,x) = M
if
Nr(T)
5
implies
6o
+
nl/'f(t,x),
holds f o r
and
5
Fi(t,x)/Mi
ko
5
ko
for
(t,x)
E
i s the constant i n Lennna 2.4.
QT ,
Then
5 60,
t
(5.3)
-1 ko
llf(t)1I2 + t
E
[O,T].
i n t o account, we have
1
0
I(Lf(T1
- r(f,f)(T)ll
2 dT 5
C
2
[Ifoll
Indeed, i n t e g r a t i n g (2.5) over
Qt
and t a k i n g (2.4)
Shuichi KAWASHIMA
70
t
Ilf(t)112 +
(5.4) where
t i v e constants, R(F)
c > 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
Discrete Velocity Models of the Boltzmann Equation
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),
<
e(j)
>
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)
<
[XVl’f,
f >
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.
Discrete Velocity Models of the Boltzmann Equation
[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 .
This Page Intentionally Left Blank
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)
(<(r+t) + g ( r - t ) ) +
=
-
[aiL(s, 0)
@(s,O)]ds
r-t
3.
Proof o f Theorem We s h a l l show the theorem by a c o n t r a d i c t i o n .
Suppose t h a t
u i s a n o n - t r i v i a l C 2 - s o l u t i o n o f ( 3 ) , ( 4 ) ; and then g ( x ) ro, rl be p o s i t i v e numbers such t h a t g ( x ) ?! 0
*
0.
Let
if ro rl.
We f i r s t show
(7)
v (r,t) L 0
for
r L ro + t, t 2 0.
Take to (>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
Let 0 <
E
< 1.
Suppose v ( r , t )
=
-
E
f o r some r, t w i t h 0 < t
L e t tl be t h e f i r s t p o s i t i v e t f o r which v(rl. ( L ro + tl).
3;
=
-
E
f o r some r1
Then, s i n c e
U(s,O)
- T(s,O)
by f = 0, and s i n c e
a contradiction:
r 2 ro + t.
tl)
s to.
-
5L E
2
= 0
0, we have, by ( 8 ) and ( 6 ) w i t h r = rl,
-
1
E;
note
4
L 0 and v ( r , t )
2
-
E
t = tl,
for 0 < t s t
Hence v ( r , t ) 2 0 f o r 0 < t < to, r L ro+ts i n c e
E
1' i s arbitrary.
Kytiya MASUDA
90
By step by s t e p , we can conclude that v ( r , t ) 2 0 f o r r 2 ro + t , t 2 0. l t T h u s , by (6) and ( 7 ) , v ( r , t ) 2 k < ( r - t ) + T(T+r-t, T ) d T ; note -
g
2 0
T
and
0 2
0.
Integrating both sides of the above inequality with
respect t o r over the interval: t+rO5 r 5 t + r l , we get t k ( t ) 2 CI++ 5 ( T + r , T ) dr) d.r (9) rO where '1 k(t) = v(r + t , t ) dr; C 1 = G(s) ds.
J o (Ir1
1
jrl
'0
rO
By the Schwarz inequality,
rO
sc*
i"
'0
$ ( P +r,
T)
dP
rO since with some positive constant C2. Hence, by (61, k ( t ) 2 C1 + C3
-;r~ k(r)'
dT,
t
2 0.
( c1 , c 3 ; positive
0
constant 1.
If k ( t ) i s a solution of the ordinary d i f f e r e n t i a l equation d
k(t) /
then we have k ( t )
d t = C3 2
k(t)*/
k ( t ) , by
( t + l ) ; k(O) = C1,
the comparison theorem.
By a simple calculation we know t h a t Hence k ( t ) theorem.
blows u p in f i n i t e time:
k ( t ) blows
u p in f i n i t e time.
a contradiction. This proves t h e
91
Quasi-Linear Wave Equations
REFERENCES
C l ] John, F.,
Blow-up o f s o l u t i o n s f o r q u a s i - l i n e a r wave equations
i n t h r e e space dimensions.
[2] Kato, T.,
Comm. Pure Appl. Math. 34, 1981, pp. 29-51.
Blow-up o f s o l u t i o n s o f some n o n l i n e a r h y p e r b o l i c equations.
Corn. Pure Appl. Math., 33, 1980, pp. 501-505. [3]
Klainerman, S., Global e x i s t e n c e f o r n o n l i n e a r wave equations.
Corn.
Pure Appl. Math. 33, 1980, pp. 43-101.
[4] Klainerman, S. and Mqida, A.,
Formation o f s i n g u l a r i t i e s f o r wave
equations i n c l u d i n g t h e n o n l i n e a r v i b r a t i n g s t r i n g . Appl. Math.,
[5]
Van Wahl, W.,
Comm. Pure
33, 1980, pp. 241-263. Lp decay r a t e s f o r homogeneous wave equations.
Z. 120, 1971, pp. 93-106.
Math.
This Page Intentionally Left Blank
L e c t u r e N o t e s in Num. Appl. Anal., 6, 93-105 (1983) Recent Topics in Nonlinear PDE, Hiroshima,1983
A Kinetic Approximation of Entropy Solutions of First Order Quasilinear Equations
T e t s u r o MIYAKAWA Department of Mathematics, Hiroshima University Hiroshima 730, Japan
T h i s paper deals w i t h a new method f o r c o n s t r u c t i n g g l o b a l weak s o l u t i o n s o f t h e Cauchy problem f o r general f i r s t o r d e r q u a s i l i n e a r equations o f c o n s e r v a t i o n t y p e i n s e v e r a l space v a r i a b l e s :
where s u b s c r i p t s i n d i c a t e d i f f e r e n t i a t i o n .
Using t h e v a n i s h i n g v i s c o s i t y method
under some r e g u l a r i t y and boundedness assumptions on Ai and B, Kruzkov [6] proved t h a t f o r each
uo
in
Lm(Rn)
t h e r e i s a unique f u n c t i o n
u(x,t)
i s a s o l u t i o n o f (M) i n t h e f o l l o w i n g sense:
(i) u (ii)
is in
u(-,t)
(iii)
(E)
uo
in
T > 0.
L ~ ~ ~ as ( R t ~J. )0.
The i n e q u a l i t y n l l + t / u - k l d x d t + i1 =1 I J ~ g n ( u - k ) [ A ~ ( x , t , u ) - A ~ ( x , t , k ) ] +id~ x d t
-
n A ~ . ( x , t , k ) + B ( x . t , u ) ] ~ d ~ d 't>
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
= <
c = 0
s
= <
be t h e f a m i l y o f s o l u t i o n o p e r a t o r s o f t h e
and p u t
L e t the nonlinear operator
We now d e f i n e f o r small
t}
K(t,s)
h > 0
be d e f i n e d by
approximate s o l u t i o n
u
h
by
.
Tetsuro MIYAKAWA
96 h
[t/hl
I1
u (x,t)
= (K(t,
n
h[t/hl)
K(Jh,(J-l)h)uo)(x)
j=l
i
where [a] denotes t h e g r e a t e s t i n t e g e r i n a € R
THEOREM.
Assume (A.1)-(A.3)
e n t r o p y s o l u t i o n o f (M) w i t h
and l e t
u(.,O)
h
u (-,t)
+
uo
= uo.
.
Our r e s u l t i s t h e f o l l o w i n g
be i n
L"(Rn).
Let
u
be t h e
Then in
u(.,t)
1
1 Lkoc(Rn)
as
h
-+
0
u n i f o r m l y i n t 2 0 on every compact s u b i n t e r v a l .
I n what f o l l o w s we prove t h i s r e s u l t under t h e a d d i t i o n a l assumption t h a t uo
i s l o c a l l y o f bounded v a r i a t i o n i n t h e sense o f T o n e l l i and Cesari ([4],[7]).
The passage t o t h e case o f general
2.
uo
i s discussed i n [3].
Estimates f o r approximate s o l u t i o n s
To ensure t h e convergence o f t h e approximate s o l u t i o n s estimates f o r
uh
and t h e i r d e r i v a t i v e s .
uh we need some
The r e s u l t s i n t h i s s e c t i o n a r e proved i
i n [3] i n t h e time-independent case: Ai = A (x,u),
B = B(x,u); and t h e p r o o f s
i n [3] can be a p p l i e d t o t h e p r e s e n t case w i t h no e s s e n t i a l change. The s o l u t i o n
where
fi(o,S)
z ( o ) = z(o;C)
=
f
o f t h e problem (m) w i t h
i s expressed as
1 axi
(z(o),o,S)+b(z(a),o,~), c(a,S) = c(z(a),a,S) and i denotes t h e c h a r a c t e r i s t i c c u r v e associated w i t h t h e l i n e a r
e q u a t i o n (m) such t h a t o f K(t,s),
= fo
zft) = x
and
z f s ) = y.
one can e a s i l y show t h e f o l l o w i n g
Using (2.1) and t h e d e f i n i t i o n
First Order Quasilinear Equations LEMMA 2.1. IV(x)I L r
Fix
for
T
x€Rn
0
and l e t
v
and
IC(x,t))
zr
>
lK(t,S)Vl,
/-Im
where
= <
ea(t-S)
r
>
0.
Let
p(x)
Lm(Rn).
r > 0
Choose
f o r ( x , t ) € Rnx[O,T].
(l+t-s)r
denotes t h e norm o f
To e s t i m a t e L 1-norms of
be i n
97
o5s5
for
so t h a t
Then we have
t 2 T,
Lm(Rn).
K(t,s)v,
we i n t r o d u c e t h e weight f u n c t i o n s
pr(x),
be a smooth nonnegative f u n c t i o n w i t h compact s u p p o r t i n
such t h a t i p ( x ) d x
= 1.
Fixing T
>
Rn
0, we p u t
(2.2)
where n
(2.3) and of
Mr =
br = w/Mr,
w >
0
sup{lai(x,t,E)l; i=l
i s an,y f i x e d number.
n Lq(R ), 1 ~q
I-,and
I n t h e same way as i n [3],
LEMMA 2.2.
Let
v
I n what f o l l o w s we denote by
1.1
9
t h e norm
put
and
where
w
be i n
IC(x,t)l 2 r
L"(Rn)). on
Choose
Rnx[O,T].
r > 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.
First Order Quasilinear Equations
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
$I€ Ci(Rn)),
First Order Quasilinear Equations
where
103
L(t)$ =
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
First Order Quasilinear Equations
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.
This Page Intentionally Left Blank
L e c t u r e N o t e s in Num. Appl. Anal., 6, 107-124 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1983
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);
Delay-Diffusion Equations
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
Delay-Diffusion Equations
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
Delay-Diffusion Equations
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<
e x i s t s a p o s i t i v e constant and
U(E)
D=X. J
-1
E2€‘ ( j 1 )
y = y 2 ~ ’ t O ( ~ ’ ) where ,
defined i n (4.6),
55
y2
Then t h e r e
< c 0 such t h a t f o r each E ~ ( O , E ) , P P t h e e q u a t i o n ( 4 . 5 ) has a F l o q u e t exponent
E
i s one o f t h e r o o t s o f (4.11) and
i s as i n Theorem
E~
E2.
A
j
is
H.
Main theorems
From Lemma
Theorem
A
in
54 n e x t theorems i m m e d i a t l y f o l l o w :
E.
C o n s i d e r t h e e q u a t i o n (4.1) under t h e assumptions ( A l ) , (A3).
Let
D2
be an
mx m
(82) and
orthogonal m a t r i x w i t h non-negative
121
Delay-Diffusion Equations
elements
X
and l e t
d e f i n e d i n (4.6).
j
c(O,E
E
9
periodic solution
Theorem
j - t h eigenvalue o f t h e operator
If f o r the matrix
E2 = 0 X
2 j y2, t h e r e e x i s t s a c o n s t a n t
has a p o s i t i v e r o o t that for
be t h e
),
and
U=U(E)
p(t;E)
D=D2c2
A~(R)
t h e e q u a t i o n (4.11)
O<E <E such q' q P t h e s p a t i a l l y homogeneous E
t o (4.1) i s unstable.
C.
C o n s i d e r t h e e q u a t i o n ( 4 . 1 ) u n d e r t h e same assumptions i n Theorem Let
i. be
a p o s i t i v e number and l e t
o f the operator boundary.
GN(c')., where
0
A2(?)
8.
be t h e second e i g e n v a l u e
i s a bounded domain w i t h smooth
Fix the diffusion coefficients.
I f f o r the matrix
E2 = XD
the
e q u a t i o n ( 4 . 1 1 ) has a p o s i t i v e r o o t , t h e n t h e r e e x i s t s a p o s i t i v e constant
9
E ~ < E such t h a t f o r E h(O,cr), ~ = U ( E ) and t h e domain P w i t h t h e c o r r e s p o n d i n g e i g e n v a l u e h 2 ( R ) = k2, the spatially
homogeneous p e r i o d i c s o l u t i o n
Remark
i s unstable.
.
As mentioned i n solution
p(t;E)
p(.;-c)
$ 4 , i t i s o f t e n t h e case t h a t t h e b i f u r c a t i n g
i s s t a b l e ( w i t h r e s p e c t t o e i t h e r s p a t i a l l y homogeneous
o r inhomogeneous p e r t u r b a t i o n ) a t l e a s t n e a r t h e b i f u r c a t i o n p o i n t . However, even i n such a case, we see f r o m Theorems i f we f i x
E
and change
D
homogeneous p e r i o d i c s o l u t i o n
and P(.;E)
0 and
and
C
that
9 appropriately then t h e s p a t i a l l y may e v e n t u a l l y l o s e i t s s t a b i l i t y .
T h i s shows t h a t t h e s t a b i l i t y r e g i o n o f L ( f o r w h i c h becomes s m a l l e r when
B
P(.;E)
i s stable)
a r e changed i n t h e above manner.
Yoshihisa MORITA
122
§6 A p p l i c a t i o n
We s h a l l apply t h e theorems i n
55 t o t h e e q u a t i o n i n §I.
L e t us c o n s i d e r t h e equation:
As mentioned i n
51, f o r any f i x e d p o s i t i v e d i f f u s i o n c o n s t a n t
bifurcates a t
1.1'0
from t h e steady s t a t e
v(t,x)
I0;
t h e b i f u r c a t i n g 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 . a p p l y i n g Theorems
and
R a 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
any bounded domain p(t;E)
d
and By
6 and C, howewver, we s h a l l see t h a t t h e
d e s t a b i l i z a t i o n o f t h e p e r i o d i c s o l u t i o n occurs i n t h e sense o f Remark in
15.
To check t h a t Theorems us c a l c u l a t e t h e r o o t s
y2
6 and C a p p l y t o t h e p r e s e n t case, l e t o f (4.11).
I n t h i s case t h e e q u a t i o n (4.11)
i s given by (6.1
( 1 + (;
where
)y;
d = +,(n)d2E2;
[13 ; 551.
+ 2 ( d2 -
n
3n2
+ -zb )y2
t
d;
-
d2 = 0,
t h e readers w i l l f i n d t h e e q u a t i o n (6.2) i n
Thus (6.2) has a p o s i t i v e r o o t
y2
for
0 < d2 < n / 5 , which
ensures t h e occurrence o f 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 . I n Fig A i n the
we i l l u s t r a t e t h e s t a b i l i t y and i n s t a b i l i t y r e g i o n s
( p , \ d)-parameter space. (Note t h a t t h i s f i g u r e i s v a l i d o n l y
2
Delay-Diffusion Eqoations
i n a s u f f i c i e n t l y small neighborhood o f
123
Since
( p , X2d) = ( O , O ) . )
p = p ( ~ ) i s expanded as
L-i =
the curve
n
i n Fig. A
L - i ~ E )=
3n - 2 ~ lo €2 +
has s l o p e
2n 3n-2
0(~3),
a t the origin.
0 ( Fig. A
References
[ l ] G.E.
H u t c h i n s o n ; C i r c u l a r Causal Systems i n Ecology, Ann.
N. Y. Acad.
50 ( 1 9 4 8 ) : 221-246. Sci. -
[ 2 ] G.S. { 1
Jones; The E x i s t e n c e o f P e r i o d i c S o l u t i o n o f
+ f ( x ) 1 , J . Math. A n a l . A p p l .
5
f ' ( x ) = -of(x-1)
( 1 9 6 2 ) : 435-450.
[3] J. Kaplan and J . Yorke; On t h e S t a b i l i t y o f P e r i o d i c S o l u t i o n s o f a D e l a y D i f f e r e n t i a l E q u a t i o n , S I A M J . Math. Ana. [ 4 ] S. Chow and J. M a l l e t - P a r e t ;
J. D i f f e r e n t i a l E q u a t i o n s
26
fi ( 1 9 7 5 ) :
262-282.
I n t e g r a l A v e r a g i n g and B i f u r c a t i o n ,
( 1 9 7 7 ) : 112-159.
[5] J. Hale; Theory 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 s , S p r i n g e r , New
Yoshihisa MORITA
124
York - Heidelberg-Berlin, (1977). [6] N . MacDonald; Time Lags in Biological Models, Springer, New York-
Heidel berg-Berl in, (1 978).
[7] K. Yoshida; The Hopf Bifurcation and its Stability for Semilinear Diffusion Equations with Time Delay Arising in Ecology, Hiroshima Mathematical J.
11 (1982):
321-348.
[8] J. Lin and P.B. Kahn; Phase and Amplitude Instability in Delay-
Diffusion Population Models, J. Math. Biology
13 (1982): 383-393.
191 B.D. Hassard, N.D. Kazarinoff, Y.H. Man; Theory and Applications of
Hopf Bifurcation, London Math. Soci., Lecture Note
[lo]
41 (1981).
C.C. Travis and G.F. Webb; Existence and Stability for Partial
Functional Differential Equations, Trans. Amer. Math. SOC.
200
(1974):
394-418.
[ll] T. Erneux and M. Herschkowitz-Kaufmann; Bifurcation Diagram of a Model Chemical Reaction Bull. Math. Biol.
-
I, Stability Change of Time-Periodic Solutions,
fl (1979):
21-38.
[I21 K. Maginu; Stability of Spatially Homogeneous Periodic Solutions of Reaction-Diffusion Equations, J. Differential Equations
fl (1979):
130-138. C131 Y. Morita; Destabilization of Periodic Solutions Arising in DelayDiffusion Systems in Several Space Dimensions, received in Japan J. Appl. Math..
L e c t u r e N o t e s in Num. Appl. Anal., 6, 125-142 (1983) Recent Topics in Nonliiieav PDE, Hiroslrinia, 1983
On Some Nonlinear Dispersive Systems and the Associated Nonlinear Evolution Operators
S h i n n o s u k e OHARU* and Tadayasu TAKAHASHI* * 'Mathematics Department, Hiroshima University, Hiroshima 730, Japan **National Aerospaee Laboratory. Tokyo 182, Japan
1. I n t r o d u c t i o n I n t h i s p a p e r we a r e concerned w i t h t h e i n i t i a l - b o u n d a r y v a l u e p r o b l e m f o r the nonlinear dispersive equation (1.1)
ut + ( @ ( t , x , u ) ) x + I/l(t,x,u)
(1.3)
u(t,a+) = u(t,b-)
-
0 < t < T, a < x < b,
uxxt = 0,
O < t < T ,
= 0,
where T > 0, ( a , b ) i s a ( p o s s i b l y i n f i n i t e ) s u b i n t e r v a l o f t h e r e a l l i n e R, @
and $I i n ( 1 . 1 ) a r e r e a l - v a l u e d f u n c t i o n s d e f i n e d on [O,T]x(a,b)xR,
i s a g i v e n i n i t i a l f u n c t i o n , and t h e terms u ( t , a + ) , the values limxzau(t,x),
limxtbu(t,x),
u(t,b-)
uo i n ( 1 . 2 )
i n (1.3) stand f o r
respectively.
A s p e c i a l case o f ( 1 . 1 ) i s t h e l o n g wave e q u a t i o n (1.4)
Ut
+ ux + uux -
Uxxt
= 0
w h i c h was p r o p o s e d as an a l t e r n a t i v e t o t h e Korteweg-de V r i e s e q u a t i o n Ut
+ ux + uux + uxxx
by Benjamin, Bona and Mahony [ 2 ] .
= 0
See a l s o Benjamin [l] and P e r e g r i n e [12] f o r 125
126
Shinnosuke OHARU and Tadayasu TAKAHASHI
t h e d e r i v a t i o n o f (1.4). These equations a r e well-known as mathematical models f o r l o n g waves o f small amplitude and, as seen from t h e papers by Bona and Bryant [3],
Bona and Dougalis [4],
and Medeiros and Miranda
[lo],
Iwamiya e t a l . [ 6 ] , Medeiros and Menzala [9]
many works have been devoted t o t h e study o f
equations o f t h e type ( 1 . 4 ) . I n t h i s paper we t r e a t t h e problem (1.1)-(1.3)
in
1 t h e Sobolev space Ho(a,b) i n terms o f n o n l i n e a r o p e r a t o r t h e o r y and e s t a b l i s h e x i s t e n c e and uniqueness theorems f o r t h e s o l u t i o n s by a p p l y i n g a g e n e r a t i o n theory o f nonlinear evolution operators.
2
L e t A be t h e d i f f e r e n t i a l o p e r a t o r d/dx in L (a,b)
and l e t A be t h e one-
2
1 dimensional Laplace o p e r a t o r d e f i n e d by Av = A v f o r v E H,(a,b)
n H2 (a,b).
F u r t h e r , we d e f i n e two composition o p e r a t o r s G ( t ) and Y ( t ) i n LL(a,b) by [ O ( t ) v l ( x ) = @(t,x.v(x))
and
[ y ( t ) v l ( x ) = $(t,x,v(x)),
r e s p e c t i v e l y . Then t h e problem ( 1 . 1 ) - ( 1 . 3 )
x E (ah),
can be r e f o r m u l a t e d as t h a t of
1 1 f i n d i n g f o r a g i v e n i n i t i a l f u n c t i o n uo i n Ho(a,b) an Ho(a,b)-valued f u n c t i o n
u ( * ) on [O,T]
1 s a t i s f y i n g u ( 0 ) = uo, u ( - ) E C ([O,T];
1 Ho), and
1 where u ' ( . ) stands f o r t h e s t r o n g d e r i v a t i v e i n Ho(a,b) o f u ( * ) . Now equation (1.5) i s r e w r i t t e n as a standard t y p e o f e v o l u t i o n equation s i n c e ( I
2
a bounded i n v e r s e on L (a,b).
Namely, d e f i n i n g operators A ( t ) , t E
A(t) = (A
-
-
A ) has
[O,T], by
I ) - l [ A Q ( t ) + Y(t)],
1 we w r i t e ( 1 . 5 ) as t h e e v o l u t i o n equation i n Ho(a,b) (1.6)
u ' ( t ) = A(t)u(t)
and reduce t h e problem ( 1 . 1 ) - ( 1 . 3 )
Now,
t o t h e Cauchy problem f o r (1.6).
u s i n g t h e method o f l i n e s as t r e a t e d i n M a r t i n [8], we f i n d r e g u l a r
s o l u t i o n s o f (1.6) and c o n s t r u c t t h e s o l u t i o n o p e r a t o r s f o r (1.6).
The f a m i l y
o f s o l u t i o n o p e r a t o r s then g i v e s r i s e t o an e v o l u t i o n o p e r a t o r P = {U(t,sf:
0 5 s 2 t 2 T I on HA(a,b) and t h e e v o l u t i o n o p e r a t o r U p r o v i d e s s o l u t i o n s o f t h e
Nonlinear Evolution Operators
127
1 problem (1.1)-(1.3) i n t h e sense t h a t g i v e n i n i t i a l f u n c t i o n uo i n Ho(a,b) f u n c t i o n u ( t , x ) = [U(t,O)uo](x)
the
i s a unique s o l u t i o n o f (1.1)-(1.3).
The same approach as above i s employed i n [ll] f o r t h e case i n which Q ( t ) :Q i s time-independent and Y ( t ) :0. F u r t h e r , we n o t e t h a t e v o l u t i o n equations o f t h e t y p e (1.5) a r e a l s o t r e a t e d i n [13] and [14] by Showalter. Our technique p r o v i d e s a u n i f i e d method f o r s o l v i n g n o n l i n e a r d i s p e r s i v e equations as mentioned above i n e i t h e r case o f O i r i c h l e t and p e r i o d i c boundary c o n d i t i o n s . I n f a c t , we s h a l l show v i a a p p r o p r i a t e t r a n s f o r m a t i o n s o f unknown f u n c t i o n s t h a t our r e s u l t can be a p p l i e d t o t h e long wave e q u a t i o n (1.4) w i t h v a r i o u s time-dependent boundary c o n d i t i o n s . Moreover, our r e s u l t as w e l l as t h e argument used i n t h e p r o o f can be e a s i l y adapted so as t o cover t h e case i n which (a,b) i s t h e whole l i n e R and c o n d i t i o n (1.3) i s r e p l a c e d by t h e p e r i o d i c boundary c o n d i t i o n u ( t , x ) = u ( t , x + l ) f o r ( t , x ) E ( 0 , T ) x R . S e c t i o n 2 c o n t a i n s assumptions on t h e f u n c t i o n s
$J
and $ and discusses
t h e uniqueness o f s o l u t i o n s o f t h e problem (1.1)-(1.3). S e c t i o n 3 deals w i t h b a s i c p r o p e r t i e s o f t h e o p e r a t o r s A ( t ) and t h e c o n s t r u c t i o n o f approximate s o l u t i o n s o f (1.6). S e c t i o n 4 i s devoted t o t h e c o n s t r u c t i o n o f an e v o l u t i o n 1 o p e r a t o r on Ho which y i e l d s s o l u t i d n s o f (1.6). F i n a l l y , i n S e c t i o n 5, o u r r e s u l t i s a p p l i e d t o (1.4) w i t h nonhomogeneous boundary c o n d i t i o n s .
2. Uniqueness o f s o l u t i o n s I n t h i s s e c t i o n we make t h r e e assumptions on t h e f u n c t i o n s
@
and J,
appearing i n e q u a t i o n (1.1) and then d i s c u s s t h e uniqueness o f s o l u t i o n s o f t h e i n i t i a l -boundary v a l u e problem 1.1)-( 1.3). k k I n what f o l l o w s H denotes t h e Sobolev space H (a,b)
f o r each nonnegative
0 . i n t e g e r k ; i n p a r t i c u l a r , H i s t h e o r d i n a r y Lebesgue space L2 :L2(a,b). i n n e r p r o d u c t and t h e norm of H
k
a r e denoted by
(a,.),
and
I.lk.
The
respectively.
k For k 2 1, H can embedded i n t o t h e space Ck-l z C k - l (a,b) o f k - 1 times 1 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 f u n c t i o n s on (a ,b). The Sobolev space Ho(a ,b) i s
128
Shinnosuke OHARU and Tadayasu TAKAHASHI
simply denoted by
1 Ho.
L e t X be a Banach space. C([O,T];
X) i s w r i t t e n f o r t h e space o f X-valued
k For each p o s i t i v e i n t e g e r k we w r i t e C ([O,T];
continuous f u n c t i o n s on [O,T].
X)
f o r t h e space o f X-valued f u n c t i o n s which a r e k 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
on [O,T]. We now assume t h a t t h e f u n c t i o n s $(t,x,C)
and $(t,x,S)
s a t i s f y three
c o n d i t i o n s l i s t e d below:
( I ) The f u n c t i o n s
$, $x, $c, QXc,
and f o r each r > 0, $
5
$ and @
5
a r e continuous on [O,T]x
and I# a r e bounded on [O.T]
5
1 (11) For each v E Ho, $(t,-,v(-))
E H1 f o r t
x
(a,b)
E [O,T]
x
(a,b)xR,
[-r,r].
1 and t h e H -valued f u n c t i o n
1 E L 2 f o r t E [O,T] $ ( t , . , v ( * ) ) i s o f c l a s s C([O,T]; H ) , w h i l e $ ( t , . , v ( * ) ) 2 2 and t h e L -valued f u n c t i o n t -+ $ ( t , - , v ( * ) ) i s o f c l a s s C([O,T]; L ) . t
-+
(111) There a r e nonnegative continuous f u n c t i o n s al and a2
L e t uo be g i v e n i n
on [O,T]
1 Ho. A r e a l - v a l u e d f u n c t i o n u :u ( t , x ) on [O,T]
i s c a l l e d a s o l u t i o n o f (1.1)-(1.3) (2.1)
u(0,x)
= uo(x)
for
+ I#(t.x,u(t,x))w(x) 1
t E (0,T) and w E Ho.
x
(a,b)
i f i t s a t i s f i e s the following conditions:
x E (a,b);
and
for
such t h a t
+ utX(t,x)w'(x)]dx
= 0
Nonlinear Evolution Operators
129
1 (2.2) and ( 2 . 3 ) t o g e t h e r i m p l y t h a t t h e Ho-valued
Conditions (2.1), f u n c t i o n u ( t ) :u ( t , . )
i s o f c l a s s C'([O,T];
Hh) and u ' ( t ) = u t ( t , . )
for
1 where u ' ( t ) stands f o r t h e s t r o n g d e r i v a t i v e i n Ho o f u ( t ) .
t E (O,T),
Condition (2.4) states t h a t u s a t i s f i e s the equation
(2.5)
ut + ( $ ( t , x , u ) ) x
+ $(t,x,u)
-
0 < t < T, a < x < b.
utxx = 0,
As mentioned i n t h e I n t r o d u c t i o n , l e t A be t h e d i f f e r e n t i a l o p e r a t o r d/dx from H Av =
1
L 2 and l e t
into
A be t h e one-dimensional Laplace o p e r a t o r d e f i n e d by
A 2 v for v E Ho1 rl H 2 . I t i s easy t o see t h a t A s a t i s f i e s t h e r e l a t i o n
(2.6)
f o r v E H' and w E H i ,
(hv,dO = - (v,Aw)o
-
and t h a t ( I
A ) has a bounded i n v e r s e ( I
-
on
A)-'
L2. For each
t E [O,T],
we
d e f i n e composition o p e r a t o r s @ ( t ) and Y ( t ) by
1 1 f o r v E H o , r e s p e c t i v e l y . By c o n d i t i o n ( 1 1 ) , @ ( t ) maps H i i n t o H and Y ( t ) maps 1 2 1 Ho i n t o L f o r t E [O,T]. Moreover, s i n c e sup I v ( x ) I 5 l v l l f o r v E Ho, i t a<x
(2.7)
I@(t)vl
f o r t E [O,T]
-
@(t)vzIo
1 and vi E Ho w i t h
vi
Il
Here Mr i s t h e c o n s t a n t
I r , i = 1,2.
d e f i n e d f o r each r > 0 by
We then see t h a t a f u n c t i o n u ( t , x ) on [O,T] i f and o n l y i f t h e f u n c t i o n u ( t ) :u ( t , . )
(2.9)
u(0)
= uo;
(2.10)
u(.)
E C ([O,Tl;
1
x
(a,b) i s a s o l u t i o n o f (1.1)-(1.3)
s a t i s f i e s the f o l l o w i n g :
1 1 H o ) and u ' ( t ) E Ho
n
H
2
f o r t E (0,T);
Shinnosuke OHARU and Tadayasu TAKAHASIII
130 (2.11)
( I - A ) u ' ( t ) + A @ ( t ) u ( t ) + Y ( t ) u ( t ) = 0 f o r t E (0,T).
I n view o f . t h i s we o b t a i n t h e f o l l o w i n g uniqueness theorem. P r o p o s i t i o n 2.1.
1 Given i n i t i a l f u n c t i o n uo i n Ho, t h e r e e x i s t s a t most one
s o l u t i o n o f t h e i n i t i a l - b o u n d a r y value problem (1.1)-(1.3). Proof. Let u i ( - ) ,
i = 1,2,
be any p a i r o f s o l u t i o n s s a t i s f y i n g ul(0) 1 H o ) , t h e r e i s a number r
~ ~ ( =0 uo. ) Since u i ( - ) E C([O,T]; Iui(t)Il
I r f o r t E [O,T]
Then, u s i n g (2.6),
f o r t E (0,T).
and i = 1,Z.
( 2 . 7 ) and (2.11),
WP
Set w ( t ) = ul(t)
-
=
0 such t h a t
u 2 ( t ) f o r t E [O,T].
have
S o l v i n g t h i s d i f f e r e n t i a l i n e q u a l i t y , we o b t a i n w ( t ) :0.
3. The e v o l u t i o n e q u a t i o n For each t E [O,T],
(3.1)
1 2 we d e f i n e an o p e r a t o r A ( t ) from Ho i n t o H: n H by A(t) = (A
-
I)-'[A@(t) + Y(t)].
The e q u a t i o n (2.11) i s then w r i t t e n as (3.2)
u ' ( t ) = A(t)u(t),
0 < t < T,
q.e.d
Nonlinear Evolution Operators
and t h e problem ( 1 . 1 ) - ( 1 . 3 )
i s reduced t o t h e Cauchy problem f o r t h e e v o l u t i o n
1 e q u a t i o n ( 3 . 2 ) i n Ho (see P r o p o s i t i o n 3.2 below). I n t h i s s e c t i o n we d i s c u s s fundamental p r o p e r t i e s o f t h e o p e r a t o r s A ( t ) and c o n s t r u c t approximate s o l u t i o n s o f ( 3 . 2 ) . ( i ) L e t r > 0 and Mr t h e number d e f i n e d by (2.8).
Lemma 3.1.
l A ( s ) v - A(t)Wll
(3.3)
-
5 M,~v
( i i ) L e t ai, a ( t ) = (a,(t)
(3.4)
W[0
+ I @ ( s ) v - @ ( t ) V l g + I'+'(s)V
1 . and v, w E Ho w i t h l v I 1 5 r and
f o r s , t E [O,T]
t E [O,T].
Then
lwll
-
YY(t)Vlg
Ir .
i = 1,2, be t h e f u n c t i o n s g i v e n i n c o n d i t i o n (111) and s e t
+ a2(t))/2 for t
E [O,T].
Set B ( t ) = [ A @ ( t ) O+ Yl(t)OIO f o r
Then (A(t)v,v)l
5 cr(t)lvl:
Proof. L e t t E [O,T]
+ O(t)lvll
for
t E [O,T]
1 and v E H o .
1 and v, w E Ho. By (2.6) and (3.1) we have
( A ( t ) v ,w)l = (AA(t)v
-
h @ ( t ) v - Y(t)v,w)O + (AA(t)v,hw)O
= (@(t)v,hw)O - ( Y t ) v
This, t o g e t h e r w i t h (2.7), t E [O,T]
implies
i ) . To show t h e second a s s e r t i o n , l e t
1 and v E Ho. Then we have
( @ (t ) v ,Av
lo
.h =
J,
,do.
$ ( t ,x , v ( x ) ) v ' ( x ) d x
Shinnosuke OHARU and Tadaynsu TAKAHASHI
132
and
From c o n d i t i o n (11) and t h e e s t i m a t e (3.3) i t f o l l o w s t h a t A ( t ) v i s 1 continuous i n H i w i t h r e s p e c t t o ( t , v ) E [O,T]xHo. Also, c o n d i t i o n (11) implies t h a t
i s continuous on [O,T].
Moreover, u s i n g (3.1) and ( 3 . 3 ) , we
ibtain the following result. 1 1 P r o p o s i t i o n 3.2. L e t uo E Ho and l e t u ( * ) be an Ho-valued continuous
f u n c t i o n on [O,T]
s a t i s f y i n g the i n t e g r a l equation
Then t h e f u n c t i o n u ( t , x )
on [O,T]
= [u(t)](x)
gives a unique s o l u t i o n
o f t h e i n i t i a l - b o u n d a r y value problem (1. I ) - ( 1 . 3 ) .
In t h e r e s t o f t h i s s e c t i o n we w r i t e Br = ( v E Ho; 1 lrrl =
Lemma 3.3.
IvI1
max a ( t ) O<=tST
i
and
for
Id]
=
r
,
0,
max B ( t ) . O
0, t h e r e i s a number l o = l 0 ( r ) E ( O , l / l ~ l ) such
F o r each r
t h a t f o r each t E [O,T],
:r l
E (O,l0)
1 . and v t Ho w i t h l v l l
5 r , t h e r e e x i s t s an
1 e l e m e n t v \ E Ho s a t i s f y i n g
v,
(3.5)
- \A(t)v,
= v
and
P r o o f . L e t r > 0, K = s u p { I A ( t ) w l l ;
t E
[O,Tl,
l / K } . Given t E [O,T],
Xo = m i n { l / l a l , 1/2Mr+1,
Gw = v + \ A ( t ) w
for
w E Br+l},
1 E (O,i0) w E H.,
and l e t
and v E Br, d e f i n e
1
Then
/GwI1
L jVil
+ \ l A ( t ) w l l i r + XK
<
r + 1
for
w E Br+l
and (3.3) y i e l d s
lGwl
-
Gw211 2 I M r + l l W 1
-
w2I 1 <
1
2 1 ~ 1- w2 I1
for
'1 * '2
E Br+l*
T h e r e f o r e t h e a p p l i c a t i o n o f t h e c o n t r a c t i n g mapping p r i n c i p l e imp1 i e s t h a t t h e r e e x i s t s an e l e m e n t v x E Br+l e s t i m a t e ( 3 . 4 ) t h a t , ,v
w i t h ,v,
-
s a t i s f i e s (3.6).
X A ( t ) v X = v. I t now f o l l o w s f r o m t h e q.e.d.
Shinncquke O H A R U and Tatlnraiii T X K A H A S H I
124
Making use o f t h e above lemmas, we can c o n s t r u c t a p p r o x i m a t e s o l u t i o n s t o ( 3 . 2 ) i n t h e f o l l o w i n g sense ( c f . [ 5 ] ,
1 z E Ho, r = e 2 ' a 1 T ( l z l l t [ B I T ) , and l e t
P r o p o s i t i o n 3.4. L e t s E [O,T), Kr
~
[ 7 ] and [ e l ) .
~ u p i I A ( t ) w l ~ ; t ~ [ O , T ] , w ~ B , } . F u r t h e r , l e t X o ( r ) be a number as s p e c i f i e d
i n Lemma 3.3 and { c n 1 a n u l l sequence o f p o s i t i v e numbers such t h a t cn < i O ( r ) / 2 for n
>
n 1. Then f o r each n 2 1 t h e r e e x i s t s a p a r t i t i o n { s = to < ty <
' tn
= T I o f [s,T] N(n) following properties: .*
-
1 and a n Ho-valued f u n c t i o n u n ( - ) on [s,T]
for
ti
(ii)
u n ( s ) = z and i u n ( t ) - u n ( ~ ) l , 5 K r l t
(iii)
u n ( t ) E Br
(iv)
un i s l i n e a r on each [t;-l,t;]
5
E,
for
- TI
f o r t, T E [s..T].
t E [s,T].
) = A(t:)un(tl)
-
with the
1 5 k 5 N(n).
(i)
t;-l
...
A(tI)un(t;)I1
and for
t E (tnk_l.t:).
5 cn f o r t E [tk-l,tk] n n and 1 5 k 5 N ( n ) .
f o r 1 2 k 5 N(n). P r o o f . L e t t: t:
,-.-,t i
i n [O,T]
=
s and u (t:) n
= z. Suppose t h a t a n o n d e c r e a s i n g sequence
and a sequence o f e l e m e n t s un(t:),-.-,
un(tF) i n
Ho1 have
been chosen such t h a t u n ( t n ) E B f o r 0 5 j 5 k . We f i r s t d e f i n e t"k1 E [ t t , T ] ~r i n t h e f o l l o w i n g manner: I f t; = T, we p u t tltl = T and un(tFtl) = u (t'). n k n n If T, we d e f i n e t"k1 = m i n { t k + hk, T I , where h; d e n o t e s t h e supremum o f
tt
a l l h E (0,5,]
such t h a t
Nonlinear Evolution Operators Note t h a t t;
< t"k1 by ( 3 . 3 ) .
135
Next, a p p l y i n g Lemma 3.3, we d e f i n e un(t;+l)
1
t o be an element o f Ho such t h a t
n N o t i n g t h a t tk+l -
t l C hkn : cn,
we i n f e r from (3.9) t h a t
Thus we o b t a i n a sequence { ( t F , ~ ~ ( t ; ) ) li n~ ~ [O,T] ~
x
Br.
= T f o r some i n t e g e r N(n) 1 . Assume on t h e We t h e n demonstrate t h a t tn N(n) n n T f o r a l l k c 0. Since I u n ( t . ) - u n ( t k ) I 1 1 K r I t y - t:l f o r contrary t h a t t i J 1 j , k i 0 by ( 3 . 8 ) , t h e sequence {(t;,un(tE))}k20 i s Cauchy i n [ s , T ] * H O . Hence,
by ( 3 . 3 ) , t h e r e i s a number d E ( 0 . 5 1 such t h a t
f o r a l l (t,w)E[O,T]xH,!,
k
L
w i t h It
- t nk I : C
and I w
0. By t h e d e f i n i t i o n o f h l , t h i s i m p l i e s h:
c o n t r a d i c t s t h e f a c t t h a t h; t h a t tl(n)-l
< (ti,)
= ti+l - t:
+
0 as k
on [s,T]
n f o r t E [tk-l.t;]
un(tk)II n
- un(t:-,)I1
and f o r a l l
6 f o r a l l k ' 0, which --f
<*.
Therefore, i t i s concluded
1
d e f i n e an Ho-valued
by
and 1 '1 k
:N ( n ) . Then
i t i s easy t o check t h a t {tEJo,k,N(n)
and un(-) s a t i s f y ( i ) , ( i f ) , ( i i i ) , ( i v ) and ( v i ) . Since t: Iun(t;)
,K r c
= T f o r some i n t e g e r N(n) z 1.
n n Using t h e f i n i t e sequence '(tk,un(tk))}l,k,N(n), f u n c t i o n un(.)
=/
-
i Krhl-,
for 1
< k
- t;-l
n : hk and
! N(n), p r o p e r t y ( v ) f o l l o w s from ( 3 . 7 ) ,
i n d t h i s completes t h e p r o o f o f t h e p r o p o s i t i o n .
q.e.d.
Shinncxuke OHARU arid T:I~:I~.?RII TAKAHASHI
136
4. Construction of the evolution operator In this section we construct a nonlinear evolution operator on Ho1 which provides the solutions of the evolution equation (3.2). By a nonlinear evolutior operator on Ho1 i.s meant a family U = IU(t,s); 0 5 s C t 5 T I o f nonlinear operators from HA into itself such that (El) U(s,s) for 0
t
s
r
and ( E 2 ) for s
' T;
continuous on [s,T]
and z
E [O,T)
=
I, u(t,s)U(s,r)
E
1 U(t,s)z Ho,
=
U(t,r)
i s strongly
with respect to t. Assume that conditions (I), (11) and (111) are satisfied.
.-1_I_4irerEc
Then there exists a nonlinear evolution operator U on HA with the following properties: t A(T)U(r,s)zdi ( i ) U(t,s)z = z +
i
for 0
'
=
IU(t,s); 0 I s
s :t
5
T and
S
( i i ) Given uo E Ho,1 the function u(t,x)
=
[U(t,O)uo](x)
2
t 5 TI
z E H01
is a unique solution
of the initial-boundary value problem (1.1)-(1.3)
(iii)
IU(t,s)zll
t t exp(1 S~ ( r ) d i ) [ ~ z ~ ~S +fi(i)dr] \ for 0
T (iv) Let ro
'
0 and r
exp( [Tf,,(T)d,r)[ro+ \ot(~)di].
=
' s 2
t
5
1 T and z F: H0'
If zi E Ho1 and Iz. 1 I1
Ir 0'
'0 i = 1,2, then
for 0
'
s
Proof.
'
t
.
T and k
0,l.
=
1 Let r and z E Ho.
Let s E [ O , T )
Proposition 3.4, and let
f
I
0, fio(r) a number as specified in
" ! be a null sequence of positive numbers such that
be the function constructed n ' 0(r)/2 for n 1. For each n 1 , let un( in Proposition 3 . 4 . Further, let { t ~ l l c k c N ~ be n l the partition of [s,T] a )
Lorresponding t o un( .). We first prove that the sequence 'un(-)i i s Cauchy in C([O,T];
1
H o ) . In view
137
which shows t h a t t u n ( - ) } i s Cauchy i n C([O,T];
1
Ho).
Define (4.1)
U
t ) = limn,un(t)
for
t E [s,T].
Then we i n f e r w i t h t h e a i d o f P r o p o s i t i o n 3.4 t h a t
f o r t E [s,T].
Also, we deduce form P r o p o s i t i o n 3.4 ( v i ) t h a t
t l u ( t ) I , i eup(J u.(T)dT)[lzll
+ ~ B ( T ) ~ T ]f o r
t E [s,Tl.
5
S
We then c o n s t r u c t t h e aimed e v o l u t i o n o p e r a t o r U by d e f i n i n g U ( t , s ) z = u ( t ; s,z) where u(.;
for
s E [O,T],
t E [s,T]
and
1 z E Ho,
s,z) denotes t h e l i m i t f u n c i t o n u(-) o b t a i n e d through (4.1) f o r s
and z . F i r s t we see f r o m t h e above-mentioned p r o p e r t i e s o f u(.;
s,z)
t h a t Ll has
p r o p e r t i e s ( i ) and ( i i i ) . A s s e r t i o n ( i i ) f o l l o w s f r o m ( i ) and P r o p o s i t i o n 3.2. A s s e r t i o n ( i v ) i s deduced from ( 3 . 3 ) ,
( i i i ) , and t h e f a c t t h a t
Shinnosuke OHARU and Tadayasu TAKAHASHI
138
for 0 5 s 5 t S
I
T, zi
E Ho,
i n d e n t i t y U(t,s)U(s,r)
i = 1,2, and k = 0 , l .
= U(t,r)
1 and z E Ho, u ( t ; s,z)
F i n a l l y , the evolution
i s o b t a i n e d by observing t h a t f o r each s E [O,T)
i s a unique s o l u t i o n o f ( 3 . 2 ) . The p r o o f i s thereby
complete.
q.e.d.
Remark 4.2.
1 Let uo E Ho. Then t h e f u n c t i o n u ( t . x )
= [U(t,O)uo](x)
satisfies
the integral equation
f o r ( t , x ) E [O,T]
x
(a,b),
where G(x,()
i s t h e Green's f u n c t i o n o f t h e problem
(A - I)w = v
in
2
L and K(x,C)
=
- G5 ( x & )
G(x,S) = y(a,b)[e
f o r x,
6
E (a,b),
namely:
e - I ~ - E I ( 1 + e - 2 ( b - a ) ) + e (x+S)-2bl
-(x+S)+2a
I
and
and y(a,b) = [ 2 ( 1 - e-
2(b-a)
)]-
1
.
1 I n view o f t h i s , we see t h a t i f uo E Ho
then u s a t i s f i e s equation (1.1) p o i n t w i s e on (0,T)
x
(a,b).
il
2 C ,
Further r e g u l a r i t y
p r o p e r t i e s o f t h e s o l u t i o n u a r e obtained by means o f t h e above-mentioned i n t e g r a l equation; i n p a r t i c u l a r , i f t h e n o n l i n e a r f u n c t i o n s @I,
$ and t h e
i n i t i a l f u n c t i o n uo a r e Cm, then so i s t h e s o l u t i o n u. Remark 4.3. Assume t h a t @ ( t , x , c ) = $ ( t , x + l , C ) and $(t,x,S) f o r (t,x,c)
E [O,T]x
(a,b)xR.
= $(t,x+l
,5)
Then, by a s l i g h t m o d i f i c a t i o n o f t h e argument
developed above, we can prove t h e e x i s t e n c e and uniqueness o f s o l u t i o n s o f t h e p e r i o d i c i n i t i a l -boundary v a l u e problem ut
+
(@I(t,X,U))x + $(t,x,u)
-
uxxt = 0,
0 < t < T, x E R,
Nonlinear Evolution Operators
139
where uo i s a g i v e n i n i t i a l f u n c t i o n s a t i s f y i n g t h e p e r i o d i c i t y c o n d i t i o n uo(x) = uo(x+l) f o r x E R.
5 . A p p l i c a t i o n t o problems w i t h time-dependent boundary c o n d i t i o n s Our r e s u l t s can be a p p l i e d t o t h e l o n g wave e q u a t i o n (1.4) w i t h t i m e dependent boundary c o n d i t i o n s . Here we g i v e two examples of such a p p l i c a t i o n s . 1 ~Example 1. L e t h E C ([O,T])
and c o n s i d e r t h e i n i t i a l - b o u n d a r y - v a l u e
problem (see [3]):
1 where t h e i n i t i a l f u n c t i o n vo i s g i v e n i n H ( 0 , ~ ) and s a t i s f i e s t h e c o m p a t i b i l i t y condition vo(0) = h(0). L e t (a,b) be t h e h a l f l i n e ( 0 , m ) and d e f i n e @(t,x,C) = 5 + h ( t ) e - x + ( C + h ( t ) e - x ) 2 / 2 , f o r (t,x,C)
E [O,T]
x
(0,m)
x
$(t,x,S)
= 0
R . Then i t i s c l e a r t h a t these f u n c t i o n s s a t i s f y
c o n d i t i o n s ( I ) and ( 1 1 ) o f Theorem 4.1. Also, i t i s easy t o check t h a t
+
c o n d i t i o n (111) i s s a t i s f i e d f o r y ( t ) = h + ( t ) and a 2 ( t ) E 0, where h ( t ) = max{h(t),OI f o r t E [O,T].
Therefore, Theorem 4 . 1 ensures t h e e x i s t e n c e o f
a s o l u t i o n u c u(t,x) o f the equation ut + (Q(t,x,u))x such t h a t u(0,x) t
= vo(x)
-
-
uxxt = 0,
h(0)e-x f o r x E [O,-)
0 < t < T, 0 < x < and u(t,.)
E (0,T). Set v ( t , x ) = u ( t , x ) + h ( t ) e - x f o r ( t , x ) E [O,T]
m,
1
E Ho(O,m) x [O,m).
for Then v i s
Shinnosuke OHARU and Tadayasu TAKAHASHI
140
a unique s o l u t i o n o f t h e problem ( 5 . 1 ) and t h e H'(0,m)-valued
function
v(t) E v(t,-) satisfies
where [ p ( t ) ] ( x ) = h ( t ) e - x f o r ( t , x ) E [O,T]
f o r t E [O,T],
B(t) = {
Ji
[h(t)e-x + h(t)2e-2x]2dx11/2
1 Example 2. L e t h, g E C ([O,TI) problem (see
for
x
( 0 , ~ ) and
t € [O,T].
and c o n s i d e r t h e i n i t i a l - b o u n d a r y v a l u e
[4]):
-
w t + w x + wwx (5.2)
= 0,
Wxxt
0 < t < T, 0 < x c 1,
W(0,X) = wo(x). w(t,O)
O < X < l ,
= h ( t ) , w(t.1)
= g(t),
where t h e i n i t i a l f u n c t i o n wo i s g i v e n i n
0 < t < T, 0 < x < 1,
H 1 ( 0 , l ) and i s assumed t o s a t i s f y 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 s w o ( 0 ) = h ( 0 ) and w o ( l ) = g ( 0 ) . I n t h i s case, l e t (a,b) be t h e u n i t i n t e r v a l ( 0 , l ) and d e f i n e O(t,x,5)
=
5
+
g(t)x
+
q(t,x,S) f o r (t,x,c)
F: [O,T] x [0,1]
x
h(t)(l
- X+I( 5
= g'(t)x
+ g(t)x + h ( t ) ( l -x)I2/2,
+ h'(t)(l -x)
R. Then c o n d i t i o n s ( I ) and (11) a r e o b v i o u s l y
s a t i s f i e d , and c o n d i t i o n (111) i s v e r i f i e d w i t h a l ( t )
= (h(t)
-
g(t))'
and
u 2 ( t ) :0. Therefore Theorem 4.1 i m p l i e s t h a t t h e problem ( 5 . 2 ) admits a unique s o l u t i o n w such t h a t w ( t ) Iw(t)
-
=
w(t,.)
E H 1 ( 0 , l ) f o r t E [O,T]
I t q(t)I,
5 e x p ( Z j ( h ( 7 ) - g(T))+dT)[lwo
0
-
and
t q(o)I, + J 0 ~ ( ~ ) d ~ 1
where [ q ( t ) ] ( x ) = g ( t ) x + h ( t ) ( l - x ) f o r ( t , x ) E [O,T]x[O,l]
f o r t E [O,T], and 1 O(t) = f o r t E [O,T].
[(g(t)
0
- h(t))(l +g(t)x+h(t)(l
- x ) ) + g ' ( t ) x + h ' ( t ) ( l -x)]2dx11/2
141
Nonlinear Evolution Operators References
T. B. Benjamin, Lectures on n o n l i n e a r wave motion, L e c t u r e s i n A p p l i e d Mathematics, Vo1.15, 3-47, AMS, Providence, R . I . ,
1974.
T. B. Benjamin, J. L. Bona and J . J . Mahony, Model equations f o r l o n g waves i n n o n l i n e a r d i s p e r s i v e systems, P h i l o s . Trans. Roy. SOC. London, Ser. A, 272(1972), 47-78. J . L. Bona and P. J. Bryant, A mathematical model f o r l o n g waves generated by wavemakers i n n o n l i n e a r d i s p e r s i v e systems, Proc. Cambridge P h i l o s . SOC.,
731 1973), 391-405.
J . L. Bona and V . A . Oougalis, An i n i t i a l - and boundary-value problem f o r a model e q u a t i o n f o r propagation o f l o n g waves, J. Math. Anal. Appl., 75( 19BD), 503-522. T. Iwamiya, Global e x i s t e n c e o f s o l u t i o n s t o nonautonomous d i f f e r e n t i a l equations i n Banach spaces, Hiroshima Math. J., 13(1983), 65-81. T. Iwamiya, S. Oharu and T. Takahashi, On t h e semigroup approach t o some n o n l i n e a r d i s p e r s i v e equations, L e c t u r e Notes i n Num. Appl. Anal.,
V01.1,
95-134, Kinokuniya Bood S t o r e Co., Tokyo, Japan, 1979. N. Kenmochi and T . Takahashi, Nonautonomous d i f f e r e n t i a l equations i n Banach spaces, Nonl i n e a r A n a l y s i s , TMA, 4 ( 1980), 1109-1121. R. H. M a r t i n , J r . , D i f f e r e n t i a l equations on c l o s e d subsets o f a Banach space, Trans. Amer. Math. SOC.,
179(1973), 399-414.
L. A. Medeiros and G. P. Menzala, Existence and uniqueness f o r p e r i o d i c s o l u t i o n s o f t h e Benjamin-Bona-Mahony equation, S I A M J. Math. Anal., 8 ( 1977), 792-799.
[lo]
L. A . Medeiros and M. M. Miranda, Weak s o l u t i o n s f o r a n o n l i n e a r d i s p e r s i v e equation, 3 . Math. Anal. Appl., 59(1977), 432-441.
[ll]S. Oharu and T. Takahashi, A c l a s s o f e v o l u t i o n equations i n H i l b e r t space
and one-parameter semigroups o f n o n l i n e a r Fredholm o p e r a t o r s , prepri'nt.
142
Shinnosuke OHARU and Tadayasu TAKAHASHI
[12] D. H. Peregrine, C a l c u l a t i o n s o f t h e development o f an undular bore,
J . F l u i d Mech., 25(1966), 321-330. [13] R. E. Showalter, Existence and r e p r e s e n t a t i o n theorems f o r a s e m i l i n e a r Sobolev e q u a t i o n i n a Banach space, S I A M J . Math. Anal., 3(1972), 527-543. [14] R . E. Showalter, Sobolev equations f o r n o n l i n e a r d i s p e r s i v e systems, Appl. Anal.,
7( 1978), 297-308.
L e c t u r e N o t e s in Num. Appl. Anal., 6, 143-154 (1983) iji Noiiliiiear PDE, Hirosllinza, 1989
Receiit Topics
Nonstationary or Stationary Free Boundary Problems for Perfect FIuid with Surface Tension
Hisashi OKAMOTO* Department of Mathematics, University of Tokyo Tokyo 113, Japan
1.
Introduction We c o n s i d e r a f r e e boundary problem f o r a c i r c u l a t i n g p e r f e c t
f l u i d , which i s a model f o r a f l o w around a c e l e s t i a l body.
We c o n s i d e r
o n l y t h e f l o w i n a p l a n e through an equator o f a c e l e s t i a l body. t h e f l o w i s regarded as a two-dimensional one. s i m p l i c i t y , assumed t o be a u n i t c i r c l e i n
The equator
r
Hence is, for
R L . We assume t h a t t h e f l u i d
i s i n c o m p r e s s i b l e , i n v i s c i d and i r r o t a t i o n a l .
We a l s o assume t h a t s e l f -
g r a v i t a t i o n o f t h e f l u i d i s neglected and t h a t o n l y t h e g r a v i t a t i o n due t o the inside o f
r
i s taken i n t o account.
Then t h e s t a t i o n a r y problem
t o be considered here i s f o r m u l a t e d as f o l l o w s : PROBLEM ( S ) .
r
Find a function
V
and a c l o s e d Jordan c u r v e y
s a t i s f y i n g t h e c o n d i t i o n s ( 1 . 1 ) - ( 1 . 5 ) below.
r ,
(1.2)
v = o
on
(1 -3)
V = a
on Y ,
~~
*
Partially supported by the Ffijukai.
143
outside
Hisashi OKAMOTO
144
1
(1.4)
71VVl
2
+
on Y
Q t uKy = unknown c o n s t a n t
,
Qy
Fig. I
Here we have employed t h e f o l l o w i n g n o t a t i o n .
r
Uy : a doubly connected domain between
r
Q : a given f u n c t i o n d e f i n e d o u t s i d e Q =
I'
t h e e x t e r n a l pressure
t e r n a l f o r c e ''
, o , wo
: t
"
t h e p o t e n t i a l o f t h e ex-
Ky : t h e c u r v a t u r e of
V
,
i s what i s c a l l e d a stream f u n c t i o n f o r t h e f l o w .
This problem i s s t u d i e d i n [5] and [6]. considered i n Imai [4].
S i m i l a r problems a r e
The r e s u l t s i n [5] a r e sumnarized as f o l l o w s .
Q = Q o ( r ) depends o n l y on t h e d i s t a n c e from t h e o r i g i n , then a c i r -
c l e yo
o f radius The r a d i u s
(S).
2 T(ro
y
a r e p r e s c r i b e d p o s i t i v e constants,
The f u n c t i o n
If
( see F i g . I ) ,
,
u : the surface tension c o e f f i c i e n t , a
"
and y
-
1 ) = wo.
ro w i t h t h e o r i g i n as i t s c e n t e r i s a s o l u t i o n o f ro o f
yo
i s determined by (1.5),i.e.,
The corresponding stream f u n c t i o n
V
ro > 1
i s g i v e n by
,
Free Eoundary Problems for Perfect Fluid
145
a
This solution ( called a t r i v i a l solution ) e x i s t s f o r any However, there e x i s t s a countable s e t
{anl;=l
> 0.
with the following
properties.
1 ) For any a
If
{an)m fixed, we have a solution f o r
sufficiently
Q
Qo.
close to
2)
4
an
4
, then the point a n i s a bifurcation p o i n t , i . e . ,
we have a branch o f nontrivial solutions emanating from ( a n , O ) .
( Of
Q = Qo(r) i s fixed ) .
course
I n p a r t i c u l a r the property 2)
implies t h a t we have a solution with-
out O(2)-symmetry i n s p i t e o f the O(2)-symmetry o f the data.
The r e s u l t s above seem t o indicate t h a t the t r i v i a l solution i s n
a*.
and t h a t i t loses s t a b i l i t y a t
0 < a < a* :inf a
stable f o r
n
To investigate the s t a b i l i t y problem we have to consider the nonstation-
ary problem corresponding t o ( S ) , which i s s t a t e d as follows. Find a time-dependent closed
PROBLEM ( N S ) . {
(r,e)
V(t,r,e)
E
R2 ,P
(1.10)
r = y(t,O)
= P(t,r,B)
1 (1.9)
;
azv
,05 e
< 2n
Jordan curve y ( t ) =
1 and functions V
s a t i s f y i n g the conditions (1.6)
r
-
=
(1.13) below.
in
QT,-y
in
Q
T,Y '
Hisashi OKAMOTO
146
(1.13)
IfiY(,)I
uo
.
The c o n d i t i o n (1.8) i m p l i e s t h a t f l u i d p a r t i c l e s on the f r e e
REMARK.
The equations
boundary remain on t h e boundary throughout t h e motion.
(1.9) and (1.10) are t h e E u l e r equation w r i t t e n i n terms o f t h e stream function
V.
F o r s i m p l i c i t y we have p u t
( g:constant )
Q o ( r ) = -g/r
As f o r the i n i t i a l values yo and Vo we assume t h a t yo c5+a 1 1 27I 2 ( S 1, 7 6 yo(e) de TI = u0 , I[ yo - ro[~,, a 1
t
-
A V =~
o
i n Q~ ,
vOJr
= 0,
voly0
= a.
Our goal i s t o show the unique e x i s t e n c e o f a s o l u t i o n o f PROBLEM (NS).
The e x i s t e n c e i s assured l o c a l l y i n time i f t h e i n i t i a l curve
belongs t o
CA
f o r some
h > 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 ) )O
We n e x t s o l v e a D i r i c h l e t problem ( l . & , 71, (1.8)* regarding
147
y = y
as a g i v e n boundary.
U
and
Denoting by
Vu
below )
(1.12) f o r
V.
the solution
thus obtained, we s o l v e (1.9) and (1.10) which a r e t h e Cauchy-Riemann equations.
Then we d e f i n e a mapping
F(Y~,u) where
Ku(t)
tion for
yo
a
-
xie(P
OK
i f and o n l y i f
X
with
Y, = yo + u.
Then
gu
y,
yu(t,e)
The f u n c t i o n
g,
Ru(t)
i s a solu-
Hence o u r task i s t o f i n d
=
R
by
i s d e f i n e d by
yo(el
+
vu( t) .
u(t,e)
:I R , , ( ~1) I
is so d e f i n e d t h a t
instead o f
For a s u f f i c i e n t l y small
yu(t).
on [O,T]
t h i s e q u a l i t y we e a s i l y see t h a t write
yu
F.
we d e f i n e a f u n c t i o n
(2.2)
Observe t h a t
F(yo,u) = 0.
We b e g i n w i t h t h e d e f i n i t i o n o f E
by
i s the curvature o f y u ( t ) .
a zero p o i n t o f
u
F
+
gu(t). aYU
yu(t,8)T(t,8)d8 :oo.
Observe t h a t
:0.
From
Here and h e r e a f t e r we g,
i s t h r e e times
Hisaahi OKAMOTO
148
continuously d i f f e r e n t i a b l e
and t h a t
= 0.
gu(0) = (O g,',)
We n e x t use
t h e f o l 1owing PROPOSITION 2.1.
There e x i s t unique
fu and
satisfying
Vu
in
AVu = 0
(2.3)
QT,,
:QT *YU
vu
(2.4)
on
= 0
& jo2n Ga V( t , l , e ) d e
[O,t]xr
z 0.
akvu Furthermore we have
( k = 0,1,2 REMARK.
fu F C2([0,T]),
E
C([O,T];CsCa(~)).
derivatives o f PROPOSITION 2.2.
5-k+0.
t1
("(
onto "(t)
20 , we can
S i m i l a r r e l a t i o n s h o l d f o r t h e time
Let
be t h e f u n c t i o n i n t h e preceding p r o p o s i t i o n .
Vu
such t h a t a 2ev u+ s aq u = O r1 m 1 a r ao u
- - +a 2-V- Uq
atar
(2.10)
Furthermore we have q,(t,-,-) NOTATION.
C
Vu.
Then we have a u n i q u e . q,
(2.9)
E
).
Using a canonical p u l l - b a c k from
r e g a r d Vu
(2.8)
-(t,-,.) at
E
C
4+a
.
= o
in
QT,u
149
Free Boundary Problems f o r Perfect Fluid
Of course
Z
i s a Banach m a n i f o l d .
We f i n a l l y d e f i n e a mapping
Then
yu
F :
Z x
X
Y
+
solves o u r problem i f and o n l y i f
F(yo,u)
t h e e x i s t e n c e of a zero p o i n t o f t h e mapping smooth and c l o s e t o THEOREM 2.1.
such t h a t f o r any
u
in
3.
X
satisfying
= 0.
We can show
if yo is s u f f i c i e n t l y
F
ro, More p r e c i s e l y we have
Put h = 8 105 + $a-B). 15
6
by
E Z with 0 F(yo,u) = 0.
y
Then t h e r e i s a p o s i t i v e c o n s t a n t
11 y -
r
11 A
< 6
we have a unique
Outline o f the proof. We prove THEOREM 2.1
due t o Zehnder [9]. derivative of NOTATION 3.1.
by 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
Hence we f i r s t d e r i v e a c o n c r e t e expression o f t h e
F. We d e f i n e a l i n e a r o p e r a t o r
1
Hv =
n#O where we have p u t f o r
n
E
r: R = n
( - i R ) v eine n n
Z\IOl
+ rgn
n ro - rgn
Then we can d e r i v e t h e f o l l o w i n g formula:
H
by t h e e q u a t i o n below.
150
Hisashi OKAMOTO
Of course we a l s o need
DuF(yo,u)
a l i z e d i m p l i c i t f u n c t i o n theorem. DuF(yo,u)
However, we o m i t t h e expression f o r
because o f i t s complexity.
DUF(rO,O), t h e s i t u a t i o n f o r
(yo,u) # (ro,O) t o use a gener-
for
Although we t r e a t o n l y
DuF(yo,u)
i s very s i m i l a r .
The expression (3.2) ensures t h a t an i n v e r s e o f t h e o p e r a t o r DUF(rg,O) e x i s t s and i s continuous i n some sense.
Indeed we can v e r i f y
t h i s by t h e H i l l e - Y o s i d a t h e o r y o f semi-groups o f o p e r a t o r s as f o l l o w s . Observe f i r s t t h a t
(3.4)
DuF(ro,O)w = f
w(o,e)
E 0
aw , -(o,e) at
E
i s r e w r i t t e n as
Y
:o
To s o l v e t h i s h y p e r b o l i c e q u a t i o n we p u t
. v =
$.
Then (3.3) and (3.4)
a r e e q u i v a l e n t t o an o p e r a t o r e q u a t i o n below.
(3.5)
We c o n s i d e r (3.5) i n t h e f u n c t i o n space Here and i n what f o l l o w s
Ht ( S 1 )
we w i l l show t h a t t h e o p e r a t o r
M E H3/2(S1)/Rut2(S')/R.
means a Sobolev space on S 1
.
Now
Free Boundary Problems for Perfect Fluid
A -
0
[
generates a Co-semigroup i n
151
1
a i o a0
L
2b
M.
I f we p u t
t h e n i t i s s u f f i c i e n t t o show t h a t
A.
generates a semi-group i n
M.
2 . Denoting t h e canon c a l L - r n n e r p r o d u c t by ( , ) , we o b t a i n L v,v) 0
(3.6)
f o r any
--jo n3 ‘nl 2 n+O r i Rn
1
-
=
v =
1 vneine
H
6
3/2
1 (S )/R,
nfO as a H i l b e r t space equipped w i t h an
Hence we can r e g a r d H3/2(S’)/lR inner product
(w,v),
canonical LL-inner p r o d u c t . w i t h an i n n e r p r o d u c t ( LEMMA 3.1.
2 1 L (S )/R
z -(Low,v), w h i l e We regard
[;I)
[:]
The o p e r a t o r
A.
M
r (w,z),
i s equipped w i t h a
as a H i l b e r t space equipped
+ (v,y).
Now we have
i s a maximal d i s s i p a t i v e o p e r a t o r i n
hence generates a Co-semigroup i n
M.
M,
I t s spectrum i s composed o n l y o f
eigenvalues which l i e on t h e imaginary a x i s . With t h e a i d o f t h i s lemma we can a l s o show t h a t t h e o p e r a t o r generates a Co-semigroup i n of eigenvalues.
where
an
M
and t h a t i t s spectrum i s composed o n l y
More p r e c i s e l y , we have
a(A) =
I A,
6
or
i s a b i f u r c a t i o n p o i n t g i v e n i n [5,
=
A
; k
71.
E
Z \{O}
}:
I n particular,
i n f (l+nRn)’/*an, we see t h a t A i s d i s s i p a t i v e i f and nzl o n l y i f 0 < a < a** and t h a t a t a = a* ( :i n f a ) some eigenvalues n n>l 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
L e c t u r e N o t e s in Num. Appl. Anal., 6, 155-196 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1983
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
<
m
1. Note t h a t Ho(e)= Lp(&) p a r t i c u l a r l y .
L e t robe a f i x e d p o s i t i v e constant such t h a t an C I x r > 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)
$2
Let
+
E
IAUl-,(n-1
)/z.in
= <
6.
be a p o s i t i v e c o n s t a n t w i t h 0 <
3$+(3m +
7 ) ~+ ] 3[n/2]
+ 6.
E
5
1
, and
in
an i n t e g e r w i t h
I f n = 3, then t h e r e e x i s t p o s i t i v e constants
,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
160
a and A0 having t h e f o l l o w i n g p r o p e r t i e s :
f
2%
~2
([I),-)x
5 ) s a t i s f y f o r some
I f $o
6 with 0 < 6
,i
2$+2
E,,
5
(z), o1
E$,*~'(;)
and
60
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, t h e Problem (M.P) has a s o l u t i o n
u
E
Cm " ([G,-)
x
$) s a t i s f y i n g
(Uniqueness). C3([0,-) I*ulm,o,o
i) a r e
x
= <
6 1 and
Remark 1.2.
There e x i s t s a small c o n s t a n t 6 , > 0 such t h a t i f u, v
E
two s o l u t i o n s o f Problem (M.P) f o r t h e same data k i t h
I"lm,o,o (1)
2 1, then u
= v.
For 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 , see 54.2 and Mizohata
[6l. (2)
Since t h e n o n l i n e a r f u n c t i o n F i s d e f i n e d o n l y i n [ 0 , m ) ; Ihl
x
5
I
A E
1 I , we always assume t h a t I A U ~ ~ = ,< ~1, , when ~ we c o n s i d e r a
s o l u t i o n u o f Problem (M.P).
One o f t h e d i f f i c u l t i e s i n t h e p r o o f i s t h a t the l o s s o f d e r i v a t i v e s occurs a t each s t e p i n t h e i t e r a t i o n .
E s p e c i a l l y n o t e t h a t t h e n o n l i n e a r term F a l s o
depends on t h e d e r i v a t i v e of o r d e r 2 w i t h r e s p e c t t o time t i n our problem.
For
t h e Cauchy problem we can overcome such a d i f f i c u l t y by reducing a f u l l y n o n l i n e a r e q u a t i o n t o a q u a s i l i n e a r equation, f o l l o w i n g Oionne [I] (see a l s o Klainerman and Ponce [3] and Shatah [ l o ] ) .
For t h e i n i t i a l boundary v a l u e problem, however,
such methods a r e n o t a p p l i c a b l e . case n = 3.
Furthermore, t h e l o s s of decay occurs i n t h e
I n o r d e r t o overcome such d i f f i c u l t i e s , we s h a l l make use o f t h e
s o - c a l l e d Nash-Moser technique.
Our s t r a t e g y f o l l o w s Klainerman [ 2 ] and Shibata
Nonlinear Wave Equation in Exterior Dnmain
161
[12] (see a l s o Rabinowitz [9]).
A u n i f o r m decay e s t i m a t e and an L 2 - e s t i m a t e f o r a l i n e a r i z e d problem w i l l p l a y an i m p o r t a n t r o l e i n t h e p r o o f .
I n p a r t i c u l a r , t h e r e s u l t s o f decay estimates
a r e new and a r e proved i n t h e same way as Shibata [12] and Tsutsumi [13].
Tools
used i n a p p l y i n g t h e Nash-Moser technique, such as an i n t e r p o l a t i o n i n e q u a l i t y between a f a m i l y o f c e r t a i n semi-norms and a proper smoothing o p e r a t o r , a r e t h e same as those used i n Shibata [ll,121. Now we g i v e a well-known example, i.e.,
" t h e n o n l i n e a r v i b r a t i o n equation":
Example.
I n t h e course o f t h e p r o o f below a l l constants w i l l be s i m p l y denoted by C. In particular, C =
C(*,.--,*) w i l l denote a c o n s t a n t depending on t h e q u a n t i t j e s
appearing i n parentheses.
52. Uniform Decay Estimate.
In t h i s s e c t i o n we s h a l l
show 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:
(2.1)
t_:iu
= f
u = o
in
[O,m)
x
R,
an
[a,-)
x
an,
Throughout S e c t i o n 2 we always assume t h a t t h e data $o, $1 and f o f t h e e q u a t i o n ( 2 . 1 ) a r e so n i c e f u n c t i o n s t h a t a l l t h e i r norms and semi-norms appearing below a r e bounded.
We d e f i n e u j ( x ) ( j 2 0 ) s u c c e s i v e l y by
162
Yoshihiro SHIBATA and Yoshio TSUTSUMI
i
q x ) =
U l ( X ) = *,(x),
$O(X)*
We s h a l l say t h a t t h e data $o, $1 and f o f t h e equation (2.1) s a t i s f y 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 i f u . ( x ) = 0 on an ( j = 0, 1, 2,e.e).
J
I t i s known t h a t i f q0
E
C"(C),
q1
E
Cm(T2) and f
E
Cm([O,-)
x
C)
satisfy the
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 , then Problem (2.1) has a unique solution u
E
Cm([O,-)
x
5 ) (see Mizohata [ 6 ] ) .
F o r t h a t s o l u t i o n we s h a l l show
t h e f o l l o w i n g u n i f o r m decay estimate:
Theorem 2.1.
L e t n 2 3.
1 . 1 ( 4 ) i s s a t i s f i e d f o r n.
Assume t h a t R i s "non-trapping",
L e t q0
E
Cm(E),
a1
E
Cm(E) and f
s a t i s f y the compatibility condition o f order i n f i n i t y .
E
i.e., Assumption Cm([O,-)
(1) i f n
p and p' a r e p o s i t i v e numbers (p may be i n f i n i t y ) such t h a t
2 '-'(l--) T P
Cm([O,-)
1 + P P
1,=
x
E)
Then, t h e s o l u t i o n u ( t , x )
5 ) o f (2.1) s a t i s f i e s t h e f o l l o w i n g estimates:
E
x
4 and >
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) = <
c
11 fllD
e-6t
where C and 6 depend o n l y on a,.b,
i f n i s even and n
2)
+
I I ~ ~I,I I t~ 2 0,
n and n;
4, then t h e r e e x i s t two constants C, 6 > 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 <
lom
e x p ( - m kt) U(t)f d t ,
Then we have (A
+ &T
k)G(k)f =
Hence, we have f o r k
(2.2)
; Im k
i f n i s odd and n 5 3,
5 I,
if
We define t h e Laplace t r a n s f o r m o f U ( t ) by ..I*
E C1
E
fl f 0-
F ( k ) f = (A + fl k ) - ' f
k
E
0-.
n i s even and n 2 4.
<
0 )
165
Nonlinear Wave Equation in Exterior Domain
2 1 2 where ( A + k ) - g denotes t h e s o l u t i o n u o f ( A + k ) u = g i n a, u = 0 on aa. Taking t h e i n v e r s e Laplace t r a n s f o r m o f (2.2), we o b t a i n (2.3)
U(t)f =
Z a L i
im-"" - m - & i o
e x p ( m k t ) (A + fl k ) - ' f
dk,
Thus, Theorem 2.3 f o l l o w s from a r o u t i n e c a l c u l u s i f we p r o p e r l y
f o r any 6 > 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
<
t
<
t
m
,
V(0) = VO' E
D(A),
f
E
V ( t ) = U(t)Vo + (see, e.g.,
C1(R1 ;:It),
c
U(t
Mizohata [6]).
-
As i s w e l l known, we have t h e r e p r e s e n t a t i o n S ) f ( S ) ds
Therefore, we see from Theorem 2.3 t h a t f o r t h e data
Q0, 9, and f s a t i s f y i n g t h e assumptions o f Theorem 2.2 t h e s o l u t i o n u o f (2.1)
satisfies
i f n i s odd and n 2 3,
e-6t, t
t ) -n+l ,
if n i s even and n 2 4.
Here we have used the inequal it y
j
t
P(t
-
5)
1 t s ) - ~ds
C(q) ( 1 + t)-',
q > 0,
Nonlinear Wave Equation in Exterior Domain
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
Yoshihiro SHIBATA and Yoshio TSUTSUMI
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
<
1.
(Q. E. 0.)
169
Nonlinear Wave Equation in Exterior Domain
2.3.
P r o o f o f Theorem 2.1.
The p r o o f o f Theorem 2.1 i s e s s e n t i a l l y t h e
same as t h a t o f S h i b a t a [12] and Tsutsumi [13]. By t h e Seely technique we extend q o ( x ) , $ , ( X I and f(.,x)
C“-functions.
We denote t h e extended f u n c t i o n s by j b ( x ) , Tl(x)
respectively.
L e t u,(t,x)
from n t o Rn as and F(.,x),
be t h e smooth s o l u t i o n of t h e problem
F o r any i n t e g e r N 2 0 Theorem 2.9 g i v e s
where
(q(1-i)
,
1+ a
(a >
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,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
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
Nonlinear Wave Equation in Exterior Domain
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 = (
= aji(t,x)
for a l l (t,x) E
E
Lo,-)
Rn and a l l ( t , x )
x E
5.
[O,-)
x ?i,
.
Yoshihiro SHIRATA and Yoshio TSUTSUMI
172
Then i f f o l l o w s t h a t f
E
d.i s
a s t r i c t l y hyperbolic operator.
EL-' (an i n t e g e r L 2 2), we have a unique s o l u t i o n u
[6]).
E
Thus, i f
FL (see Mizohata
By u s i n g Theorem Ap. 1 i n Appendix we can prove t h e f o l l o w i n g lemma
concerning an LL-estimate i n t h e same way as Shibata [ l l , 121.
L e t n 2 3.
Lemma 3.1.
L e t L be any i n t e g e r w i t h L 2 0.
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 d depending o n l y
Assumption 3.1 holds.
on n and n such t h a t i f f o r some n > 0 f
E
EL'',
Assume t h a t
t h e unique s o l u t i o n u
E
l,4
ZLt2
Im, +n,l
i 1,
Iw~,,o,o
5 d and
o f (3.1) s a t i s f i e s the following L
2
-
as t ima te:
Considering a s o l u t i o n u o f (3.1) as a s o l u t i o n o f
u(0,x) = (atu)(o,x)
= 0
i n n,
2 we have t h e f o l l o w i n g theorem concerning a decay e s t i m a t e and an L - e s t i m a t e by Theorem 2.1,
Lemna 3.1 and Theorem Ap.1 i n Appendix.
Theorem 3.2. Lemma 3.1.
L e t n 2 3.
L e t d be t h e p o s i t i v e constant described i n
Assume t h a t Assumption 3.1 holds.
K = L + 3[n/2]
+ 6.
Assume t h a t a l l norms of f and
t h e unique s o l u t i o n u
E
gK o f
an i n t e g e r L 2 0, p u t
For
A
(3.1) w i t h t h e data f
E
below a r e bounded.
EK-l s a t i s f i e s t h e
Then
Nonlinear Wave Equation in Exterior Domain
173
f o l l o w i n g estimates: ( 1 ) Suppose t h a t n 2 4. L e t p and p * be p o s i t i v e numbers such t h a t 1 +1 , = 1. I f 14 Im,O,O = q ( l - $ ) > 1 and ,1 1, then < d and 1 4 m , n - 1 P P -+I-$
z
(2)
Suppose t h a t n 1 3 .
2 n-1 -(l--) = 1 and 2 P and 14 lm,l+a,l 5
Here, f o r p =
54. 4.1.
m
1 1 - t -* = 1. P P 13 then
we d e f i n e
P-2
L e t p and p * be p o s i t i v e numbers such t h a t L e t a > 0 be s u f f i c i e n t l y small.
2 n-1 = 2 , p' = 1 and $I--)
P
I f I ~ j l ~2 , d ~ , ~
n-1 = -
2 .
I t e r a t i o n Scheme. Smoothing Operator.
A l i n e a r operator J,(e)
(e
_L
1 ) having the
f o l l o w i n g p r o p e r t i e s was constructed i n Shibata [12].
Lemma 4.1.
L e t e , k and p be r e a l numbers w i t h e 2 1, k 2 0 and 1 5 p
-
Then there e x i s t s a l i n e a r operator Sl(e) following properties:
w i t h t h e parameter e having t h e
m.
174
Yoshihiro SHIBATA and Yoshio TSUTSUMI
(1)
f o r any i n t e g e r N 2 0 and any f u n c t i o n u
ISl(e)ulp,k,N
= <
C(Pik,N)
and f o r any i n t e g e r i 2 0
(2)
f o r any i n t e g e r
and ( a l u ) ( o , x ) = 0
IUlp,k,N
and any f u n c t i o n u
EpyN with Iu) p,k,N
p,k,N
<
<
( i = O,.**,N-l)
f o r It1 5 1 and y ( t ) = 0 f o r It1 2 2 . $(el,e2)u
E
(U\
= 0;
Furthermore, we choose a f u n c t i o n v ( t )
(4.1)
:p’N
I
(a$l(e)u)(o,x)
N 20
with
E
= v(tej’)
E
C;(R
1 ) such t h a t f o r y ( t ) = 1
For 81 => 1 and e 2 ~
1 we, p u t
S,(e,)u.
By Lemma 4.1 we have t h e f o l l o w i n g theorem.
Lemna 4.2.
(1)
Let 1 5 p 5
-, e , 1 1
and e2 2 1.
Then,
f o r any i n t e g e r N 2 0, any r e a l number k 2 0 and any f u n c t i o n u
with I u ( p,k,N
,EpSN
<
IS2(el~e2)Ulp,k,N= < C(pSkYN) IUlp,k,N and f o r any i n t e g e r i 2 0
(a:S2(el (2)
with
,e,)u)(o,x)
=
o
;
f o r any i n t e g e r N 2 0, a r e a l number k 2 0 and any f u n c t i o n u
IuI Pik,N
<
-
and (a;u)(O,x)
= 0
(i= O,...,N-l)
€EpYN
Nonlinear Wave Equation in Exterior Domain
(3)
175
f o r any i n t e g e r M, N w i t h M > N 2 0, any r e a l numbers k , m w i t h
€ 3pyN
k > m 2 0 and any f u n c t i o n u
w i t h IuI
p,m,N
<
-
and ( a & ) ( o , x )
= 0
( i = 0,
1 ,..*,N-l)
4.2.
Compatibility Condition.
S i n c e t h e n o n l i n e a r term F a l s o depends on
t h e d e r i v a t i v e of o r d e r 2 w i t h r e s p e c t t o t i m e t i n o u r problem, we have t o pay s p e c i a l a t t e n t i o n t o 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 .
I n t h i s s e c t i o n we s h a l l
i n t r o d u c e t h e c o m p a t i b i l i t y c o n d i t i o n , f o l l o w i n g S h i b a t a [11? 12). we l e t u
E
Cm([O,m)
x
For s i m p l i c i t y
E ) and p u t
f ( t , x ) = n u + F(t,x,hu),
By t h e i m p l i c i t f u n c t i o n theorem i t f o l l o w s t h a t t h e r e e x i s t s a s u f f i c i e n t l y small p o s i t i v e c o n s t a n t d’ such t h a t i f
then t h e r e e x i s t f u n c t i o n s (4.3)
V. E
J
uJ. ( x ) = v ~ ( x , ~J;@,(x),
4’:”
( j 2 2 ) w i t h v.(x,O)
J
= 0 and
~ J X - ’ @ ~ ( X( )6,J - 2 f ) ( 0 3 ~ ) )
f o r a l l i n t e g e r s j 2 2. Thus, we i n t r o d u c e 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 i n t h e f o l l o w i n g form.
L e t d’ and v . be t h e same as i n (4,2) and (4.31, J We s h a l l say t h a t t h e data @ o ( x ) y +,(x) and f ( t , x ) s a t i s f y t h e
D e f i n i t i o n 4.1. respectively.
176
Yoshihiro SHIRATA and Yashio TSUTSUMI
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 N i f $o, $1 and f s a t i s f y t h e f o l l o w i n g two conditions :
4.3.
1.1.
I t e r a t i o n Scheme.
Let
% be
a p o s i t i v e c o n s t a n t described i n Theorem
be s a t i s f i e d f o r t h e d a t a 40,
L e t 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
$1 and f o f (M.P).
It1 2
We choose a f u n c t i o n y ( t )
It( 2
1 and y ( t ) = 0 f o r
Put
u,(x) = $,(x) and u . ( x ) ( j 2 2) a r e f u n c t i o n s c o n s t r u c t e d
where u o ( x ) = $,(x), i n 84.2.
2.
C;(R 1 ) such t h a t y ( t ) = 1 f o r
E
J
q Yf
Note t h a t v i s determined o n l y by $o,
and F.
By D e f i n i t i o n 4.1
and (4.3) i t f o l l o w s t h a t
-
a:(f
(1Jv + F ( t , x , A v ) ) ) I t +
f o r j = 0, l,...,m solution (4.4)
u
-
=
= 0
2 and t h a t v = 0 on [0,-)
x
an.
Putting w = u
o f (M.P), we see t h a t w s a t i s f i e s
tJw +
G(t,x,Aw)
= g
w(0,x) = (atw)(o,x)
in
= 0
[O,m)
x
R,
i n n,
where
1,
1
(4.5)
G(t,x,Aw)
=
(1
-
2 r)(dxF)(t,x,Av
+ rAw)(Aw,Aw) d r ,
-
v for a
Nonlinear Wave Equation in Exterior Domain
g = f
-
177
w
( u v + F(t,x,Av))
E
Em-’
I
Thus, we d e s c r i b e o u r i t e r a t i o n scheme f o r s o l v i n g t h e problem (4.4), f o l l o w i n g Klainerman [2] and Shibata [ 1, 121.
F i r s t we d e f i n e wo by t h e s o l u t i o n
of i n [O,-)
i w o = g
i n R.
= 0
wo(O,x) = (atwo)(o,x) Put
NOW we s h a l l d e f i n e a l r e a d y determined.
i
are ( p 2 0). F o r t h e moment we assume t h a t wo,~..,w P P L e t B be a f i x e d c o n s t a n t w i t h B > 1. L e t E be t h e p o s i t i v e
c o n s t a n t d e f i n e d i n Theorem 1 . 1 ( 3 ) . ,=E
(4.6)
r
,
Put
e . = BJ. J
We define t h e smoothing o p e r a t o r S . by J
s.u = J
(4.7)
s,
and
s,
operator
2
by
(4.8)
.j
where
P
w P
2
s,(ej)u,
if n
$- ( e j y e g ) u ,
i f n = 3,
4,
a r e t h e l i n e a r o p e r a t o r s d e f i n e d i n 54.1.
=xw+
(daG)(t,x,S
Aw )Aw.
P
P
We d e f i n e e: and eg- ( j 2 0 ) by J
e’ = (dAG)ft,x,Aw.)Ai j
J
j
-
(d,G)(t,x,S.nw.)& J J
j’
W e define the l i n e a r
Yoshihiro SHIBATA and Yoshio TSUTSUMI
178 e
C A
-
= G(t,x,Awjtl)
j
G(t,x A .) ' J'
-
(d,G)(t,x,Awj)Aij.
Put
+
e = e:
(4.9)
-1
(j 2 0).
e
We d e f i n e E j (j 2 0) by
Put
gp =
-
(Sp
-
Sp-l)Ep-l
-
SP e P-1 - (Sp - Sp-l)G(t,x,Awo)
( P 2 1).
F i n a l l y we d e f i n e \j by t h e s o l u t i o n o f
P
i P (o,x)
= ( a \j )(o,x)
=
t P
o
i n Q.
Thus, we can s u c c e s i v e l y determine two f u n c t i o n sequences I w {
i P 1.
P
1 and
Note t h a t
fwPtl
(4.13)
+ G(t,x,Awptl)
= g t (1
For wo and
i J.
-
Sp)G(t,x,AWo)
+ (1
-
Sp)Ep
+
ep.
(j 2 0) we have t h e f o l l o w i n g i m p o r t a n t lemma, which w i l l be
proved i n t h e n e x t s e c t i o n .
Lemma 4.3.
Assume t h a t Assumption 1.1 holds.
Theorem 1.1 be s a t i s f i e d . i n Theorem 1.1.
L e t a,
Then, wo and
E,
E, m
i . (j 2 0) J
L e t a l l assumptions i n
and A0 be p o s i t i v e constants d e f i n e d
s a t i s f y t h e f o l l o w i n g estimates:
Nonlinear Wave Equation in Exterior Domain
(1)
Suppose t h a t n 2 6.
(accordingly then Awo, AGj
= E
L e t 0 = max (4[n/21+7,
P+
3[n/2] + 6 ) . IE /I CP ( [ 0 , m ) x 4
z)
179
m+l) and
f
= 20 t [n/2] t 2
I f a and h0 a r e chosen s u f f i c i e n t l y s m a l l ,
and
4 f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-1) w i t h 0 < 6
(2) and
Suppose t h a t 4 5 n 2 5.
T=26 + l ( a c c o r d i n g l y
s m a l l , then !two,
AGj
E
=
-T IE n C
L e t B = max (3[n/2]+6, mtl) = max (12, mtl)
3[n/2]
t
6 = 'i:+ 12).
([0,-)
x
6)
' t
If a and 6 o a r e s u f f i c i e n t l y
and
f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-2) w i t h 0 < 6
(3) Suppose t h a t n = 3. integer w i t h r
i I [ t+
Let
60.
IJ
= ~ / 7and B =
( 3 m + 7 ) ~ ]( a c c o r d i n g l y
Ifa and 6o a r e chosen s u f f i c i e n t l y s m a l l , then
71 +
%=
Awe,
t
(m + 2 ) ~ . L e t
3[n/2] E
t
6
= yt
4 P € ! f \ C ([0,m)
z 60.
be an
9). x
5)
and
f o r 6 s a t i s f y i n g t h e i n e q u a l i t i e s o f t h e d a t a i n Theorem 1.1-3) w i t h 0 < 6 2 do.
65. P r o o f o f L e m a 4.3 and Main Results. We s h a l l now p r o v e Lemma 4.3 by an i n d u c t i o n argument. same n o t a t i o n s as i n 64.
We s h a l l use t h e
- L e t L denote an i n t e g e r and k denote a r e a l number.
180
For
Yoshihiro SHIBATA and Yoshio TSUTSUMI
t h e moment we always assume t h e f o l l o w i n g assumptions: '
[A.5.1]
Awo
E
V
N
zLflCL([O,m)
E) and
x
i f n 2 6,
w
CA.5.21
ko,.-.,kp
[A.5.3]
i f Ihl
a r e a l r e a d y determined and Lemma 4.3 h o l d s f o r
ko,**-,
* P '
z s1 ,
then f o r any i n t e g e r s N and L w i t h N ' 0
and
%
O 5 L 5 L ld,G(-Y*J)Im,O,L N
where
y=
<
C(n,L,N)
<
m,
+ 4[n/2] + 7 if n 2 6 and ';J =
+ 3[n/2] + 6 i f 3 5 n
Let
We s h a l l f i r s t prepare several lenmas t o prove Lemma 4.3. s u f f i c i e n t l y small p o s i t i v e c o n s t a n t and e s p e c i a l l y Noting t h a t
~ - E - B -L T.
and -B+o'i:)
T
T
=
5 if n
5.
T
be a
= 3.
if n = 3, we can prove t h e f o l l o w i n g
lemna i n t h e same way as Klainerman [2] and Shibata [ll,121.
Lemma 5.1.
Assume t h a t [A.5.1
-
31 h o l d .
For w . ( j = O,l,...,p+l) J
we have
the following: Iv
(a)
hwj
(b)
if n
E
-'i L IE fl c ([(I,-) x 6) ;
2 4, then
'"J12,O,L
+
I"jlb(n),c(n),L
= <
C6
f o r -8+L
lAwj1b(n),c(n),L
= <
C s e J.
-B+L 1AwJ12,0,L
+
-'I,
f o r -B+L &
T
-
and 0 5 L 2 L,
Nonlinear Wave Equation in Exterior Domain
ISjAwj12,0,L
+
IsjAwJ1 b(n),c(n),L
= <
181
for L >
C(L) a ejBtL
where b ( n ) = 4 and c ( n ) = ( n - 1 ) / 4 i f n 2 6, and b ( n ) =
m
T,
and c ( n ) = (n-1)/2 i f
Y
i f n = 3, then f o r -@+uL 2
< C6
2,0,L
=
m,k,L
=
<
ca
f o r k-B+sL
-T,
_i -1, rv
for - B t d 2
T
and 0 5
L 5 L, h
f o r k-B+oL 2 for L >
12,0,L
= <
C(L) 6 9 j B + O L
IsjAwj Im,k,L
= <
C(k,L) 6 e t - B t a L
ISj"j
0 2 L 5 L and 0 5 k 5 1-E,
1, >
-
1-E o r L > L ;
i f n 2 4, then
(d) I(1
for k
T,
-
sj)Awj12,0,L
+
I(1
-
Sj)Awjlb(n),c(n),L = < Csey'+L J
for
o
N
L
L,
where b ( n ) and c ( n ) a r e t h e same as i n ( b ) ; (e)
I
i f n = 3, then
-
'j)"j
Im,k,L
= <
c 6 et-B+oL
for 0
z k 5 1-E and 0 5 L z r .
By choosing 6 s u f f i c i e n t l y small we assume t h a t :
rA.5.41 < C6
IAwjlm,o,o
5 C l A w j 12,0,[n/21+1
=
lA'jl-,o,o
5 Cl~'jl2,0,[n/21+1
= <
z a1 1
From Theorem Ap.1,
rA.5.1
-
Cs 5 %
( j = O,l,...,p+l), ( j = O,l,-..,p).
41 and Lemmas 4.1 and 4.2 we have the following
Yoshihiro SHIBATA and Yoshio TSUTSUMI
182
1emma.
Lemma 5 . 2 .
For e . ( j = O,--.,p) J
Assume t h a t Assumption 1.1 holds.
we have
the following: &
ej
rACL([O,-)
E)
E
]E
lej12,k,L
5
cs
<
c s e k-(1t7E)B+oL
lejll,k,L
=
(a)
x
;
(b)
n,
Proof.
for
‘jk-36toL
j
By t h e d e f i n i t i o n o f e j
i n t h e case o f n = 3.
Since e
j
= e*
5 k 2 2 ( 1 - ~ ) and 0 5 L 5 L,
i f n = 3,
f o r 0 5 k 5 1-E and 0 jL 2 L,
if n = 3.
T+B
-
(a) i s clear.
+
e:-, j
So, we s h a l l prove ( b ) o n l y
we have o n l y t o prove t h a t ( b ) holds ~
f o r e: and e - - , r e s p e c t i v e l y . However, we s h a l l prove o n l y f o r e: because we J j J can prove f o r e’- i n t h e same way. 2 Since dG ,
j
t,x,O)
= 0, i t f o l l o w s t h a t
lo{I, 1
ej =
l (d,G)(t,x,r’(S.Aw. 3 x
By Lemma 5.1,
(S.AW. J J
+ r(l
J
J
+
r(1
-
-
S.)Awj))dr’ J
S.)AW.,(l
J
J
-
1
Sj)AW
j’
Aij) d r .
Theorems Ap.1 and Ap.2 we have f o r k w i t h k-6 2
T
and k 5 1-E
183
Next by L e w a 5.
le;ll
,k,L
= c
, Theorems
Ap
c I(' [Awjlm,o,L
+ Ihw. 2,0,LI(1 J
-
sJ.
Thus, t h e l a s t i n e q u a l i t y i n ( b ) i s proved.
(Q. E. 0.)
By L e n a s 4.1,
Lemma 5.3.
4.2, 5.2 and
ej
=
BJ we have t h e f o l l o w i n g lemma.
Assume t h a t Assumption 1.1 h o l d s .
Then,
' U r v
(a)
Ep
E
EL/7CL([0,-)
x
5)
;
(b 1 JEpl~,n-l,t
2
' /Eplq,n-l,L 3-4-
= c
c5
2
for -&+L
5
-T,
if n 2 6 ,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
184
IEpl2,n-1 ,L
+
lEp11 ,C,L 2
< C 63
IEp12,k,L
=
IEpll,k,L
= <
C6
3
< C 63
IEpll,k,L
=
z -T,
for - 2 ~ t L
< C 63
=
for
T+B
k and k-38toL 2 -'I, i f
f o r k - ( 1 + 7 ~ ) ~ + o2L
ek-(1t7E)B+oL p
-T,
if 4
n
zn 2 5,
= 3,
if n = 3 ;
f o r k - ( 1 + 7 ~ ) @ + o L2
T,
0 2 k
z 1-E
and
'u
O ~ L L L , i f n = 3.
N
N o t i n g t h a t l - ~ - ( l t 7 ~2 )T,~ 2 ( 1 - ~ ) - 3 8 2 Lemmas 4.1,
T
and -3BtoL 2
4.2 and 5.3 the f o l l o w i n g lemma.
L e m a 5.4.
Assume t h a t Assumption 1.1 holds.
Then,
T~
we have by
Nonlinear Wave Equation in Exterior Domain
3 k-3B+aL ' ~ ) ~ p l 2 , k , L= < C(k,L) 6 ep+l
1('p+l
I(sp+l
-
Sp)Epll,k,L
By Assumption 1.1,
Lemna 5.5. (a)
= <
3 k-(1+7~)4+0L C(k,L) 6 ep+,
[A.5.1]
and [A.5.3]
E
'rl - i C ir ([0,-)
E
,
x
f o r k 2 T+B and L 2 0, i f n = 3,
f o r k 2 0, and L 2 0, i f n = 3.
we have t h e f o l l o w i n g lemma.
Assume t h a t Assumption 1.1 holds.
G(t,x,Awo)
185
Then,
;
(b)
By Lemmas 4.1, 4.2 and 5.5 and Theorems Ap.1 and Ap.2 we have t h e f o l l o w i n g 1emma.
Yoshihiro SHIBATA and Yoshio TSUTSUMI
186
Combining Lemmas 5.2, fact
eo
=
5.4,
5.5 and 5.6 and u s i n g Lemmas 4.1,
4.2 and t h e
1, we have t h e f o l l o w i n g lemma.
Lemma 5.7.
Assume t h a t Assumption 1.1 holds.
90' gp+l
(a)
E
E"nC"([O,-)
x
5)
Then,
;
(b)
I n o r d e r t o use Theorem 3.2, we have t o e v a l u a t e t h e c o e f f i c i e n t s o f t h e
:.tj
operator
d e f i n e d i n (4.8).
A . = (d,F)(t,x,Av) 3
+
Noting t h a t (d,F)(t,x,Av)
Put
(d,G)(t,x,SjAwj). = 0 for
It1 2 2,
we have t h e f o l l o w i n g lemma by
Nonlinear Wave Equation in Exterior Domain
Lemma 5.1,
[A.5.1
Lemma 5.8. $o,
-
41, Theorems Ap.1, Ap.2 and Ap.3.
c
L e t L be a p o s i t i v e c o n s t a n t d e f i n e d i n [A.5.2].
$1 and f s a t i s f y a l l assumptions i n Theorem 1.1.
holds.
< C6
=
f o r -B+L 2
-T
and 0 5
L e t t h e data
Assume t h a t Assumption 1.1
Then we have t h e f o l l o w i n g :
IA014,d,L 4
187
L 5 L,
Yoshihiro SHIBATA and Yoshio TSUTSUMI
188
cy
l.
~
[ A p t 1 12,1tE,L
62 el+E-B+oL P+l
f o r 0 5 L 5 L.
I n p a r t i c u l a r , choosing 6 s u f f i c i e n t l y small, we have t h e f o l l o w i n g :
Here d i s a p o s i t i v e constant g i v e n by Lemma 3.1.
Proof.
We s h a l l g i v e t h e sketch o f t h e proof for,%
P + l o n l y i n t h e case
o f n = 3. Since (dhG)(t,x,O) we have by L e m a 5.1,
= 0,
2 (dAG)(t,x,O)
Theorems Ap.1,
= 0 and f o r It1 2 2
Ap.2 and Ap.3
( d X F ) ( t r x , h v ) = 0,
Nonlinear Wave Equation in Exterior Domain
1+E -8 Noting t h a t 7
-T
, we
have f o r
l+E 2 -8toL
2
189
T
F i n a l l y we have by Lemma 5.1
From the above lemmas we can complete the p r o o f o f Lemma 4.3.
Proof of Lemma 4.3. First
We s h a l l prove Lemma 4.3 by an i n d u c t i o n argument.
we assume f o r the moment t h a t r A . 5 . 1
have by Lemas 5.7, 5.8 and Theorem 3.2 t h a t
- 41 hold.
Then, i f n 2 6 , we
190
Yoshihiro SHIBATA and Yoshio TSUTSUMI
I"ptl12,O,L
I"ptl14,n-JL 4
2
t 6 max(1,
We have used t h e f a c t
2 4[n/2]
e -~+L+3[n/21+6) P+l
t
7 a t the l a s t i n e q u a l i t y .
by choosing 6 so small t h a t max { C(L)s ; 0 holds f o r
I n t h e same iptl,
eply 1
IL
zT 1 -5 1 we
I n particular,
see t h a t Lemma 4.31)
way i t i s c l e a r t h a t under t h e assumptions CA.5.11
and EA.5.31 Lemma 4.3(1) h o l d s f o r
i0,By
t h e way, we see by t h e assumption on
t h e data i n Theorem 1.1, Theorem 3.2, Theorems Ap.2 and Ap.3 t h a t [A.5.1]
5 . 31 h o l d .
and [A.
Therefore, an i n d u c t i o n argument g i v e s Lemma 4.3(1).
i n t h e same way we o b t a i n Lemma 4.3(2) f o r 4 2 n 5 5. F i n a l l y , f o r n = 3 we s h a l l v e r i f y t h a t under t h e assumptions [A.5.1 Lemma 4.3(3) h o l d s f o r 1 +
E
2 B +
T
iptl. By Theorem
3.2,
Lemmas 5.7,
-
41
5.8 and t h e f a c t t h a t
we have
We have used t h e f a c t s t h a t 1+€ -B+o 5
-T
and t h a t I + E - ( ~ - E ) B + C5 J 0 a t the
second i n e q u a l i t y and t h e l a s t i n e q u a l i t y , r e s p e c t i v e l y .
Thus, we have
-.I
I n p a r t i c u l a r , by choosing 6 so small t h a t max { C(L) & 2 ; 0 2 L obtain
L 1 5 1 we
Nonlinear Wave Equation in Exterior Domain
191
Next, by Theorem 3.2, Lemmas 5.5 and 5.8 we have <
("ptl
L,l-€,L
=
+
,2,1+~-6 P+l
6'1
t
c ( L ) 6 3 [ e l + ~ - ( 1 + 7 ~ ) 8 + o ( L + 3 [ n / 2+4 1 1 P+l el+~-36+o(L+3[n/2]+4) P+1
eAl7-6 max (1, e
I+~-B+o(L+3[n/2]+6) P+ 1
+
<
-
qL1 &3[
( 1+E ) / 2 P+ 1
- 6+0 ( L+3 [n/ 2]+4) 1
1+~-36
1
ep+l
el+~-(1+7~)6+o(L+3[n/2]t4) P+l
+ e 2( 1+~)-46+o(L+3[n/2]+6) P+l
-6+0 5 - -T a t t h e f i r s t
We have used t h e f a c t s t h a t 1 + ~2 6+r and t h a t inequality. <
2 By t h e way, s i n c e z ~ + o ( 3 [ n / 2 ] + 4 ) - & 6
<
Oand 1 + 3 ~ - ( 3 - ~ ) 6 3+ ~ ( 3 [ n / 2 ] + 6 )
0, we have
( b 7 ~6+0 ) (L+3 [n/2]+4) 5 -1 2( 1 + )-46+a ~ (L+3[n/2]+6)
-
E-
- ( 1- E ) 5
( 1+E ) B+oL
-
Thus, i t f o l l o w s t h a t
By t h e Sobolev imbedding theorem and ( 5 . 1 ) we have
Therefore, by i n t e r p o l a t i n g between (5.2) and (5.3) we have
for 0
z k 2 1-E
and 0
L
IT.
Thus, s i n c e -.~6+o([n/2]+1) 2 0, we o b t a i n by
(5.4) cu
for 0
(5.5) Ifwe choose 6 so s m a l l t h a t max
{
C(L) '6
;0
k 5 1-E and 0 5
5L
zy
L 5 1.
15 1, t h e n (5.5) and
Yoshihiro SHIBATA and Yoshio TSUTSUMI
192
(5.1)’ give
L e n a 4.3(3) f o r ;p+l.
Since we can prove i n t h e same way as t h e
case o f n 2 6 t h a t Lemma 4.3(3) h o l d s f o r
wo
and t h a t [A.5.1]
and [A.5.3
-
41
T h i s completes t h e p r o o f o f
hold, an i n d u c t i o n argument g i v e s Lemma 4 . 3 ( 3 ) . Lemma 4.3.
(9. E. D.)
P r o o f o f main r e s u l t s .
Put
m
Then, from Lemnas 4.1
-
3, Lemmas 5.1
-
6 and (4.6) we e a s i l y see t h a t u = v + w
i s t h e d e s i r e d s o l u t i o n o f (M.P) ( f o r d e t a i l s , see Klainerman [2] and Shibata
[ll,121).
Furthermore, we can prove the uniqueness o f t h e s o l u t i o n o f (M.P)
by t h e energy method i n t h e same way as Shibata [12].
(Q. E. D.)
Concluding Remarks.
(1)
When n = 3, we used t h e c u t - o f f f u n c t i o n i n time.
The authors do n o t know whether we can prove w i t h o u t i t f o r n = 3 i n t h e same way as Klainerman and Ponce [ 3 ] and Shatah (2)
[lo].
We can a l s o o b t a i n t h e analogous r e s u l t s f o r t h e mixed problems o f t h e
n o n l i n e a r Klein-Gordon equation and t h e n o n l i n e a r Schrodinger e q u a t i o n i n t h e same way (see, e.g.,
TsuTsumi [13]).
56. Appendix. I n t h i s s e c t i o n we s h a l l s t a t e several theorems which p l a y an i m p o r t a n t r o l e i n the p r e s e n t paper.
Theorem Ap.1.
Let
For t h e i r p r o o f , see Shibata [ll,121.
p = Rn o r a. L e t
and f and g be f u n c t i o n s from [O,-) @,
x
and
JI
be f u n c t i o n s f r o m ,g t o
R1
& to IR1 . Assume t h a t a l l semi-norms o f
$I,f and g appearing below a r e bounded.
k and
@
L e t M and
N be nonnegative i n t e g e r s ,
m be nonnegative numbers and p and q be r e a l numbers w i t h 1 5 p, q 2
m.
Nonlinear Wave Equation in Exterior Domain
Then,
Furthermore, i f F(t,x,O)
Theorem Ap.3.
Let
= 0, then
o0,
$1 and f be t h e data o f (M.P)
such t h a t a l l semi-
193
YoRhihiro SHIBATA and Yoshio TSUTSUMI
194
Let
norms appearing below a r e bounded. i n 54.
L e t H(t,x,x)
E Wm([O,m)
x
5
x
% and {
1x1
v ( t , x ) be t h e same as those d e f i n e d
2 1 1).
I f H(t,x,O)
= 0, then
References P. Dionne, Sur l e s p r o b l i m e de Cauchy hyperboliques b i e n poses, J . Analyse Math., c21
10 (1962). 1-90.
S. Klainerman, Global e x i s t e n c e f o r n o n l i n e a r wave equations, Corn. Pure
Appl. Math,, 33 (1980), 43-101.
S. Klainerman and G. Ponce, Global s m a l l amplitude s o l u t i o n s t o n o n l i n e a r e v o l u t i o n equations, Comm. Pure Appl. Math., 36 (1983), 133-141.
P. Lax and R. P h i l l i p s , S c a t t e r i n g Theory, Acad Press, 1967.
R. B. Melrose, S i n g u l a r i t i e s and energy decay i n a c o u s t i c a l s c a t t e r i n g , Duke Math. J . , 46 (1979), 43-59. r61
S. Mizohata, Quelque problemes au bord, du t y p e rnixte, pour des equations
hyperboliques, S h i n a i r s u r l e s Gquations aux derivees p a r t i e l l e s , C o l l i g e de France, (1966/67), 23-60. [71
A. Moser, A r a p i d l y convergent i t e r a t i o n method and n o n - l i n e a r d i f f e r e n t i a l equations, Ann. Scu. Norm. Pisa, 20(3) (1966), 265-315, 499-535.
r 81
J . Nash, The embedding problem f o r Riemannian m a n i f o l d s , Ann. Math., 63 (1965), 20-63.
r91
P. H. Rabinowitz, P e r i o d i c s o l u t i o n s o f n o n l i n e a r h y p e r b o l i c p a t i a l d i f f e r e n t i a l equations
II , Corn.
Pure Appl. Math.
, 22
(1969), 15-39.
J . Shatah, Global e x i s t e n c e o f small s o l u t i o n s t o n o n l i n e a r e v o l u t i o n equations, J . D i f f e r e n t i a l Eqs.
, 46
(19821, 409-425.
Y. Shibata, On t h e g l o b a l e x i s t e n c e o f c l a s s i c a l s o l u t i o n s o f mixed
Nonliiiear Wave Equation i n Exterior Domain
195
problem f o r some second o r d e r n o n - l i n e a r h y p e r b o l i c o p e r a t o r s w i t h d i s s i p a t i v e term i n t h e i n t e r i o r domain, Funk. Ekva., [12]
25 (1982), 303-345.
Y . Shibata, On t h e g l o b a l e x i s t e n c e o f c l a s s i c a l s o l u t i o n s o f second o r d e r
f u l l y n o n l i n e a r h y p e r b o l i c equations w i t h f i r s t o r d e r d i s s i p a t i o n i n t h e e x t e r i o r domain, [13]
Tsukuba J . Math., 7 ( 1 ) (1983), 1-68.
Y . Tsutsumi, Global s o l u t i o n s o f t h e n o n l i n e a r Schrodinger e q u a t i o n i n
e x t e r i o r domains, t o appear i n Corn. P. 0. E. [14]
Y . Tsutsumi, Local energy decay o f s o l u t i o n s t o t h e f r e e Schrodinger
e q u a t i o n i n e x t e r i o r domains, t o appear i n 3. Fac. S c i . Univ. Tokyo, Sect.
IA, Math. [15]
B. R. Vainberg, On t h e a n a l y t i c a l p r o p e r t i e s o f t h e r e s o l v e n t f o r a c e r t a i n c l a s s o f o p e r a t o r - p e n c i l s , Math. USSR Sbornik, 6 ( 2 ) (1968), 241-273.
[16]
B. R. Vainberg, On e x t e r i o r e l l i p t i c problems p o l y n o m i a l l y depending on a s p e c t r a l parameters and t h e a s y m p t o t i c behaviour f o r l a r g e t i m e o f non s t a t i o n a r y problems, Wath. USSR Sbornik, 21(2) (1973), 221-239.
[17]
B.
R. Vainberg, On t h e s h o r t wave asymptotic behaviour of s o l u t i o n s o f
s t a t i o n a r y problems and t h e a s y m p t o t i c behaviour as t
-+
-
o f solutions o f
n o n s t a t i o n a r y problems, Russian Math., Surveys, 30(2) (1975), 1-58.
[la]
W. von Wahl, LP-decay r a t e s f o r homogeneous wave equations, Math. Z.,
120 (1971), 93-106.
196
Yoshihiro SHIBATA and Yoshio TSUTSUMI
Lecture Notes in Num. Appl. Anal., 6,197-210 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1983
Diffusion Processes and Partial Differential Equations
Kazuaki TAIRA
The purpose of t h i s paper i s t o s t u d y i n r i m a t e connections between second-order d i f f e r e n r i a l o p e r a t o r s and Markov p r o c e s s e s .
The paper i s
d i v i d e d i n t o two c h a p t e r s . Let
D
N
be a connected open s u b s e t of IR
.
The following r e s u l t i s
well-known by t h e name of t h e s t r o n g maximum p r i n c i p l e f o r t h e L a p l a c i a n A =
z
-a 2.2
i=i ax
.
"If
u
E
C2(D),
Au
2
0
in
D
and
u
t a k e s i t s maximum
i
a t a p o i n t of
D,
then
u
i s a constant."
The purpose of Chapter I i s
t o r e v e a l t h e underlying a n a l y t i c a l mechanism of propagation of maximum ( s h a r p maximum p r i n c i p l e ) f o r degenerate e l l i p t i c o p e r a t o r s of second o r d e r , e x p l a i n i n g t h e above r e s u l t .
The mechanism of propagation of maximum
i s c l o s e l y r e l a t e d t o t h e d i f f u s i o n phenomenon of Markovian p a r t i c l e s .
Chapter I1 i s devoted t o t h e semigroup approach t o t h e problem of c o n s t r u c t i o n o f Markov p r o c e s s e s i n p r o b a b i l i t y theory.
I t is well-known
t h a t by v i r t u e of t h e c e l e b r a t e d Hille-Yosida theorem i n t h e t h e o r y of semigroups, t h e problem of c o n s t r u c t i o n of Markov p r o c e s s e s can be reduced t o t h e s t u d y of boundary v a l u e problems f o r d e g e n e r a t e e l l i p t i c o p e r a t o r s of second o r d e r .
S e v e r a l r e c e n t developments i n t h e t h e o r y of 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 have made p o s s i b l e f u r t h e r p r o g r e s s i n t h e s t u d y of boundary v a l u e problems and hence of t h e problem of c o n s t r u c t i o n of Markov
197
Kazuaki TAIRA
198 processes.
The details will be published i n the forthcoming book "Diffusion Processes and Partial Differential Equations" (Academic Press).
I.
A SHARP MAXIMUM PRINCIPLE FOR DEGENERATE ELLIPTIC OPERATORS
91.0 Introduction Let A
be a second-order differential operator with real coefficients
such that
where the coefficients aiJ , bi
satisfy :
2 are C -functions on JRN all of whose derivatives of
1" ail
order
2
2
are bounded in lRN and the matrix
N
positive semi-definite in ,JR 2"
bi
,
assuming that
(aiJ) i s aij = aji. N
are C1-functions on lRN with bounded derivatives in IR
.
In this chapter we shall consider the following
D
PROBLEM.
Then determine
&a
connected open subset of IRN
connected, relatively closed subset D(x)
containing x , such that 2 if u E C (D),
then u The set D(x)
Au:
0 & I D,
M throughout D(x)
.
sup u = M < D
is called the propagation set of x
-
x E D.
of and
D,
u(x) = M ,
in D .
We shall give a coordinate-free description of the propagation set D(x)
in terms of subunit vectors, introduced by Fefferman-Phong [31 in
studying the subellipticity of second-order differential operators with non-negative principal symbols.
Diffusion Processes and Partial Differential Equations
11.1
Statement of R e s u l t s [ 3 ] , we say t h a t a tangent vector
Following Fefferman-Phong
(
N 2 L yj q j )
j =1
where
N
2
i,j=l
for all
aij(x)q. q.
axes so t h a t t h e m a t r i x
,
Xr > 0,
Z r ~ dx .
TI =
j=1 J
1 J
at
D
(ai')
Artl
=
So r o t a t e t h e c o o r d i n a t e
is diagonalized at
... =
A
=
N
0
where
r = rank ( a i j ( x ) ) .
L
X =
yj
j=1
for
i f and only i f
Ao
+(t) = If
; i (~ y(t)
i s subunit f o r
)
is s u b u n i t f o r
:(t)
a ax
is subunit
j
is c o n t a i n e d i n t h e e l l i p s o i d of dimension
X
A s u b u n i t t r a j e c t o r y is a L i p s c h i t z p a t h
d
(ai' (x)) = (Xi 6 i j ) ,
x :
N
Then i t i s e a s i l y seen t h a t a t a n g e n t v e c t o r
T>
E
j
Note t h a t t h i s n o t i o n i s
x.
independent of t h e p a r t i c u l a r c o o r d i n a t e c h a r t .
...
X =
N
Z
i s t h e c o t a n g e n t s p a c e of
TZD
X1 > 0.
199
Ao
at
at
Ao
y(t),
y : [ t ,t ]
1
y(t)
+
D
such t h a t
f o r almost every
- ;(t)
so is
2
t.
; hence s u b u n i t
t r a j e c t o r i e s a r e not oriented.
We l e t
X
=
O
N
.
N
L
(bl-
Z
i=1
aaij
-ax) " axi j=1 j
which i s c a l l e d t h e d r i f t v e c t o r .
Note t h a t
s u b p r i n c i p a l p a r t of t h e o p e r a t o r
A
i s t h e so-called
Xo
i n terms of t h e theory of 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 , and t h a t i t i s i n v a r i a n t l y d e f i n e d a t t h e p o i n t s where t h e m a t r i x Adrift
(ai')
i s degenerate.
t r a j e c t o r y i s a curve B(t) = x o ( e ( t ) )
on
9 : [tl,t2]
[t,,t,I,
+
D
such t h a t
r :
Kazuaki TAIRA
200
and t h i s curve i s o r i e n t e d i n t h e d i r e c t i o n of i n c r e a s i n g
t.
Our main r e s u l t i s t h e following
THEOREM 1.1. T h e p r o p a g a t i o n in D -
of a l l p o i n t s
y
E
D
set
of
D(x)
x
which can be j o i n e d t o
in x
D
is t h e
ClOSUKe
b~ a f i n i t e number
of s u b u n i t and d r i f t t r a j e c t o r i e s . ( ai j )
Theorem 1.1 t e l l s us t h a t i f t h e m a t r i x
i.e,, i f
r = rank (ai' (x))
neighborhood of
x,
= N
,
i s non-degenerate a t
then t h e maximum propagates i n an open
but i f t h e matrix
is degenerate at
(aii)
t h e maximum propagates only i n a "thin" e l l i p s o i d of dimension and i n t h e d i r e c t i o n of
Xo.
x,
x,
r
then ( c f . (1.1))
Now w e s e e t h e reason why t h e s t r o n g maximum
p r i n c i p l e h o l d s f o r t h e Laplacian
A .
In [ 9 ] , Stroock and Varadhan c h a r a c t e r i z e d t h e s u p p o r t of t h e d i f f u s i o n process corresponding t o t h e o p e r a t o r
(which i s t h e c l o s u r e of t h e
A
c o l l e c t i c n o f a l l p o s s i b l e t r a j e c t o r i e s of a Markovian p a r t i c l e with generator
A)
and, a s one of i t s a p p l i c a t i o n s , they gave a (not coordinate-
f r e e ) d e s c r i p t i o n of t h e propagation set. We can prove t h a t o u r propagation s e t
D(x)
c o i n c i d e s with t h a t of
Stroock-Varadhan [ 9 ] : THEOREM 1 . 2 .
the -
~D
closure i n
The p r o p a g a t i o n set
-
of t h e p o i n t s
$(t),
D(x) t
2
@ Theorem 1.1 c o i n c i d e s
0
where
1 a p a t h f o r which t h e r e e x i s t s a p i e c e w i s e C - f u n c t i o n such t h a t --
@ : [O, t] J, : [ O , t ]
+.
&
D
N
+.
1R
Diffusion Processes and Partial Differential Equations REMARK 1.1.
s u b s e t of
By Theorem 4.1 of [ 9 ] , w e see t h a t
D(x)
201
is the largest
having p r o p e r t y (*) i n some g e n e r a l i z e d s e n s e ( s e e a l s o [ 6 ] ,
D
Chap. V I , Theorem 8 . 3 ) .
In t h e c a s e where t h e o p e r a t o r
A
i s w r i t t e n a s t h e sum of s q u a r e s
of v e c t o r f i e l d s , H i l l [ 5 ] gave a n o t h e r ( c o o r d i n a t e - f r e e ) d e s c r i p t i o n of t h e propagation s e t , although h i s proof w a s n o t complete.
H i l l ' s result
i s completely proved and extended t o t h e non-linear c a s e by Redheffer [ 7 ] . Now suppose t h a t t h e o p e r a t o r
A
i s w r i t t e n as t h e sum of s q u a r e s
of v e c t o r f i e l d s :
(1.2)
2
A =
Yk
+
Yo
k= 1 where
2 a r e r e a l C - v e c t o r f i e l d s on BN and
Yk
.
N f i e l d on IR
Yo
1 is a r e a l C -vector
As a byproduct of Theorem 1 . 2 , w e can prove t h a t o u r
propagation set
D(x)
c o i n c i d e s w i t h c h a t of H i l l [51.
Before s t a t i n g
t h e r e s u l t , r e c a l l t h a t H i l l ' s d i f f u s i o n t r a j e c t o r y i s a curve
Itl,
t2]
+
such t h a t
D
they may be t r a v e r s e d
H i l l ' s diffusion t r a j e c t o r i e s a r e not oriented ;
i n either direction.
Hill's
with
Yk,
Yo
increasing
i n s t e a d of t
6 :
drift
t r a j e c t o r i e s a r e defined s i m i l a r l y ,
but they a r e o r i e n t e d i n t h e d i r e c t i o n of
.
We can prove t h e following
THEOREM 1.3.
(1.2).
Suppose t h a t t h e o p e r a t o r
Then t h e p r o p a g a t i o n set
closure i n ~-
D
of a l l p o i n t s
y
D(x) E
D
of
A
i S w r i t t e n i n t h e form
Theorem 1.1 c o i n c i d e s w i t h t h e
which can be j o i n e d t o
x & a finite
number of H i l l ' s d i f f u s i o n and d r i f t t r a j e c t o r i e s . --REMARK 1.2. (cf.
Theorem 1 . 3 i s i m p l i c i t l y proved by Stroock and Varadhan
[ 8 ] , Theorem 5.2 ; [ 9 ] , Theorem 3 . 2 ) , s i n c e t h e support o f t h e
Kazuaki TAIRA
202
diffusion process corresponding to the operator A
does not depend on the
expression of A .
11.
SEMIGROUPS AND BOUNDARY VALUE PROBLEMS
52.0
Introduction
Let D
aD and be the space of real-valued continuous functions on D = D’
let C(5)
be a bounded domain in lRN
with smooth boundary
an.
strongly continuous semigroup {TtItLo of bounded linear operators on C(5) is called a Feller semigrouE on D if {TtI satisfies the following
A
condition :
It is known (cf. [ 2 ] ) that there corresponds to a Feller semigroup
-
on D
a strong Markov process jC on
P(t,x,dy)
-
(T,It20
D whose transition function
satisfies :
It is just the semigroup property: Tt+S
=
Tt * Ts which reflects the
Markov property that the future is independent of the past for a known present.
For a Feller semigroup {TtIt20 on D , Ttf Ul f = lim tJ.0
-
define
f
t
provided that the limit exists in the infinitesimal generator of
C(%).
The operator
Ul
is called
{TtI and its domain will be denoted by
The celebrated Hille-Yosida theorem in the theory of semigroups states that a Feller semigroup {TtItLO on D is completely characterized
D(07).
203
Diffusion Processes and Partial Differential Equations
01
by i t s i n f i n i t e s i m a l g e n e r a t o r
.
Under c e r t a i n c o n t i n u i t y hypotheses concerning t h e t r a n s i t i o n f u n c t i o n such as
P(t,x,dy)
-1
lim tJ.0
I,,-,,>,
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 <
5
E~
y(t)
for almost every
an ordinary Euclidean ball of radius
The operator
(A.1)
207
C
1
1
BE(x', P )
on E j
A
about
P
& elliptic near
c BLo(xt, C1
x'
.
is the following
L3
and there exist constants
such that for sufficiently
> 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 <
sufficiently
p >
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
<
E;
5
1
and
Ci
>
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.
L e c t u r e N o t e s in Num. Appl. Anal., 6, 211-219 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1985
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)<+m}, IsI=n ( ( g l l (fi+a): QT
ID:flF)+
[O,T11
Atusi TAN1
212
Using l o c a l coordinates, i t i s n o t d i f f i c u l t t o d e f i n e such spaces f o r f u n c t i o n s d e f i n e d on t h e boundary o f
n.
The same n o t a t i o n
w i l l be used
f o r t h e spaces o f v e c t o r f u n c t i o n s , t h e norm o f a v e c t o r supposed t o be equal t o t h e sum o f norms o f a l l i t s components. CL
= 1, n o t a t i o n s such as
(Ll
a r e used.
' ~ ' X , Q ~
the set o f a l l functions
q(p,e)
For t h e Halder exponent
By ~ ~ ~ ~ ( ( O , - ) ~ ( ( O , m ) we ) , mean
which i s d e f i n e d on
(0,m)
n-times
x (0,m),
p a r t i a l l y d i f f e r e n t i a b l e and i t s n - t h o r d e r d e r i v a t i v e s a r e l o c a l l y L i p s c h i t z continuous t h e r e .
-
The m o t i o n o f general f l u i d s
One phase f r e e boundarv Droblem.
2.
i s described by t h e system o f d i f f e r e n t i a l equations: D Dt p = -Pv*v,
[
(1)
L v = v * P+ p f ,
Dt PI3 - Ds = Dt i s the density,
Here p=p(x,t) f l u i d a t time
a t the point
t
t e r n a l f o r c e s and t h e entropy
p
e=e(x,t)
P = (-p + p ' v - v ) I + 2 ~ 0 ,
v.(KvB) +p'(V.V) 2 t 2 p D : D . v=v(x,t)=(v1,v2,v3) x=(x1,x2,x3),
f=f(x,t)
i s a v e c t o r o f ex-
i s t h e a b s o l u t e temperature.
S, t h e c o e f f i c i e n t s o f v i s c o s i t y
c i e n t of heat conduction
i s the velocity o f the
u and
p',
The pressure
are given functions o f the variables
K
p,
e
a/ax2, a/ax3); D/Dt = a / a t t ( v - 0 ) ; I
(=aS/ae) > O , 2 p t 3 p ' ~ O . v = (a/axl, i s t h e i d e n t i t y m a t r i x o f o r d e r 3;
D =D(v)
Djk = b(avj/axk+ avk/axj),
s a t i s f y i n g the conditions
11,
K,
S,
i s a m a t r i x w i t h elements 3
D:0 =
D
p,
and t h e c o e f f i -
j,k=1,2,3;
D
j,E=1 j k j k '
We s h a l l be concerned w i t h t h e f o l l o w i n g problem f o r t h e equatibns ( 1 ) . One phase free boundary prubZem : When t h e general f l u i d occupying a domain
n c R 3 a t t h e i n i t i a l t i m e f l o w s , i f t h e r e does n o t occur t h e phase t r a n s i t i o n , i n t o a f i e l d d e f i n e d by o u t e r f o r c e o u t e r heat c o n d u c t i v i t y
Ke(x,t)
f ( x , t ) , o u t e r pressure pe(x,t), 3 and o u t e r temperature ee(x,t) ( x € R ),
determine t h e subsequent m o t i o n o f i t . Denoting by
n(t)
t h a t t h e boundary o f r(t).
n(t)
consists o f the r i g i d p a r t
Then our problem i s made up o f f i n d i n g t h e domain
function
(p(x,t),
v(x,t),
e(x,t))
t h e boundary c o n d i t i o n s on
E
n(t)
and t h e
s a t i s f y i n g (1) i n J ~ ={ ( x , t ) ~ I ~ ~
x r n ( t ) , t e ( 0 , T ) l (T>O), the i n i t i a l condition
(3)
t, we suppose and t h e f r e e p a r t
t h e domain occupied by t h e f l u i d a t t i m e
z
v ( x , t ) = D (non-slip condition),
e=31(x,t),
Equations of Motion of General Fluids
and on
213
1
{r(t)ltE[O,T]
Pn = -p n, e
(4)
Kve-n = he(e
- ee(x,t))
and t h e e q u a t i o n
D
m F ( x , t ) = 0,
(5) where
n=n(x,t)
tion
F(x,t)
i s t h e u n i t i n n e r normal v e c t o r a t x ~ r ( t ) and t h e funci s d e f i n e d such as r ( t ) = I x e R3 I F ( x , t ) = 0).
Throughout t h i s paper we assume t h a t 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 s a r e v a l i d even i f t h e y a r e n o t w r i t t e n down e x p l i c i t l y . Our f i r s t r e s u l t i s t h e f o l l o w i n g .
Suppose ( i )r, I : E C ~ +( a~ e ( 0 , 2 ) ) , d i s ( r , O >0;(&) x c2+a(52) - I0 < % 5 po(x) S T , , 0 6 5 6 ( x ) 2 0 ); -0- 0 0
Theorem 1. ([4,5])
voJe,le clfa(3il
(Po,
x
cZia(~)
Then there e x i s t s a unique solution ( P, v, c2+a, l+a/2 2+a,l+rr/2 t o B1+a@T x,t @T')XC~,t
of ( 1 ) % ( 5 1 , which belongs
8)
nT,! (p*>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<) 0 0
- wI]
=0
tx
= 01 x
0,3
- wI
a d j u g a t e m a t r i x of &(xO,tO,w;i<)
o f det[@x
t3
5 ' = ( E ~ , $ ) E I R ~ and v € C 1
+ c,'),
1
there exists a positive
and
[O,T],
f;(j)'s
w i t h p o s i t i v e imaginary
parts. ( b ) garantces t h e p o s s i b i l i t y f o r t h e c o n s t r u c t i o n o f t h e r e g u l a r i z e r of (11.) i n t h e h a l f space R 3, , from which, t o g e t h e r w i t h t h e p a r t i t i o n o f u n i t y , f o l l o w s t h e ' s o l v a b i l i t y o f t h e a u x i l i a r y l i n e a r i z e d problem ( 1 1 ) : There e x i s t s a unique s o l u t i o n $ ~ C ~ + a y 1 c a / 2 ( ~ To )f ( 1 1 ) s a t i s f y i n g t h e O't0
\I$\] L2' 5 [C1(T,M1)
+ C2(T,M1)M2](T"*
+T1+a/2),
QT
where
C1
and
C2
increase monotonically i n
T
and
M1
and
C2+0
as
T+O. I f we choose t h e c o n s t a n t M2 and To i n such a way t h a t M2 > > C (T,M ) t M f o r any p o s i t i v e number M and f o r such M2, [C1(TO,M1) + 1 1 + M](T;/'t T01+a/2) 2 M1, C,(To,M1)M2 B M, t h e n 56% We denote To by 0 T f o r simplicity.
.
3".
Next we c o n s t r u c t t h e sequence
twn(xo,tO)l
of successive approximate
s o l u t i o n s as f o l l o w s :
i
wo(xO,tO) =
w ( x ,t ) n
o
OEG~ I
i s d e f i n e d as a s o l u t i o n
0
2" i m p l i e s t h a t
Then t h e r e s u l t i n
GT.
belong t o
wn (n=0,1,2,.-.)
A p p l y i n g t h e estimates i n
2"
W = W ~ -GT. ~E
a r e w e l l d e f i n e d and
t o t h e e q u a t i o n concerning
we o b t a i n
W,,-W~-~,
I I W ~- W n - l / l
(12) where
of ( 1 1 ) assuming
C3
w(xo,tO)
-+
0 as
T
+
0.
c ~ { T , M ~ * MIIWn-l ~) T h e r e f o r e t h e sequence
u n i f o r m l y i f we choose
T I E (O,T]
- w " - ~ I I QT ( 2+a 1
{wn(xo,tO)l so as t o s a t i s f y
converges t o C3(T',M1,M2)
<
1.
Atusi TAN1
216
Then
6=w4+e0,
v = w ' + v o (w'=(w1,w2,w3)),
i s o u r d e s i r e d s o l u t i o n o f (6)
Q,
~ ( x , , t o ) = p O ( x O ) exp[-
jioi d ~ ]
The uniqueness o f t h e s o l u t i o n o f (6)
(9).
*(9) f o l l o w s from t h e uniqueness o f t h e s o l u t i o n o f ( l o ) , which i s proved by t h e f a c t t h a t two s o l u t i o n s supposed t o e x i s t s a t i s f y t h e i n e q u a l i t y analogous t o (12).
4".
The unique s o l u t i o n o f t h e o r i g i n a l f r e e boundary problem (I)% ( 5 )
can be o b t a i n e d by t h e formulae (P(Xlt1, v(x,t),
e(x.t).
x n ( t ) ) =nx;;
The p o s i t i v i t y and boundedness o f
t %(X0,tO), and
p
i(xo,to),
8x0,to),
n).
e a r e obvious from o u r c o n s t r u c -
t i o n method. I n our approach, i t i s c r u c i a l t o check t h e complementing
Remark 4.
condition (b) stated i n 2".
The boundary c o n d i t i o n ( 3 ) imposed on
c can
be replaced by some s l i p - c o n d i t i o n f o r v e l o c i t y K ( P ~ ) - T = v . T (K=K(p,B) > O )
v.n=O,
and t h e c l a s s i c a l boundary c o n d i t i o n s f o r temperature ve.n = g ( x , t ) where
n=n(x)
and
T=T(X)
or
ve-n
- e,(x,t))
=0
a r e t h e u n i t i n n e r normal and t h e u n i t t a n -
X ~ Z ,r e s p e c t i v e l y
gential vectors a t
- K'(e
( c f . [7]).
I n t h e case
K r 0, t h e
problem i s s t i l l open. I t i s n a t u r a l and p l a u s i b l e , t o t h e m u l t i phase f r e e boundary Droblem. present author, t h a t t h e movement o f one f l u i d a c t s upon those o f o t h e r s ,
= 3.
SO t h a t
i n t h i s s e c t i o n we c o n s i d e r t h e r n u l t i phase f r e e boundary,problem
a r i s i n g from t h e movement o f a f i n i t e number o f nonmiscible general f l u i d s . Since t h e discussions on more than t h r e e such f l u i d s a r e t h e same as those f o r t h e two f l u i d s , we o n l y study t h e two phase f r e e boundary problem. L e t n [resp. a boundary
r
W]
be a bounded o r unbounded domain i n R
[resp. y];
t h e d i s t a n c e between
p o s i t i v e ; t h e e x t e r i o r boundary [resp. u ( t ) ] w [resp. O S u(t)
and
y
assumed t o be r i g i d .
[resp. n]
with
i s supposed t o be Denoting by u ( t )
t h e domain o f t h e f l u i d a t t i m e t which i n i t i a l l y occupies
, then
R - ~ ]
and t h e f u n c t i o n s
defined on
r is
r
3
u(t)
o u r problem c o n s i s t s o f f i n d i n g t h e domain (p,
v, e )
d e f i n e d on u ( t ) and
s a t i s f y i n g t h e equations
(p*,
v*,
w(t),
8*)
Equations of Motion of General Fluids
i
(13)
~[E-v D
P = [ - p + u ' ( V * ~ ) ] l +2uD,
=V.P+pf,
p0
s
[-] DDt
217
=v.(KVB) + p ' ( V . V ) 2 +2pD:D
aT= { ( x , t ) ~ ~ ~ l x ~ w t( et )(o,T)I, ,
in
[A]* ?* Dt
= -p*v.v*,
+ 2u*D*,
P* = [ - p * + u * ' ( v - v * ) ] I
p*f*,
,
D* = D(v*),
= v . ( ~ * v ~ * ) + ~ * ~ ( v ~ v * ) ~ + ~ ~ *D*, D*: in
J; = { ( x , t ) E R
4
I x € u ( t ) %t E (O,T)I,
the i n i t i a l conditions
t h e boundary c o n d i t i o n s
v = v*,
e = e*, (16)
Pn(x,t) = P* n ( x , t )
Kve.n(x,t)
v*=O,
n(x,t),
=K*ve*.
e*=el
,
on
rT,
and t h e e q u a t i o n
(17)
DDt F ( x , t )
on
yT.
[A]* & +
( t ) i s t h e boundary of u ( t ) , F(x,t) (v*-v), 3 y ( t ) = { x € R 1 F(x,t) = O I and n ( x , t ) i s a u n i t normal v e c t o r x ~ y ( t ) pointing i n t o the i n t e r i o r o f w(t).
Here i s such as at
= 0
=
The second theorem i s as follows. Theorem 2. ([61)
(ii) P (,;
(poJ
v;,
Suppose (i)
voJ eo) E C'+~(JI 0 );
E CI*(;)
x
x
r,
y e C B f a ( a € (0,1)1,
c Z i a ( i ) x c ' + ~ ( z ) (0
C2+a(;)
x
c2+a (a) -
(o<$,
-
disIr, y 1 > 0; 5 Po, o < e 5 e < ii j 4o= 0
,
o<~*~.o*
Atusi TAN1
218 (0 < p,
p*
<;=constant,
0 < 0,
e*
T h i s theorem can be proved by almost t h e same procedure as t h a t o f Theorem 1, so t h a t here we g i v e o n l y one remark.
As i n one phase f r e e boundary problem, we t r a n s f o r m (13)*(17)
by t h e
c h a r a c t e r i s t i c t r a n s f o r m a t i o n and w r i t e them i n a s h o r t e r f o r m ( c f . (10)) can be s o l v e d by t h e same formulae as t h a t i n 2, 1" [ b ( x o , t o ) , b*(xo,to) - wa = = ( x
1:
0
,t ,w;v)w+fi(xo,to,w)
i n QT=wx(O,T),
0 0
at0
i n q ? = u x (o,T),
(18) = +(XO,tO,W,W*)
W*
w * = (0,
a, A,
h1-
0;)
on YT
9
rT.
on
a r e t h e same as those i n (19) and &*, Here B, a r e obtained by r e p l a c i n g a l l t h e q u a n t i t i e s i n 01, dj, B,
B , B*, 4* 4 w i t h corre-
sponding t h e a s t e r i s k q u a n t i t i e s , As a l r e a d y s t a t e d i n 2,2",
i t i s essential
4
i s w e l l s e t o r n o t . Moreover i t i s yT 3 3 - s u f f i c i e n t t o examine i t i n t h e case of W C R , , U C R - , W A U = { X ~ , ~ = O } . t P u t t i n g W(xo,tO) = (w(xOptO),w*(x~,-x0,3,to)) ( x b = ( x ~ ,, x,,,)), ~ we c o n s i d e r whether t h e boundary c o n d i t i o n on
\
t h e f o l l o w i n g problem as t h e a u x i l i a r y l i n e a r i z e d one corresponding t o (11):
+ Q =G)w;sot , ,xia( 0
[i.e..
a*(Xb 3-xo , 3 , to0 ,w*;vl
we deduce t h e problem concerning w*
0
0
'V2,
i n R!
0
)
$+
-v3)
t o t h e one i n
W,]3
and we can a s c e r t a i n , a f t e r c o n s i d e r a b l e l e n g t h y c a l c u l a t i o n s , t h a t t h e complementing c o n d i t i o n f o r t h i s problem i s t r u e . we do as before.
Once t h i s is checked,
Eqiiatioiis of Motion of General Fluids
219
References
[I]
T. Beale,
The i n i t i a l v a l u e problem f o r t h e Navier-Stokes equations
w i t h a f r e e surface,
Comm. Pure Appl. Math., z(1981), 359-392.
A f r e e boundary problem f o r compressible
[2] P. Secchi and A. V a l l i ,
v i s c o u s f l u i d , L i b e r a Univ. Trento, UTM
[3]
B. A.
COJlOHHHKOB,
Pd3peUIHMOCTb 3 a H a Y I I 0 ZBIIXeHIIA BfI3KOR
H e C K U M a e M O f i KHJlKOCTli, M3B.
[4]
AH,
A. Tani,
cep. M a T . ,
100,1982.
41
OrpaHHVeHHOfi
(1977),
CBO60JlHOtl R O B e p X H O C T b M ,
1388-1424.
On t h e f i r s t i n i t i a l boundary v a l u e problem o f compressible
viscous f l u i d motion, Publ. RIMS, Kyoto Univ.,
[S] A. Tani,
f l u i d motion. J. Math. Kyoto Univ.,
[6] A. Tani,
9(1977), 193-253.
On t h e f r e e boundary v a l u e problem f o r compressible v i s c o u s
21 (19Sl), 639-859.
Two-phase f r e e boundary v a l u e problem f o r compressible
viscous f l u i d motion, t o appear i n J . Math. Kyoto Univ.
[7] A. Tan?, The i n i t i a l v a l u e problem f o r t h e equations o f t h e motion o f compressible viscous f l u i d w i t h some s l i p boundary c o n d i t i o n , i n p r e p a r a t i o n .
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Lecture Notes in Num. Appl. Anal., 6, 221-239 (1983) Recent Topics in Nonlinear PDE, Hiroshima, 1983
Scattering of Solutions of Nonlinear Klein-Gordon Equations in Higher Space Dimensions
Masayoshi TSUTSUMI' and Nakao HAYASHI* * *Department of Applied Physics, Waseda University Tokyo 160. Japan **Department of Mathematics, Waseda University Tokyo 160, Japan
1. Introduction.
The scattering theory for nonlinear Klein-Gordon equations has been developed by many authors (e.g. Segal [lo], Strauss 1121 -[151
,
Reed [ 9 1
)
.
In this paper our aim is
to extend recent results of Strauss [ 1 4 1 on low energy scattering.
In order to nention more precisely, let us write
the nonlinear Klein-Gordon equation (NLKG)
a2
u
-
A U
+ m 2u + l u ~ p - ~=u 0,
(KG)
a2
v
-
Av
+
x e m d , t s IR at together with the corresponding linear equation
atl
m 2v
=
0,
x
where m is a positive constant and
f
d
md, t s n , 3.
The problem con-
sidered here is to find conditions under which there exist (free) solutions u+
-
of (KG) and a (perturbed) soltuion u of
(NLKG) such that Ilu(t) where
11
- IL
with (KG):
,
- u+(t)lIe--+
-
o
as
t -2
m
,
denotes the energy norm naturally associated
Masayoshi TSUTSUMI and Nakao HAYASHI
222 ( 5
11
a
(w(t)';iiw(t))I\,)
Then the wave operators
a (U+(O), -u
-
:
at 2
.
R+ may be defined by
-
a
( u ( 0 ), p ( O ) ) ,
(0))-
and the scattering operator S
s
by
(U-(O), ;a i i U - ( O ) ) d
:
We say that solution
a
;iiU+(O))
.
u of (NLKG) (or of (KG)) is of finite
energy if
Itu(t)lle is finite for any
(u(t), &u
2 d t)) E H1 (Dd) x L (lR
x L 2 (IRd
(U+(O),
)
t, or equivalently, if The space HI (lRd
for any t.
)
is thus called the space of finite energy.
)
In [13]-[15] Strauss proved that the wave and scattering operators can be defined on a neighborhood at the origin in the space of finite energy provided that The upper bound 1
4 + d-l,
1
4
4
+ a 5 p 5 1 + d-l
.
however, does not seem optimal since
the corresponding results for the nonlinear Schrodinger equation iut + A u
-
=
0 hold
for
1
+
4 d -< p < 1
+
4
(see
[151). The nonlinear Schrodinger equation is the non-relativistic limit of (NLKG) (see [18], [lg]) and hence
we may expect
4 that the results for (NLKG) hold for the same range 1 t a
1
5p
+ d-2 . Our purpose is to show it:) 4
In general useful methods by which we attack nonlinear hyperbolic problem are energy estimates, Lp-Lq (decay) esti; mates for linear problem
and estimates of nonlinearity in
various function spaces (e.g., Sobolev spaces, Besov spaces). The methods employed in this paper are the same.
The difficul-
ty is the suitable choice of the spaces in which solutions of (IJLKG) lie.
t) In case d = 3, one of the authors has been investigated in t171.
223
Nonlinear Klein-Gordon Equations
2. Results.
'1
We begin by introducing function spaces: Let s c IR and d p < m . By H S r P ( I R ) we denote the usual Sobolev space
of fractional order s of all LP-functions on IRd such that
where
F and F-l are Fourier transform and inverse Fourier
transform, respectively, and
Ilflb We write and
=
(jlf(x) lPdx ) l/P
HSr2(7Rd) simply by HS(7Rd
for s > 0,
s = [s]
integer less than s and
+
o
).
Let
1
5 p,q
<
m
, where [sl denotes the largest d The Besov space BSrq(IR ) P d S ( 7 R ) of rapidly decreasing
0< u < 1.
is the completion of the space functions in the norm
where
u k ( x ) = u(x+k) (see [l] )
.
Our main results are as follows: Assume that
Theorem 1.
-&
4
1 + d-1 < p 5 2 and p2 = 2(d+l)/(d-1).
and
+
4 -<
for
d
1
d-1
p< 1
+
4 for 3 ~d
d-2
2 9. Let p1
=
28
(d+l)(p-1)/2
(a) For any free solution u- of finite energy there exists a unique solution u of (NLKG) on IR such that u
E
Lm (IR ; H1 ( IRd ) ) ( 7 Lpl ( (-m, TI
x
IRd ) n Lp2 (
(-ml
TI x
IRd
)
for any finite T, with
a
-UEL
at
m
2
(IR; L (IRd)),
and Ilu(t) (b)
- u-(t) -eI]
0
as a.e.t
+-m
.
If [[u-(t)l[e is sufficiently small, then
u
satisfies
Masayoshi TSUTSUMI and Nakao HAYASHI
224
u
E
Lpl (IR
x
IRd
solution u+
)
,-, Lp2 (IRt
x
md )
I
and there exists a unique
of (KG) such that
Ilu(t)
-
u+(t)
Assume that
Theorem 2 .
0
2
as a.e.t
<
Furthermore
.-+m.
4
p < 1 + d-2 for d = 9,10,
and (3d-1)(ail) - j { d + l ) (d-3) (d2-10d 2 (d-1) (d+l) Let 13 = p
-
4 (p-1) -(dtl)p 2 + 4(d+l)p
-
+
clI
E~
+
5 ) for d z l l .
being a suf-
(3d+7)
ficiently small positive number, and
s =
.
Zd(B-l)/(d+l) (p-B) (p-1)
Then the assertions (a), (b) in Theorem 1 hold valid. Further-
and
If we consider the Cauchy problem for (NLKG) with initial data U(X,O) = f(x),
U(X,O) = g(x),
XE
d IR
I
we obtain as a by-product of the proofs of Theorems 1,2 Theorem 3
Suppose that all the hypotheses of Theorem 1 (or
of Theorem 2 ) on p are satisfied. Then, for any initial data (f,g) E H1 (IRd 1 x L 2 (IRd ) the Cauchy problem for (NLKG) has a unique solution u such that u E Lm([O,T]; H1(IRd 1 ) n LP1([O,T]
x
IRd
) ,q
with
a
-UE
at
m
L ([O,T]; L2(IRd)),
for any T > 0
.
Lp2f[0,T] X lRd)
6
Nonlinear Klein-Gordon Equations
Remark 1.
When the initial data are assumed to be only of
finite energy, the question of uniqueness of weak solutions to the Cauchy problem for (NLKG) has not been completely solved. (see [ 4 1 ,
[ 81).
ness question:
Theorem 3 gives a new result on the unique-
for 3
5 d 5 10, solutions to the Cauchy problem
for (NLKG) are uniquely determined by their initial data of 4
Remark 2 .
.
1 5 p < 1 + d-2
finite energy provided that
When solutions are radial, Theorem 1 holds for
.
4 < +d-1
4 p < 1 + d-2 In order to prove this assertion, we may use the same device
all d 2 3
under the assumption 1
as that in [ 4 1 .
3 . Proofs
We start with Lp-Lp' estimates of solutions of (KG), which were obtained by Brenner
[ 2
I (see also
be the solution of (KG) with Cauchy data a t V(x,o) = g(x), X E Rd . Then v(t) = a
[
5 1 :
,
[
7I 1.
v(x,O)
!&E(t)f
=
Let v
f (x);
+ E(t)g
where
- sin ( t J \ 5 1 2 +
E(t) = F - l ,/1512 Lemma 1.
Let
+
m2 ) F
.
2
u > 0, 1 < p' < 2,
1 + -, 1 P
=
1.
Then
where
Here and in the sequel the letter c stands for various positive
225
Masayoshi TSUTSUMI and Nakao HAYASHI
226
constants depending only on known quantities. If necessary, we indicate dependence of constants on quantities a , B I
* - -
by
c ( a ,8 , * * . I . The following lemma was obtained by Strichartz [16], and is useful for the study: Lemma 2.
1 d Assume that (f,g) B H ( W
provided that
4
+ d- 5
2
p 5 2
) x
2 d L (IR
Then
).
+ d-2 -. 6
An extension of Lemma 1 has been established by Marshall [ 6 1 : Lemma 3 .
Let
1 1 q’r
r = { If
1 1 -)f q’r
(-
r
be the triangle (--)
and
: o ( - <1 - , -1 - - <12 - < (d-l)r - - 2- } . r - 2 s <
1
+
1
-
dt)”‘
1 ?,
1 r
( -
1 + -, r
1 q
1 2
2 dr
= l), then
< c I)(f,g)/Ie .
The next two lemmas concern Besov space estimates of nonlinearity and is essentially due to Brenner-von Wahl [ 31. We omit the proofs.
.
4 4 Lemma 4 . Let 1 + < p < 1 + d-2 Let p1 = (d+l) (p-l)/2, d-1 1 + -, 1 = 1, p = 2{d(p-1) - (p+3)I/{d(p-l)+(p-5) 1 and 0 < ol< p p1 p1 1 d d Then for every u E H (IR n Lpl(W
.
provided that Lemma 5 . Theorem 2 .
Let
q 2 l/p
d 2 9.
. Assume that p satisfies the condition in
Let 6 be the same as in Theorem 2,
Nonlinear Klein-Gordon Equations
where
E
is a positive number so small that
that
q
1. l/p
.
(Note that
0< u2< p
227
u2
+
E
C
P , provided
p - p > 1 under the
assumptions in Lemma 5).
In order to prove our theorems we study the property of the o:,erator Pv :
Lemma 6.
Suppose that all the hypotheses of Theorem 1 on p
holds valid.
Let
in IR such that Lpl(I xIRd
)
and
V O I
we have
< -
where
I
C
be any (not necessarily bounded) interval
.
Then, for any
U E L ~ ( I R ; H ~ ( I R ~ (1 ))
Masayoshi TSUTSUMI and Nakao HAYASHI
228
since C
H
(d-1)(pl-2)/2pl
arpi
Choose
d (IR
u
for any
)
a
>
0 and
on
I t-T 1
Since B ~ : ~ , ~ ( w ) ~ P1
2 1.
12q
< m
,
we get
so that u +
(0)
E
2
€
P,
<
then, making use of Lemma 4 , we have
By the well-known singular integral inequality
we obtain
where
Then, from (9) we have u =
(d-l)p-d-3} (dp-d-4) (d-1)t (d+l)p-d-5 1
'
It is easily seen that under the hypotheses of the lemma (5
defined by (11) satisfies ( 7 ) ,
we obtain the assertion. Lemma 7. satisfied.
(8)
and (10)
. Therefore Q.E.D.
Suppose that all the hypotheses of Lemma 5 are Let
s =
2d(B-l)/(d+l) (p-B) (p-1) and I be any
Nonlinear Kiein-Gordon Equations
229
(not necessarily bounded) interval in W , such that V E I. Then for any u E Lm(IR;H1 ( W d ) ) 0 L"(Ix IRd) we have
The proof of Lemma 7 can be done by exact1 manner as in the proof of Lemma 6
the same
if we use Lemma 5 instead
we get Corollary of Lemma 7. Under the same assumptions of Lemma 7 , we have
Lemma 8.
Suppose that all the hypotheses of Theorem 1 (or
Theorem 2 ) on p are satisfied. Then, for every u,v
n Lp2 (I x Wd
E
Lpl(I x W d )
we have
By virtue of Lemma 1, we have
Prqof.
p2
1
- -d-1
t
<
C
It-TI
d+l
V
where
l/p2 + 1/p;
=
II
1
lar integral inequality
the singu-
Masayoshi TSUTSUMI and Nakao HAYASHI
230
Q.E.D. We now proceed the proofs of our results. integral equation of the form (IE)" u(t) = zE(t)fd + E(t)g- +
t
Consider the
E(t-r)lUfT)[P'lUfrfdT
V
= As
u-(t) + (P"U) (t)
.
usual we regularize the problem in such a way that we obtain
global existence for the smooth solution u .(t) of the regularized m 1 problem. Let { h . } j=l be a sequence of Cm- functions on IR 1
such that
h,(O)
=
0,
c being a positive constant independent of j,
I
U
(16)
H.(u) 3
Let
hj (u)
(fjJ), g:j)
h.(s)ds 2
0
and (17)
=
E
j
'Ue
IR,
lulp-lu pointwisc as
1
$(Ed
(j)I g-(1)) (fas
0,
1
x
Ci(lRd 1
1-
.
+-
such that
(f-,g-) strongly in H 1 ( Rd 1
i
X
L2 (IRd 1
It has been established in [ 3 1 that there ex-
-*+a.
ists a unique solution u.(t) of the regularized equation 3 t (1) + E(t-T)h.(u(T))dT -E(t)f<j) + E(t)g(IE);, u(t) = d dt 3 v
u
satisfying solution u
j
u<j) (t) +
E C1 (IR j satisfies
;
(P", ju' (t),
Hk (Wd
))
for some large k
. The
Nonlinc-
. Klein-Gordon
Equations
a2 u . - A u . + m 2u + h.(u.) = -
(18)
at2
3
j
3
1
0,
1
XE
23 I
IRd ,
t e IR,
with
From Lemma 2 we see that for any interval I C n , (t) - u-
where
Q,
+0
as j
j
+
m
(i = i,2)
.
We have
where M
=
M(r) is a continuous nondecreasing function on [ 0 , m )
such that M ( 0 )
=
0.
Moreover, for any compact interval I on IR
we have
If v is sufficiently near to
for any T 2 Proof.
v
,
- m ,
we have
where constants c does not depend on j and v
The first two assertions (20), (21) come from the well-
.
Masayoshi TSUTSUMI and Nakao HAYASHI
232
-known energy identity. We now prove (22). Take I so that v E I. Making use of Lemma 6 (or Corollary of Lemma 7 ) , we have
If we choose the length of I so small and j , so large that the
has a positive root, then we obtain (25)
(1
I
where
l/P1 IIuj(t)llP1 dt) P1
<
-
for any
Yo
j
2 j,
I
Yo denotes the least positive root of (24). Lemma 8
yields
We can assume that c YE-'
< 1/2 -
.
sufficiently small I which contains for some T
Hence we have (22) for v
.
The assertion (23)
1. v may be established in the same manner. Let
vl" iR be fixed. The solution u.(t) satisfies I
Nonlinear Klein-Gordon Equations
233
E ( t - r ) h . ( u . (.r))dT 3
3
A
Lemma 2 g i v e s
(i = 1 ' 2 ) .
+ rij)
< c M(llff-,g-)lle
Hence, i n much t h e same way a s above, w e see t h a t t h e r e e x i s t s a i n t e r v a l Il ( u l
E
11) on which
where c i s a p o s i t i v e c o n s t a n t independent o f j . t h e l e n g t h of I1 b u t n o t on v1 i t s e l f . f o r any compact i n t e r v a l I
c depends on
Thus w e conclude t h a t
and any T, ( 2 2 ) and ( 2 3 ) h o l d v a l i d ,
respectively.
Q E.D.
From Lemma 9 w e see t h a t t h e r e e x i s t a subsequence of {u.} 3 ( a l s o denoted by {u.) ) and a f u n c t i o n U E L m ( l R ;H1 ( Rd ) ) n 3
~ ~ L P 2 ( I x I Rw d i)t h
LP1(IxlRd)
that
u . --ju 3
Lpl(I x
md
)
weakly s t a r i n L m ( m ;H1(lR and i n Lp2 ( I x IRd )
s t a r i n L m ( l R ; L2 (IRd ) )
If
s u c L m ( R ; L 2 ( X I d ) ) such
.
with
d
)
1 and weakly i n
ap j d@ d I
weakly
Moreover u s a t i s f i e s t h e e s t i m a t e s
v is s u f f i c i e n t l y near t o
- O D ,
w e have
Masayoshi TSUTSUMI and Nakao HAYASHI
234
(29)
Note that the constant c does not depend on
v
,
To show that u solves (NLKG) in the sense of distribution is accomplished by the standard manner (see Reed [ 8 1 )
.
We
have < u , $ > =
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,
<
vn
f o r any T ( 1. vn )
s u f f i c i e n t l y near t o
I
where - m
,
p" = ( p - l ) / p l
w e c a n assume
.
If w e t a k e T
Nonlinear Klein-Gordon Equations
237
Hence
which tends to zero as
- m because of (32)
vn-
from ( 2 6 ) - ( 2 9 ) with replacing exists a function u Lp2 ( (--,TI
as
x lRd )
vn+
-00
,
with
Lm(lR
&u
more
u
d
u
by
uv n
, we
Hence,
see that there
: H1(md 1 ) p, L p l ( (--,TIx ztd L a ( = : L2 (lRd
E
1)
)
and strongly in Lp2
weakly star in
n
such that
1 u + u weakly star in Lm ( R: H ( R d ) n'
weakly in Lpl ( (--,TI x lRd
d u> dt vn
u
.
(
(-a
L m ( W : L2 (Rd ) )
,TI
.
x
)
,
lRd )
:
Further-
satisfies (NLKG) in the sense of distributions and is
the unique solution of the integral equation t E(t-r) lu(-r)lP-'u(T)d-r (IE)u(t) = u-(t) +
j
-m
in (-m,T].
Indeed we have
which tends to zero as v
.
j- m .
Masayoshi TSUTSUMI and Nakao HAYASHI
238
It can be easily shown by the same argument as in Strauss [15] (see also Tsutsumi[l71) that Ilu(t) t*-m.
- u-(t)lle
0
as
u is a desired unique solution of (NLKG).
Thus
The assertion b) may be established by the same argument as in [171. The last assertion of Theorem 2 follows from Lemma 3, Lemma 7 and (29) Theorems 1,2
(
or (30)
This completes the proofs of
).
. References
1. Berq,J. Lofstrbm, J.
:
Interpolation Spaces, Berlin-
Hedelberq-New York, Springer
1976
.
2. Brenner, P. : On the existence of global smooth solutions of certain semi-linear hyperbolic equations, Math
2.
167
(1979) 99-135. 3. Brenner, P., von Wahl, W.
:
Global classical solutions of
nonlinear wave equations, Math.
2.
176
(1981), 87-121.
4. Glassey, R.T., Tsutsumi, M. : On uniqueness of weak solutions to semilinear wave equations, Comm. in Partial Differential Equations
1
(1982), 153-195.
5. Marshall, B. Strauss,Wi and Wainger, S.
:
Lp-Lq estimates
for the Klein-Gordon equation, J. Math. Pures Appl. (1980)I 417-440 6. Marshall, B.
:
59
. Mixed norm estimates for the Klein-Gordon
equation (1981) preprint. 7. Pecher, H.
:
Ein nichtlinearer Interpolationssatz und
seine Anwendunq auf nichtlineare Wellengleichungen, Math. 2. 161
_ .
8.
(1978), 9-40.
Reed, M.
:
Abstract Nonlinear Wave Equations, Lecture
Notes in Mathematics, No. 507, Springer-Verlag, BerlinHeidelberg-New York, 1976.
Nonlinear Klein-Gordon Equations
9.
239
Segal, I.E. : The global Cauchy problem for a relativistic scalar field with power interaction, Bull. SOC.
91
Math. France
.
10.
(1963), 129-135.
Dispersion for nonlinear relativistic
:
equations, 11, Ann. Sci. Ecole Norm. Sup. (4) I (1968) 459-497. 11.
Strauss, W. A. : On weak solutions of semi-linear hyperbolic equations, Anais. Acad. Brazil, 42 -
Ciencias
(19701, 645-651.
.
12.
:
Nonlinear invariant wave equations. In:
Lecture Motes in Physics, vol 73, 197-249 (1977) Springer-Verlag
. .
13.
:
Everywhere defined wave operators.
In:
Nonlinear Evolution Equations, 85-102, Academic Press, New York, 1978.
.
14.
:
Nonlinear scattering theory at low
energy, J. Functional Analysis
.
15.
:
5
(1981) 110-133.
Nonlinear scattering theory at low
energy, Sequel, J. Functional Analysis 16.
43
(1981) 281-293.
Strichartz, R. S. : Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J.
17.
Tsutsumi, M.
:
44 (1977), 705-714.
Scattering of solutions of nonlinear Klein-
-Gordon equations in three space dimensions. in J. Math. SOC. Japan) 18.
-.
:
(to appear
.
Non-relativistic approximation of nonlinear
Klein-Gordon equations in two space dimensions. (to appear in Nonlinear Anal.) 19.
Hayashi, I$.,
Tsutsumi, M.
:
Nan-relativistic approxi-
mation of non-linear Klein-Gordon equation,
preprint.
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