RECENT TOPICS IN NONLINEAR PDE IV
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NORTH-HOLLAND
MATHEMATICS STUDIES
160
~-~~~
Lecture Notes in Numerical and Applied Analysis Vol. 10 General Editors: H. Fujita (Meiji University) and M. Yamaguti (Ryukoku University)
Recent Topics in Nonlinear PDE IV
Edited by
MASAYASU MIMURA (Hiroshima University) TAKAAKI NlSHlDA (Kyoto University)
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Lecture Notes in Numerical and Applied Analysis Vol. 10 General Editors H. Fujita Meiji University
M. Yarnaguti
Ryukoku University
mtorial Board
H. Fujii, Kyoto Sangyo University M. Mimura, Hiroshima University T. Miyoshi, Yarnaguchi University M. Mori, The University of Tsukuba T. Nishida, Kyoto University T. Taguti, Konan University S. Ukai, Osaka City University T. Ushijirna, The University of Electro-Communications
PRINTED IN JAPAN
PREFACE
The f i f t h Meeting on Nonlinear Partial Differential
Equations (PDEs) was held a t Research I n s t i t u t e of Mathematical I n s t i t u t e , Kyoto University from January 6 t o January 9, 1988. The topics f o r the meeting was the theory and applications of nonlinear PDEs i n mathematical physics, reaction-diffusion theory, biomathematics and i n other applied sciences.
There
were 18 speakers who gave outstanding presentations on recent
bvorks i n analysis, computational analysis of nonlinear PDEs and their applications.
This i s the volume of the proceedings
of this meeting.
We express our gratitude t o the contributors of t h i s meeting:' t h e i r presence made the meeting so successfull.
Mas aya s u M i mu r a Takaaki Nishida
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CONTENTS
Preface.. ...............................................................................
v
Yoshikazu GIGA: A Local Characterization of Blowup Points of Semilinear Heat Equations.. ...............................................
1
Shuichi KAWASHIMA and Yasushi SHIZUTA: The Navier-Stockes Equation Associated with the Discrete Boltzmann Equation ...... 15 Shigeo KIDA, Michio YAMADA and Kohji OHKITANI: Route to Chaos in a Navier-Stokes Flow ............................................ 31 Akitaka MATSUMURA and Takaaki NISHIDA: Periodic Solutions of a Viscous Gas Equation.. ................................................ 49 Takeyuki NAGASAWA: On the One-dimensional Free Boundary Problem for the Heat-conductive Compressible Viscous Gas.. .... 83 Mitsuhiro T. NAKAO: A Computational Verification Method of Existence of Solutions for Nonlinear Elliptic Equations.. ........... 101 Hisashi OKAMOTO: Degenerate Bifurcations in the Taylor-Couette Problem ......................................................................... 121 Shigeru SAKAGUCHI: Uniqueness of Critical Point of the Solution to the Prescribed Constant Mean Curvature Equation Over Convex Domain in R 2 . ....................................................... 129 Takashi SUZUKI: Symmetric Domains and Elliptic Equations ......... 153 Seiji UKAI: On the Cauchy Problem for the KP Equation................ 179 Atsushi YOSHIKAWA: Weak Asymptotic Solutions to Hyperbolic 195 Systems of Conservation Laws ............................................ Nobuyuki KENMOCHI and Irena PAWLOW: The Vanishing Viscosity Method in Two-Phase Stefan Problems with Nonlinear Flux Conditions...................................................................... 211
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Lecture Notes in Num. Appl. Anal., 10, 1-14 (1989) Recent Topics in Nonlinear PDE lV, Kyoto, 1988
A Local Characterization of Blowup Points of
Semilinear Heat Equations Dedicated to Professor Tosihusa Kimura on his sixtieth birthday Yoshikazu Giga Department of Mathematics Hokkaido University Sapporo 060, JAPAN 1. Introduction This note is essentially based on my work with Kohn [11.121. Here we apply our method to obtain a local version of our results in [11.121 so that we explain a crucial idea of our methods
.
We are concerned with the blowup of solutions of semilinear heat equation (1.1) ut where
D
-
AU
- IUIP-' u
=
o
in
D x (0.~1
Rn and p>l. There are many
is a domain in
examples where blowup occurs spatially inhomogeneously [l-4,6-8,11,12.14,15].A point non-blowup point if
u
a E D
is called g
is locally bounded in a
(parabolic) neighborhood of (a,T). Otherwise a is called a blowup point. Our goal is to distinguish blowup points and non-blowup points by the asymptotic behavior of the solution
u
as
t
-+
T. This problem is studied in 1
2
Yoshikazu GIGA
[11.12] when the Dirichlet boundary condition
is
u=O
imposed. Their result reads: Thorem 1.
Suppose that
or D=Rn. Suppose that solution
of
D
convex with
on
u=O
aR
u
gnJ
(1.1) on D x (0,T’) for every
lim (T-t)’ u(a+y(T-t)‘I2 , t) t+T implies that
a E D
boundary
C2
8
= 0.
=
bounded
T’
a non-blowup point provided
(1.2) 1 < p < (n+2)/(n-2)
Then
that
n i 2.
~f
In this note we prove a local version of this result assuming an upper bound on the rate of blowup. denote the open ball of radius u
Main Theorem.
6>0.
Bb(a)
Rn centered at a.
of
(1.1)
on
Q8(a,T)
Assume that
that
(1.4) lim(T-t)’ u(a+y(T-t)1’2. t-+T
equals zero and that for each uniform
in
g solution
=B8(a)x(T-h2, T) for some
Suppose
8
Let
for
lylic.
Then
a
t)
00,
the
convergence
is a non-blowup point.
Since the converse i s trivial, this characterizes non-blowup points. p . 2 9 8 1 so the limit
This answers the conjecture in [ 9 , (1.4) equals
*B8
provided that
a
Blowup Points of Semilinear Heat Equations
is a blowup point and that p p=(n+2)/(n-2).
3
satisfies (1.2) or
Compared with Theorem 1, there is an
advantage in the Main Theorem, since it is a local result
so it does not matter what boundary conditions are imposed. Unfortunately the upper bound (1.3)is only proved by imposing boundary conditions and restrictions on initial data
uo
or
p.
For example under the assumption of
Theorem 1 with (1.2) the estimate (1.3) holds for uorO [lo]. When
D
is bounded, the restriction (1.2) can be removed
if we assume Auo+u0
0
2
in addition [ 6 1 ; see also [3,7]
for the Neumann boundary condition. So even when
p
violates (1.21, there is the situation where the Main Theorem is applicable. The proof of the Main Theorem is similar to that of Theorem 1. The first ingredient in the proofs of both Theorems is to show that that
lu(x.t) I(T-t)’
a
is a non-blowup point provided
is small uniformly near
a.
The
second ingredient is to show that the limit (1.4)=0 leads to the uniform
smallness of
lu(x,t) I (T-t)’
near
a.
For
this we use similarity variables as in [9-121 and prove various a priori estimates for rescaled functions via energy relations. Since there are no boundary conditions in the Main Theorem, the proof of this step differs from that of Theorem 1 although the flavor is the same. We give below, mainly, the proof of the second step of the Main Theorem since the first step 1s a local result and proved in [11,12]. The upper bound (1.3) avoids some
Yoshikazu GIGA
4
technical difficulties, so the proof is simpler than that of Theorem 1. The assertion in the Main Theorem holds for more general equations f(u)-lulP-lu
ut-Au-f(u)=O
whenever
grows at most as 1 u 1 9 for some
q
However, we only write the proof for (1.1) to clarify the main idea.
2. Rescaled functions and energy identities
To study
u
and suppose that
8 Since
u
=
=
a point (a,T) we recall rescaled
For simplicity we take (a.T) = ( O , O ) .
functions [lo].
(2.1) w(y,s)
near
u solves (1.1) on (-t) 8 u(x,t), x
Q,=Q,(O,O).
(-t)ll2y. -t
=
b=1
We set e- S ,
=
l/(p-l). solves (1.11, w
(2.2) pws - v.(pVw)
+
solves
8pw - plwlP-'w
=
0, p(y)
=
e-lYlZ/4
on
w1
=
t(y.s); lyt<es/2,
S>O).
We shall derive integral identities by multiplying (2.2) with
w
ws and integrating by parts. They involve
or
"the energy
"
on a ball of radius R
Blowup Points of Semilinear Heat Equations
Proposition 2.1. Suppose that (2.4) M
=
w
solves ( 2 . 2 )
5
on
W1 with
sup Iw(y,s)l < w1
Then w
for
Here
s>o.
l&,(s)l
(2.7) with
satisfies
C
S
2 C exp(-s / 8 ) , s
depending only
on
where
do
Bs
1, i
gnJ
M. p
Proof. Multiplying (2.2) with parts over
2
w
=
1,2,
n.
and integrating by
yields
i s the surface area element. Using (2.3)and
we now obtain (2.5) with
Yoshikazu GIGA
6
Similarly, multiplying (2.2) with Bs
parts over E ~ ( s )=
ws
and integrating by
yields (2.6) with
[ ( 3 ( I ~ w l ~ + B l2w)l -
Iwlp+'+w
p+1
aBS
@)pdu s ar
(cf.[9, p.3121). It remains to prove ( 2 . 7 ) .
Since (1.1) is parabolic,
( 2 . 4 ) implies
(2.8)
l m v l i M',
(2.9)
Wr
with some
=
Iw,~
M' depending only on
If
(2.10) t(y,s); l y i
on
(1=1,2).
r>e-1'2,
M,n,r and
p [9,
we see easily
< s . s L 1) c
Since p=exp(-lyl2 / 4 ) , &i
M'(l+lyl)
{(Y,s); lyl < resi2, s > -2Qn r}, r < 1
Proposition 1'1.
of
i
wr
( 2 . 7 ) now follows from definitions
0
Remark. The Identity (2.6) is found in [9, p.3121. (2.5)
a global version is proved first in [lo, p.81.
For The
proof is almost the same as [9,101 although (2.5) is not explicitly written in the literature. We just reproduce the proof for the reader's convenience.
7
Blowup Points of Semilinear Heat Equations 3.
A small integral bound on rescaled functions We shall prove that integrals of w
energy
Es[w](s)
Theorem 3.1.
is small when
Suppose that w
upper bound (2.4). For every s+=s.(&.p,n,M)tl
and C
lB
1
Iwl’pdy
5
is sufficiently large.
s
solves (2.2) on W1 with an E > 0 , there are constants
C(p,n)
such that
implles
E(sl):= Es [w](sl) < E (3.1)
=
is small if the
Gal" for
s L sl,
S
where
s1
is an arbitrary number with
sits+.
We begin by two lemmas on differential inequalities.
that
z(s)
and
E(s) are real-valued C
functions on an interval
[so.
w).
Assume that
Lemma 3.2.
(3.2)
Suppose
dz/ds L - c l E ( s )
(3.3) dE/ds i b 2 ( s ) ,
with some
c2ze - & , ( s )
s 2 so
Q>l. ci>O, giL0 ( 1 = 1 , 2 )
Then Qim E ( s 1 s+w
+
z(s)>O
exists and is nonnegative.
Proof. By (3.3) and (3.4)we see a=Qirns+=E(s) is
1
and
Yoshikazu GIGA
8
either
-m
or finite.
(including - - I .
Then
since (3.4) holds.
s
The inequality (3.2) now implies that finite time.
Suppose that
y(s)
function on an interval [ s o , - 1 .
l$I2ds i A , S
for
A i
z would blow up in
This leads to a contradiction, so
Lemma3.3 (t121).
(3.6)
were negative
-c1E(s)-hl(s) would be greater
-a/2 for sufficiently large
than
u
Suppose that
i
a L 0.
is a nonnegative
C1
Assume that
dy/ds
=
0
1
with positive constants
Then there is a constant
ci(i=1,2)
depending only
C
on
p > 1.
ci
such that --
P r o o f . By ( 3 . 5 ) we see
either for
y(s) i A"
or
clyp i
i
+
c2A1-a
a > O . This leads to
Applying ( 3 . 6 ) and taking a=1/2p yield fs+1y2p(t)dr < CA with
C
independent of
A.y and
s.
Applying this with
p
Blowup Points of Semilinear Heat Equations
9
(3.6) to an interpolation inequality
9 = l/(p+l)
yields (3.7) since A i 1.
0
Proof of Theorem 3.1. Using Holder's inequality and Jpdy <
with
m,
we see
independent of
c'
w
yield (3.2) and (3.3)with
and
s.
So (2.5) and (2.6)
E(s)=Es[wI(s),
Applying Lemma 3.2 with (2.7) yields
Integrating (2.6)over (sl,-) now yields 0
(3.9)
s
Iw,~
s1 Bs
m
2 pdy ds, i E(sl)+S E2(s)ds. S 1
The identity (2.6) also deduces
Applying (3.8) to (2.5) yields
and s =l. 0
Yoshikazu GIGA
10
(3.11) clyPfl - yi i 4E(s)
with
cl>O
for every E(sl)<&
+
depending only on 8 > 0
there i s
s,,
p =
y(s)
and
n.
=
z(s)'/'.
s
2 1
By (2.7) we see
s,,(&,p,n,M)>l such that
with s12s,, implies that the right hand sides of
(3.9)-(3.11) is dominated by sLsl>s,,.
for all
28
Estimates ( 3 9 ) and (3.11) now yield (3.5) and
(3.6) with
and s 0 = s l . Applying Lemma 3.3, we
c2=l
A=2E,
now obtain (3.1) from 4.
E1(s).
3.7).
Proof of the Main Theorem We may assume
instead of Let
and
T=O by a translation of
We may also assume
coordinates. v(x,t)
a=O
6=2 by considering
(2/&)26u(2x/6,4t/62)
=
u.
w
be the rescaled function around
0
defined
by (2.1). The assumption that the limit (1.4) equals zero i s equivalent to (4.1)
and the convergence i s assumed uniform for lylrc
This
lmplies that Qim Es[wl(s)
(4.2)
=
0.
s+m
where
Es[w](s)
is the energy defined by (2.3). Indeed,
11
Blowup Points of Semilinear Heat Equations
since (2.2) is parabolic, (4.1) implies
aim mv(y,s) S+=
=
o
uniformly for IylSc (cf. [131). Since (1.3) implies (2.4), we have a bound (2.8) for
Vw
on Wr defined by (2.9).
Applying Lebesgue’s convergence theorem now yields (4.2), since Spdy
is finite and (2.10) holds.
The crucial step is to obtain the uniform smallness of
lu(x,t)l(-t)’
near ( 0 , O ) ; this is not trivial since
this does not follow directly from (4.1). We shall prove that for every small ID0
on some (parabolic) neighborhood of (0.0). Let a rescaled function around w6(y,s)
(4.4)
This
wG
=
w b be
&,lbl
(-t) 6 u(x,t). ~=(-t)’/~y+6. -t=e- S
solves ( 2 . 2 ) on
W1
since
161s1
.
and 6=2.
Since (1.3) Implies (2.41, we apply Theorem 3.1. Estimates (3.1) and the Gagliardo-Nirenberg inequality [ 5 , p.271
l/n yields
=
u/2 + (l-u)/q, 1 > u > 0 ,
12
Yoshikazu GIGA
with
?I(&)
provided that
Es
continuous in
sits*
1
[w61(s1)<&
~)(E)JO
for some
as
6
(cf. Lemma 2.3 in
and a ball
Bp(0)
[lo]),
such that
& +
0,
s12sI where
6. Since it is easy to see
independent of
is
6 and
independent of
is
sc
Es[w6](s)
is
by (4.2) there
Es [w6l(s1)<~ for
1
6€Bp(0). We now conclude from (4.5) that
This leads to (4.3)since
w6 is determined by (4.4).
The uniform small bound (4.3) implies that (0.0) is a non-blow up point.
This follows from a following local
bound on the browup rate so the proof of the Main theorem
is
ROW
complete.
Lemma 4.1.
Let
u
solve
with
for some l
g@
K>O.
There is a constant y=y(K.p,n)
Blowup Points of Semilinear Heat Equations
13
implies that ( 0 , O ) is a non-blowup point. The proof is found in [11,121 so is omitted. References 1. L.A. Caffarelli and A. Friedman, Blow-up of solutions
of nonlinear heat equations, J. Math. Anal. Appl., to appear. 2.
X-Y. Chen and H. Matano. Convergence, asymptotic periodicity and finite-point blow-up in one-dimensional semilinear heat equations, preprint.
3.
Y.-G. Chen, On blow-up solutions of semilinear parabolic equatlons, Doctor's dissertation, University of Tokyo, 1988.
4.
Y.-G. Chen and T. Suzuki, Single-point blow-up for semilinear heat equations in a non-radial domain, preprint .
5.
A.
Frhdman, Partial Differential Equations, Holt,
Rinehart and Winston. New York, 1969. 6.
A. Friedman amd B. Mcleod, Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34(1985), 425-447.
7. H . Fujita and Y.-G. Chen, On the set of blow-up points
and asymptotic behaviours of blow-up solutions to a semilinear parabolic equation, to appear in Analyse Mathhatique et Applications. published by Gauthier Villars. Paris 1988. 8.
V.A. Galaktionov and S.A. Posashkov, The equations
Yoshikazu GIGA
14
ut
=
uxx +u6 .
Localization and asymptotic behavior of
unbounded solutions, preprint No. 97, Keldysh Institute of Applied Mathematics, 1985(in Russian). See also Diff. Uraven. 22 (1986). 1165-1173. 9.
Y.Giga and R.Kohn, Asymptotically self-similar blowup of semilinear heat equations, Corn. Pure Appl. Math. 38 (1985). 297-319.
10.
Y.Giga and R.Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987). 1-40.
11.
Y.Giga and R.Kohn. Removability of blowup points for semilinear heat equations, Proc. of EQUADIFF87, to appear.
12.
Y.Giga and R.Kohn, Nondegeneracy of blowup for semilinear heat equations. preprint.
13.
0.A.Ladyzenskaja. V.A.Solonnikov and N.N. Ural'ceva,
-Linear and
Quasilinear Equations
of Parabolic
Type,
Amer. Math. S O ~ . ,Providence, 1968. 14.
C.E.Mueller and F.B. Weissler. Single point blowup for a general semilinear heat equation, Indiana Univ. Math.J. 34(1985), 881-913.
15.
F.B.Weissler, Single point blowup of semilinear initial value problems, J.Differentia1 Equations 55(1984), 204-224
Lecture Notes in Num. Appl. Anal., 10, 15-30 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
The Navier-Stokes Equation Associated with the Discrete Boltzmann Equation
S h u i c h i KAWASHIMA
*
*
and
**
D e p a r t m e n t of A p p l i e d S c i e n c e F a c u l t y o f E n g i n n e r i n g 36 Kyushu U n i v e r s i t y Fukuoka 812 , J a p a n
1.
**
Yasu s h i SHIZUTA
D e p a r t m e n t of M a t h e m a t i c s Nara Wanen's U n i v e r s i t y Nara 6 3 0 , J a p a n
Introduction T h i s i s a summary o f o u r r e c e n t work [6] c o n c e r i n g t h e
N a v i e r - S t o k e s e q u a t i o n d e r i v e d from t h e d i s c r e t e Boltzmann e q u a t i o n . W e c o n s i d e r a m o d e l of g a s whose m o l e c u l a r v e l o c i t i e s a r e
r e s t r i c t e d t o a set of
m
constant vectors
v 1 , ~ - * , v i n IR". rn The d i s c r e t e Boltzrnann e q u a t i o n d e s c r i b i n g t h e e v o l u t i o n of t h e g a s is w r i t t e n i n t h e form aFi
at +
(1.1)
Here
vi-v x F i =
Fi = F i ( t , x ) (1 5 i
m
z
j ,k,k=l 5
m)
(A~-F~F,
-
l s i ~ m .
i s t h e d e n s i t y d i s t r i h t i o n of
g a s m o l e c u l e s w i t h t h e v e l o c i t y vi a t t i m e t t 0 and p o s i t i o n .. n x c R The c o e f f i c i e n t A ;: i s a positive constant i f t h e
.
expression
(vklvll) + (vi,v.)
sion; otherwise,
..
Ail
A1' = Aij = Aji Ilk kll kll
I
corresponds to a n o n t r i v i a l colli-
i s zero. and
A ;:
..
I t i s assumed t h a t = At;
15
for any
i, j , k, 1 .
Shuichi KAWASHIMA and Yasushi SHIZUTA
16
W e a l s c assume t h a t
Aij kil
+
for some
0
i, j , k , i l , which i m p l i e s
t h e e x i s t e n c e of n o n t r i v i a l c o l l i s i o n . W e r e w r i t e (1.1) by u s i n g v e c t o r n o t a t i o n .
...,Fm) ' (1.2)
Put
F = (F1,
and V3 = d i a q ( v l .
- * - , v ,), m7
3'
where the superscript
t
the j-th cmponent of
v..
i - t h e q u a t i o n of
... , Q m f F , F )) '.
5
j
5
n,
d e n o t e s t h e t r a n s p o s e and
vij
is
W e d e n o t e t h e r i g h t m e m b e r of t h e Qi(F,F)
(1.1) hy
Let
1
x = (xl,-.
and
,xn).
put
Q(F,F) = (Q1(F,F),
Then (1.1) c a n be w r i t t e n
as (1.33
Ft +
n
.
Z V'Fx
j=l
t
j
= Q(F,F),
(or
x . ) denotes the p a r t i a l differen3 (or x . ) t i a t i o n with respect t o t 1 The p u r p o s e of t h i s n o t e i s t o s t u d y t h e h y d r o d y n a m i c a l where t h e s u b s c r i p t
.
e q u a t i o n s ( s u c h a s t h e E u l e r and t h e N a v i e r - S t o k e s e q u a t i o n s ) d e r i v e d from t h e d i s c r e t e Boltzmann e q u a t i o n ( 1 . 3 ) by a p p l y i n g t h e Chapman-Enskog method.
A general recipe f o r deriving these
hydrociynamical e q u a t i o n s w a s g i v e n b y G a t i g n o l [ l ]
.
S h e also
s u c c e e d e d i n o b t a i n i n g a n e x p l i c i t form of t h e E u l e r e q u a t i o n . I n t h i s n o t e , we s h a l l c i v e a n e x p l i c i t form o f t h e N a v i e r - S t o k e s e q u a t i o n and t h e n f o r m u l a t e a s u f f i c i e n t c o n d i t i o n t h a t g u a r a n t e e s t h e e x i s t e n c e of a u n i q u e g l o b a l s o l u t i o n t o t h e N a v i e r St:okes e q u a t i o n . 2.
T h e Chapman-Enskoq
method
I n o r d e r t o e x p l a i n t h e Chapman-Enskog m e t h o d , i t i s c o n -
Discrete Boltzmann Equation venient t o introduce a s m a l l parameter
E
17 and r e w r i t e
> 0
( 1 . 3 ) i n t h e form
+
Ft
(2.1)
n
.
Z V’Fx
j=1
The Chapman-Enskog
A.
W e d e n o t e by
a
expansion t h e s p a c e of summational i n v a r i a n t s .
Lm c o n s i s t s of
Note t h a t
Q ( F , F )> = 0
f o r any
F
dard inner product of
,$ ( d l
-E1 Q ( F , F ) .
=
j
j-
E
basis
J,
E
mm
where
<
,
>
lRml
Em.
Let
dim
T/L =
satisfying
w
Let
denotes t h e stand
$(I) ,
and l e t
F
of t h e d i s c r e t e Boltzmann e q u a t i o n
be t h e moment v e c t o r o f
with respect t o t h e
F
{$(l)l---lJ,(d)], that is, w = ( w , , . . . , ~ , ) ~
x
(2.2)
i x a b a s i s of
W e consider a solution
(2.1).
vectors
= < $ ( k ) ,F),
with
l < k < d .
I n t h e Chapman-Enskog t h e o r y , it i s assumed t h a t
F
h a s an
e x p a n s i o n of t h e form OD
(2.3)
Z tzpF(’).
F =
p=o
p
Here
2
0 , depend on
(t,x)
o n l y t h r o u g h t h e moment
v e c t o r o r i t s s p a t i a l d e r i v a t i v e s , namely, t
of
0
)
for
F(O)
p
t
0.
= F(’ )(axw a : la1
Moreover, it i s assumed t h a t t h e components
a r e a l l p o s i t i v e and t h a t
F(P)
E
%
w h e r e u r n ’ d e n o t e s t h e o r t h o g o n a l complement of e n t l y , w e have
for
??.
p
2
1,
Consequ-
Shuichi KAWASHIMA and Yasushi SHIZUTA
18
We s h a l l d e r i v e t h e e q u a t i o n s f o r we t a k e t h e i n n e r p r o d u c t of
(2.1) with
p
L
0.
( J ( ~ ) ,1 i k
substitute (2.3) i n t o t h e resulting equations.
First,
r: d , and
This yields
Next, we s u b s t i t u t e ( 2 . 3 ) i n t o (2.1) and c a l c u l a t e t h e t i m e d e r i vatives
FLp)
by u s i n q t h e e x p r e s s i o n s
and t h e e q u a t i o n s ( 2 . 5 ) . p o w e r s of
Q(
- ,-
)
W e consider
know t h a t
F(')
I t i s known =
2
i s r e g a r d e d as a b i l i n e a r form. F'O)
and
F(l)
( 2 . 6 ) 0 t o g e t h e r w i t h (2.4)0.
is a Maxwellian.
We write
From ( 2 . 6 ) 0 we F ( O ) = (MI, *
( ( 1 1 ) t h a t ( 2 . 6 ) 0 h o l d s i f and o n l y i f
(loqMl,-*-,logMm)T
%
E
.
l ~ q F ( =~ ) C u a $(
--
,Mm)
l o g F (0)
T h e r e f o r e we c a n w r i t e
e)
R= 1 for some
u
=
(ul,--.,ud)T
p r e s s e d i n t e r m s of
Here
0)
Then, e q u a t i n g t h e c o e f f i c i e n t s of
Q ( F ( o ) , F ( 0 ) ) = 0,
D e t e r m i n a t i o n of
1;.
= F ( P ) ( aEw;
on b o t h s i d e s of t h e r e s u l t i n g e q u a t i o n , w e h a v e
t
!2.6)o
Here
F(')
M(u)
u
c
I R ~ .
6
d
IR
,
C o n s e q u e n t l y , F(')
n a m e l y , we h a v e
i s t h e v e c t o r w i t h components
i s ex-
F(O) = M(u).
T.
19
Discrete Boltzmann Equation M . ( u ) = exp
(2.7)
$1''
i n which stitute of
u.
F'O)
Ll C
,
= M(u)
into (2.4)0
w
t h a t t h i s mapping
u
t h e i n v e r s e mapping
w
w
-+
u = u(w).
w
w(u)
sub-
a s a function
is the vector with
A simp1 c a l c u l a t i o n shows
1 s k s d.
+
I$(~). We
and r e g a r d
w ( u ) , where
=
components {$(k) ,M(u)),
=
l s i s m ,
d e n o t e s t h e i - t h component o f
We c a n write
mapping by
(el) ,
U ~ $ J ~
i s a d i f f e o m o r p h i s m and t h e r e f o r e
u
exists.
W e denote t h i s inverse
Then we a r r i v e a t t h e e x p r e s s i o n
F(O)
M(u(w)) I which g i v e s a u n i q u e s o l u t i o n o f t h e problem ( 2 . 6 ) o l
(2.4)0.
Next w e shall d e t e r m i n e with (2.4)1. (2.8)
by s o l v i n g ( 2 . 6 ) 1 t o g e t h e r
F(l)
The e q u a t i o n ( 2 . 6 1 1 t a k e s t h e form
2Q(M(u) , F ( l ) ) = G ' O ) ,
W e define the linearized collision operator t o t h e Maxwellian
where
M(u)
-
*
,M,
(u))
.
W e r e c a l l h e r e some p r o p e r t i e s of
i s g i v e n by
%.
N(LU )'=
Then ( 2 - 8 ) i s r e w r i t t e n a s
L U ( s e e 131).
symmetric and n o n n e g a t i v e d e f i n i t e .
Au'I2
corresponding
a s follows:
hU = diag(M1 ( u ) ,
c i d e s with
Lu
Lu
I t s n u l l space
i s real N(LU)
Hence t h e o r t h o g o n a l complement of
A -12' U
'.
coinN(LU)
Now we examine t h e r i g h t s i d e
Shuichi KAWASHIMA and Yasushi SHIZUTA
20 of
(2.11).
T a k i n g t h e i n n e r p r d u c t of
k < d , and u s i n g
and hence
P
2
6
hence
A;1/2F(1)
the expression
In fact, 1
k
On t h e o t h e r h a n d ,
U
E
N(Lu)'.
I *
( 2 . 6 1 1 means
Therefore, the
= -L;1A-1/2G(o), U
F ( l )=
where
5 = 0 (see [ 2 ] ) .
-A1/2 u Lu -1A -1/2G(0) u
denotes the
';L
Thus w e o b t a i n
with
G(O)
g i v e n by
T h i s e x p r e s s i o n c a n be r e d u c e d t o
(2.9) together with t h e r e l a t i o n s
u
AU
and
On t h e o t h e r h a n d , w e h a v e
because
P.1/2L-1 C.
G (0)
aM(u) /au,
,
= AU$ ( 1 )
d, gives
E
other.
u
This implies
,4-1'2F(1)
where w e u s e d t h e f a c t t h a t
.. d ,
d.
5
N(LU)'.
Lu a t
reduced r e s o l v e n t of
(2.9).
I.
of l i n e a r inhomogeneous e q u a t i o n s c a n be s o l v e d
system ( 2 . 1 1 ) uniquely as
1
G (O)
. m L and
F(')
w i t h ( F ' O ) = M ( u ) ) , w e know t h a t
(2.4)0
for
= 0
($(k) , G C 0 ) )
$(k), 1
(2.9) w i t h
v a n i s h e s on
L-l U
t o (2.9)
V.
I
commute w i t h e a c h
-1 1 / 2 + ( ' ) Lu A u
N(LU).
= 0
1
Therefore, applying
we o b t a i n t h e d e s i r e d e x p r e s s i o n
I ,
for
(2.12)
< 11
.
The E u l e r and t h e N a v i e r - S t o k e s e q u a t i o n s
W e substitute
terms o f (2.13)
t
+
i n t o ( 2 . 5 ) and n e g l e c t t h e
This g i v e s t h e Euler equation:
O(F).
w
F'O) = M(u(w))
n
.
Z f'(w)x
j=1
= 0,
j
Discrete Boltzmann Equation where 5
On t h e o t h e r hand, i f we s u b s t i t u t e
d.
(with
and ( 2 . 1 2 ) of
i s t h e v e c t o r w i t h components ( Q f k ) lVIM(u)>l
f7(w(u))
1 s k
2
+
w t
i n t o ( 2 . 5 ) and n e g l e c t t h e terms
u = u(w))
n
n
Z fJ(w)x = j=1 j
GiJ ( w ) = B
Here
i j
E
C {Gil(w)wx i,j=l
(u (w)) Dwu ( w )
BiJ ( u )
with
Dwu(w) d e n o t e s t h e J a c o b i a n m a t r i x o f W e substitute
w = w(u)
d e n t v a r i a b l e from (2.15)
A
0
w
( u ) = ut +
0 A ( u ) = DUw(u)
where
F ( O ) = M(u(w))
we o b t a i n t h e N a v i e r - S t o k e s e q u a t i o n :
O(E ) ,
(2.14)
21
u(w)
given below; with respect t o
i n t o (2.13) and change t h e depen-
t o u. T h i s y i e l d s n . C A’(u)ux = 0 , j=1 j A J ( u ) = Dufj[w(u))
and
.
S i m i l a r l y , by
t h e same change o f d e p e n d e n t v a r i a b l e , w e c a n t r a n s f o r m (2.14) into (2.16)
A0 (u)ut
+
n
C AJ(U)ux
j
j=l
n = E
1 {BiJ(u)ux}xi.
i,j=l
j
The c o e f f i c i e n t m a t r i c e s i n ( 2 . 1 5 ) and ( 2 . 1 6 ) a r e g i v e n e x p l i c i t l y a s follows.
For
n -1
w = ( W ~ ~ - . * , W ~ E) S
ZAJ(u)wj
and
,
..
~e
put
B ( u , o ) = ZB”(u)w.w.,
m a t i o n s a r e t a k e n o v e r all
w
A(u,w) = 1’ where t h e f i r s t two sum-
V ( w ) = ZV’W.
1 3 j = 1, ..*,n, w h i l e t h e l a s t sum-
.
Shuichi KAWASHIMA and Yasushi SHIZUTA
22
m a t i o n i s t a k e n Over a l l and ( 2 . 1 7 )
that
(2.18)
V(;)
i , j = l,.**,n.
= diag(vl-w,*--,vm.u)
I t f o l l o w s from ( 1 . 2 )
,
From t h e s e e x p r e s s i o n s we c a n e a s i l y v e r i f y t h e f o l l o w i n g p r o -
u
p e r t i e s for e v e r y
d R :
E
(2.2011
Ao(u)
i s r e a l symmetric and p o s i t i v e d e f i n i t e .
(2.2012
A’(u),
1
(2.20)
Bij(u)’ = Bji(u)
j
n , a r e r e a l symmetric.
5
holds f o r
i,j = l,---,n.
Further-
more, B ( u , w )
i s r e a l symmetric and n o n n e g a t i v e d e f i -
n i t e f o r any
’li E
Here we u s e d t h e f a c t t h a t tive d e f i n i t e f o r e v e r y
u
S
L;’ E
n -1
.
i s r e a l symmetric a n d n o n n e g a d
IR
.
T h e s e o b s e r v a t i o n s a r e summa-
r i z e d as f o l l o w s . Theorem 2 . 1 .
The N a v i e r - S t o k e s e q u a t i o n d e r i v e d f r o m t h e
d i s c r e t e Boltzmann e q u a t i o n ( 2 . 1 ) a s t h e s e c o n d order a p p r o x i m a t i o n of t h e Chapman-Enskog e x p a n s i o n i s w r i t t e n i n t h e c o n s e r v a t i o n form ( 2 . 1 4 ) .
w
to
By c h a n g i n g t h e d e p e n d e n t v a r i a b l e f r o m
u , t h i s e q u a t i o n c a n be p u t i n t o a symmetric s y s t e m
1 2 . 1 6 ) whose c o e f f i c i e n t matrices s a t i s f y t h e p r o p e r t i e s (2.20)1-3. Remark.
The E u l e r e q u a t i o n ( 2 . 1 3 ) c a n be t r a n s f o r m e d i n t o
a symmetric h y p e r b o l i c s y s t e m ( 2 . 1 5 ) by c h a n g e o f t h e d e p e n d e n t variable.
T h i s r e s u l t i s d u e t o G a t i g o n o l [l]
.
Discrete Boltzrnann Equation 3.
P r o p e r t i e s of t h e N a v i e r - S t o k e s e q u a t i o n
A.
Entropy f u n c t i o n
23
W e have shown t h a t t h e N a v i e r - S t o k e s e q u a t i o n ( 2 . 1 4 )
is
t r a n s f o r m e d i n t o a symmetric system ( 2 . 1 6 ) by change of t h e T h e r e f o r e , w e c a n a p p l y Theorem 2 . 1 of [ S ]
dependent v a r i a b l e .
and c o n c l u d e t h e e x i s t e n c e of a n e n t r o p y f u n c t i o n
More c o n c r e t e l y , w e have
Navier-Stokes e q u a t i o n ( 2 . 1 4 ) .
.
~ ( w ) of t h e
m
m
N o t e t h a t t h e f i r s t term on t h e r i g h t s i d e of
(3.1)1 i s e x a c t l y
t h e Boltzmann H-function e v a l u a t e d a t t h e Maxwellian The a s s o c i a t e d f l u x
cqz(w),.*-,qn(w))
F = M(u).
i s g i v e n by
A s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t o u r e n t r o p y f u n c t i o n
satisfies
n
= €
C
n {{UrBiJ(U)Ux)]x
i ,j = l B.
- E
J
i
C (uX,,Bij(u)ux
i,j=l
1
j
).
Hydrodynamical b a s i s and i s o t r o p y c o n d i t i o n W e have a l r e a d y v e r i f i e d t h a t t h e m a t r i x
(2.19)2 (u,w)
E
B(u,w)
g i v e n by
i s r e a l symmetric and n o n n e g a t i v e d e f i n i t e f o r a n y IRd x S n - l ;
we c a l l t h i s m a t r i x
B(u,w) t h e v i s c o s i t y
m a t r i x of t h e Navier-Stokes e q u a t i o n ( 2 . 1 6 ) .
Since
B(u,w)
is
g i v e n e x p l i c i t l y , we c a n e a s i l y d e t e r m i n e i t s n u l l s p a c e . Lemma 3.1.
For e v e r y f i x e d
(u,w)
E
IRd x
the null
Shuichi KAWASHIMA and Yasushi SHIZUTA
24 space
of t h e v i s c o s i t y m a t r i x
N(B(u,u))
B(u,w) c o i n c i d e s
with t h e space
T h i s lemma shows t h a t t h e n u l l s p a c e p e n d e n t of
u
i
b u t it d p e n d s on
Ed
N(B(u,w)) n-1
i n general.
S
w c
i s inde-
This causes a serious d i f f i c u l t y i n solving the i n i t i a l value T o resolve t h i s d i f f i -
problem f o r t h e N a v i e r - S t o k e s e q u a t i o n .
c u l t y , w e assume t h e e x i s t e n c e o f what we c a l l a h y d r c d y n a m i c a l
%, .
iusis of
Definition. il
A
{ ~ ( 1,.-. ) ,y(d)\
basis
of
-2 i s c a l l e d
h y d r o d y n a m i c a l basis i f t h e r e e x i s t s a p a r t i t i o n
the integers
{ I , I I } of
such t h a t t h e following t w o p r o p e r t i e s
{l,...,d)
hold: the vectors
(3.4)
I*'
,
The v e c t o r s
(3.4),
v ( w ) ~ l ( ~k ) E, I,
belong t o
%
f o r any
sn- 1 V ( W ) $ ( ~ ) ,k
w
pendent f o r any
E
E
Sn-l,
II, are l i n e a r l y indei f t h e y are r e g a r d e d a s
elements of t h e quotient space
Bm/*a.
The l a t t e r c o n d i t i o n ( 3 . 4 ) * means t h e f o l l o w i n g : F o r e v e r y fixed
u
I:
and o n l y i f
sn -1 , t h e r e l a t i o n zk = 0
be r e a l c o n s t a n t s a n d
E.
for
k
In
c
11,
where
I-
II
and
II,
Sn -1.
zk
E
9 ,h o l d s
if
are assumed t o
i s t h e summation w i t h r e s p e c t t o
Note t h a t t h i s c o n d i t i o n i m p l i e s
II. k
for
CIIzkV(u)J' -(k)
V ( U ) $ ( ~ )&
'w
W e s h a l l show t h a t t h e e x i s t e n c e o f a h y d r c d y n a m i c a l basis
Discrete Boltzmann Equation of
%
25
and t h e f o l l o w i n g i s o t r o p y c o n d i t i o n a r e e q u i v a l e n t t o
each other. Isotropy Condition. viscosity matrix
The n u l l s p a c e
B(u,w)
Theorem 3.2.
N(B(u,w))
d o e s n o t depend on
(u,w)
of t h e Rd
E
E
sn-I.
%,
I f t h e r e e x i s t s a hydrodynamical basis o f
t h e n t h e Navier-Stokes e q u a t i o n ( 2 . 1 6 ) corresponding t o e v e r y
,
fixed basis
s a t i s f i e s Isotropy Condition.
$J(~)
Conversely, i f t h e Navier-Stokes e q u a t i o n ( 2 . 1 6 ) p o n d s t o some basis t i o n , t h e n t h e space
,
Jl(')
cr/z
- .,
Qfd)
)
s a t i s f i e s I s o t r o p y Condi-
a d m i t s a hydrodybamical basis.
W e f i r s t c o n s i d e r a s p e c i a l N a v i e r - S t o k e s e q u a t i o n which
Proof.
c o r r e s p o n d s t o a hydrodynamical b a s i s
";3n .
W e write
6
and
=
[$('I
The n u l l s p a c e =
{z
E
N(G(G,w))
IRd; zk = 0
depend on
(G,w)
for E
lRd
k x
of
f o r t h e unknown and t h e v i s -
B(;,w)
c o s i t y m a t r i x of t h i s Navier-Stokes e q u a t i o n . (*)
which corres-
Then w e have:
c o i n c i d e s w i t h t h e space E
It),
9
and hence it d o e s n o t
sn-'.
T h i s i s e a s i l y v e r i f i e d by u s i n g Lemma 3 . 1 and ( 3 . 4 )
1,2'
Next we c o n s i d e r t h e N a v i e r - S t o k e s e q u a t i o n which c o r r e s ponds t o an a r b i t r a r i l y f i x e d b a s i s
V/'L. In t h i s case,
we write
u
Y = { $ (1), . . . I ,,,(d)]
and
and t h e v i s c o s i t y m a t r i x , r e s p e c t i v e l y . b a s i s of
WL
from
? to
Y.
of c o o r d i n a t e s of v e c t o r s i n
B(u,w)
of
f o r t h e unknown
W e s h a l l exchange t h e
This induces the transformation
%.
L e t u s d e n o t by
matrix t h a t represents t h i s transformation.
S
the
Then, by s t r a i g h t -
Shuichi KAWASHIMA and Yasushi SHIZUTA
26
forward c a l c u l a t i o n s , w e f i n d t h a t
= Su
T h i s combined w i t h t h e p r o p e r t y ( * )
L)S.
Hence
= S-'%.
sn -1
.
and
B ( u , w ) =S1fc(;,
shows t h a t
d o e s n o t d e p e n d on
N(B(u,u))
N(B(u,w))
( u , ~ )E IRd
x
The p r o o f of t h e f i r s t h a l f o f t h e t h e o r e m i s c o m p l e t e .
T o p r o v e t h e s e c o n d h a l f , w e c o n s i d e r a basis { q ~ ( ~ ) , . - . ,
dA and
of
(d))
t h e corresponding Navier-Stokes equation.
W e d e n o t e i t s v i s c o s i t y m a t r i x by n u l l space
W e write
N(B(u,w))
31, ,, f o r
d o e s n o t d e p e n on
{eril,*--,ed)
*n;,
{ e 1 , - * - , e r ) and
respectively.
e
Put
.
{ 11
+-his
,.. ,;
(d))
+
.
{r
C.
Local s o l u t i o n s
l,...,d?
= dim
2 ol
ek.
1
i
S"-l.
and l e t
2,
be t h e bases o f
-
x
k
and d , where
Then we c a n show t h a t
,
i s a hydrodynamical b a s i s of
namely, i t s a t i s f i e s ( 3 . 4 ) TI=
( u , w ) c IRd
$ (k) - L 2d =lekLQ(L),
d e n o t e s t h e L-th component of
k P,
r
Let
N(B(u,u)).
and assume t h a t t h e
B(u,w)
1,2
with the p a r t i t i o n
I = {l,-..,r},
F o r t h e d e t a i l s , see [6].
We c o n s i d e r t h e i n i t i a l v a l u e p r o b l e m f o r t h e N a v i e r - S t o k e s equation (2.14) (3.5)
,
with the i n i t i a l condition
W(0,X) = w o ( x ) ,
Our N a v i e r - S t o k e s system ( 2 . 1 6 )
x
E
"a
.
e q u a t i o n ( 2 . 1 4 ) c a n be p u t i n t o a symmetric
by c h a n g i n g t h e d e p e n d e n t v a r i a b l e from
and t h i s symmetric s y s t e m ( 2 . 1 6 ) if
n
IR
w
to
u,
s a t i s f i e s Isotropy Conditions
a d m i t s a h y d r o d y n a m i c a l basis ( s e e Theorems 2 . 1 a n d 3 . 3 ) .
T h e r e f o r e , a p p l y i n g Theorem 3 . 1 o f f o 11owing :
[51
,
we c a n c o n c l u d e t h e
Discrete Boltzmann Equation
27
';31/t
Suppose t h a t t h e s p a c e
Theorem 3.3. d y n a m i c a l basis.
a d m i t s a hydrou * v
Then t h e r e e x i s t s a d i f f e a n o r p h i s n
which c o n v e r t s t h e N a v i e r - S t o k e s e q u a t i o n ( 2 . 1 6 )
i n t o t h e normal
form of t h e symmetric h y p e r b o l i c - p a r a b o l i c system i n t h e s e n s e of
[5]. C o n s e q u e n t l y , we know t h a t t h e N a v i e r - S t o k e s e q u a t i o n
( 2 . 1 4 ) c a n be t r a n s f o r m e d i n t o a c c u p l e d system of a symmetric
h y p e r b o l i c system and a symmetric s t r o n g l y p a r a b o l i c s y s t e m ,
w
by c h a n g i n g t h e d e p e n d e n t v a r i a b l e from
to
u.
Therefore,
a p p l y i n g Theorem 2 . 9 of [ 4 ] , w e c a n show t h e e x i s t e n c e of a l o c a l s o l u t i o n t o t h e i n i t i a l v a l u e problem ( 2 . 1 4 ) , Theorem 3.4.
wo
-
w
w
Let
basis.
E
(3.5)
(with
s
for
[n/2]
E
+
D.
Then t h e r e e x i s t s a p o s i -
C o [ [O,Tol ; H S ( I R n ) )
w(t,x)
has a u n i q u e s o l u t i c m
= 1)
a p p r o p r i a t e f u n c t i o n space over E
2.
and suppose t h a t
lRd
such t h a t t h e i n i t i a l v a l u e problem ( 2 . 1 4 )
To E
a d m i t s a hydrcdynamical
be a c o n s t a n t vector i n
Hs(lRn)
t i v e constant
%
Suppose t h a t
(3.5).
[O,To]
.
x
i n an
R n ; we have
w
-
,
w
S t a b i l i t y c o n d i t i o n and g l o b a l s o l u t i o n s
I n t h e p r e v i o u s p a p e r s [ 3 ] , [71, we s o l v e d t h e i n i t i a l v a l u e problem f o r t h e d i s c r e t e Boltzmann e q u a t i o n ( 1 . 3 ) g l o b a l l y i n t i m e i n a neighborhood of a g i v e n Maxwellian state, by assuming t h e f o l l o w i n g c o n d i t i o n s S t a b i l i t y C o n d i t i o n DB. f o r some Also,
A
R
and
w
B
S
Let
n-1
.
.
J, E
Then
%?.
and
A$ = V(w)J,
J, = 0 .
i n [ 4 ] , [7], w e s t u d i e d t h e i n i t i a l v a l u e problem
28
Shuichi KAWASHIMA and Yasushi SHIZUTA
f o r s y s t e m s which are w r i t t e n i n t h e form ( 2 . 1 6 ) and s a t i s f y I s o t r o p y C o n d i t i o n , and p r o v e d t h e e x i s t e n c e of a g l o b a l ( i n
t i m e ) s o l u t i o n under t h e following c o n d i t i o n . S t a b i l i t y Condition and
= 0)
NS.
0
> A ( u ) z = A(u,w)z
Let
z
E
f o r some
N(B(u,w))
X
E
(i.e., B(u,w)z and
R
w
Sn-l.
E
z = 0.
Then
By u s i n g t h e e x p l i c i t e x p r e s s i o n s o f t h e m a t r i c e s g i v e n
we c a n show t h e e q u i v a l e n c e o f t h e s e
i n ( 2 . 1 7 ) and ( 2 . 1 9 ) 1 , 2 , two s t a b i l i t y c o n d i t i o n s Theorem 3 . 5 .
I f S t a b i l i t y C o n d i t i o n DB f o r t h e d i s c r e t e
Boltzmann e q u a t i o n ( 1 . 3 ) h o l d s , t h e n t h e N a v i e r - S t o k e s e q u a t i o n
u
s a t i s f i e s S t a b i l i t y C o n d i t i o n NS f o r e v e r y
(2.16)
E
IRd
.
C o n v e r s e l y , i f S t a b i l i t y C o n d i t i o n NS f o r ( 2 . 1 6 ) h o l d s f o r
some
u
E
R d , t h e n ( 1 . 3 ) s a t i s f i e s S t a b i l i t y C o n d i t i o n DB.
Now we assume t h e e x i s t e n c e o f a h y d r c d y n a m i c a l basis of
y2q
and S t a b i l i t y C o n d i t i o n DB for ( 1 . 3 ) .
I n t h i s case, t h e
Navier-Stokes e q u a t i o n ( 2 . 1 6 ) i s transformed i n t o t h e normal form of t h e symmetric h y p e r b o l i c - p a r a b o l i c
s y s t e m and s a t i s f i e s
S t a b i l i t y C o n d i t i o n NS (Theorems 3 . 3 and 3 . 5 ) . o r i g i n a l Navier-Stokes given b
(3.1)1.
equation (2.14)
Also, the
has an entropy function
T h e r e f o r e , Theorems 4 . 3 and 4 . 4
[ 4 1 are
of
a p p l i c a b l e s t r a i g h t f o r w a r d l y to t h e problem ( 2 . 1 4 ) ,
(3.5),
and we o b t a i n t h e f o l l o w i n g g l o b a l e x i s t e n c e Theorem. Theorem 3 . 6 .
Suppose t h a t
%
admits a hydrodynamical
b a s i s and t h a t t h e d i s c r e t e Boltzmann e q u a t i o n ( 1 . 3 )
satisfies
Discrete Boltzmann Equation
29
S t a b i l i t y C o n d i t i o n DB. (i) I f
wo
- -w
i s small i n
Hs(Rn), where
t h e n t h e i n i t i a l v a l u e problem ( 2 . 1 4 ) a unique global solution s p a c e over [ O r - ) solution
w(t,x)
x
-
as
(ii) A s s u m e t h a t
wo
s 2
[n/21 + 3.
t *
t
-+
- w
-
E
= 1) h a s
-
w e C0 ([O,=);HS(Wn))
-.
i s ma11 i n
Then t h e s o l u t i o n
v e r g e s , i n t h e s e n s e of as
w
converges, uniformly i n
w
constant s t a t e
(3.5) (with
i n an a p p r o p r i a t e f u n c t i o n
w(t,x)
Rn; we have
,
+ 2,
S t [n/2]
HS-2(IRn),
w
This
to the
Hs(Rn) n L 1 ( R n )
w(t,x) to
x e Rn,
.
obtained i n at the rate
,
where
(1)
con-
t-"l4
a.
P
References. R.
G a t i g n o l , "Thborie C i n g t i q u e d e s G a s
2 Repartition
Discrete d e V i t e s s e s " , L e c t u r e N o t e s i n Phys.,
V e r l a g , Berlin-Heidelberg-New
36, Springer-
York, 1975.
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 " .
ed
, Springer-Verlag,
second
N e w York, 1 9 7 6 .
S. Kawashima, G l o b a l e x i s t e n c e and s t a b i l i t y of s o l u t i o n s
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 , Recent T o p i c s i n N o n l i n e a r PDE, L e c t u r e N o t e s i n N u n . Appl. A n a l , 6 , Kinokuniya, Tokyo, 1983, 59 - 8 5 . S . Kawashima, Systems o f a h y p e r b o l i c - p a r a b o l i c c o m p o s i t e
type, w i t h a p p l i c a t i o n s t o t h e e q u a t i o n s of magnetohydrodynamics, Dectoral T h e s i s , Kyoto U n i v e r s i t y , 1984. S . Kawashima and Y.
S h i z u t a , On t h e normal form of t h e
Shuichi KAWASHIMA and Yasushi SHIZUTA
30
symmetric h y p e r b o l i c - p a r a b o l i c
s y s t e m s associated w i t h
t h e c o n s e r v a t i o n l a w s , t o a p p e a r i n T6hoku Math. J . (61
S . Kawashima and Y. S h i z u t a ,
The N a v i e r - S t o k e s e q u a t i o n
i n t h e d i s c r e t e k i n e t i c t h o e r y , t o a p p e a r i n J. M6c. Th6or. A p p l . [71
Y.
.
S h i z u t a and S . Kawashima, S y s t e m s of e q u a t i o n s of h y p e r -
b o l i c - p a r a b o l i c t y p e with a p p l i c a t i o n s t o t h e discrete Boltzmann e q u a t i o n , Hokkaido Math. J .
,
1 4 (1985) , 249
- 275.
Lecture Notes in Num. Appl. Anal., 10, 31-47 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
Route to Chaos in a Navier-Stokes Flow
Shigeo KIDA Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan,
Michio YAMADA Disaster Prevention Research Institute, Kyoto University, Uji 611, Japan and
Kohji OHKITANI Department of Physics, Faculty of Science, Kyoto University, Kyoto MM, Japan
Abstract A route to chaos is investigated numerically in the motion of an incompressible viscous fluid which is governed by the Navier-Stokes equation with an external force. The highsymmetry is imposed on the velocity field. The complexity of the flow is characterized by the multi-periodicity of the temporal variation of the velocity field. By examining the frequency power spectrum of the energy and the orbit of the points of state we found the following scenario to chaos: steady -+simply periodic -+
+
doubly periodic
-+
triply periodic
non-periodic (chaotic) motions. The difference of the spatial complexity between the
chaotic motion and the fully developed turbulence is discussed.
31
Shigeo KIDA. Michio YAMADA and Kohji OHKITANI
32
1. Introduction
The pattern of fluid motion is qualitatively different depending on the value of the Reynolds number. Generally, when the Reynolds number is small, the laminar flow with simple spatio-temporal structure is realized. As the Reynolds number is increased, the
flowbecomes more and more complicated and finally to turbulence. Landau' conjectured the following scenario on the process of complexity of the structure of the velocity field with increasing Reynolds number: steady
-+
doubly periodic (quasi-periodic with two fundamental frequencies)
--t
-, . . .
--t
simply periodic
--t
- + n-periodic
non-periodic (chaotic) motions. The turbulence is regarded as the limiting state
of the multi-periodic motions. Newhouse, Ruelle and Takens,' on the other hand, proved
that a triply periodic motion in a dynamical system is unstable to a small Cz perturbation while a quadruply periodic motion is to a small Cooperturbation. Therefore, if such perturbations are compatible with the Navier-Stokes equation, the triply and/or quadruply periodic motions may not be observed in real fluid systems. Since then, extensive researches have been made on the transition to turbulence, especially the existence of the triply periodic motion in various fluid systems as well as other dynamical systems. It has been found that there are many different routes to chaos depending upon the kinds of the dynamical systems. A t the same time the triply, quadruply and even quintuply periodic motions were observed in the Rayleigh-Bknard convection3 and in the Taylor vortex flow.' The triply periodic motion was also observed numerically for a model equation to the Taylor vortex flow6 and for many other simple dynamical systems" In this paper we present a process of complexity found by solving the full Navier-Stokes equation numerically. 2. N u m e r i c a l Simulation of the Navier-Stokes e q u a t i o n
We consider the motion of an incompressible viscous fluid in a cyclic box of period 2 7 ~
Route to Chaos in a Navier-Stokes Flow
33
with an effective steady force. The motion is governed by the Navier-Stokes equation and the continuity equation, which are written in the Fourier representation as
and
Here Z and G are respectively the Fourier transforms of the velocity u(z) and the vorticity W(Z)
= rot u(z);
and W(Z)
=
C
-
~ ( kexp ) [ik zj,
-
Iki I , l k a l , l k i l l t N
where
N
is an integer. The subscripts in (1)
(5)
(3) denote the vector components. The
summation convention is assumed for the repeated subscripts.
rjkl
is an alternating tensor,
iXlm(k) is the inverse Fourier transform of the product u~(z)um(z), v is the kinematic viscosity of fluid, and k = [kl.The time argument is omitted for brevity. The high-symmetry' is impoeed on the velocity field in order to save the memory capacity and the computation time. Equations (1)
-
(3) are solved numerically starting
with a suitable initial condition. The nonlinear terma are calculated by the pseudo-spectral method. The aliasing interaction is suppressed by cutting off the Fourier components at a maximum wavenumber k,,
=
N.8 The time integration is performed by the Runge-
Kutta-Gill scheme. In order to imitate a constant energy mpply at lower wavenumbers the following Fourier components of the velocity are kept to be constant all the time:
34
Shigeo KIDA, Michio YAMADA and Kohji OHKITANI
and similar conditions for iil and ii3. We made several rum with different values of viscosity. The calculation was continued until the effects of the initial condition were thought to fade away. The parameters used in the simulation are listed in Table I. The integer N is chosen so that the truncation effects at the maximum wavenumber
$N
may not be significant.
Different kinds of turbulence are conveniently compared with each other for the same values of Taylor’s micrescale Reynolds number Rx, which is expressed in terms of the energy & ( t ) , the enstrophy Q ( t ) and the kinematic viscosity u as
where c ( t ) = 2 uQ ( t ) is the energy dissipation rate. In the numerical simulation the micro-scale Reynolds number fluctuates in time. In figure 1 is plotted the mean Reynolds number against l / v . It is seen that Rx is proportional to l / u for small l / v , while it is proportional to
0for large l / u . These asymptotic
dependences on the viscosity of the micro-scale Reynolds number are the manifestations of the characteristics of the laminar and turbulent velocity fields, respecti~ely.~ The patterns of the fluid motion are different for different values of viscosity. For large values of viscosity v > u c ( x 0.012) the steady motion is realized, whereas for viscosity
smaller than v, the velocity field fluctuates in time. The temporal behavior of the velocity field becomes more and more complicated as the viscosity is decreased. By decreasing the viscosity gradually we found that the following scenario of the temporal complexity in the velocity field: steady
-+
periodic
+
doubly periodic
-P
tripfy periodic
-P
non-periodic
(chaotic) motions. In the following sections the characteristics of the respective types of motins will be discussed in terms of the temporal variation of the energy (in 53) and the phase portrait of the orbit of the points of state (in 54). 3. Time Series and Frequency Power Spectrum of Energy
Route to Chaos in a Navier-Stokes Flow
35
In figures 2 is shown, for several values of viscosity, the time! series of the kinetic energy
& (t) per unit mass of fluid for later periods of the long runs. The energy oscillates regularly in time for v = 0.011, 0.008 and 0.0069. The amplitude of the willation is constant for v = 0.011, while it is undulated for v = 0.008 and 0.0069. The energy for v = 0.0065, on
the contrary, fluctuates in a very complicated manner or irregularly. In order to see the spectral behavior of the variation of energy we made the Fourier transformation of the energy by using M(= 2'')
numerical data taken at every time
step. In figures 3 is plotted, for the corresponding cases of figures 2, the frequency power spectrum r ( w ) averaged over a number of samples calculated in different periods of time. The spectral interval Aw = 2 r/M A t and the maximum frequency (the Nyquist frequency) n /At are 0.01917 and 157.1, respectively. The Fourier component at zero frequency is
suppressed because it is irrelevant to the variation of energy. The spectrum for v = 0.011 has two distinct peaks at
w1
= 2.575 and at its harmonic
2w1, and a very weak peak at 3Ul (figure 3(a)). It was confirmed by the parabola fitting to
the inverse of the spectrum that these peaks certainly represent line spectra (see Appendix). At the same time their frequencies were determined within error of several percents of Aw. The motion in this case is therefore in a simply periodic state. There are scores of distinct peaks observed in the spectrum for v = 0.008 (figure 3(b)). We have checked that all of their frequencies are expressed by sums and differences of two fundamental frequencies
w1
= 2.574 and
wp
= 0.326, which implies that this motion is
doubly periodic. The second frequency wp represents the undulation of the amplitude of
energy which is evident in figure 2(b). The spectrum for v = 0.0069 ia composed of a lot of peaks (figure 3(c)). We can identify two fundamental frequencies w1 (= 2.581) and third fundamental frequency at
w3
w2
(=0.295). There appears the
(= 0.068), which cannot be expressed by a sum or
difference of simple multiples of w1 and wl. We have checked that all the frequencies can be expressed
M sums
and differences of three fundamental frequencies w 1 , w2 and w3. This
36
Shigeo KIDA, Michio YAMADA and Kohji OHKITANI
is therefore in a triply periodic state. Figure 3(d) shows the power spectrum for v = 0.0065. There are too many peaks observed. We counted the number of the peak spectral components. It amounts to 187 in the frequency domain 0 5 w 5 8 , which is nearly a half of the total number of the spectral components therein. Such many peaks are consistent with the assertion that the spectrum in this case may be continuous and therefore the motion is chaotic.
4. Orbit and Poincar6 Section of Points of State The multi-periodicity of the motion can be well distinguished by looking at the orbit of the points of state (or the attractor) in the phase space. The attractor may be a fixed point, a limit cycle, a (twa-) torus, a threetorus or an strange attractor for a steady, a simply periodic, a doubly periodic, a triply periodic or a chaotic motion, respectively. Figures 4 are examples of the orbits of the points of state projected onto a subspace spanned by three Fourier components of vorticity,
(20,20, lo), z1(10,20,10) and i& (0,30,10) for cases of
(a) simply periodic, (b) doubly periodic and (c) chaotic motions. A limit cycle is evident in figure 4(a) which demonstrates that the motion is in a simply periodic state. The orbits shown in figures 4(b) and 4(c) are dispersed so much that we cannot see their structure. In figure 5(b) is plotted a Poincard section of the orbit for u = 0.008 intersected by a plane perpendicular to the w1(10,20,1O)-axis which is shown in figure 5(a). The Poincar6 section seems to lie on a closed curve, which implies that the orbit is running over a twotorus and that the motion is in a doubly periodic state. Figures 6 are the same plots for
v = 0.0065. The orbit is scattered on the Poincard plane, which is consistent with that the orbit is on a strange attractor and the motion is chaotic. 5 . Chaos to Turbulence
The chaotic motion we have considered in the previous sections does not necessarily implies that the flow is 'turbulence'. The chaotic motion exibits the temporal irregularity,
Route to Chaos in a Navier-Stokes Flow
37
whereas turbulence has both spatial and temporal irregularity. Figurea 7 are the snapshots of the stream lines for the cases of (a) the steady motion ( u = O.l), (b) the chaotic motion (u= 0.oOSS) and (c) the turbulent motion
(Y
= O.ooO5). Here are shown the components
of the velocity on a plane perpendicular to the za-axis. For
Y
= 0.1 a single big eddy
exists steadily and the stream limes are simple. It has still a simple structure even in a chaotic state as seen in figure 7(b). For very small viecceity (u = O.OOOS), however, as shown in figure 7(c), the velocity field, which is composed of a lot of eddies of different sizes, becomes very complicited.
The micro-scale Reynolds number Rx is about 3, 40 and 200 for v = 0.1, 0.0065 and 0.0005, respectively. The above result tells
u8
that when Rx
= 40, the flow may be chaotic
but not turbulent. Incidentally it ia known by many experiments that various similarity laws for the fully developed turbulence hold for Rx 2 80." The skewness of the velocity derivative, for example, shows different dependences of the Reynolds number below and above Rx
= 80.
l2
It is one of the interesting problems to investigate what quantities
characterize the transition around this critical Reynolds number. 6. Summary and Discussion
The process of the temporal complexity of the velocity field we have found in the present study is summarized in figure 8. As the Reynolds number is increased, the temporal behavior of the velocity field undergoes the following sequence of transition: steady (S) -+ periodic (P)
+
doubly periodic (QP,)
-+ triply
periodic (QP3)
+
chaotic (C)
motions. The critical Reynolds numbers shown in figure 8 were estimated by examining the Reynolds number dependences of the variance of the variation of the energy and the Lyapunov exponent? We observed the re-laminaliiation of the velocity field; a periodic motion appears again at larger Reynolds numbers. Now we would like to make several comments. First, it is hard to distinguish highly multi-periodic motion from non-periodic onea numerically because of finite resolution. For
Shigeo KIDA, Michio YAMADA and Kohji OHKITANI
38
example, although we regarded the power spectrum in figure 3(d) as continuous, it may be possible to be a collection of a number of line spectra with many fundamental frequencies.
We cannot therefore deny the possibility of existence of more than triply periodic motions by the present numerical simulation. Second, the existence of the triply periodic state is concluded only within the numerical resolution because it is poaclible that the spectrum in figure 3(c) may have any invisible broadband component. Third, if a triply periodic state does ever exist, it implies that the perturbations in the theorem by Newhouse ct uL2 may
not be generic for the Navier-Stokes equation. Fourth, there is a fear that the constraint
of the spatial periodicity or the high-symmetry imposed on the velocity field might inhibit such perturbations that destroy the triply periodic motion. It is left for a future problem what happens if we release the condition of high-symmetry. Acknowledgments The numerical simulation was done on the FACOM M382-VP200 system in the data processing center of Kyoto University. This work was partially supported by Itoh Science Foundation and the Grant-in-Aid for Scientific Research from the Ministry of Education in Japan. APPENDIX
The Power Spectrum of A Periodic Function
We describe here the typical form of the power spectrum of a periodic function obtained by the Fourier transformation over a finite domain of period. Let us express a real periodic function of period 2 n / n by a Fourier series as
C m
j(t)=
A,, exp [ i m n t ] ,
(A.1)
m= -m
where Am's are complex numbers. Since j ( t ) is real, A,,, = A,',
where
' denotes the
complex conjugate. The Fourier transform of f f t ) over a finite period, 0 5 t 5 T, is calculated to be
Route to Chaos in a Navier-Stokes Flow
39
where w, = 2nn/T, n being an integer. The Fourier transform m = -00,.
. .,
00.
T(w,)has an infinite number of pole singularities at w,
I- IZ,
The power spectrum, f(wn)
= mR,
therefore, behaves around wn = mn as
Thus the inverse of the power spectrum is approximated by a parabola, the axis of which lies at the frequency
mn.
In figure 9 is shown the inverse of the power spectrum around the fundamental frequency w1 for v = 0.011 (cf. k u r e 3(a)). The solid curve is a parabola which is determined in such a way that it goea through the loweat two points of data and is in contact with the abscissa. It is seen that the other points of data in this figure are also lying in the neighborhood of the parabola, which suggests that it is a line spectrum of a periodic function broadened by Fourier transformation over a finite period. The frequency of the line spectrum is given by the axis of the parabola, which is 2.575.
References
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Oxford, Pergamon, 1959. S. Newhouse, D. Ruelle and F. Takens, Comm. Math. Phys. 64, 35 (1978).
J. P. Gollub and S. V. J. Beneon, Fluid Mech. 100, 449 (1980); A. Libchaber, S. Fauve and C. Laroche, Physica 7D,73 (1983); R.W. Walden, P. Kolodner,
A. Passner k C.M. Surko, Phys. Rev. Lett, 53, 242 (1984).
M. Gorman, L. A. Reith and H. L. Swinney, Ann. N.Y. Acad. Sci. 357, 10 (1980). H. Yahata, Prog. Theor. Phys. 64, 782 (1980), 09, 396 (1983). H. T. Moon, P. Huerre and L. G. Redekopp, Phys. Rev. Lett. 49, 458 (1982); P. Davis and K. Ikeda, Phys. Lett. 100A,455 (1984).
Shigeo KIDA, Michio YAMADA and Kohji OHKITANI
' S. Kida, J. Phys. SOC. Japan, 64,2132 (1985).
' S. A. Orszag, Stud. Appl. Math. 6 0 , 293 (1971). S.Kida, M. Yamada and K. Ohkitani, (in preparation). 'I'
S. Kida, and Y. Murakami (in preparation).
I 1 -4. S.Monin
and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence,
MIT Press, Vol. 2, 1975. l2
S. Tavoularis, J.C. Bennett and S. Corrsin, J. Fluid Mech. 88, 63 (1978).
l/u
N At
10 5 1 / u 5 500 128 0.02
1000 256 0.01
2000
512 0.002
300 200 100
60 YO
4
30 20 10
10'
1o 2
1 o3
llv Figure 1. Dependence on viscosity of the averaged micro-scale Reynolds number.
42
Shigeo KIDA, Michio YAMADA and Kohji OHKITANI
Figure 2. Time series of the energy. (a) (c)
Y
v = 0.0069.(d) v = 0.0065.
= 0.011. (b) Y = 0.008.
43
Route to Chaos in a Navier-Stokes Flow
n
-= 3 -m -N
-0
m
n
-r-
n
u d
cl
v
-a i -In
-3.3 -m
3"
--N
-0
--
a
Figure 3. Frequency power spectrum of energy. (a) Y = 0.011.
(b) u = 0.008. (c) u = 0.0069.(d)
Y
= 0.0065.
-
N
44
Shigeo KIDA, Michio YAMADA and Kohji OHKITANI
21(0,30,10)
/
/
(4
G1(10,20,10)
G1(20,20,10)
Figure 4. Projection of the orbit of the point of state onto a threedimensional subspace of state. (a) v = 0.011. (b) v =
0.008. ( c ) v = 0.0065.
G1 (20,20,10)
Route to Chaos in a Navier-Stokes Flow
3
I
8
I
I
,
t
I
45
#
&
,
#
I I
I
.a?
................ .
-
O 1 --!!
,0 ?
4
8
1
-
...... ;... . .L..-.... i-". ............... ............' 1 l *
1
-
-1 -
-2
8
-
: ....... ..... ! :/::-7
s 0:
1s
I
(b) .. ..
2n
1
" " " " " " " ~ " "
-.
.
.
,
Figure 5. (a) The projection of the orbit of the points of state onto a threedimensional subspace of state together with a Poincark plane for v = 0.008. (b) The Poincard section of the orbit intersected by the plane shown in (a).
Figure 6. The same as figure 5 for Y = 0.0065.
Shigeo KIDA, Michio YAMADA and Kohji OHKITANI
46
3 --II 32
X1
31
31
G=
G=
Figure7. Stream lines on plane $r
I X1,ZZ I
and
x3
3
31
Gr
1 = z7r
for (a) u = 0.1,
(b) u=0.0065 and (c) v = 0.0005.
3 -7r 32
31
mn
Route to Chaos in a Navier-Stokes Flow
S
P
,
0 0
.QPn 115 34
86 26
,
QPs,
c
41
... p ...
. . . . . .
c llv RA
200 52
Figure 8. The process of temporal complexity.
x lo7
3
2
24,
1 F 4
1
0 2.50
2.55
2.60
2.65
CJ Figure 9. The parabola fitting of the inverse of the frequency power spectrum of the energy for v = 0.011.
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 10, 49-82 (1989) Recent Topics in Nonlinear P D E IV, Kyoto, 1988
Periodic Solutions of a Viscous Gas Equation Dedicated to Professor Hiroshi FUJITA on the occasion of his sixtieth birthday
Akitaka MATSUMURA
and
Kyoto University Department of Mathematics Kitashirakawa, Kyoto Japan 606
Kanazawa University Department of Mathematics Marunouchi, Kanazawa Japan 920
5
Takaaki NISHIDA
Introduction
1
We consider the one-dimensional motion of viscous gas on a finite interval.
When the gas is assumed barotropic or polytropic,
it is known by Kanel'[3] and Kazhikhov[61 that the initial boundary value problem with fixed boundary has the unique global in.time solution which decays to the constant equilibrium state as time tends to infinity.
While in the case the time-dependent external
force or the piston acts on the gas, the obtained global in time solution has only the bound which depends on time (Kazhikhov[61, ItayaI21) so that the asymptotic behavior in time is not known. Here we consider the isothermal gas motion under the time-dependent external force or under the moving boundary condition (piston problem).
We will show that if the external force or the piston
motion is bounded with respect to time, then the bounded solution exists globally in time, and that if the force or the piston motion is periodic in time, there exists at least a periodic
49
Akitaka MATSUMURA and Takaaki NISHIDA
50
solution with the same period.
Last we show several examples of
computer simulation for the periodic problems which suggest some stability properties of'the periodic solution and then we remark on the periodic piston problem for the viscous heat-conductive gases.
5
2
Formulation and Theorem The one-dimensional motion for the viscous gas is well
formulated by the following system of equations in Lagrangian mass coordinate.
(2.1)
i
-
U V tt
t- P u X x
0
.
UUX
( - I x t f
=
V
Here
v = I/P
is the specific volume,
p(v)
is the pressure,
u
is the velocity,
u
is the viscosity coefficient and
is the external force, and the suffix
t
or
x
differentiation with respect to the variable.
p =
f
denotes the partial In what follows we
will assume that the gas is isothermal, i.e., a
(2.2)
p
=
-
I
a
is a positive constant,
V
and that the viscosity coefficient is constant (2.3)
u
=
constant
>
The system of equations ( 2 . 1 )
Q
0
,
is considered on a fixed domain
in the Lagrangian mass coordinate
Viscous Gas Equation
51
with the boundary conditions on the boundary
where
x =
0 and
x = 1
is the given function (piston velocity).
u,(t)
Since the left boundary is fixed, the transformation between Euler coordinate (r,S)
and Lagrangian mass coordinate
(t,x)
is
given by
,
T = t
(2.6)
5 =
I
.
X
0
v ( t , x ) dx
Thus the time dependent external force has the form X
f
(2.7)
=
f(t,
0
v(t,x)
dx
1
.
We treat two initial boundary value problems. (i)
The external forcing problem, i.e., with
the system (2.1),(2.5),(2.7) (ii)
.
0
u,(t)
The piston problem, i.e., the system (2.1),(2.5),(2.7) with
f(T.6)
5
0
.
The initial data
are supplied, where we may assume
f
(2.9)
(2.10)
c-1
1
0
V0(x) dx
=
vo(x)
c
1
(2.11)
and the compatibility condition
I
for a positive constant
C
,
Akitaka MATSUMURA and Takaaki NISHIDA
52
Here
L 2 function on i0.11
is the Sobolev space of
/IJ
first derivative also belong to L 2 Theorem _____ 1
If the external force
respect to
L
T
0
,
0 6 5
5
1
whose
.
f = f(?,<)
is bounded with
together with the first
derivatlves, then the external forcing problem ( i ) with ( 2 . 8 ) has the unique global in time solution, which is bounded
(2.11)
in
tir-norm, i.e.,
Theorem 2 x =
If the piston does not collide the fixed boundary 0
and does not go to p l u s infinity, i.e., X ( t )
XO-l
1 t
I
C
0
<
u l ( s ) ds
Xo
,
for a positive constant where
X ( t )
Xo
,
is the piston path, then the piston problem (ii)
witn ( 2 . 8 ) - ( 2 . 1 1 )
has the global solution, which is bounded in
H 1 -norm, i . e . ,
Theorem
3
X(t)
If the external force ( 2 . 7 ) is periodic with respect to
t
or the piston motion
,
then the problem (i)
or (ii) with (2.9) has a time periodic solution respectively with the same period.
( v ,
u ) ( t , x )
Viscous Gas Equation 1
53
External Force Problem
3
(
V
t
-
t t O (3.2)
=
U
u(t,O) = 0
0
,
0 5 x 5 1
.
,
u ( t , l ) = 0
,
t
2
.
0
When the initial data ( 3 . 3 ) satisfies ( 2 . 9 ) - ( 2 . 1 1 )
f(r,S)
and the force
is bounded together with the first derivatives, the
exists for any
global solution of ( 3 . 1 ) - ( 3 . 3 )
t 2 0
. Kanel"31
Therefore it is sufficient for the proof of Theorem 1 to obtain the bounded estimate of the solution in the norm of It is easy to see by the equation ( 3 . 1 )
f
(3.4)
1
v ( t , x ) dx
=
0
I
1
and ( 2 . 9 ) =
vo(x) dx
1
0
.
The energy conservation law for ( 3 . 1 ) gives the following.
Multiplying ( 3 . 1 ) by
vx/v
and integrating it on
have (
I
u v - ( X ) 2 0 2 v
vX - dx
)t
t
V
It follows from these two equalities
I
1 avx
0
2
-dx v3
(0.1)
, we
H1
.
54
Akitaka MATSUMURA and Takaaki NISHIDA (
J
2
1 1
u
uv
0 2
2
v
- u 2 - - x
(3.5)
I
+
2
vx 4 v2
t-
t
a(
=
I
1 u
ux2 av 2 - ( - t & ) d x v3 0 2’ v
-
v
1
-
l o g v ) d x jt
1
0
u f t -U V X 2v
f
dx
.
Let u s define 1 u2
( 3 . 6 ) E(r)’
-
1
5
0
2
-
2
uuv
+
2v
vx
2
.
t a ( v - 1 - l o g v ) d x
~
4 v2
It is easy t o see that
I
1
2
1 u2 t
-(
0 12
2
vx ) -
(3.7)
“2
2
1 5
E(t)’
5
+ a ( v - l - l o g v ) d x
u2 t
0
2
vx vla( t
v
- 1
-
6
l o g v ) dx
.
Proposition 3.1 (3.8)
s u p E(t)’
(3.9)
sup
(3.10)
t2O
where Proof
inf
v
( v-I.
u
tr0,05x51
I
,
C
5
tzo
c-1
L
fl,,
C = C(E(0)2.
First w e note that
,
inf v o , v )
05x51
1
because of the positivity of v
f ( t .
J
X
v dx) 0 and ( 3 . 4 ) .
is independent of
1
If\,
5
Using the boundary
condition and (3.4) w e know
V
and also we have
V
t
.
Viscous Gas Equation
V
55
V
Thus the equality (3.5) can be estimated a s (3.11)
dE(t)' t
- I -u x t x
dt
4
8 V
2 3 d x
v
flm
C
5
V
2
-
Now we compare the first and the second terms in the left hand side of the inequality (3.11). that there exists a point x o (3.12)
V(t#X0(t))
It follows from the equality (3.4)
I
=
xo(t)
such that
.
1
Therefore we have the estimate
Thus we get
(3.14)
i
5
=
v
1/2
( 8 % d x )
d
I
5
t
V
ux 2
I
u2 dx
lu(m2
5
d
I
I
v3 dx
.
dx
V
Furth rmore let us compare two quantities: (3.15
X
=
f
v 2 TX d V
Using (3.13) we have
x
,
Y
-
2 I + d x . V
2 X
Akitaka MATSUMURA and Takaaki NISHIDA
56
Let us consider the function
Since the function y
, the inverse function of
0
2
is monotone increasing with respect to
G
x
exists and has the
= G(y)
estimate
X
and
is monotone decreasing with respect to
Flx)
X .
Then the estimate (3.16) can be rewritten as G-'(x)
(3.19)
Y
2
G-'(x)
By using monotonicity of
and the inequality (3.7), i.e.,
~
X
X
5
,
cE') , c = 1 2 / v 2
we obtain from (3.19)
G-' ( c E 2 )
(3.20)
G-'(X) x
5
-
cE2
X
X
S
Y
.
T h u s by summing (3.14) and (3.20) and using (3.18) we have
G-l ( c E 2 )
&-I (cE2) F2
~
cFL
CE
2 5
r 8
,
5
-
X
dx
2
2
t
i----j-dx" x
V
2 J u 2 t -
"x
2 t a(
v
- 1 - log v 1 d x
V
2
t
a
(
1
t
I
Z
2
d
x
.
)
3
V
Therefore we arrive at the main estimate.
(3.21)
dE2
-
aU
3
t
conclude from (3.21
)
Viscous Gas Equation (3.22)
SUP E 2 ( t )
The bound ( 3 . 9 ) for inf
(3.23)
C ( lfIm2, E ( 0 ) ' .
6
tro
57 a.
u
-
)
is obtained from ( 3 . 2 2 ) and ( 3 . 1 3 ) .
v v
.
C-'
2
t20,06XSl
Last it is not difficult to obtain d
- ( E(t)2 t
if we choose
5
4
ux2 dx )
2
dt
a and v
t
u
Z
X
dx )
Q.E.D.
small.
2
0
,
0 5 x 5 1
9
u(t.1)
(4.2)
u(t.0)
=
(4.3)
v(0,x)
= vo(x)
0
,
.
= U,(t)
-
u(0.x)
When the piston velocity
,
uo(x)
t
,
2
0
.
0
6 x
2 1
-
is given, the piston
ul(t)
problem is to solve ( 4 . 1 ) - ( 4 . 3 )
with ( 2 . 9 ) - ( 2 . 1 1 ) .
The equation
and the boundary condition ( 4 . 2 ) gives the piston path 1
(4.4)
t
Piston Problem
t
(4.1)
( E(t)2
V
=
1 t
1
8
t
0
U l ( s ) ds
>
X(t)
t
0
Now we make a transformation of the unknown variables for
Akitaka MATSUMURA and Takaaki NISHIDA
58
the piston problem to obtain one similar to the external force
problem.
Let us define
(4.5)
lI(L,X)
u l ( t )
x
X ( t )
0
-I
=
a n d rerjlace the velocity
U l ( t )
=
L‘ t
by
u
2.
u l ( c l
2.
u - ( - t - ( -
- -
X ( t )
,
v ( t , x ) dx
u
in the equation (4.1).
d
u l ( t ) )
X ( t ) 2
d t
)
X ( t )
J
satisfies the boundary conditions.
14.71
=
‘L
0 ,
u ( t . 1 )
=
v ( t . x )
0
‘:‘he function u”(t,O)
.
X
dx
.
0
Furthermore let us define the functions
and rewrite the system (4.6) with respect to W
(4.9)
X
c m t
-
2 X ( t )
w
+
( L ) x
1
(m,
. We
u)
have
0
=
1
uw
-( 2 ) x
=
x(t)
rn
m
-
X(t)-
dul d t
x
I
0
m(c,x)
where they satisfy the same conditions as the external force
problem : 1
(4.10)
_r
(4.11)
r(t.0)
dx
m(t.x)
=
1
0 =
0
1
w ( t , l )
, = 0
,
dx
,
Viscous Gas Equation
59
and also "the external force" has the form (4.12)
f
dul(t) x
X(t)-
a.
0
dt
m(t.x)
.
dx
Since the existence of global in time solution to the piston problem (4.1)-(4.3) with (4.4) is known by ItayaL21, it is sufficient for Theorem 2 to obtain the a priori estimate in H 1 norm of the solution for (4.9) uniformly with respect to t Proposition (4.13)
4.1
x0-l
2
0
.
If the piston path satisfies the condition
<
x ( t )
(
x0
and its velocity u,(t)
for a positive constant
XO
is bounded together
= dX(t)/dt
with the first derivative, then the piston problem ( 4 . 1 ) (4.4) with the condition (2.9)-(2.11) has the estimate (4.14)
inf
v
t
C-1
,
t~0,05xsl
is independent of Proof
t t 0
.
Similarly to the external force problem it is not
difficult to obtain the inequality for the solution
(m,
W )
of
(4.9). I (
(4.15)
- w 2 2
t ax2( m
- 1 - log
m
)
dx j t
Multiplying the second equation of (4.1) by
t
mx/m
I
uw Xm
dx
and using the
Akitaka MATSUMURA and Takaaki NISHIDA
60
first equation of ( 4 . 1 ) , urn
we have
wm
am
( I2 - 2 dx dul
m
x
dt
m
0
I ( - L J
=
It follows from ( 4 . 1 5 ) ( 4 . 1 6 ) 1 -3
i
I
w
2
t
2
a
2
I 2
t
Xm
2m2
(4.16)
)t
dx
Xm
m d x t -
2 w x t - ) d x x3m
u wm 1 x x2m
that for any
a
x
-
(
m
-
~
,
au m x ~ o 2m
auwm m ) Xm
g
t d x~ )t -
~
2
1 +---
(4.17)
x2
Xm
i x
<.
1-1d'
2aX2
dx 2xm3
1
1-1
dt
Now
we can choose
and
the second terms on the left hand side of ( 4 . 1 7 ) are
definite,
and
a
so
small that the integrals of
consequently the same argument of
5
the
3
dx
m
.
first positive
can apply
Q.E.D.
to it.
5
§
mX
Periodic Solutions In the following we prove the existence of periodic solutions
o n l y €or the piston problem ( 4 . 9 1 ,
since the external force
problem (3.1) can be treated similarly and is easier than the In order to simplify the treatment of the condition
piston problem. v
>
(S.1)
0
or
>
rn
v
=
0
en
we introduce the change of variable or
m
=
e"
The piston problem has now the form
respectively
.
Viscous Gas Equation - -
n
e-"
x(t)
w(t,O)
I Proposition
5.1
2 w x
=
0
en d x
0
=
w(t.1)
1 =
61
=
0
1
Let us assume the condition of Proposition 4.1.
Let the piston path
X ( t )
and so
u,(t)
with respect to time with period T a periodic solution for (5.2)(5.3)
= dX/dt
be periodic
Then there exists at least 5.4) with the same period
which satisfies
It has th
If the piston motion is small, i.e.,
/ X I Z I m is small, then
the periodic solution is unique and stable. The proof of Proposition is composed of three parts. (i) (ii)
discretization with respect to the space variable. energy estimates to apply Leray-Schauder fixed point theory.
Akitaka MATSUMURA and Takaaki NISHIDA
62
(iii) uniform estimates with respect to the discretization.
I
Discretization Let u s discretize as follows.
Consider
the
system of linear ordinary differential1
for
w )
with periodic inhomogeneous
In,
terms,
equations
which
is
a
linearization of discretized problem of ( 5 . 2 ) - ( 5 . 4 ) .
W . 1
a(ni+l ,t
-ni)
u
-
r, x
W i + ]
-wi
-
- (
Ax
w . 1
-w.
1-1
)
AX
AX
where =
VO(t)
'
(5.10)
Lemma ___
5.2
.-
T
0
1
V
C
expfni's))
Ax d s
is required.
T
=
0 !=I
Let
and
h = (hl,"',hN)
smootn and periodic in ?i (5.11 1
and
WN(t) = 0
I
i-1
r 0
hi(s) ds
?!:ex there exists
=
with period
t =
C 2 ( ~ , A x )2
... , g N - l ) . Assume
(gI, T
be
.
0
I
such that the linear system
IS.8)(5.9)(5.10)
has a unique periodic solution
the same period
T
which satisfies
(n,
V )
with
Viscous Gas Equation
Proof
63
After deriving the necessary estimates for the periodic
solutions to (5.8)(5.9) we will notice that the requirement (5.11) is the necessary and sufficient condition for the existence of periodic solution to the linear inhomogeneous ordinary differential equations (5.8)(5.9) with periodic force terms. Thus
we
derive
solution (n. w ) (5.9) by
,
wiAx 1
.
first the estimates Multiplying (5.8) by
N
=
N C ( anihi i=l
"i.t
-
(5.15)
0
the
=
Ax B;
periodic
fi;
.
-ni-l)
1.t
Since
t wigi
i t 1 -wi Ax a(ni
=
the
periodic and
i = 1,2."',N
t
) Ax
z
i =O
witl
U(
-w.
2 Ax
Ax
-
0
0
(
AX
W .
0
,
aniAx
N-1 )t
for
and adding them, we have
i= 1 , 2 . " ' , ~ - 1
__ z ( a n i 2 t w i 2 2 i-1
(5.13)
(5.12)
HalanayLlI.
t
i
-
, u
wi+l
- (
Ax
0
-wi
Ax
= 1,2.'",N-l
solution of
,
i= 1,2."',N
homogeneous
* + - w 1. - w 1. - 1 Ax
I
t
adjoint
equation
Akitaka MATSUMURA and Takaaki NISHIDA
64
has the similar estimate to ( 5 . 1 4 )
(5.15)
i= 1,2,"',N. ii
Witl
a = wo
= w.
Using ( 5 . 1 5 ) a
0
0
=
1.t
Thus
=
i = l,.?;",N-l,
,
,
we have
.
i = N-1,"*.1
a
n .
,
constant
=
,
i = 1,2,"',N
,
a
w .
=
,
0
and
i = I.Z,"'.N
.
the necessary and sufficient condition for the existence of
periodic solution to ( 5 . 8 ) ( 5 . 9 ) :s
h . = 0
from this we see
n.
(5.16)
,
= 0
g:
a
with
is the inhomogeneous term
(h, f )
orthogonal to ( 5 . 1 6 ) which is equivalent to ( 5 . 1 1 ) . I f we remember the boundary condition for
w
and so we use the
inequality
in ( 5 . 1 4 ) ,
we know
Substituting the equation ( 5 . 8 ) in the equation (5.9) we have
nit1
Multiplying to
-ni
Ax
Ax
, we can obtain the estimate.
i = O.!."'..V-I ?:
Z - ( 2
and adding the results with respect
nitl
Ax
-ni )Lt
Ax
t
a
- E 2
(
n i t l -"i 2
Ax
1
Ax
65
Viscous Gas Equation
Using ( 5 . 1 4 ) we have after integration with respect to
Multiplying the equation (5.9) by
wiAx
,
nitl
-n.
t
i- l . Z , " ' , N - l
,
and adding them, we have N
( Z
(5.20)
1
-
i=I 2
w i z ( t ) A x )t
5
C
N C
is1
(
Ax
.
wi t wigi) Ax
Here we remember the following lemma. Lemma5.3 period
Let the function
y(t)
0
2
,
z(t)
be periodic with
T and satisfy the differntial inequality : dy(t) 6
dt
z(t)
.
Then we conclude that max y ( t )
OdtsT
I
d
T
-I T
O
y(t) dt
t
I
T 0
Iz(t)l
dt
If we apply this lemma to the inequality ( 5 . 2 0 ) estimates ( 5 . 1 7 ) ( 5 . 1 8 ) ,
To remove the term
. and use
the
we have the inequality
In(s)lm
in the right hand side of ( 5 . 2 1 )
need the following cosideration.
we
Akitaka MATSUMURA and Takaaki NISHlDA
hh For
our
solution
(n.
v )
of (5.8)(5.9)
the
requirement
(5.10) i s a l w a y s s a t i s f i e d if w e add a s u i t a b l e c o n s t a n t t o
Then t n e r e e x i s t s
t o
E [O.Tl
s u c h that
.T
X
i-1
exp(n,(ro))
and so t h e r e e x i s t
i,
BX
,
j,
=
1
such that
Usinq ( 5 . 2 2 ) w e easily h a v e
Applying lemma 5.3 we have
Especially we h a v e
to t h e inequality (5.18) a n d u s i n g (5.14),
w
Viscous Gas Equation
gl,
max I h ,
C
6
67
OstiT
Q.E.D.
Periodic Solution of Nonlinear Ordinary Differntial Equations
11
We turn our attention to the discretized problem for n = (nl,n2,
... , n N )
,
w = (w ,w 1 2'
"'
of the equation (5.2),
'wN-l)
which is a system of nonlinear ordinary differential equations
_ -
"i.t
e-'i
w.
X(t)2
(5.23
-
-wi-l Ax
e - n . w I. - w1.- 1 1
j
=
,
0
=
g
i
,
i= 1,2,"'.#
.
,
i = 1.2,"'.N-l
AX
where
the boundary condition (5.24)
= 0
W,,(t)
and
w,(t)
=
are imposed,
0
and the condition (5.25)
N
c
exp(ni(t))
Ax
is required.
1
=
i=l
Let the piston path and be periodic in
X(t) t
satisfy the condition of Proposition 4.1
with period
T
.
We want to solve the
discretized piston problem (5.23)(5.24)(5.25)
for any
N
in the
ball BH = (
In, w ) ( t )
= (nl(t),*.'.nN(t),
w,(t),
bounded continuous and periodic in
... 9 w N - l ( t ) ) t
,
with period
T
,
Akitaka MATSUMURA and Takaaki NISHIDA
68
with the norm
Proposition
-
5.4
Ho(
max OstzT
For any
X
L e t us denote
=
Mo =
s u c h that the periodic nonlinear
ordinary equation ( 5 . 2 3 ) ( 5 . 2 4 ) ( 5 . 2 5 ) Proof
.
1
c M
w)(t)l
there exists a constant
N
h', T )
/X.ullm,
I(n.
I
T
T
O
-I
pistor, and define
X(t)
has a solution in dt
.
B
the mean position of the
Rewrite the system (5.23) in the form : w . -w.
1-1
1
Ax
--
hi
(5.27) a(nitI 1 r t
Ax
-ni)
-
v
v - ( AX
- w. -w.
w i + l -wi
1-1
1
Ax
AX
i
where
,
i = 1,2,"',N
=
1,2,"',N-l
)
=
,
gi
1
Viscous Gas Equation
69
Here we also suppose
and the condition ( 5 . 2 5 ) T
N Z exp(ni(s)) 0 i=l
I
(5.30)
is replaced by A x ds
T
=
The original problem ( 5 . 2 3 ) ( 5 . 2 4 ) ( 5 . 2 5 ) periodic solution of ( 5 . 2 6 ) - ( 5 . 3 0 )
L
Since
solution
by
of
we can apply lemma 5 . 2 to
inhomogeneous" system ( 5 . 2 7 ) .
operator of ( 5 . 8 ) ( 5 . 9 )
side
us
Let
denote
the
and the problem ( 5 . 2 7 ) -
L-I
is equivalent t o operator equation:
(5.30)
To obtain a fixed point of the equation ( 5 . 3 1 )
A -
for
apply Leray-Schauder Theorem in the form (Vejvoda[9 Theorem
(
Leray-Schauder
)
K
Let
open convex set in a Banach space mapping from (i)
.
1
=
and so the right hand
is defined in ( 5 . 2 8 )
"linear
A
in the case
satisfies the condition ( 5 . 1 1 ) ,
(5.27)
the
is equivalent to the
F
K
X
.
a
,
B
):
non-empty
and
F
bounded
be a continuous
Suppose
is compact.
(ii) There exists a unique point (iii) F
B
[ O , l ] into
be
we
1
K
xo
is Frechet differentiable at
has the inverse in (iv) F ( x , A ) f x
for any
L(B,B)
x e aK
.
such that
(xo.O)
and for any
Then there exists a fixed point
x1
c
and
F(xO,O)
r
x0.
- Fx(x0,O)
A E [0,1) such that
-
.
70
Akitaka MATSUMURA and Takaaki NISHIDA the following we derive a priori estimates f o r t h e
in
solution of our operator equation (5.31) t o guarantee
remaining
for
x0 = (0.0)
the way we see that conditlons
are
A = 0
p c a r i o i i s solctior. for ( 5 . 2 7 ) - ( 5 . 3 0 ) , (5.32)
-
n . 1.t
Ax
a(e-"itl
(5.333
w .
1 . t
-e-"i)
e-n.
w. -w. I
Ax
unere
fl
Multiply
is
1-1 )
uiAx
by
=
and
respectively and sum up them.
N
L -
j=1
Since
w)
the
1s
i = 1.2."',N U
-
X Ax
f
I
i
,
(
-wi
e-"itl
Ax
i = 1,2."',N-l
given by (5.26).
(5.33)
t
-
Ax
,
0
=
1-1
1
X2
(n,
it satisfies the equation
e - n 1. w . - w .
-
o u r case and the
in
If
easy t o see.
( 1 ~ ) .On
oe-"i
w . ~
X
1
(5.32)
We have
-w.
AX
1-1 ) 2 A x
5
by
a(e"i
-])Ax
,
Viscous Gas Equation Consider (5.33) after substituting ( 5 . 3 2 ) 'it1
( u
-"i
Multiply it by
a(e-"itl
-
)t
Ax
-e-"i)
71 in it. wi,t
X Ax
(witl - w i )
-
i
f
X
X
and sum up with respect t o
. We
i
have N-1
nitl - n .
P
z
2
i=l
w . n i t 1 -"i 1 , Ax X Ax
1 -
-
1)2
( - (
Ax
N - I a ( n i t l - n i ) e - n it1 . -e-"i
- z
(5.37)
Now
we
0
(5.371 by Q
Z
X Ax
denote
the second term on the left
,
hand
side
of
i-e.,
Z
i-1
Ax
Ax
N-1 n i t l -n Z ( i)2 i=l Ax
=
Ax
N-1 nitl - n i e - n i t 1 -e-"i
-
Ax
i=l
I
1
Ax
exp(-(nit8(nitl-ni)))
0
dB Ax
.
Using this notation we estimate the second term on the right hand side of (5.371, i.e., N-1 I
(5.38)
C
i=l
If], 5
-
X
f i nitl - n i
-
X
Ax
N-1 nitl - n i E l Ax i=l
I
Ax
1
1 (
0
exp(-(nit6(nitl-ni)))
d6
)'I2*
72
Akitaka MATSUMURA and Takaaki NISHIDA € 0
5
lfl, 2 -
t
EX2
where we have u s e d the inequality : H- I
Z
i =I
I
Ax N-1
Z
c
1-1
N-l
Z
5
i=l
r,
J
I
exp(-(nit8(nitl-ni)))
d8 )
1
0
d8 AX
exp(8nitlt(l-8)ni)
1 @ exp(nitl)
(1-8)
t
e x p ( n i ) dB A x
0
I N-1 - Z
>
0
(
2 i=l
exp(nitl)
t exp(ni)
)
Ax
.
1
S
Similarly we have from (5.35)
I
(5.39)
N-I
u w. n , ' I l t l - n i A x l
z -
i-1
x2
5
/u,lm2
cQ+-E-
EaX
AX
3
e-"i w . - w . ( 1
x
1-1
Ax
and aiso we have from (5.25)
.
(5.40)
max ~ n I . i
Therefore
we conclude from (5.34)(5.36)-(5.40) for small
t
(5.41
S-1
a~ I(---(
i-1 )
H t
I:-
Q"~
5
2 nitl
2 ,ie-"i
-n. 1)2
-
X
Ax w. -w.
( 1
a u w i. n i. t 1 -
1-1
1 2 Ax
24
Ax
-"i
) c Ax
a
0
)
2 Ax
Viscous Gas Equation
By the way we notice here that if uI = 0
,
X
73
Q
0
and so it follows from ( 5 . 4 1 ) that
, then n = 0
,
0
and
w = 0
and
f
=
that the condition (ii) is satisfied. Thus we have for any
[0.1]
from ( 5 . 3 5 ) ( 5 . 4 0 ) ( 5 . 4 1 )
m
and by ( 5 . 2 5 ) we have
Then using lemma 5.3 we have
and so we have
Therefore we
can take
Mo
= C
,
which guarantee the Q.E.D.
(iv).
I11
condition
Uniform Estimates With Respect To
N
We want to obtain similar estimates to the continuous case. Let
Akitaka MATSUMURA and Takaaki NISHIDA
74
N-I f
au2 n i t l
(-(
i-1
2
- -a u w 1. n i. t 1 - " i
-ni)2
X
Ax
) Ax
Ax
Since we have ( 5 . 4 0 ) , N-1
P Thus
: Z ( i=l
nitl
-n. 5
Ax
Q
exp(ln1,)
Q
5
.
exp(a'/')
in the similar way to the continuous case in 5 3 using
function
G ( y ) = y e x p ( y'")
(5.42)
P
we have
G(Q) , namely
4
the
G-'(P)
.
Q
5
Therefore ( 5 . 4 1 ) gives the following for some constants
U
and
c.
dt
Integrating it with respect to
J
(5.44)
T
G-'(cE2(t))
0
If we multiply (5.43) by G - ' ( c E ~ ) ~t
(S.45)
5
c
( 1 t
dt
we have
t 5
,
c G - ' ( C E ~ ) ~ cvG-lx
If.u1l,2
C-l )
If, u 1
C ( 1 t
c-lx
is
property (5.45)
proved G-lx
we have
5
by G-l
the explicit for large
,
.
we have for any
Cl
Cl
t
t
c1
Here we use the inequality for a constant
which
m2
.
C2
expression
.
(3.17)
and
the
Using this inequality in
Viscous Gas Equation If we notice
G-lx
75
o and integrate it, we have
)
Thus using lemma 5 . 3 and ( 5 . 4 3 ) we have
Therefore we have obtained the uniform estimate for
,
lnlm
We can redo the estimate using this maximum norm estimate to obtain max
OStbT
(5.48)
E2(t)
If,
c
5
Ax dt
" i t 1 -"i)2
, f
0 1.1
UllOJ
2
Ax
Ax
9
Last estimate can be obtained by multiplying ( 5 . 3 3 ) by - (
witl
Ax
N-1
I:
1 - (
ill 2
witl
-Wi
-w
Ax
iPt
Ax
, i.e.,
wi - w i - l )
Ax
c -( - - (
-wi
Witl
Ax
2X
ill
Ax
i.1
1
N - 1 ue-"i t
-
wi
Ax
lfIm2
Ax
Thus we have by using ( 5 . 4 7 ) ( 5 . 4 8 ) N-1
C
max
(5.49)
OStbT i l l
I
T N-1
I:
0 ill
1 witl
-(
Ax
2 1
{ - (
Ax
w
Ax
-wi)2
i t 1 -wi
Ax
-
,
wi - y i - l
Ax
- w ~ - ~l )2
Ax
)
I2
Ax
1
'
Akitaka MATSUMURA and Takaaki NISHIDA
76
At
last using these uniform estimate (5.47)(5.48)(5.49)
take
the
limit
as
N
+
Proposition 5.1 as follows.
,
=
x
= iA.r
subsequence
t
ni(t)
2
w(N).(t,x))
, respectively.
0
, 0
,
i = 1,2,"',N,
5 x
4
1
As
the functions
j
-L
m
obtain
2k,
=
which coincide
(n(N),
and have the uniform estimates (5.47)(5.48)(5.49), a subsequence
to
, N
,
wi(t)
of (5.32)(5.33) on the mesh points
I.?."',N-I,
I
(n(N)(t,x),
in the strip
with the discrete solution
a
We can construct continuous and
piecewise linear functions k = I,?,"'
along
m
we can
x
w('))
=
(i-l/Z)Clx
satisfy
we can select
such that
the limit of which satisfies
ar.d
is
a weak solution of our piston problem (5.2)-(5.4).
By using the ellipticity of the second equat on of (5.2) with
respect to the space variable we obtain
,
Viscous Gas Equation
77
Therefore since the second equation of ( 5 . 2 ) can be regarded a linear parabolic equation with respect to
, it follows from the
w
energy estimate using Friedrichs mollifier in
and
(n, w )
t
that
becomes a strong solution of ( 5 . 2 ) - ( 5 . 4 ) .
This
completes the proof of Proposition 5 . 1 .
§
I
6
Remarks Numerical Computations The nonlinear differential equations ( 5 . 3 2 ) ( 5 . 3 3 ) can be
discretized with respect to time to give a simple explicit finite difference scheme for the variables AX
n/N
=
-
vk
,
y
-
v(kAt,(i-l/z)Ax)
At
u
r
:
(v. u)
At/Ax
=
constant
,
J I:
=
u(kAtriAx)
=
ul((ktl)At)
, ,
Ax
=
0
,
U kN t l
,
We carried out several computations using the scheme ( 6 . 1 ) - ( 6 . 3 ) ,
78
Akitaka MATSUMURA and Takaaki NISHIDA
when the piston velocity is given by (6.4)
ul(&)
a sin
=
W t
I
and the initial data are taken from the followings: (5.5)
vo(x)
Example
1.
u
1
=
b cos j x
f
,
= 0.01
,
a = 1.0 ,
uo(x) w = 1.0
Computer solutions converge to
Example 2 .
LI
, u
= 0.01
= 1.0
,
,
c sin j x
(period
=
j
2n),
= 1,2,"'
N
= 256
.
periodic function with the
a
.
c
same 2n period in
=
w = 2.5
(period = 4 n / 5 ) ,
.
N = 256
Computer solutions converge to a two-periodic function with the period 8n/5 in
t
u
,
Example 3 .
-
0.01
.
a = 1.0
,
w
-
-
4.0 (period
n/2),
-
N
.
256
Computer solutions converge to a three-periodic function with the period 3 n / 2 in Example 4 .
u
.
t
, a
= 0.01
=
0.5 ,
w = 4.0
(period
=
N = 256
n/2).
,
Computer solutions converge to a periodic function with the same n / 2 period in Example 5.
u
t
,
= 0.01
a
-
.
3.0
,
w =
4.0 (period = n / 2 ) ,
.
N = 256
Computer solutions converge to a two-periodic function with the period n in Example 6 .
u
t
= 0.1
.
.
a = 1.0 ,
w =
4.0 (period
=
n/2),
N
256
=
.
Computer solutions converge to a periodic function with the same n / 2 period in Example
7.
u
= 0.0025
t
,
.
a = 1.0
,
w = 4.0
(period
=
n/2),
N
= 512
Computer solutions converge to a three-periodic function with the period 3 n / 2 in
t
.
These numerical examples suggest that in some cases
.
Viscous Gas Equation
79
(Examples 2,3,5,7) the periodic solution with the same period as the piston which was obtained in our theorem i s not stable or at least not a global attractor.
11
It should be clarified.
Remark on the periodic piston motion of the general gas
The one-dimensional motion of viscous and heat-conductive gases is governed by the system
where we assume that the equation of state for the internal energy e
of the gas is given by
(6.7)
e = e(v,S)
,
and so
ae
ae
-=
- p ,
aV
- = e as
.
Here we notice that the polytropic gas is a special case
We
consider the piston problem for (6.6)(6.7) with the
boundary
conditions (6-9)
u(t.0)
= 0
I
u(t.1)
=
U 1 ( t )
,
If the gas is polytropic, then the initial boundary value problem (6.6)(6.8)-(6.10)
has a unique global in time solution
for each smooth initial data such that
Akitaka MATSUMURA and Takaaki NISHIDA
80
It is proved by Itaya[Z] and Kazhikhov[61.
_ Remark _6.7
Let the piston velocity
period T
I
(6.11)
ul(t)
be periodic with
and
T 0
u,(t) dt
=
0
-
Then the piston problem (6.6)(6.7)(6.9)(6.10) periodic solution with the same period
does not have any
T except the piston
velocity vanishes identically. Assume that there exists a periodic solution the same period T
( V , U . @ ) with
for the periodic piston motion 1 6 . 6 ) ( 6 . 7 ) ( 6 . 9 )
We use the energy conservation law (the third equation
(6.10).
of ( 6 . 6 ) )
and an energy equality by Okada-Kawashima [a].
e (6.13)
E(v,u
-
e = e
Integrate the energy conservation law with respect to in
[O,Tl
X
[O,ll
and use the periodicity with respect to
and the boundary condition (6.9)(6.10). (‘5.14)
T
:
0
t .
P l t , l ) U l ( t )
t
We have
dul(t) LJ l o g v ( t . 1 ) ___ dt
d
t
=
O
.
x
t
Viscous Gas Equation Integrate the equality ( 6 . 1 2 )
o *
o
e
ve
T1 PUl(t) 0
in the same region.
,
We have
V
-
p(t,llul(t)
because of ( 6 . 1 4 ) and ( 6 . 1 1 ) . u = 0
81
9 = constant
-
IJ l o g v ( t . 1 )
dUl(t)
-d
t
dt
-
O
.
Thus we obtain
, and
v = constant
. Q.E.D.
References
[TI
A. Halanay, Differential Equations, 1 9 6 6 , Acad. Press, New York
[21
N. Itaya, Some results on the piston problem related with fluid mechanics, J. Math. Kyoto Univ., 2 3 ( 1 9 8 3 ) , 6 3 1 - 6 4 1
[31
J. I. Kanel', On a model system of equations f o r one-dimensional
gas motion, Differential'nye Uravnenija (in Russian
),
4 (7968)
721 -134 [41
J. I. Kanel', Cauchy problem for the dynamic equations for a viscous gas, Sibirskii Mat. Zh., 20 ( 1 9 7 9 ) , 2 9 3 - 3 0 6
151
A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of the initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 ( 1 9 7 7 ) , 2 7 3 - 2 8 2
[61
A.
V. Kazhikhov, To a theory of boundary value problems for
equation of one dimensional nonstationary motion of viscous heat-cionductive gases, Boundary Value Problems for Hydro-
82
Akitaka MATSUMURA and Takaaki NISHIDA dynamical Equations
(
in Russian
)
, No.50,
(1981), 37-62, Inst.
Hydro-dynamics, Siberian Branch Akad., USSR. [7] J. Leray and J. Schauder, Topologie et equations fonctionelles Ann, Sci. Ecole Norm. Sup., 51 (1934): 45-78 181
M. Okada and S. Kawashima, On the equations of one-dimensional
motion of compressible viscous fluids, J. Math. Kyoto Univ., 23 (1983), 55-71 191
0. Vejvoda, Partial Differential Equations: Time Periodic
Solutions, 1982, Martinus Nijhoff Publ. The Hague
Lecture Notes in Num. Appl. Anal., 10, 83-99 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
On the One-dimensional Free Boundary Problem for the Heat-conductive Compressible Viscous Gas TAKEYUKI NAGASAWA
Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama 223, Japan'
1. Introduction. We consider the one-dimensional motion of the fluid, which satisfies the equations of state of the polytropic ideal gas, with the prescribed stress on the boundary and with adiabatic ends. By use of the Eulerian coordinate system, the motion of the gas is described as the free boundary problem by the following three equations corresponding to the conservation laws of the mass, moment and energy Pr
cvp(eI
for
(5,T ) E [ l 1 ( ~ ) , & ( ~ )x]
+ .e,)
+( P V ) ~
0,
= -Rpeve
+
IJV;
+
[0,oo), with the initial condition
and the boundary conditions
Present address: Mathematical Institute, T6hoku University, Sendai 980, Japan
83
Takeyuki NAGASAWA e,(eI(T), 7)
d el ( T )- V(e,(T),T),
dT
(z =
--oO
= 0,
< e,(o) < Cz(0) < $00
1, 2). ( p , v , a), unknown functions, represent the density, the velocity, the
absolute temperature of the gas; ( R ,p , c v l K ) , given positive constants, stand for the gas constant, the coefficient of viscosity, the heat capacity at constant volume and the coefficient of heat conduction respectively. P ( T ) ,given function, represents the outer pressure.
el ( T ) and &( 7 )are curves defining free boundary.
In discussing this problem it is convenience to transform the above equations in the form described by the Lagrangian mass coordinate system that is denoted by ( 5 . t ) . We may assume, without loss of generality, the initial value p o of p satisfies is given by
/(
rtifo)
p o ( ( ) d [ = 1. Then the relation between ( ( , r )and (s,t)
1(0)
For the sake of notational simplicity, for example, we write the function
~ ( ( ( x), t ; t ) as z(x, t ) . By simple calculations, we know that the functions 1
-. v.0) satisfies the system of equations P
(1.2)
,
ui= (-RB+p?) U
.: -11
cvel = -R-ev, + p- U + U
I
($)=
for ( z . t ) E [O?11 x [0, m), with the initial condition
ijrld thr. borinda.ry conditions
(1.5)
(-R! + a %U ) ( 0 , t ) = (-R! + p 2U)
(lit) = - P ( t ) ,
(u
=
Free Boundary Problem for the Heat-conductive Compressible Viscous Gas
e,(o,t) = e,(i,t)
(1.6)
85
= 0.
P ( t ) is a given C'-function. From the (1.2) and (1.5), it follows that g'vdx
=
1
1
vodx.
Since our system is invariant in adding any constant to u , without loss of generality we may assume the above integrals are zero:
On our problem, Kazhykhov [2] showed the global existence of solutions for P ( t ) E 0. We considered the case of P ( t ) > 0, and established the existence theorem in [4]. In this paper, we mainly discuss the large-time behavior of solution. The solution behaves in different ways in response to signP(t). For example, for
P ( t ) = P > 0 we have a trivial solution (1.8)
u(x,t) = ii,
+,t)
= 0,
qX,t)= 8
corresponding the initial data u ~ ( x=) I,
~ ~ (= 2 0, )
eo(z) = 8,
where ii and 8 are positive constants satisfying the relation Pii = Re.
(1.9)
On the other hand, for P ( t ) G 0 there exists a trivial solution
(1.10)
(
:>
u(x,t) = G(1 + t ) , v(x,t) = ii x - -
, B(r,t) = e
corresponding the initial data u o ( 2 )= u,
vo(x) = u
(x - -:) ,
fJO(X) =
8.
Takeyuki NAGASAWA
86
For this
and
e are positive constants satisfying the relation
We must pay attention to the large difference between the behavior of U(I.
t ) . It may be explained in the following way. Since the specific volume u and
the absolute temperature
must be positive by the physical reason, the signature
of the velocity gradient u, at the boundary is determined by the difference
e
between the inner pressure R- and the outer pressure P ( t ) (see (1.5)). The 21
velocity gradient v , governs the constriction of the gas through (1.1). Therefore
if P ( t ) is positive, then the solution shall converge the stationary state, and
e
R - and P ( t ) balance each other (1.9). If P ( t ) is non-positive, then U
v , at the
boundary is always positive and the specific volume u shall grow to infinity. In $2, we shall mention the global existence of the solution for any C1function P ( t ) in Holder class. In 333 - 5, the mathematical analysis on the above conjecture will be developed. Though all results will be stated in the terminology of the Lagrangian mass coordinate system, it is easy to rephrase in the Eulerian one. For function spaces
and B;'"
we should refer to [3, Eq. (2.2)
- (2.6)]. And other spaces W'**(O, 1) etc. are commonly used one. From now on,
C and C(.) etc.
denote positive constants depending on (their argument(s) and)
possibly the initial and boundary data. For convenience we frequently denote different constants by the same symbol C even in the same sentence.
2. Global existence.
The existence of the temporally local solution with u > 0, B
> 0 to this
problem and its uniqueness are proved in a way similar to Tani's argument [8] in Holder class
x H+za x H+za for some
TO > 0 provided the initial
B ~ ) to HI+" x H2+0 x H 2 + a . We shall establish the global data i u ~ , u ~ , belongs
existence of the solution [411where the theorem was proved under the assumption
Free Boundary Problem for the Heat-conductive Compressible Viscous Gas
87
P ( t ) > 0. However we can eliminate this restriction by virtue of the argument in this section.
Theorem 1 ([4]). Assume that the initial data
(uo,v0,80)
belonging to
HI+" x H2+" x H2+" satisfies the compatibility conditions with (1.5) - (1.7), and that P ( t ) is a C1-function. Then there ezists a solution (u,u,O) to the problem
(1.1) - (1.7) globally in time and uniquely in the class . . I?;+.+" x IT++" x H$+". Moreover both u and 8 are positive.
T>O
Since a unique solution exists locally by Tani's theorem [8],we have to get a priori estimates for the solution
We proceed the argument under the assumption u
> 0 and 6 > 0. Reading
[4, 531 carefully, we find that to show (2.1) and (2.2) we only need .1
Before proving (2.3), we give two relations of our system.
Lemma 2.1. We have
1' 1'
L1(v2
(2.5) =
+ R8 - P(r)u)dxdT +
pu d s -
u
1'
1' 1' 1' p o d s-
uo
vodfdz
v dtdt.
Proof. We multiply (1.2) by u and add the result to (1.3). Integrating over [0,1] x [0, t], we have (2.4) by virtue of (1.5) and (1.1).
Takeyuki NAGASAWA
88
The integration of (1.2) over [0, x] and (1.1)yields
We integrate this equation over [O, 11 x lO,t]. To integrate by parts the first term of the left-hand side, we need (1.1). The result is (2.5). I Proof of Theorem I. First we shall show
u)(z,~)dz5 d ~C(T) for
05t
I T.
I
b y (1.1) and (1.7), it is expressed as rz
rl
where
By use of the relation
G(z,()= G ( ( , x ) ,we have
Therefore w e integrate both sides of (2.5) with respect to time variable, and then get
(2.7)
Free Boundary Problem for the Heat-conductive Compressible Viscous Gas
89
Noting that G(z, <) is non-positive and that u ( x ,t ) is positive, we have
from (2.4), (2.5) and (2.7). Then an application of Gronwall’s lemma gives rl
rt
Noting (2.4), we get (2.6). Next we shall show (2.3). We calculate the second term of the second side of (2.7). With the the help of the well-known formula
we obtain
The last equality follows from Plancherel’s theorem, and a(t) E ( 0 , l ) is a function of t defined by
1 I,,4<,W& 1
= 0.
Since u is positive, a ( t ) is well-defined by the above relation. Now we get from (2.6) - (2.9) (2.10)
Integration by parts yields
Takeyuki NAGASAWA
90
Therefore (2.5), (2.4), (2.6) and (2.10) give
1' for any
u(s,t)ds 5 C(E,T)
E
+&l 1
u2(z,t)dz
> 0. We take sufficiently small and then use (2.8) to get rl
Noticing (2.4) again, we get (2.3). -4s stated above, in the same way as [4, 53, Lemma 3.2 ff.] we obtain (2.1)
and (2.3). Now we complete the proof.
3. The case of P ( t ) -+
p > 0.
In this section our aim is to show that the global solution converges to the state (1.8) - (1.9) when P ( t ) does to the positive constant P for arbitrary initial data. Recalling (2.4), the proof of the convergence seems to be difficult if we do not assume the convergence of integral of P'(t) on [O,oo). Hence we impose the
assumption
Theorem 2 ([4]). Assume that (3.1) holds and that the f i m i t of P ( t ) as
t
00
is positive. Then the limit
(3.2) ezists, and the solution ( u , v , e ) converges to a stationary state
\\-'-*(0.1) a3 t -+ and
00,
where
(ii,8)
(zL,O,8) in
is a root of simultaneous equations (1.9)
Free Boundary Problem for the Heatconductive Compressible Viscous Gas
91
The convergence i s dominated b y
for some
C > 1 and X > 0 which depend on R,
j i , c y , K,
P,
1
IP'(t)(dt and
initial data but not on t . Proof. Since we can see the complete proof in [4,f34 - 51, we shall only give a n outline here. Without loss of generality, we may assume
Therefore from (2.4) and (3.1), we have a uniform estimate with respect to t :
ll(vz + + 8
(3.4)
Hence the limit (3.2) exists, and ti,
e and (1.3) by 1 - -. e
(3.5)
u)(z,t)dz5 C.
8 are well-defined. We multiply (1.2) by
v
Combining these relations and (l.l),we get the identity
dt
+ e V ( t ) = ( P - P (t ))-d
dt
1
1
tL(z,t)dZ
0
after some calculations. Here U ( t ) and V ( t ) are non-negative functions of t defined by
With the help of (l.l),the integration of (1.2) over [O,z]yields an equation of
Takeyuki NAGASAWA
92
The variation-of-constantsformula gives the expression of u :
where f ? ( x , t )arid
y ( t )are given
y ( t )= exp
by
{ -i1' P
P(r)dr}.
o
By virtue of (3.3), y(t) behaves like an exponential function. Combining (3.3) - (3.6), we can obtain a uniformly estimate
(3.9)
C-' 5 u ( z ,t ) 5 C for
( 5 ,t
) E [0,11 x [0,m).
From t.his we can deal with (1.2) and (1.3) as uniformly parabolic equations with respect to v and 6 respectively. The energy method yields
(3.10)
and
The equation (1.2) can be rewritten as (3.12)
Calculating the energy
(3.13) and
(3.13)
d
-at( v - P + R ( ; )
1"
2
3'
v -p-
d r , we have
Free Boundary Problem for the Heatconductive Compressible Viscous Gas
93
By use of (3.10) - (3.14), it follows from (3.7) that u ( z , t ) + ii
(3.15)
as
t
-+ 00
uniformly in
z E [O,1].
It is a better property than (3.9). We again use the energy method with the help of (3.15). Then we obtain
q X ,t ) + 8
as
t
-+ 00
uniformly in
II(. - U ,v , e - $)~l:,~+ o L/)v:,
Bs
+ e:,)dsdT
t
z
-,
E [0, 11,
00,
5 c.
In what follows, we may proceed the argument under the assumption that ( u , v ,0) are close to ( f i , O , 8 ) in W'p2(0,1). ThereforeU(t) and Y ( t ) are equivalent
and /I(u,, 8,)lli respectively
to II(u - U ,v , 6 -
(11 . 112 is an L2(0,1)-norm).
Define
1'
+ c1
W(t)=U ( t ) If C1 and
{p(logu), -
dz
+ C2(Pl(t)+ G2(t)).
C2 are sufficiently small, then we can show
(3.16)
+
+
dW dt ( t ) C--'(W(t) V ( t ) ) 5 C F ( t )
by (3.5), where .F(t) is a function dominated by
Integrating (3.16), we get the decay rate for II(u - ii, u z , v ,8 -
(1.2) and (1.3) by
8)lli. We multiply
and BIZ respectively, and integrate
/~~ use of that for them. Then we can estimate the decay rate for ~ ~ ( v , , f l , )by
}}(a- 21, ti,, v , 8 - 8)ll$, and get our desired result.
I
Takeyuki NAGASAWA
94
We can also show a similar result when the outer pressure mined t.ogether with the location of free boundary, i . e .
P(.)is deter-
P(.)= P(!;(t)) (see
[51).
4. The case of
P ( t ) G 0.
For P ( t ) G 0, our problem is a model of motion of the gas that is put into a
vacuum. Okada [7]and Kawashima [l]considered the asymptotic behavior of the solution of this problem and showed that the solution constructed by Theorem 1 converges to the state ( l . l O ) , if the initial data and the ratio between R and
c v satisfy some smallness conditions. U'e can, however, show the convergence theorem without any smallness condition by use of a similar technique to 53. Theorem 3 ( [ S ] ) . Let (ii,e) be a positive root of simultaneous eqwztions (1.11) and
T h e n there e z i d constants C
> 1 and X > 0
which depend on R, p ,
CV,
n and
initial data but not o n t such that the solution ( u , w , e ) t o the problem (1.1) -
(1.7) satisfies the estimate
/I (-
- u, v(1, t ) - u
(. - ;)
Remurk. A positive root (ii, fi and
ii =
(1
2
5 C(1+ t)-!
1,2
e) of the above simultaneous equations exists;
8 are given by
[
,q x ,t) - e)
(J36--6cvp),
Free Boundary Problem for the Heat-conductive Compressible Viscous Gas
95
Proof. The idea of proof is that we transform the problem to one similar to U
in the previous section. First we change an unknown function u -+ 4 = and then change a variable t
(1
-t
+ t)'
t^ = log(1 + t ) . Thus we can rewrite (1.1) - (1.3)
as
(4.3)
(4.4) Here we use the notation
f
to mean
for a function f(z,t)of z and t. However, to avoid complicated notation, in what follows, we write again (;,b,
8,t^)
as ( u , v , 8, t ) . Moreover we introduce a
new unknown function (4.5)
Remark that w belongs to
. .
H$+.+" if (u,v)does to
T>O
Using w ( z , t ) , we can deduce (4.2)- (4.4)as follows:
(4-7)
+ w = (-R!+p*)
Wt
u
z
,
n
x
H$+".
Takeyuki NAGASAWA
96
Initial and boundary conditions (1.4) - (1.7) are deduced
Since the original problem (1.1) - (1.7) has the solution in
n B$++" If++.+" x
T>O
xHi.+". the reduced problem (4.6) - (4.11) also has a global solution in the same class. Moreover both u and B are positive.
If we use w(r.t ) instead of v ( z , t ) ,the boundary condition (1.5) with P ( t ) f 0 will he transformed into (4.10) which is the same type as that in $3. We can
improve Theorem 2 that can be applied to the problem (4.6) - (4.11) to show
for some C > 1 and X
> 0. Making use of the original time variable and
unknown functions, the above estimate is tuned into the assertion of Theorem
3. For details, we should refer to [6]. I
5.
The case of P ( t ) < 0. When P ( t ) is negative, it is not easy to find a trivial sdution like (1.8)
or ( l . l O ) , even for P ( f ) E P
< 0. We find difficulty here in studying the
asymptotics in this casc. The results of Thtorexns 2 and 3 enable us to infer that for this case the specific volume u would grow faster than for P ( t )
0.
Unfortunately the author do not know how to prove this conjecture on u ,
1'
b u t \ ~ h e 1 1P ( t ) ti
=P
ds and L 1 ( u 2
< 0, we can show the lower bound of the growth rate of
+ B)dx, which are faster than that of
To see this, we need the following lemma.
1'
u di
for P ( t ) f 0.
Free Boundary Problem for the Heatconductive Compressible Viscous Gas
97
Lemma 5.1. There e z i d s a t o 2 0 such that
Proof. First we assume that
By use of (2.4) with P ( t ) = P
< 0 and u > 0,
8
> 0, we find
the existence of
C > 1 such that l ( v 2
+ e + U)(z,t)dt 2 c-1
for all t 2 0.
Therefore making use of (2.5), we have the assertion. Next we assume
If there exists a C
> 1 such that
L'(v2
+ e + u ) ( z ,t>dz2 c-1
for all t 2 0,
then we have the assertion by the same reason as the prebious case. Otherwise, there exists a sequence
{tn},,E~
such that &(tn) + 0 as n -+
we have
> c-' > 0 for some n.
1
00,
where
Takeyuki NAGASAWA
98
Theorem 4. Let ( u , v , O ) satisfy (1.1) - (1.7) and u > 0
,8 >0
with
P ( t ) E P = const. < 0. T h e n there ezists a positive c o n d a n t C (> 1) such that
Proof. It follows from (2.4) with P ( t )
P < 0 that if the first estimate in
the t.heorem holds, then so does the second, and vice versa.
By the previous lemma, we may assume
from the beginning, and two relations in Lemma 2.1 yield
where y(t) =
ltI'
u(z,.r)dzd.r
+ 1.
It follows from these relations that y(t)
and
I'
1c-yt3+ 1)
u(z,t)dz= y'(t) 2
Soting (2.1) with P ( t )
c-' (y(t))'13
2
c-yt2
+ 1).
P, we obtain our result. I REFERENCES
[l]Iiawashima, S., Large-time behavior of solutions t o the free boundary prob-
l e m for the equations of a viscous heat-conductive gas, preprint.
[2] Iiazhykhov, A . V., S u r le solubilitk globale des problkmes monodimensionn,els a.uz valeurs initiales-limite'es pour les e'quations d 'un gaz visqueuz et
caiorzfhe, C . R. Acad. Sci. Paris SCr. A 284 ( 5 ) (1977), 317-320.
Free Boundary Problem for the Heat-conductive Compressible Viscous Gas
99
[3] Nagasawa, T.,On the one-dimensional m o t i o n of the polytropic ideal gas non-fized on the boundary, J. Differential Equations 65 (1) (1986), 49-67.
[4]Nagasawa, T.,O n the outer pressure problem of the one-dimensional polytropic ideal gas, Japan J. Appl. Math. 5 (1) (1988), 53-85.
[5]Nagasawa, T.,Global asymptotics of the outer pressure problem of free t y p e , Japan J. Appl. Math. 5 (2) (1988) (to appear).
[6]Nagasawa, T.,O n the asymptotic behavior of the one-dimensional m o t i o n of the polytropic ideal gas with stress-free condition, Quart. Appl. Math. (to appear).
[7] Okada, M . , Free boundary value probZems f o r the equations of one-dimensional m o t i o n of compressible viscow fluids, Japan J. Appl. Math. 4 (2)
(1987), 219-235. (81 Tani, A., O n the free boundary value problem f o r compressible viscous fluid m o t i o n , J. Math. Kyoto Univ. 21 (4) (1981), 839-859.
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 10, 101-120 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
A Computational Verification Method of Existence of Solutions for Nonlinear Elliptic Equations
Mitsuhiro T. NAKAO Faculty o f Science, Kyushu University Hakozaki , Fukuoka 812, Japan 91.
Introduction
In the author's report t 4 3 , we proposed a numerical method for automatic proof o f the existence of weak solutions for certain linear elliptic boundary value problems by computer.
And its
extension to the more general linear case will be described in [ 5 1 . The main techniques in these works consist of the verification method by computer for the existential condition of solutions based on the infinte dimensional fixed point theorems, i.e. Schauder's and Sadovskii's theorems.
In order to realize them, we
used the properties of the solution for Poisson's equation and the results of error estimates for the finite element approximation as well as the method of interval arithmetic.
In the present paper,
we formulate a numerical verification method which can be applicable to nonlinear elliptic boundary value problems.
Further
we provide some computational examples which seem to be difficult to prove theoretically but were verified in the computer by the use of that technique. In the following section, we describe the boundary value problem considered and the fixed point formulation f o r the existence of solutions.
In Q13, we define the concepts of rounding
101
Mitsuhiro T. NAKAO
102
and round ng error for function space, using the projection into certain f nite element subspace, which are similar to those in 1 4 1 .
in
5
.
we present a general algorithm, based upon the idea in the
prev nus section, to construct the set satisfying the vefification cond tions by Schauder's fixed point theorem.
We attempt. in the
last section, to verify the concrete problem as an application o f the preceding arguments.
Also we consider about the method to
prove the local uniqueness of solution f o r the problem.
Formulation o f the p oblem
92.
Let R be a bounded convex domain in R n , 1 i n i 3 , with piecewise smooth boundary and p
< -.
et p be a fixed real number, 2 i
First we set up the to lowing assumption.
Al.
When q = p o r q
(1
- -1 ) - 1 , for any P
E
LqtR), the
problem :
in R , (2-1)
on a R , has n unique solution 6 E W.
q,o
tQ)
n W'(R) 9
and the estimates
(2-2)
is a positive constant and Wm(n) denotes the usual q Lq-Sobolev space o f m-th order on R and W' (R) implies the h o l d , where
q
q.0
subspace of W'tR) whose element vanishes on 9
an.
We wil
usual ly
and LPtRt etc.. and simp y denote suppress the symbol R in W'(Q) P by W-' and L p , respectively, from now on. rote that f o r a given p > P
103
Solutions for Nonlinear Elliptic Equations
1 the truth or falsehood of A 1 depend on the shape of the domain
and the dimension n. p and R.
When n
In case of p =
2,
1, A 1
A1
is always true for arbitrary
holds for each convex polygonal or
polyhedral domain ( C 1 1 ) . We consider the following nonlinear Dirichlet problem :
(2-3)
Here, let f satisfy the hypotheses as follows : A2.
f(.,u,Vu) E Lp
A3.
For any bounded subset U o f W'
.
for each u E W i , o
f(-,U,VL!) is a l s o
PtO,
bounded set in Lp. f is the continuous map from W'
A4.
Now for
PI
0
to Lp.
-
E Lp let G$' denote the solution o f (2-1).
operator G : Lp
Then the
is compact because of the assumption A 1 W' PPO and the compactness of the imbedding W2 Wb. Therefore, from P A 4 , map Gf : W i W' defined by Gf(.,u,Vu) E W' for any P,O PlO P,O u E Wb,o i s also compact. Thus by the use of Schauder's fixed
point theorem, i f , for a non-empty, bounded, convex and closed set
c c
w;,o ,
(2-4)
Gf(.,L',VU) c U ,
then there exists a weak solution u E W'
P,O
n W i for
(2-3)
in C
.
Mitsuhiro T. NAKAO
104
Rounding and verification conditions
53.
Analomusly in C 4 1 , we take a finite element subspace Sh for 0
< h <
wq,o*
such that h' c W' for all 1 < q,o 1 < q < m d e f ne PhU E s h by 1
(3-1)
(V
q
<
-.
F o r each u
E
u-P U),VV) = 0, h
where (6,$) implies JQ6 x ) df ( x ) d x . T h e n we assume that Sh has the following approximation property :
for u E W -
q,o
n W'
<
1
q'
<
q
m,
where C(l) is a positive constant independent o f h and Iul 9
means wq
the semi-norm o f u on W'(i-2)
defined by
4
Furthermore, suppose that for the solution of ( 2 - 1 ) (3-3 1
IBI - i
w;
where C(') c(2) i q
ci2)ntnLs,
is a positive constant and i t can b
9
2
q
for q = p and
q
= (1 -
1 -1 -)
P
.
taken naturally
ke now have the following Lp error estirna Lemma 1.
F o r each u
where q = ( 1
-
1 -1
-)
P
.
E
W*
P.0
n Wi P P
Solutions for Nonlinear Elliptic Equations Proof.
-
L)-1
.
Let
= u-Phu.
Consider the problem (2-1) for q
For each (). E Lq and P by part and ( 3 - 1 ) (1
105
8
E S h , we have f r o m integration
Choosing $ as $ = Phb, ( 3 - 2 ) and ( 3 - 3 ) yield
Thus we have
(e.$)
i C~”C‘2)hllVt:IILp11$~lLq q
.
The conclusion is now from (3-2) and the duality for norms. Hereafter, we use a notation T u
=
Gf(.,u,Vu) for each u E
the rounding We now define, for the subset U of W ’ P,O’ Wr;,o. R(TL!) c Sh and the rounding error RE(TU) c W’ as P*O (3-4)
R(TU) = (uh E
Sh
; uh= PhTu, u E U ) ,
and (3-5)
RE(TU) = ( d E W ’ * !dHWi i Ci1)hlTUIW2 P,O ’ P P
and
respectively, where ITUIW: means the supremum of W 2 semi-norms for P P all u E U. Then by ( 3 - 2 ) and Lemma 1 we have
Hence, we obtain the following verification condition.
Mitsuhiro T. NAKAO
106 Lemma 2 .
I f U is a nonempty, bounded, convex and closed
such that subset of W' P?O
R(TL')
3-7)
+
RE(TU) c U ,
n W; then ( 2 - 3 ) has a solution u E W' P,O 94.
in U.
Computing procedures for verifica 1 on
I n this section, we propose a computer alrtori thm to obtain the set which satisfies the sufficient cond tion ( 3 - 7 ) of verification for the existence of solutions t o the problem ( 2 - 3 ) . We use an iterative method t o generate such a set. Let ( d i ) ,
1 i j i M,
be a basis of Sh and
denote the set of
all linear comb nations of (6.1 with interval coefficients. J
Further R + imp1 es the set of nonnegative real numbers. set f o r a
E
44-1
[a1 E
)
A l s o we
R* dllW: i a and A#ll,p
i
Cqhal,
P 1
- L)-l* P
* * *
,
and a.
E
\ow we define the itera ive sequence (u(~)), i = 0 , 1 , similar to that in [ 4 1 .
First, for i
(0)
0, let uh
R* be appropriately chosen and set U ( O ) = uLo)+ l a o l .
we determine u h( ~ ) E
and .'4-3)
GI
and a i E 'R
as follows :
€
Sh
For i 2 1,
Solutions for Nonlinear Elliptic Equations where
C
(') ('). = Cp P cP
107
Here, in general we interpret uh =
M j1 = l~
~
E
6 1 as
(4-4)
uh =
(0
M
2
E Sh ;
j=1
aj6j,
a,
1 i
E Aj,
j i
M).
Therefore we have
Note that, in almost all cases, there will be no other means of estimates of the interval value in the right hand side o f (4-2) but to overvalue as illustrated in the next section.
Thus u L i ) is
determined by an interval vector solution for the system of linear equations (4-2) with interval right hand side.
Further the value
Il*-.IILp in (4-3) implies the supremum for all d
E
.Ai)
We now set
tail.
+
Then the following properties hold.
For the sequence L! (i)- - uh(i)
Lemma 3.
+
CCci], i = 0 , 1 ,
defined by (4-2) and (4-3) with any initial value have
L'(i-l).
.Lo)
...
and a d , we
R(TU ( i - 1 ) 1 c Uh(i)
and
RE(TU(~-~))c [ail. Proof.
If
4,
E R(TU(i-l)),
then there exists u E L' ( i - l ) such
that for arbitrary v E Sh COG
h
,Vv) = (V(Tu),Vv)
d
,
~
Mitsuhiro T. NAKAO
108
where we have used the fact that -A(Tu)
f(-,u,Vu).
But the last right hand side should be contained to (Vuii),Vv) by
huh
Hence, we have
(4-2).
Next, for any d E RE(TU
E u(~).
h
(i-1))
from ( 3 - 3 ) (4-6
)
p C ( 2 )
P
P
.
hll f ( ,U ( i - l ,Vu ( i - l
)nLP.
Similarly, we have
< 4-7)
H#HLp < C P C 9 hr~f(.,U(i-l),VU(i-l)) HLP =
and ( 4 - 7 )
(4-6)
Lemma. \ow
uh
+
C
ha.
9
1
imply the assertion of the latter half in the
Thus we have completed the proof. we describe the computer algorithm to obtain the set U =
[ a ] ,which satisfies the verification condition (3-7), by the
use of the iterative sequence (L(i)).
This algorithm is quite
similar to that in 1 4 1 .
let
We take parameters
E
> 0 and d > 0, usually
iteration number x!
then we stop the iteration and set
E i 6.
If for an
Solutions for Nonlinear Elliptic Equations (4-8)
Uh
(N)+
=
H
Uh
j=1
109
I-l,llbdj
and
a = aN
(4-9)
+
6.
Further, we again compute (4-2) and (4-3) for U = uh 5
is, choose uh E
-
GI
G
and
la], that
E R + such that
( V u h , W k ) = (f(*,U,VU),6k),
(4-10)
+
1 i k
S
M,
and
a = cphHf(.,u,vu)nLp. 5
(4-11)
Then, from the Lemma 2 and 3, we have the following verification conditions. Theorem 1
Suppose that
(4-11), respec ively.
and
G
are defined by (4-10) and
G
i
If
5
uh c uh
(4-12)
Gh
and
a,
then there exists a solution u E W 2 n W 1 for (2-3) in uh + [a], P P,O where uh c uh means that each coefficient interval in Gh is 5
included in the corresponding interval in uh.
55.
Examples o f numer cal verification
We provide some numeri a1 examples for verification in two deimensional case according to the procedures described in the previous section. Let consider the problem
Mitsuhiro T. NAKAO
110
-Au = Ib:,b21ui
+
Cf;,f>I
in
R,
on 38, shere 9 = ( 0 , l ) x ( 0 , l ) c R 2 and [b,,b,I,
If:,f21 a r e intervals
which mean that
‘iote that i t is not difficult to extend the arguments in preceding sections to the equations with interval coefficients such as ( 5 - 1 ) by the similar consideration as i n 141. \ow
let d X : 0
< xL =
xJ < x i <
1 be a uniform
partition of the interval I = ( 0 , l ) . that is, x i = i / L ,
...
.
L.
Also
set l i =
denotes the set o f
( X ~ - ~ , X and ~ )h
= 1/L.
i
0, 1 ,
When P:(Ii)
linear polynomials on I i , we define the space
,%j (x 1 by /?$Lj!x)
(5-3)
= (v E
c(I) :
1 s i i L,
v I I , E P:(Ii), 1
V(1) = 0 ) .
V(0)
And for simplicity we take the partition of y-direction as b Further define the mesh of R and Sh by 6 = b x @ @, J[:(y),
y
-
-
&x
by and Sh = , l l : t x )
respectively.
I t can be easily seen that, i n the present c a s e , the hypotheses A 1 to A4 in 9 2 are satisfied for p = 2 by the Sobolev imbedding theorem.
Therefore,
it
is sufficlent to use the result
u f arguments in the previous sections only for p
2.
1.e. the
Solutions for Nonlinear Elliptic Equations L2-theory.
111
Furthermore, i t is seen that we can take the constants
in (3-2) and (3-3) as Cil) = Ci2) = 1 by virtue o f the estimates we adopt the inner product on W'2 , o ( R ) ( V 6 , W ) and the associated norm is denoted by lldH$
in 141.
choose
Also
i
HA by < d , $ . >
We
he basis of/MJ(x) as the same i n [41, i.e. the fol owing
hat functions on I .
(5.4)
Since t B (x).ak(y)), i t again by { d j ) , 1
F o r 1 i j,k i L-1
f1
if k =
j,
b
if k
*
j.
1 i j,k i L-1, forms a basis of Sh, we denote S j C
M,
so
4! = (L-1)'.
Now we describe the concrete algorithms to verify the problem (5-1).
First, set uio)= 0 and a0 = 0. Let
a. 1-1
E
I
R+.
Uh (i-1)
=
Y
j A!i-l)dj
j=1
E @I
and
J
Then observe that, taking account of 6 k
2
0 and !#kllL-
Mitsuhiro T. NAKAO
112 (i-1)
Here, Huh
RL2tRk)
means the supremum of norms on the support R k
of d k for all the elements in uh( i - l ) and we have used the fact that Ulai-llHLZ= h a i m l . Yext, by the use of the estimates in C23, i t follows that for any
6 E ti3 and 1 < p <
0
where I R I implies the measure of R. From ( 5 - 6 ) and some simple calculations, we obtain
Hlai-llHL,,i Jz- a .
(5-7)
2
B y the use o f
(5-8)
1-1‘
( 5 - 7 ) we have
Uf(.,~“-”,Vu(i-l)
)HL2
= I[b:.b23(uh (i-1)
+
cai-lI ) +
r; Ibl(Iluh( i - 1 )
HL4
L Ibl(#uh (i-1) l L 4
+
+
[t:,t2inL2
ll[ai-11HL4)2 + If1
qaim1)z
+
Ifl,
where Ibl = rnax(lb.l,lb2l) and If1 = rnax(lfil,lfnl). Thus we can provide the iterative alxorithm based upon ( 4 - 2 ) (4-3)
as follows :
Stopping criteria and the final step of verification are as described in the preceding section.
and
Solutions for Nonlinear Elliptic Equations
113
Also, in the present case, projection (3-1) implies s o called A
HA-projection and the error belongs to Sh. Thus, we can replace I
W b , o in the definition ( 3 - 5 ) by Sh n H b and, for any a E R + , redefine Cal in ( 4 - 1 )
[a1
f
as follows : A
sh
(d E
; ldlHj i
a and IdBL2
ha).
i
Next, we shall consider a method to assure the uniqueness for the solution of (5-1) as well as the existence. For the time being, we fix the LB-function b
E
Cbl,b21 and the
L2-function f E [f;,fzI. Let define a nonlinear operator A : Hd
+
HA
and an element F E
Hj by = (bu',d),
(5-11)
and < F , d > Then
A
E
+
F E H'.
(f,d), respectively.
becomes a compact map and also Au
for any convex subset U of H i , the set !I + U coincides with 2U = (2ulu
-
Now, for an element K$ : HA
HA,
d
*
E
3
{u
Notice that, +
vl u,v
E U)
U).
c HA we define a compact 1ine.ar operator
HA by
= (b$u,d),
(5-12)
d E Hh.
The following Lemma can be easily obtained from the result in 131, especially Theorem Lemma
4.
4.
Let uh
E
GI
and a E .'R
Set U = uh
+
Cal.
Suppose that (i)
F o r each u E U , there exists an element #' of 2 U such that
Mitsuhiro T. NAKAO
114
Au + F = K$u + F. F o r any u ; .
(ii)
-
-
U2).
F o r each $ E 2L', K+C + F
U
Au:
(iii)
there exists $
E U.
u2
...
s.
c
= K#(Ui
AU2
such that
E 2U
holds, where S ; 2 S p implies
32.
Then there exists a unique solution u for Au + F = u in U . When we denote the relation (5-9) and (5-10) as (i)
(5-13)
(uh
a map Q,
:
Ib.,b;l
- GI
.ai)= *lb.,bpl(Uh
GI
x
R+
X
(i-1)
,a. 1-1)*
R + is defined.
Then the following result can be obtained Theorem 2.
Let Ibi,bil = [-1,11[2b;,2b21, in the sense o f
interval arithmetic, and for any tuh,a) E
G,x
GI
x R'define
+
F = u in u
tTh,G)
E
R + by
When 0 E uh
+
[a]. if 5
0
uh c uh
(5-16)
< a,
and
then there exists a unique solution f o r Au
Here, uh i uh
+
h
implies that each coefficient interval in
c
uh
la]. is
strictly contained to the corresponding interval in uh. Proof.
F i r s t , by virtue o f 0 E C
u
0
+
u
-
We shall p r o v e ( i )
E 2L,
2
then we have
uh
+
( i i i )
in Lemma
[a]. for any u
4. E
H d i f we set $
6
Solutions for Nonlinear Elliptic Equations
+
which implies K u
F = Au + F and yields (i).
+
Next, for arbitrary u l , up E U and d E
provided that
+
115
= ui
up E 2U.
+
HA
Hence, ( i i )
is obtained.
I n order to prove (iii), i t is sufficient to show that for any @ E 2C R(K$U
F)
+
+
RE(K+U
+ F)
6 U,
where R ( * - * ) and RE(*..) imply the same as in
and
(3-4)
(3-5).
replacing Tu by K u + F , respectively. @ We now have for each 0 E RCK$U + F), by the definition,
E
(b+U
f,#.)
+
J
c (2b(uhuh
+
2Caluh
+
where
1 i j i N
tal[al),d,)
^u
E
Gh
Thus we have RCK+U
(5-16) E
+
RECK U
+
+
+
#j.
from the definition
(5-14).
-
F) c uh.
F), using
(f,#j)
Cfi,f2IMjULI,
and Rj means the support of
Therefore, we obtain
Next, for any 9
+
(3-5)
and
(5-15)
Mitsuhiro T. NAKAO
116 S
h ( 1 2 b ~ L m ( ~ u h l L +~ $a)'
!flL2).
+
Since t16ALi is similarly estimated, w e have
RE(K*U
(5-17)
[GI.
F) c
4
Thus by (5-16) and ( 5 - 1 7 )
R(K U #
(5-18)
Moreover. for each d E where 6: E Id
- 3UHf <
-uh c Sh and
[ G I , we decompose i t as d =
RE(K#U
*
+
A
-
+ $2
such that
-
and 6 2
$1
< E
-
$1
$2
I62
and
-
E
+
>
F o r any E
6 2 E Cal c Sh.
the orthogonality for b i Udi
+
Gh
E and $
[GI.
F)
+
F) c
Gh
+
0.
Sh and
dl
+ $2,
when
$2
I
E Sh, by
with each other, we have $2IHb
<
E.
Therefore, i f E is sufficiently small then by (5-15) and the norm equivalency for finite dimensional spaces we obtain
Furthermore, using the Poincar; inequality, we obtain (5-21 >
8$t'RLz
-
S
d211L2+ H62!lL2
<
CE +
c
h a < ha,
provided E is taken such that E < hta
-
constant in the Poincard inequality.
( 5 - 2 0 ) and ( 5 - 2 1 )
32 E [a].
G)/C, where C is the mean that
Combining this with (5-191, i t follows that 6 is an
interior point o f uh
+
[a].
+
uh
+
T h u s , i t is seen that
[GI C
uh
+
[a] =
u
Solutions for Nonlinear Elliptic Equations and therefore, from ( 6 - 1 8 ) we get the desired result ( i i i )
117
which
completes the proof. The conclusion in Theorem 2 is also valid under
Corollary 1 .
the condition 0 E [bl,b2l instead of 0 E uh Proof.
+
la].
We denote the dependency on b in the definition
(5-12) by K+ = K+,b.
Since b/2 E Cbl,bpl f o r each b E Ibl,b21, we
have
+
F , d > = (bu2 + f.6) = (b/2.(2u)*u
+
f.6)
= <(K2u,b/2)~
+
F,#>
for arbitray u E U and # E HA. +
which assures (i)
in Lemma
This implies that
= K 2 ~ , b / 2+ F
Propositions ( i i )
4.
and ( i i i )
easily
follow by the similar argument as in the proof o f the Theorem 2.
Now we illustrate some numerical results for concrete examples of which the verifications were normally completed by the scheme (5-9), (5-10).
Case 1. Problem :
-Au = [-l,lIu’ u = 0,
+
10,71,
(x,y) E Q,
(x,y) E a R .
Execution conditions : Number of elements = 100 ( h = 0.1), M = dim Sh = 8 1 ,
Mitsuhiro T. NAKAO
118
Initial values : uk0) = a. = o Stopping 8 Extension parameters :
8
= 10 -3 ,
6 = 10-l.
Results : Iteration numbers :
N
= 7
H'-error bound : a = 0.9123 Coefficient intervals :
min ,Aj = lsjG4 max
lij<M
-A .
=
'
-
0.2492.
+
0.7690.
Here, A . and 7\. are the infimum and supremum, respectively, o f the .I
J
coefficient intervals for 6 . in u h which appears in Theorem J
Further, 0 was included in each interval and thus 0
E
uh
+
1.
la].
Case 2 . Problem :
-AU = 1-2,23u2
+
tX.Y) E Q ,
t0.71,
u = 0.
tx,y)
E
3Q.
Execution conditions : Number of elements = 225 (h = 1/15), ?(
= dim Sh = 196,
Initial values : uko) = a. =
o
Stopping 8 Extension parameters :
&
= 10-3 ,
6 = 10-1.
Results : Iteration numbers : N = 9 H'-error bound : a
0.6804
Coefficient intervals :
8. =
' max i.= l<jiH ' min
lSjSM
-
0.3154.
+
0.8290.
Solutions for Nonlinear Elliptic Equations Naturally, in this case also 0
E
119
uh.
Notice that, by Theorem 2 or Corollary 1 , above t w o results yield that the solution for Case 1 is unique on the set obtained from the execution for Case 2 .
Remark.
In these executions, we used the usual computer
arithmetic with double precision for all numerical computations. Hence, there may be some rounding errors in the calculations o f the final step ( 4 - 1 0 ) .
(4-11) and i t may be not assured, in this
sense, that the verification condition
(4-12)
is exactly satisfied,
From the author's experiences, however, as far as the problem is not s o i l l , the numerical results are sufficiently reliable at least five digits o r s o .
In order that the above results become
the mathematical theorems, we need to use some arithmetical systems with guaranteed bound, e.g. ACRITH in IBM.
Mitsuhiro T. NAKAO
120
References
[I 1
P . Grisvard, Elliptic problems in nonsmooth domains, Pitman, 1985.
[2 1
C. Johnson, A. H. Schatz 8 L. 8. Wahlbin, Crosswind smear and
n stream diffusion finite element methods,
pointwise errors Math. Comp., 4 9 131
(
E. W. Kaucher & W
987), 25-38.
L. Miranker, Self-validating numerics for
function space problems, Academic Press, 1984. 143
M.
T. Nakao. A numerical approach t o the proof of existence
of solutions for elliptic problems, t o appear in Japan Journal o f Applied Mathematics, 5 (1988). 151
M . T. Nakao, A numerical approach to the proof of existence
o f solutions f o r elliptic problems !!, in preparation.
Lecture Notes in Num. Appl. Anal., 10, 121-128 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
Degenerate Bifurcations in the Taylor-Couette Problem HISASHIOKAM O T O Department of Pure and Applied Sciences, University of Tokyo, Meguro-ku, Tokyo 153, Japan
$1. Introduction. The purpose of this paper is to explain certain bifurcation equations which describe newly discovered phenomena in the Taylor-Couette problem of the fluid motion between two concentric cylinders. The equations to be considered here are: (1.1)
EX + a + a t 2 + bz2 + CX’Z) + ( p + ez2>tz= 0, z( 6A
+ &x2 + 132)f x2 = 0)
and (1.2)
{
+ ax2 + b 2 ) f XZ = 0, z(6A + d z 2 + 2122 + t 2 2 ) + ( p + dZ’)Z2 = 0, x( €A + CY
where A, x and z are real variables, E , 6, a,b, c, d , e, B , 8 and 2 are real constants. We explain in the subsequent sections how the solutions to (1.1,2) fit the bifurcation diagram given in Tavener and Cliffe [7] in which Taylor vortices of new type bifurcating from the Couette flow are computed numerically. T h e equations (1.1,2) are derived from a certain degeneration of the equations given in Fujii, Mimura and Nishiura [l]. The equation (1.1) are considered in Fujii, Nishiura and Hosono [2], but (1.2) seems to be new. Although they considers in [1,2] a reaction diffusion system which has nothing t o do with the Taylor-Couette problem, the local structure of the bifurcation is of the same category. This is because the ofthogonal group O(2) acts on both problem.. . In this paper we announce a results in [5,6] in which we employed the singularity theoretic approach by Golubitsky and Schaeffer ( [3,4] ) to see systematically the structure of the equations. In 52 we state a precise statement of the Taylor-Couette problem. In 33 the relation between (1.1,2) and the Taylor-Couette problem is given. $ 4 is a final section for discussions. 121
Hisashi OKAMOTO
122
$2. The Taylor-Couette problem. I n this section we recall sonie newly d e w l o p p e d analysis for the nlc>chanisnlof vortex uuinber exchangt, in [7].The problem considered by thein is t o determine a fluid velocity field ( u , u, w ) and the pressure 11 which satisfy t h e fc,llowing stationary Navipr-StokPs equation (2.1-4):
(2.1 1
(2.2)
(2.3)
1 8 --(ru) r dr
R 111 += 0. 82
UJ a r e r , 8 , -~cumR is the Reyncilds number. In this paper, as in
w h e w the cylindrical coordinates are adopted. u , poririit, respectively and
"71. wc. w i i s i d r r cm1y
18,
velocity fields wliich are i n d e p r d e r i t o f t l w a z i m u tal cuudiiiate 8 and t h e t iiiir t. We haw, s u i t a b l y nondinirnsiunalized the qunntitirs so that ( 2 . 1 4 ) a r ~ satisfied in
The follow iiig boundary conciitic)ns are imposed : (2.5 j
(U,.L',.U?)
= (0,l.O)
(2.7)
I n this circumstance, t h e Couettt. flciw
( r = 1)
Taylor-Couette Problem
123
B = q2/(17 - 1). satisfies all the requirements if A = 1/(1 - 7 i 2 ) , The problem is to study the solutions bifurcating from this Couette flow. Notice t h a t the condition (2.7) makes it difficult t o examine the result,s by a laboratory experiment but there is no clificulty in performing computer simulation. In fact, [7] presents a. penetrating description of t h e stability exchange of the solutions. Let us statmebriefly the results in [7]. T h e y fix the parameter 7 = 1.0/0.615 arid let t h e riondimensional height of the cylinder r vary. It is known t h a t the primary bifurca.tion branch consists of the two-cell Taylor vortices for small value of r and t h a t it consists of the four-cell Taylor vortices for larger value of r. The number of the vortices, however, is integer a.nd l- can vary continuously. 'Therefore, there exist flows of "mixed type" when r is in a intermadiatr range. T h e hifurcation diagrams in [7] describe qualitatively how this exchange from two-cell to four cell occurs. Although they consider the excha.nge mechanism between two-cell and six-cell or four-cell and six-cell, we consider only t,he exchange of two and four, which requires the least mathematical technique to theorize. 93. Degenerate bifurcation equations. In this .sectmionwe show how (1.1,2) are derived. Our starting point is t o observe a hidden symmetry in (2.1-7). Let us consider another problem to seek ( u * , I,!*,w*,p*) which < t < r, the boundary condition satisfies (2.1-4) in 1 < T < r(, (2.5.6) and t h e periodic boundary condition on z = (, Xote that t h e height of the cylinder is double. ) We call this problem [2r]and the original problem [I?]. This n e w problem [2r'] has an a.dvantage t,hat it is 0(2)-equivsriant in the following sense: Let us define an actmionof the orthogonal group O(2) by
-r
(3.2)
-
3'( ..( r, 2 ), Z)( T, z ) 7 v.4 f', z ) , p( r, r ) ) (~U(T,~'+C~),'L~(T,~+CY);LL~(T,Z+~),~~~'T,~
if y E O(2') is a rotation with angle (3.3)
-r, r.
ct,
Y(ZL(T,2),21(T,Z),7JI(7;z),1~r,L))
-
( ~ ( -rz ,) , v ( T , -z),- w ( T , - z ) , & T ,
+a))
-i))
if 3' E O ( 2 ) is a. reflection. T h e n it is ea.sily checked t h a t the problem [2r] is 0(2)-equivariant: in other words, the governing equation is covariant with the 0(2)-action (3.2,3). We recall t h a t tho larger the symmetry group is, the simpler the equation becomes. Therefore the introduction clf [2r]is an advantage, since [r]is cova.riant. with only 5 discrete group. T h e relation between [2r]and [r]is given b y
Hisashi OKAMOTO
124
PROPOSITION 3.1. If ( u * ,~ ~ * , w * , pis* a) solution t o [2r] and if ( u * ,c * , w',p') is invariant with respect to (3.3). then it satisfies [r] in 0 < z < r. Conversely, if ( u , v , IIJ, p ) satisfies [r] and if we extend it to ( u * , z*,w*,p* ) in such a way that u*,v*,p* are even extention of u. v , p , respectively, and ZL.? is an o d d extension of ZU, then ( u*,V ,P C ~ * . ~is* )a solution to [2r]. In short, finding solutions t o [r]is equivalent to finding "symmetric" solutions t o pr]. F r o m now we consider [2r].I n the remaining part of this section, we explain how the analysis in [5] is applied t o [2r]. T h e r e is a numerical evidence t h a t , at some value of r, say J?o , the linearlized operator of (2.1-4) with (2.5,s) and the periodic boundary condition has a four tlimonsional null space spanned by 91 94 which are of the following form:
-
g3 ( r, 2 ) = ( U4(T j c o (~4 $), \$( T ) C O S ( 4 +) , W,(,?-)sin(44), P4( TI() os ( 411)j )
,
= (, CT4( r)sin (4,$) I$(,r)sin (4.14) Mr4( r)cos (4.G) , P4(r.)sin (4.2))) In these expressions Uj(T ) , V,( T'),Wj(T ) , Pj( 7') are functions of T only and y4( r. t )
are determined through a certain ordinary differential equation ( see [6] ).
Let, F = F( R I?; 'u,v, 'w, p ) be a nonlinear functional for [2r]realized i n sonie Banach space, a n d let P be a projection onto t h e 4-dimensional space spanned by 91, -,$4. T h e n the bifurcat,ion of Taylor vortices is governed in a neighborhood of ( R , rcl; 0, vo, 0 , ~ " ) by
--
where (3 is in a complement of the range of P. Since 0(2)-equivariance is reflected in this bifurcation equation, G' must, be of a special form given i n [1,5]. I n order t o explain the form in [1,5], we identify real (.r, y, z, w ) space with C 2 by [ = x iy, C = z zw. Then G niust b e of the following form: G = ( G 1 , G 2 ) ,
+
(3.4)
+
Taylor-Couette Problem
125
+
(3.5) G2 = f3c f4t2l where f,(j = 1, 2 , 3 , 4 ) are functions of R , r, ]
f i ( 0 . o ; 0,010,o) # 0, f 4 ( 0 , 0 ; 0,o. 01 0) # 0 which is considered in [I]. T h e degeneration which we mentioned at the beginning of the present paper is
(A )
(B)
f2(
0, O; O, 01 O l O)
= Ol
.f4(
Ol
O; O, O, O , 1 #
and
f4(0,0; O , o , 0,O) = 0. f i ( 0 , O ; 5,0,0,0> # 0, (C ) T h e case ( B ) is considered in [2] but ( C ) seems to be new. Since (3.4,5) is applicable to a number of problems, we think it is useful t o study (3.1.5) systematically. To this end, the machinaries by Golubitsky and Schaeffer [3,4]are easy to handle for application-oriented niathematicians. Below we suinmarize the results in [5], where a computations of normal forins for (3.4,5) via the method in [3,4] are given. In the case of ( A ) , the bifurcation equation is, when slightly perturbed, generically O( 2)-equivalent t o
where CY is a pert,urbation ( unfolding ) parameter. Not8e t h a t we only consider a n O( 2)-equivariant perturvation. By the O(2j-equivalence we mean t h a t the bifurcation equation is t,ransforrned t o one of this form by a suit,able coordinates change which preserves the O(2)- equivariance. Roughly, we can say that, (3.6) is a normal form in the ca.se of ( A ) . In the case of ( B ) , (3.7)
{
[(EX
+ CY + a l i j z + blC12 + cRe(E2<).) + ( P + el<12)f( = 0, ((SX + q# + B1CI2) f t2= 0 ,
is a norma.1 form. I n the case of (C), (3.8)
i
E( E X + a + a151’
((6X
+ hlCI’)
f
Tc = 0,
+ tilEz = U,
is a norrna.1 form. When we consider t h e problem [I?] , the solutions are invariant. with respect t o the reflection (3.3:). Therefore we obtain the bifurcation equations for [I’]by rest,rict,ing complex variables E, t.o real ones z,z . T h i s restriction produces (1.1,q from (3.7,13), respectively.
<
Hisashi OKAMOTO
116
$4. Discussions. We first reinark t h a t , when R is taken as a bifurcation parameter, there are additional splitting parameters r and q. Therrefore there is a good possibility t h a t the foHowing scenario holds true: In the (r,q ) plane there is a 1-dimensional variety where t.he critical Reynolds number of 2-cell Row and that, of 4-cell flow coincide. And on t h i s variety there a.re points at which ( B ) or ( C ) holds. If we assume t h a t these points exist and are not far from the values taken in [7], then i t is nat,ursl that, the figures in [7]m e captured by ( 1 . l i 2 ) . W e no\v show how the phenomana in [7]is explained by (1.1,2). We ckoose u and ,O suitably t o have zero setmaof (1.1). In Fig. 2-6, we give c l r a w i n e by a computer a.nd a X-Y plot,ter. These explain the bifurcation diagtanis in [7] for large I‘ ( Fig. 4.3 (D-I) ). From ( 1 . 2 ) we cibt.ain Fig. 1, which explains diagrams for smaller I? ( Fig. 4 . 3 ( : C )of [7] ). Althuugh there is a pitchpork bifurcation in Fig. 4 . 3 (A,B,C) of [7], this ca.iinot be explained by (1.2:). This ,however, niay be a consequence c,f inore globa.1 bifurcation. Except for this, our pictures fit, the diagrams i n [7] qtiite well. For the complete set of the bifurcation diagrams of ( I . l , ? ) . see [GI.
Acknowledgement. T h e author wishes t,o express his t h a n k s to Prof. H . Fujii a n d Prof. k’. Nishiura for showing [2] a.nd giving k i n d advices.
REFERENCES 1 . I< Fujii, b1 Mimura and Y. Niehiura, A ptclur-e of the g f O b 5 1 brfurration dtagram
und d i g u r r n g ryrfernr, Phgsica 5D (1982),1 4 2 . Fujii. Y. Nishiura an4 Y . Hosono, O n t h e r l r u r t u r e of m u l t i p l e r r i r t e n e e o j r l a b l e s i e l i o n a r y r o l u l i o n r in r y r f e r n s of r e o c f r o n - d i f l u r i o n e q u a f i o n r , Patterns snd W a v e s - Qualitative analysis of nonlinear differcnt,ial equations, e d s . T.Niehida. M . R4iinura and H . Fujii , North-Holland (1986),167-218. M . G ol i i hi tsk y and D . S h a e f f e r , Im.pcrjrrl bifirrcnlion in the p r c i e n c r o f #urnt n r f r y , C‘#>mni.\ l a t h Phya. 6 7 ( l 9 7 9 ) , 205-232. M (:;olut,itnky and D. Yihaeffer. “Yingiilaritivls and G r u u y o in Lifurcatiun theory. V I ? ~ 1.” 5pring.r Vrrlag. N e w York. 1985. ti Oknmi, to, 0 (‘ .? )-epu i v ar I o n i b , f i r rcof io z q ua 1 i o n 3 IU I t h tu.o 131 o d r I i n it- ra c l i u m , . ( preprint ) . H O k a n i o t o and 5 . Tavenvr, A d r g c n r r n f c 0 ( 2 ) - P q u r r ~ n r ~ r n nbf a f ~ t r c n l i o ne q u a t i o n rind i t s ( r p p l i r n l i o n l o t h e T a y l o r p r o b l c m , ( preprint ) . T a v r n e l - ari,l K . A . Cliffe. P r i m n r y f l o w e r c h a i i y a r n e c h v n i r m . 8 In T a y l o r l . . - o u e t t ~ f l o u . a p p l y i n g n u n - p u z b o u n d a r y c o n d i l t o n r , ( preprint ) . in c c o l o g i c a l tntrrncirng
?. H
3.
4.
=,. C
7.
-
l i r uusordr. O( L!-eqiiivariance. multiple bifurcatiun. Tavlor v o r t e x
\
Taylor-Couette Problem
m
128
Hisashi OKAMOTO
Lecture Notes in Num. Appl. Anal., 10, 129-151 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
Uniqueness of Critical Point of the Solution to the Prescribed Constant Mean Curvature Equation Over Convex Domain in R2 Shigeru SAKAGUCHI
Numazu College of Technology 3600 Oooka, Numazu-shi Shieuoka 410, Japan
3 1. Introduction and result Let
Q
R 2 with
be a bounded convex domain in
aQ.
smooth boundary
Give a positive constant
H.
Consider the following problems:
(1.1)
(1.2)
where
{
div Tu
c
=
u
=
div Tu
o
on an, in Q,
= 2H
Tu-v =
Tu =
in Q ,
2H
cos
1 + lVuI2)
(
on aQ,
y
-
unit outer normal vector t o a constant with of
Q
2H
IQl
'
Vu
and
aQ
and
v y
denotes the (0
<
y
<
f n ) is
= cos y laQl ( ( Q ( is the area
and laQl is the length of aQ 1.
The graph of the
solution to (1.1) describes a surface of constant mean curvature
H
with boundary
357
graph of the solution to ( 1 . 2 )
x
{O}
in R 3 , and the
describes a capillary
surface without gravity over the cross section 129
Q.
Shigeru SAKAGUCHI
I30
In [2o]
Philippin showed that the solution to (1.1)
has only one critical point when
Q
is strictly convex.
His method of proof is based on an idea o f Payne [19]. Therefore it is restricted to the Dirichlet problem and
is not applied to the problem (1.2).
In [ 5
3
Chen
showed that the solution t o (1.2) has only one critical point under the hypothesis of the existence of the solution with domain
Q
y
= 0 ( This hypothesis
means that the
is an extrenal domain studied by Giusti
[lo]).
His method of proof is based on a nice comparison Chsn & Huang [ 4 1
technique found in [ 4 1 ( The
solution viith
y =
0
and the result of
is strictly convex ) and
the method of continuity with respect to contact angle y. On the other hand, many results concerning concavity properties
(
especially, convexity of level sets
of
solutions to elliptic boundary value problems over convex domain in R n ( n (
2 2
1 were obtained by various authors
see ~ 1 1 , ~ 2 1 , ~ 1 1 1 , ~ 1 2 3 , ~ 1 3 ~ , ~ ~and ~ 1 [161 , ~ 1 5 1 ,
especially see Kawohl [121 and its references
).
To illustrate their methods, we consider the Dirichlet
problem A u
where
Q
=
-1
in Q
and
u = 0
on
an,
is a bounded convex domain in Rn .
- E . Then v
satisfies
Put
v
=
Constant Mean Curvature Equation over Convex Domain in R2
-
A v =
av av
where
on aQ.
(
lvv12 + 3
v-l
131
in Q ,
denotes the outer normal derivative of
v
Kawohl 1111 and Kennington [131 improved Korevaar
' s convexity maximum principle and proved the convexity
of
v.
Their proofs are based on convexity maximum av principles combined with the fact that on a 9 av ( This fact and strict convexity o f Q guarantee that 03
v
is strictly convex near the boundary
a$? 1.
Furthermore Caffarelli & Friedman [l] showed the strict convexity of and
v
,
when the dimension
Korevaar & Lewis [16]
Rn ( n
L 2
1.
n
is equal to 2,
generalized this result to
A l l these methods need some algebraic
structures of the equation and the fact that aQ.
av av
=
03
on
In view of these illustrations, these methods do not
work for the problems (1.1) and (1.2) in order to prove convexity of level sets of the solutions, since there exists no real valued satisfying that
v =
monotone g(u)
function
g
on W
satisfies the equation with
some required structures ( see C111, [131, [l], and [16]
1.
Thus it is not known whether o r not the level sets
of the solutions to ( 1 . 1 ) and (1.2) a r e convex. Of course, the solutions to (1.1) and (1.2)
are
not a l u 2 y : ; cijnvex in Q.
themselves
132
Shigeru SAKAGWCHI
In this paper we shoiv that it is possible to treat (1.1) and ( 1 , 2 )
method of
,
in the unified manner by modifying the
Chen C51.
Introducing a method o f continuity
which is different from that o f
Let u G -
Theorem. (1.2).
Then
u
C2(
)
Chen [ 5 ] , we show
be-a solution to
(1.1)
or
has only -one critical point in Q.
Since our theorem concerns only qualitative property o f the solution, s o only under the hypothesis o f the existence of the solution we prove this theorem.
In the proof o f this theorem we see that the level sets o f the solution to (1.1) are convex when
sufficiently small and
Q
H
is
is strictly convex.
In the following sections we prove this theorem. In section 2 we introduce two families of problems indexed b y a bounded closed interval [0,1] in order to use the method of continuity. &
In section 3, using Chen
Huang’s comparison technique, we prove several basic
lemmas. Section 4 is devoted to the completion o f the proof of Theorem.
5 2. Families of problems for the method o f continuity For
t
(
0
;5,
t
1 1 , we introduce the following
133
Constant Mean Curvature Equation over Convex Domain in R'
problems: div { ( 1
(2.l.t)
where
H
+
t2\Vv12)-t Vv }
v
and
.y
2H
in
Q,
aQ,
on
0
=
=
are the constants in (1.1) and (1.2).
Since the solution to (2.2.t)
is unique up to an additive
constant, we consider the problem (2.2.t) with the condition
I
Q
v dx
= 0.
Then the solution to (2.2.t)
with this condition is unique.
Remark 2.1. (1) (2)
(2.1.1) = (1.1)
and (2.2.1) = (1.2).
(2.1.0) and (2.2.0) are linear problems.
Concerning the existence of the solutions to these problems, w e have
Proposition 2.2.
Under the --
hypothesis of the existence
of the solution to the problem (1.11, there exists a unique solution v t ('c si I to (2.1.t) satisfyinq (2.3)
s
C
f o r all
t E CO,13,
Shigeru SAKAGUCHI
I34
where
C
and
t e [0,1].
are positive
a
Furthermore, put
constants independent of ut
t vt.
=
Then
ut
satisfies div T( ut
(2.4)
=
Ut
Proposition 2.3.
2
2 tH
=
Q,
on a P . -
0
Under the hypothesis of the existence --
a of the solution to the problem (1.21, there exists unique solution Q
v t dx
0
=
vt
ut
5
I?(
)
to (2.2.t) with
satisfying the same inequality as ( 2 . 3 )
for all t c [0,13. Then
G
Furthermore, put ut
=
t
Vt.
satisfies div T( ut
)
=
T( ut ) * v = t cos
(2.5)
in Q , -
2 tH y
on
aQ,
-Proof of Proposition 2.2. Consider the problem (2.4) for t ( 0 < t < 1 I. Since 0 < tH H , taking the solution to (1.1) a s a barrier, we can get the C 1estimate of the solution to (2.4). o f quasilinear elliptic equations Trudinger [ 8 ] 1 , we ut
C2L(
a
)
Then, by the theory (
see Gilbarg &
see the existence o f the solution
to (2.4). Furthermore,
we get the inequality
since
0 < tH < H ,
Constant Mean Curvature Equation over Convex Domain in RZ
135
(2.6)
where
C
and
are p o s i t i v e constants independent o f
a
t ( F o r b r e v i t y , t h e same l e t t e r
and
C
w i l l be
a
used t o denote d i f f e r e n t p o s i t i v e constants independent o f t throughout t h i s paper).
vt
Put
satisfies (2.1.t).
By p u t t i n g
=
(1 + lVutI2
(2.7)
b(x)
we r e g a r d
vt
.
= t-'ut
vt
Then
I-',
as a unique s o l u t i o n t o t h e l i n e a r
e l l i p t i c D i r i c h l e t problem: (2.8)
I
d i v ( b ( x ) Vvt
e s t i m a t e s ( see [8,
Also,
Theorem 6 . 6 ,
T h e r e f o r e we g e t ( 2 . 3 ) (2.1.0)
0
i n t h e case t h a t i s an a s s u m p t i o n .
vo
on
p.981
3.7,
for
t(- ( 0 , l ) .
i s a l i n e a r problem.
a unique s o l u t i o n
Q,
in
aQ. 1,
we g e t
p . 36
1
that
c.
lVtl
Q
2H
u s i n g t h e Schauder g l o b a l
i t f o l l o w s f r o m [8,Theorem sup
t = 0,
=
Vt
I n view o f (2.61,
=
)
G c"( V
I n t h e case t h a t
Hence t h e r e e x i s t s
t o (2.1.0).
O f course,
t = 1, t h e e x i s t e n c e o f t h e s o l u t i o n This completes t h e proof.
Shigeru SAKAGUCHI
136
Proof - of Proposition 2.3.
for
t
( 0
c t
<
contact angle o f follows from
[6,
1 1.
Consider the problem (2.5)
Since
ut
t < 1 , we remark that the
is larger than
Therefore, it
I
in Finn’s book
Corollary 6.7, p. 144
that there exists a unique solution (2.5).
y.
utE C
2
to
( Q
Precisely, we modify the method o f continuity
due to Gerhardt [ 7 , pp. 169-1701, in which he proved the existence of the C 2 ( B
)
- solution to the equation o f
the capillary surface with gravity, and we show the existence of the solution to (2.5) belonging to Along the argument o f Gerhardt, let T =
1 t 1 There exists a solution ut
Since uo E 0
belongs to
T,
T
T
1.
C2(
be the set C2(
)
is not empty.
to
(2.5)!.
In view of
the hypothesis o f the existence o f the solution u l E C2(
to (2.5) with
assumptions,(l.3)
t = 1 , we see that the
and (1.41, in Giusti’s paper [ 9 1
(
see
sup lull 1 and E~ ( = 1 P sup ITul/ ) hold for a l l t C (0,l). Since .f ut dx = Q Q c! f o r t E T , using the arguments in Giusti [ 9 1 , with [ 9 , P. 5051
with to
(=
the help of the Paincare/ inequality we see that
hhere o f course
C
i s a constant independent o f
in the proof o f Proposition 2.2.
t
as
Therefore, it follows
Constant Mean Curvature Equation over Convex Domain in R2
137
f r o m t h e a p r i o r i bound f o r a s o l u t i o n due t o G i u s t i ( see [ 9 , Theorem 2.2, p .
5241 1 t h a t
(2.10) Furthermore, u s i n g t h e a p r i o r i estimates f o r the g r a d i e n t due t o G e r h a r d t ( see [ 7 , Theorem 1.1, p . 1583) we g e t
C, Thus, c h o o s i n g a smooth v e c - o r f - e l d Gerhardt [7,
p.
for
1691
t
for a l l
T.
a-(p> as i n
-f
a ( p ) = (1 + I p I 2
p
such
that a-(p>
(2.12) and (
te
aaT(p)/apj T
a(p>
=
(PI
for
5 3
c,
i s uniformly e l l i p t i c .
Regard
ut
as a u n i q u e s o l u t i o n t o t h e u n i f o r m l y
e l l i p t i c problem: d i v ( a"(Vut)
(2.13)
I
a"(vut).u
J
Q
u t dx
=
2 tH
=
t c o s y on
=
i n Q,
aQ,
0.
Then, w i t h t h e h e l p o f t h e e s t i m a t e (2.111, u s i n g t h e e s t i m a t e s due t o
L i e b e r m a n & T r u d i n g e r ( see [ l a ,
Theorem 1.1, p . 5101 1 , we o b t a i n
(2.14)
Shigeru SAKAGUCHI
138
Therefore
T
is closed and it remains to show that
is open.
Let
tl belong to
In order to
T.
implicite function theorem, let P {
I
0 }
=
u dx
f dx
Q
I
=
h ds } .
aQ
R X Y 3 ( t , u)
F :
{ u
-
div( a-(Vu)) a-(vu)=v
F
1
)
) x C ' + ~ (
Define the mapping
I--+ (
the
P
C2+a(
{ (f, h ) c Ca( B
2 =
and
Y =
use
T
s
)
by
2 tH,
- t cos
y
1
Z.
F(tl, u
) = 0, we can use the implicite function tl theorem and see that there exists a uniQue continuous
Since
mapping
g
defined in a neighborhood
g :
Y,
satisfying
U
-+
g(tl) = ut Since
1
and
g(t))
F(t,
= 0
U
for
of
tl,
tC- U .
satisfies (2.111, in view of (2.12) and the tl continuity of g there exists a neighborhood u - c u of
u
tl
such that
shows that
T
g(t)
is a solution to ( 2 . 5 ) .
is open.
Consequently, T = [0,11 satisfies
=
t-'ut
satisfies (2.2.t). regard
vt
and
for
Writing
t
G
(0,11.
b(x)
Ut
vt
Then
as in ( 2 . 7 1 ,
we
as a unique solution to linear problem: div( b(x) V v t ) = 2
(2.15)
the solution
(2.14).
vt
Put
This
H
b ( x ) V v t * v = cos y on
in
Q,
aQ,
and
n
vt dx = 0 .
Constant Mean Curvature Equation over Convex Domain in RZ
I n view o f (2.14), (see [8,
139
u s i n g t h e Schauder g l o b a l e s t i m a t e s
Theorem 6.30,
p.
1 2 7 1 1 , we g e t
v t d x = 0 , we c a n u s e P o i n c a r e i n e q u a l i t y . Q M u l t i p l y i n g t h e f i r s t e q u a t i o n o f (2.15) by vt and
Since
f
i n t e g r a t i n g t h i s over
Q,
we g e t
Vt
dx
0.
I n t e g r a t i n g b y p a r t s , we g e t
I n view o f (2.7)
Then,
and (2.111,
we g e t
i t follows f r o m S c h w a r z i n e q u a l i t y t h a t
Recall the trace estimate :
for a n y
E
> 0, where
C(E)
i s a c o n s t a n t d e p e n d i n g on
T h e n , u s i n g P o i n c a r k i n e q u a l i t y , we g e t
and c o n c l u d e t h a t
E.
Shieeru SAKAGUCHI
14 0
(2.18)
Hence, from the apriori bounds due to Lieberman
see
(
[17, Lemma 3.1, p. 2251) we obtain
sup lVtl 5 c. Q Consequently, by (2.16) we aet (2.3) for all t c (0,13. (2.19)
t = 0, (2.2.0) is a linear problem and
In the case that has a solution
vo
C"(
B 1.
This completes the proof.
Finally in this section we give Corollary of Proposition 2.2.
If Q
is strictly convex,
then level to -the -- -sets of the solution u convex for sufficiently small H . Proof.
(1.1) are
By Proposition 2.2 we see that
v
in c 2 ( I as t +o. vO The result of Caffarelli & Friedman [ l ] shows that (2.20)
Vt
5 is [
Dij
a 2laxiax J .
---+
strictly concave in Q , that is, the matrix
1
is negative definite in Q , where
Here, observing (2.20) and Lemma 4.3 of [ l ,
p. 4501, we see that there exists a
aQ
-
Dij
where
a -
neighborhood of
is strictly concave f o r all small
t.
Therefore, it f o l l o w s from (2.20) that the matrix [
Dij
71 is
negative definite in
Since the graph of
ut = t v t
Q
for small
has mean curvature
t. tU
Constant Mean Curvature Equation over Convex Domain in R2
9
141
3. A p p l i c a t i o n s o f Chen & H u a n g ' s c o m p a r i s o n t e c h n i q u e I n t h i s s e c t i o n we g i v e s e v e r a l b a s i c lemmas.
The
f i r s t t w o lemmas a r e p r o v e d b y t h e m e t h o d b a s e d o n Chen & Huang's comparison t e c h n i q u e .
solution t o (2.l.t)
Let
vt
be t h e
for t E C O , l ] .
or ( 2 . 2 . t )
I t follows
from t h e r e g u l a r i t y t h e o r y o f e l l i p t i c p a r t i a l d i f f e r e n t i a l equations t h a t
Lemma 3 . 1 .
Let
some p o i n t
p
Proof.
G
curvature
Kt(p)
consider a cylinder p
&
If
(0,11.
Vvt(p)
t h e n t h e Gaussian c u r v a t u r e
R e c a l l t h a t t h e graph o f Let
tH.
Suppose t h a t
Q,
belong
i s real analytic i n
Q.
= 0
at
Kt(p)
of
t is p o s i t i v e , i t s u f f i c e s t o show t h i s
Since
ut = t v t .
for
t
vt
p
= 0.
be a p o i n t w i t h
ut
h a s mean
Vut(p)
= 0.
As i n Chen & Huang [ 4 ] we
x3 = w(x) t a n g e n t t o
x3 = u t ( x )
at
such t h a t t h e l i n e generators are p a r a l l e l t o t h e
p r i n c i p a l d i r e c t i o n o f zero p r i n c i p a l c u r v a t u r e o f x3 = ut(x>
at p
a n d t h e n o r m a l cross s e c t i o n s a r e t h e
c i r c u l a r a r c s with c u r v a t u r e 2 tH. i t s maximal domain
Q".
Then
on t h e p l a n e w i t h i t s b o u n d a r y parallel lines.
Extend
x3 = w ( x ) t o
Q" i s an i n f i n i t e s t r i p 3Q- t w o i n f i n i t e
I t is e v i d e n t t h a t
w
also satisfies
Shigeru SAKAGUCHI
142
d i v Tw = 2 t H
(3.1)
w
where
and
i n Q"
T w * u = 1 on aQ-.
an-.
i s the u n i t outer normal vector t o
Since t h e two s o l u t i o n surfaces
ut
t h e same G a u s s i a n a n d mean c u r v a t u r e a t
and
p
have
w
and t h e
p r i n c i p a l d i r e c t i o n s o f them a r e c o i n c i d e n t ,
by E u l e r
formula ( see [31 ) t h e y have a second-ordered c o n t a c t at
p.
Then,
Furthermore,
since
ut-w
put
s a t i s f i e s an e l l i p t i c e q u a t i o n
v ~ i t h o u tz e r o - o r d e r
second o r d e r d e r i v a t i v e s a t o f [ 4 , p. 2541 (3.3)
Both
A
ut-w
t e r m s and
p, by
vanishes up t o Lemma 1 a n d Lemma 2
we h a v e f r o m t h e maximum p r i n c i p l e : and -
each o f which -----
B
have a t l e a s t ----
t h r e e components
meets t h e boundary
a(
QnQ-1.
Now we f i r s t c o n s i d e r t h e c a s e o f D i r i c h l e t b o u n d a r y condition (2.4). two cases.
F u r t h e r m o r e we d e v i d e t h e p r o o f i n t o
One i s t h e c a s e t h a t
w 5 0
other i s t h a t there e x i s t s a p o i n t Consider t h e former. Since
w
boundary
5
a(
ut
on
an, aQ
by t h e convexity o f
or
aQ"
and t h e
Q- w i t h w ( x ) > O .
Look a t t h e b o u n d a r y
QnQ") c o n s i s t s o f
o f which belongs t o
x
i n Q",
a ( QnQ-1. Q
the
a t most f o u r a r c s , alternatively.
each
The a r c s
Constant Mean Curvature Equation over Convex Domain in R2
belonging to
aQ-
143
are simply straight line segments.
Consider the components of the open set
A ( see (3.2) ) .
It follows from Lemma 3 i n [4, p. 2581 that it never occurs that a component o f
a( QnQ-1
meets
A
Therefore, by (3.3) the set
exclusively in aQ-.
has
A
at least three camponents each of which meets the boundary
an.
Here, since
ut = 0 on
an,
there are at
most only two components of A each of which meets This is a contradiction. Qo = { x E Q"
a(
QnQ,).
1
w(x)
< 0
aQ.
Next consider the latter.
1.
Put
Look at the boundary
By the convexity o f
Q
we see that
a(
QnQo)
consists o f at most four arcs, each of which belongs to aQ
or
ano
alternatively.
The arcs belonging to
are simply straight line segments. components of the open set
two components of
a ( QnQ,).
A.
= { x
Consider the
e
QnQo
I
ut(x) > w(x)}.
w = 0 on aQo, there are at most
and
Since ut = 0 on aQ
A.
aQo
, each o f which meets the boundary
This contradicts (3.3).
It remains to consider the case of the capillary surface
Since
(2.5).
boundary condition o f we
see that
a(
Q ut
QnQ-1
connected arcs, in which
is convex, observing the and the cylinder
x3 = w(x>,
consists o f at most four Tut*v - Tw*v
changes sign.
Therefore, the similar lemma to Lemma 3 in [4, p. 2 5 8 1 holds and we get a contradiction t o (3.3). completes the p r o o f .
This
Shigeru SAKAGUCHI
144
If
Lemma 3 . 2 .
at
= 0
Vvo(p)
p c Q , then
some point
the Gaussian curvature KO(p)
of the graph
(x,
vo(x>)
-at P - is positive. Proof. that
Let
p
KO(p)
Vvo(p) = O.
be a point with 0.
5
Suppose
For simplicity, by translation and p = 0
rotation o f the coordinate, we may assume that
I: Dij v
and X1
X2
0, and
>
] = diag[
(0)
X1, X2 1
vo(x) = w(x) + P(x>, where
Then
5 0.
X1 + ,I2 = 2H,
where
w(x) = vo(0)
+ X1(x1>2 + X2(x2I2
function in
Q.
Since
vanishes up to second order derivatives at 0
and
P(x)
P(x) is a harmonic
and
Furthermore, put
i s real analytic, we have from the maximum
P(x)
principle( see Lemma 1 and Lemma 2 in 1 4 1 1 : Both
(3.4)
A
and B
have at least three components ----
aQ.
each o f which meets the boundary -----
Consider the case of Dirichlet boundary condition (2.1.01.
Put
boundary
a(
Qo
=
QpQo).
xc
{
Since
QnQo)
belongs to
Q
w(x)
c
0 },
L o o k at the
is convex and X2
5
w
is a
0, we see that
consists o f at most four arcs each o f which aQ
or
alternatively.
aQo
components o f the open set Since
I
X1 > 0 and
quadratic function with
a(
R2
vo = 0
on a Q
most t w o components of
and A.
A.
= {
w = 0
x E Q on
aQ,,
Consider the Qo
1
P(x) > 0 } .
there are at
each o f which meets the
Constant Mean Curvature Equation over Convex Domain in RZ
a ( QnQ,).T h i s
boundary
145
c o n t r a d i c t s (3.4).
N e x t c o n s i d e r t h e c a s e o f Neumann b o u n d a r y c o n d i t i o n (2.2.0).
Since
condition o f
i s convex,
Q
vo
and t h e f a c t t h a t
X1
function with
o b s e r v i n g t h e boundary
>
0
and
w
i s a quadratic
aQ
X 2 5 0, we s e e t h a t
c o n s i s t s o f a t most f o u r c o n n e c t e d a r c s i n w h i c h V v o * v Vw*v ( = VP-v
changes s i g n .
i n [4,
lemma t o Lemma 3
contradiction t o (3.4).
For a l l t G -
Lemma 3 . 3 . points i n
Proof.
Therefore,
p. 2581
-
the similar
h o l d s a n d we g e t a
T h i s completes t h e p r o o f .
[0,11,
vt
does n o t have ---
maximal
Q.
Since
i s p o s i t i v e , t h e maximum p r i n c i p l e
2H
i m p l i e s t h i s lemma.
L e t t belong t o [0,1]. The s o l u t i o n v t h a s more t h a n t w o m i n i w a l p o i n t s , i f a n don l yi f there ---exists a p o i n t p C Q with Vvt(p) = 0 and K t ( p ) < Lemma 3.4.
Proof.
Remark t h a t
case o f ( 2 . l . t ) lemma 1.
aQ.
be a p o i n t with
vt
vt
-
aQ
( I n the
does n o t have m i n i m a l p o i n t on
We f i r s t p r o v e Vvt(p)
= 0
e x i s t s an open n e i g h b o r h o o d set o f
i s p o s i t i v e on
t h i s f o l l o w s from H o p f ’ s boundary p o i n t
Therefore
t h e boundary
Vvt*v
0
v,(p>
intersecting at
i f part
and U
Kt(p) of p
< 0.
Let
I!.
p
Then t h e r e
i n which the zero
c o n s i s t s o f two smooth a r c s p
and d i v i d e s
U
i n t o four sectors.
Shigeru SAKAGUCHI
146
Consider t h e open s e t
E =
I t follows f r o m Lemma 3 . 3
an.
t o meet t h e b o u n d a r y
G
G = { x
open s e t
two components.
I
Q
{
xe
I
Q
> vt(p)
vt(x)
t h a t each component o f
). E
has
A c c o r d i n g l y we s e e t h a t t h e
vt(x)
}
< vt(p)
T h i s shows t h a t
vt
h a s more t h a n
h a s more t h a n two
minimal points. N e x t we p r o v e
only i f part
"
Consider f i r s t the
'I.
case o f D i r i c h l e t boundary c o n d i t i o n ( 2 . l . t ) .
v t h a s more t h a n t w o m i n i m a l p o i n t s a n d t h e r e
that
e x i s t s no p o i n t Therefore, 3.3,
Suppose
p
with
vvt(p)
and
= 0
b y v i r t u e o f Lemma 3 . 1 ,
Lemma 3 . 2 ,
we s e e t h a t e a c h c r i t i c a l p o i n t o f
point.
Since
does n o t v a n i s h on
Vvt
3 . 1 and Lemma 3 . 2
Kt(p)
vt
aQ,
< 0.
a n d Lemma i s a minimal
then
Lemma
imply t h a t every c r i t i c a l p o i n t o f
vt
i s i s o l a t e d a n d t h e number o f c r i t i c a l p o i n t s i s f i n i t e . Hence we c o n c l u d e t h a t t h e r e e x i s t s a f i n i t e s e t o f s a y { pl,
minimal p o i n t s o f vt,
Put
so = max { v ( p . ) t J
set
Ls = { x
G
Q
follows from (3.5) manifold for
t o each o t h e r . so,
Ls
I
1
v,(x)
1 5 j 5 N c
...,
p2,
}.
s } for
t h a t t h e boundary
0 > s > so
Since
and
Kt(pj)
aLs}
pN } s a t i s f y i n g
Consider the l e v e l 0 > s > so.
aLs
It
i s a smooth
are diffeooorphic
is p o s i t i v e , i f s i s n e a r
h a s more t h a n t w o components.
On t h e o t h e r h a n d ,
Constant Mean Curvature Equation over Convex Domain in R2
i f
s
Ls
i s connected.
is n e a r t o
0,
and
aQ
This i s a contradiction. Since
aP
we c a n
is p o s i t i v e o n
extend the function
where
y
a)
vt
and
and
Vvt
set
Ls = { x
aQ
Then we s e e t h a t
does n o t v a n i s h i n
E
G
with dist(x,y)
R2
-
Q
= dist(
denotes t h e u n i t o u t e r normal v e c t o r
u(y)
a t y.
i s convex,
Q
R2 by p u t t i n g f o r x
to
i s a u n i q u e p o i n t on
and
an
to
to
I t remains t o c o n s i d e r t h e case (2.2.t). Vvt*u
x,
aLs i s d i f f e o m o r p h i c
147
R2
I
vt(x)
IR' -
b e l o n g s t o C1(
vt Q.
< s }.
Consider t h e l e v e l
Then
one component f o r s u f f i c i e n t l y l a r g e
R2 )
s.
t h e same a r g u m e n t a s i n t h e c a s e ( 2 . l . t )
Ls
has o n l y by
Therefore, we c o m p l e t e
the proof.
9
4.
P r o o f o f Theorem
I n v i e w o f Lemma 3.1, Lemma 3 . 4 ,
Lemma 3 . 2 ,
Lemma 3 . 3 , a n d
i t s u f f i c e s t o show t h a t t h e s e t o f m i n i m a l
p o i n t s o f t h e s o l u t i o n c o n s i s t s o f one p o i n t .
I=
Put
[O,l].
Devide
I1 = It
G
I I vt
h a s o n l y one m i n i m a l p o i n t i n
I2 =
It €
I I vt
h a s more t h a n t w o m i n i m a l p o i n t s i n Q
Then
I=
I1uI,
I i n t o two s e t s
and
I1 a n d 12:
Iln12 = 7 .
Q
1,
I t f o l l o w s from
1.
Shigeru SAKAGUCHI
148
Lemma 3.1, Lemma 3.2, a n d t h e i n e q u a l i t y (2.3) ( s e e P r o p o s i t i o n 2 . 2 a n d P r o p o s i t i o n 2.3 o p e n s e t i n I. belongs t o
I1
Il a n d
i s n o t empty.
j
i s an
tends t o
I2
=.
Therefore,
i s closed i n I.
I2
a sequence o f p o i n t s i n as
I2
Lemma 3.2 a n d Lemma 3.4 i m p l y t h a t
s u f f i c e s t o show t h a t
t,
1 that
such t h a t
t
H e n c e , Lemma 3 . 4
j
0 it
L e t i t . } be
J
converges t o and t h e
compactness arguments i m p l y t h a t t h e r e e x i s t s a subsequence point
p
{ t k } ,a s e q u e n c e o f p o i n t s
{pk},
and a
which s a t i s f y as
(4.1)
k
< 0.
+ a,
By c o n t i n u i t y we h a v e Vv
(4.2)
Since
Vv
t*
(p) = 0 # 0
t* f r o m Lemma 3 . 1 ,
belongs t o
I*.
on
and
K
aQ,
so
t*
(p)
2 0.
p c Q.
Therefore i t follows
Lemma 3.2, Lemma 3.4, a n d ( 4 . 2 ) t h a t T h i s shows t h a t
t,
I 2 i s c l o s e d i n I. The
p r o o f i s now c o m p l e t e d .
References
[I]
L.
A.
C a f f a r e l l i & A.
Friedman, Convexity o f
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A.
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Constant Mean Curvature Equation over Convex Domain in R2 E. 7 (19821,
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V e r l a g New Y o r k B e r l i n H e i d e l b e r g T o k y o , [ 7 ] C.
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t h e c a p i l l a r i t y p r o b l e m , Ann. S c u o l a Norm. Ser.
N
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[ 8 ] D . G i l b a r g & N.
Sup. P i s a
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Trudfnger,
Elliptic Partial
D i f f e r e n t i a l E q u a t i o n s O f Second O r d e r , S e c o n d Edition, Tokyo,
Springer
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V e r l a g B e r l i n H e i d e l b e r g New Y o r k
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[ 9 ] E. G i u s t i , B o u n d a r y v a l u e p r o b l e m s f o r n o n - p a r a m e t r i c s u r f a c e s o f p r e s c r i b e d mean c u r v a t u r e , Ann, S c u o l a Norm. Sup. P i s a S e r
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E.
Giusti,Onthe
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1213-1225. Rearrangements and C o n v e x i t y o f L e v e l
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convex domains,
I n d i a n a U n i v . Math.
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This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 10, 153-177 (1989) Recent Topics in Nonlinear PDE W,Kyoto, 1988
Symmetric Domains and Elliptic Equations Takashi SUZUKI
Department of Mathematics Faculty of Science University of Tokyo 51. Introduction.
In (111, B. Gidas, W.-M. Ni and L. Nirenberg showed a remarkable relation between the symmetry of a bounded domain 2 n c RN and that of the solution u = u(x) C C ( n ) n C o ( E ) for the semilinear elliptic equation (1.1) an
where
f(u),
-AU =
u>0
(in a ) ,
is the smooth boundary of n
Namely, suppose that a hyperplane real number
T X ,
Q
and
(on
an),
f E C1(R).
is symmetric with respect to y E RN.
with a normal unit vector set T i = { x c RN I x - y = A),
For each
and
(1.2)
Then,
--
< A,
<
A
*
<
aJ,
T
=
,
T (k
A, <
u = 0
< A*.
Furthermore, put
and
Tin
# #
if
+A*)/2
Q ( A )
= { x c n1x-y > A).
Under
these notations, we give the following Definition 1.
We say that the domain
symmetry with respect to
T
satisfied;
153
Q
has the
GNN
if the following conditions are
Takashi SUZUKI
154
(a) For each
in
A
.
flection set of ib)
For each
gonal to
*
5.
+ X,)/2
X < A
*
,
( A
*
+
x,)/2<
).< X
*
the re-
Q'(X),
TA
with respect to
(A)
in
k
(X
:.
lies in
, TX
is not ortho-
cl
3;;.
Then, Theorem 0 (Gidas-Ni-Nirenberg [ 1 1 3 ) .
In the case that
T
has the GNN symmetry with respect to the hyperplane y c RN,
with the unit normal vector
each solution
u
=
u(x)
of (1.1) has the following properties: (a) u (h)
e
Along each stream line
the vector field y . 7 ~< 0
T.
is symmetric with respect to
y,
holds on
.+ball
!i+
(x
spect to every hyper2lane
u
.;jx.y >
(1x1 < R )cRN
=
;L
the value =
T
T
starting from decreases. (1
*
subject to
That is,
0
+ X,)/21.
is GNN symmetric with re-
containing the origin so that
we have Corollary. =
i!xl < R 1
Each solution
is radial; i.e.,
Furthermore,
u'(r) < 0
u
of (1.1) on the ball
u = u(r),
where
r
=
c
1x1.
ci
( O < r < R).
This remarkable fact reduces (1.1) to the 1-dimensional problem
for the case
Q
=
I I x l < R 1.
For instance the following dia-
cjrams about radial solutions for (1.1) when
f ( t ) = ,iet
Symmetric Domains and Elliptic Equations
155
obtained by Gellfand [lo] and Joseph-Lundgren C133, describe complete profile of solutions:
l(n(2
2 < n < 1 0
10
2n
The fundamental idea of C 1 1 J is to compare the original solution
n'(x).
u
and its reflection
u
We note that the relation
A
with respect to -Au
=
f(u)
TA
on
is invariant
under such a kind of transformation. The aim of the present article is to study such properties as (a1 and (b) in Theorem 0 when the symmetric domain violates its GNN symmetry.
Namely, in 5 2 we shall describe
how non-radial solutions arise in case that lus
:
n
=
{a < [ X I < a + b l
(a, b > 0),
is an annu-
and in 53 we give
some other stream lines along which symmetirc solutions decrease their values. 52. Generation of non-radial solutions in annulus domains In [ 6 J , H. Brezis and L. Nirenberg observed the existence of non-radial solutions in (0 < R1 < R2 <
m)
in the case that
Q
= IR
N 1 < 1x1 < R21C R
for the equation
p
*
is sufficiently close to
N
*
where
s?
156 N
*
Takashi SUZUKI for
= -N + 2
N-2
N
2- 3
and
N*
ments are the following. =
-
=
Put
X
for
N
H1 ( n ) , 0
=
=
Their argu-
2.
Xm
1 Iv t HO(i2)Iv
=
v( 1x1 ) I ,
Then, there exists a minimizer
u 6 X\IO}
j
of
satis-
fying (2.1). I n fact, the Lagrange constant can be reduced t3
1 after the stretching transformation. urn X m \ ( 0 1
minimizer
j
of
Similarly, a
solves (2.1).
shown by Kazdan-Warner's argument ( ( 1 5 1 ) .
This can be
In fact, it holds
that
vt Xm,
for each
from which
-- 1 r
follows.
Now, the fact
u
X-
which actually can be shown when Motivated by this,
d N-1d drum) N-1 ai(r
j < jm,
is verified from p
*
P
uco
is close to
N
*
-
Coffnan h a s shown the generation
C.
of essentially infinitely many solutions for (2.2)
on
rl
where Tk
+ u = u2 L + l
-&U
=
la
<\XI
<
,
u >0
a + blCR2
e = 1, 2, . . . ( ( 8 1 ) .
2n be the --rotation k
8-direction:
as
(in
Q ) ,
u = 0
a + - with
Namely, for
(on b>0
k = 1, 2, ...
of independent variables in
an)
fixed, let
T h e n , f r o m N e h a r i ' s a r g u m e n t [18] it c a n b e shown t h a t a
vke Xk\
minimizer =
1, 2,...,
ing
solves ( 3 . 2 ) f o r each
j,
of
{O}
N o w t h e d e s i r e d a s s e r t i o n f o l l o w s f r o m show-
m.
# { j k I k = 1, 2 , . . . ,
(#I
a
-1
-
-t
m,
+-
unaer t h e operation fixed.
b> 0
H i s argument was r a t h e r t e c h n i c a l i n s e p a r a t i n g
finite
which h a s b e e n r e f i n e d by B.
k's,
(m c n )
jm 5 j n
case.
and
j n
a l w a y s h o l d s when
Similarly,
mln
5 jm
j,
for
1,'s
Kawohl i n u s e o f
Namely, h e s h o w e d t h a t
S t e i n e r ' s symmetrization ( [ 1 4 J ) . mln
k
j m <j n .
Here, w e n o t e t h a t
because
X n C Xm
i s obvious for
in that
k = 1, 2 ,
....
I n t h i s s e c t i o n , w e s h a l l p r e s e n t a new a r g u m e n t t o separate
jkts for finite
Consequently, t h e ap2ear-
k's.
a n c e o f a n y mode s o l u t i o n w i l l b e e s t a b l i s h e d u n d e r t h e 02eration (#). H e n c e f o r t h , w e s t u d y t h e e q u a t i o n ( 2 . 1 ) on R = {a < 1x1 < a
sence. 2,..-),
+
b 1cR2
Thus we p u t
x
=
for
1< p <
1
xk
H 0 ( n) ,
X m = { v ~ X ~= Vv(lxl)1
-
w i t h o u t loss of
= {v E
and
x~
T
= ~ V} V
eE-
( k = 1,
158
Takashi SUZUKI
W e f i r s t show t h e f o l l o w i n g p r o p o s i t i o n b y e x t e n d i n g t h e
a r g u m e n t of N e h a r i . P r o p o s i t i o n 1. -
uk
a minimizer
Since
ir
k
I,'+'(
L+
+
and
with
> 0
~7
k-
:
( U V p~+ l
t
XI
and
x
=
L
holds w i t h a c o n s t a n t
X,
t
-1
X ---+ ( - m I
is compact.
G)
for each
there exists
satisfying (2.1).
j,
= I n f ! ~ , v v l 2ilv c I,
j,
v =
of
m ,
W e d e f i n e p r o p e r lower s e m i - c o n t i n u o u s
Proof: functions
k = 1, 2 1 . . . ,
For each
xk,
,?+I
through
= 11,
its m i n i m i z e r
e x i s t s b e c a u s e t h e embedding
1
From t h e L a g r a n g e a n r n u l t i F l i e r
T h i s r e l a t i o n means t h a t
u t R .
wher-e
ilvi!
convex
z
d e n o t e s t h e p a r i n g between
(arezis [SI, e . g . ) .
The f o l l o w i n g lemma i s d u e t o S u z u k i - N a g a s a k i L e m m a 1. -~
For
f t X'
L;ro$erties are e q u i v a l e n t :
and
v c X,
[22].
the following t w o
Symmetric Domains and Elliptic Equations
v exk
(2.5)
and
f
6
159
0
aIr(v).
L e t u s admit t h e lemma f o r t h e moment.
In fact for
there exists a
that is,
T w = wI
(2.6)
k
with = f
-Aw
f
t
3?(w),
(in
= { v t X ( Tv = v )
v
because
k
w = 0
n), and
(on
an)
f = pvp.
f ~ a v ( v ) h o l d s , w h i c h r e a d s as
Therefore,
Here,
wtXk
P
f=pv
v '0
in
p r i n c i p l e w e conclude u = p "(p-')v
t
''v''Lp+l
and
xk
v >O
in
= 1. R
and
From t h e maximum p
> 0
so t h a t
s a t i s f i e s (2.1).
To complete t h e p r o o f ,
p l i e s ( 2 . 5 ) i n Lemma 1.
w e s h a l l show t h a t ( 2 . 4 ) i m -
I n f a c t from ( 2 . 4 ) w e o b t a i n
and
f o r each
of
5 e X.
Noting t h a t
and
v t D(9) = Xk,
w tXk
$ ( w ) = $ J ( v )= 0 < + -
because
w e add t h e s e t w o i n e q u a l i -
ties t o obtain
which means
f t aV(v).
The f o l l o w i n g p r o p o s i t i o n i m p l i e s t h e g e n e r a t i o n o f
,
Takashi SUZUKI
160
non-radial
solutions. Under t h e o p e r a t i o n ( # I ,
Proposition 2. holds. 2,
On t h e c o n t r a r y , e a c h
k = 1,
....
set
For t h i s p u r p o s e ,
Under t h e o p e r a t i o n
lies i n
R k.
Take
n
( # )
i n d e p e n d e n t of
c 20
1
is bounded € o r
j,
P r o o f : We f i r s t s h o w t h a t
10
--
jm
t h e r e is a ball
v*, .. -
f
E C,"(R2) \ ( 0 )
Vk =
with radius
B
whose s u i t a b l e t r a n s l a t i o n
be i t s t r a n s l a t i o n w i t h
functions
is f i n i t e ,
k
0 < 0 < -27t 1. k
= ( reie ( a < r < a + b ,
a lr
when
j k c O(1)
If1 cC
t h e independent variable
0
supp
e
I n t h i s way w e c a n c o n s t r u c t
c
lP1
- 2 O(R )
by
with B'
n ,LA
supp IPC B
c
Q
k.
B' and let
T h e n , the
a r e o b t a i n e d by r o t a t i n g
a E277
( a = 1, 2 , . . . ,
= lP1
+ ... +
lPk
e
k-1).
xk
fOK
which we have
Hence
j,t
O(1)
follows.
Now w e s h a l l show t h a t
Jm
__t.
t h i s end we take an a r b i t r a r y element
holds under v = v ( r ) t X_.
(#).
To
Then,
Symmetric Domains and Elliptic Equations
161
Therefore,
so that
3 r j Ik = 1, 2 , . . . I
The critical values in the following way.
are separate6 k This is a refinement of the results
by C. Coffnan and B. Kawahl.
Proposition 3.
Furthermore, Proof. =
j,
< 1-
For each
u(reie) with
t X1
We have
implies that v
8' = ke.
=
j, <
...
v(re i8) c. Xk' set Then, the mapping
<
1,.
?(r, 6 ' ) v
E
Xk w ? t X
is isomorphic and we have
a+b
a
2 - 1 a v 2
d r { r ? - - ) 2+ k r
I
162
Takashi SUZUKI
and
Hence (2.9)
=
1,
Jk(v) 5 J k + l ( v )
Since
Inf
Uvk
the relation
\ILPtl
J
v k e X,- a n d h e n c e
lk
f r01n
= 1.
~
we obtain ( 2 . 8 ) .
( k = 1, 2 , . . . ) ,
v k t X\IO)
There e x i s t s a minimizer such t h a t
Jk(v),
V€X\IO}
= - J k~
of
i n (2.9)
J,
Since
av
2 39
implies
j k - jm. Thus
j,
<
0.
I n o t h e r words,
...
<
jk
follows
by a n i n d u c t i o n .
J,
Now w e c a n g i v e t h e f o l l o w i n g theorem, where a f u n c t i o n
v F
x
i s s a i d t o h a v e made
sup:Llv
t
33,
- )
if
k
x L'.
Theoreml. mode
k ( = 1, 2 , . . . ,
There always e x i s t s a r a d i a l ,
s o l u t i o n for- ( 2 . 1 )
Furthermore,
on
ii
= t a <
k = 1, 2 , . . . ,
for each
t h a t i s of
1x1 < a + b }
c
s o l u t i o n s o f mode
R
2
.
k
a r i s e under t h e o p e r a t i o n a
(d)
Proof:
--+
b
> 0
fixed.
The f i r s t p a r t f o l l o w s f r o m P r o p o s i t i o n 1.
rl
Symmetric Domains and Elliptic Equations
For each finite
under
the relation
j,
If
= j,
by Proposition 2.
(#)
k' > k,
k,
then we have
j,,
- jm -
ever this gives a contradiction j,,
> j,
k' > k
for any
the minimizer of has made
k
j m arises holds for some
by Proposition 3. jm > j, = j,,
so that
vk
How-
- jm.
$? X k l , where
Hence uk
0
j, < j - .
It seems to be a quite interesting question
whether any k-mode solution of (2.1) is a minimizer of =
is
in Proposition 1. This means that uk
Jk
as far as
Remark 1.
j,,
c
163
Inf IIVVII/IlVII p+l or not. V€Xk\I0 1 L
j,
If it is true, from the
above argument we can conclude that when a k-mode solution for (2.1) arises, then there exists any a-mode solution for L = 1 , 2
,...,
k-1.
Remark 2 . function on
Let
(0, + - )
5
=
5(r) be a positive continuous
with golynomial growth order at
+-.
Then a similar fact can be proven for
Further, our arguments are valiC even for higher dimensional problems.
Takashi SUZUKI
164
Remark 3 .
In use of Lemma 1, Suzuki-Nagasaki C231 has
studied raaial and non-radial solutions on C R’
(0 <
where
A
a < 1)
I:
a
=
{a c 1x1 < 1)
for the nonlinear eigenvalue problem
is a positive parameter.
5 3 . Local profile of symmetric mild solutions in two dinen-
sional domais. Suppose that hyperplane
T.
c
c RN
When the
is symmetric with respect to a GNN
property is violated, there
may arise non-symmetric solutions for (1.1) even in simply connected domains. For instance, take the univalent func1 on D = t z c C I 1 5 1 < 1 1 with tion g R ( z ) = 1 + 5-R :+R 2 R > 1 and set 7.R = g R ( D ) c C 2 R Then c R is symmet-
-
.
r - i c with resgect to both x7)tR
..
2
.
As
x1
for the former,
while as for the latter is close to 1.
GNN
In fact, if
solutions arise as
A+O
for
and RR
x2
x
axes, where
is always
GNN
(xl,
=
symmetric, R
pro2erty is violated when 1 < R <
f(t)
=
JT non-symmetric Ae
t
.
See Weston [24],
Moseley [16] and Gustafsson [121, Nakane [ 1 7 1 . However, even for such symmetric domains without
GNN
property, there may exist many symmetric solutions for !l.l): i3.1)
-Au
=
f(u),
u > 0
(in
!;),
u
=
0
(on
32).
Symmetric Domains and Elliptic Equations Example 1.
When
f:R
u
increasing, the solution
*
u (x) = u(x
t h e o t h e r hand
t h e r e f l e c t i o n of
x
*
=
-
u(x)
( 3 . 1 ) is unique.
of
s a t i s f i e s (3.1),
)
with respect to
i s symmetric w i t h r e s p e c t t o Example 2 .
i s m o n o t o n o u s l y non-
R
--f
165
T,
x
f ( t ) = X t, 1
Take a
and
f'(0) > 0
u
All
u =
IP
1( x ) > 0
of
satis-
= 0
becomes s y m m e t r i c .
function
C2
f"(t) > 0
*
being t h e f i r s t eigenvalue.
X1
Then, f r o m t h e s i m p l i c i t y o f Example 3 .
u = u
T.
The f i r s t e i g e n f u n c t i o n
fies (3.1) f o r
On
being
and hence
under t h e D i r i c h l e t boundary c o n d i t i o n
A
*
(t > 0)
such t h a t
f
a n d c o n s i d e r t h e non-
l i n e a r e i g e n v a l u e problem -AU
(3.2)
€or
Let
X > 0.
u > 0
= xf(u),
d,
(in
$,, # g4
Further, when
v
€or
(on
be t h e set of i t s s o l u t i o n s .
i t i s known t h a t t h e r e e x i s t s a that
u = 0
n),
X > 7
and
-
f ) € (0,
= T(Q,
d,+
g4
for
0
t h e r e e x i s t s a unique minimal element
ldA #
g4.
JX. F o r
Namely,
v(x) 2 uX(x) (x
6
a)
Then, such
+ m )
K.
X
C
aa)
yX
in
.8,
holds f o r each
t h e p r o o f o f t h e s e f a c t s , see G r a n d a l l -
Rabinowitz ( 9 1 f o r i n s t a n c e .
ux
Then, o b v i o u s l y
becomes
symmetric because of i t s minimality. When " b e n d i n g " o c c u r s a t
X = 7,
t h e s o l u t i o n s are
also symmetric around t h e bending p o i n t I n t h e case t h a t t h e n o n - l i n e a r critica1;that
is,
term
(TI f
2
-).
A
i s of s u b -
Takashi SUZUKI ---c
lim f(t)/tP < t- + a
log f(t)/tb <
p < N* = N+2 when
for some
m
for some
m
N 2 3
N-2
b < 2
when
N
2,
=
t++m
there exists other symmetric solution fcr
I!
x
c
<
x. For
Zxample 4 .
f (t)
actually occurs in (3.2)
c
;i
R2
et
=
with
( [ y I).
c)t.her symmetric solution
If
ux
When
9, than
6
Suzuki.-Nagasaki [ 21 3 . Namely, for
6
L~ 41
vect.0:-
B =
I
on
-x an.
<
x,
ds
N
2,
=
uX
bending
there exists
for
h
2
=
where
Setting
have the connectivity of
holds for
n
0 <
1
u
-X
<
1.
€or
is known by
x -
= ~ ( n )
< X < 5;
whenever
denotes the outer unit normal
6,= -A
,8,
( 2 2n) and
dn”.x
2 ~ ) / l ~ l B ,# J
and
1 L<. N 5- 9.
is star-shaped with respect to the origin
instance,a more detailed structure of
=: 8 1 ( 6 -
y,
u X e d, than
x
u l ~ for
and
*
uX
in
A
-
T,
we
u plane.
U
The result is proven b y the following facts.
For the solu-
Symmetric Domains and Elliptic Equations
(u,
plies
s
Next,
S < 8n
A )
= A
gives
-A
first
< X
im-
from the Rellich-Pohozaev identity.
Jneu < 8 n
by Bandle's generalized sym-
> 0
p 2 m
metrization, where
{ I J . ~ .
the eigenvalues of
-
condition.
f(t) = et ,
of ( 3 . 2 ) with
tion
167
- AeU
A
... )
< p1 < p2
(--m
3 1=1
denotes
under the Dirichlet boundary
For details, see C213.
NOW, both
*
and
u X are symmetric by the arguments
in Example 3 . Example 5.
Under the same situation as in Example 3,
suppose furthermore that linear, i.e., holds for
lim f'(t) t -
0 < t <
-m.
f =
Let
donote the eigenvalues of let boundary condition.
X2/m),
tively. works of
we have
R ---+
m > 0
exists. m
Then,
is asymptotically
R
{XjIjzl - A
Furthermore, according to min{r,
:
in
-
Then
< A 2 -<
(O < 2
L (Q)
f'(t) < m
# A, = 1
and
+m)
under the Dirich-
A = x ( n , f) 2 Xl/m
0 < X 2 Xl/m
- . -*
and
Al/m
# ,8, = 2 ,
holds. < A <
respec-
In more details, the following diagram holds by the H. Amann, A. Ambrosetti and P. Hess ((13, c 2 1 ,
(the case
x2/m
<
7)
Takashi SUZUKI
168
From the same reason, those solutions are symmetric in the
o
case of
< A < min{T, ~ ~ / m ) .
Taking these examples in mind, we ask the following question: Does each symmetric solution T,qn
its maximum on symmetric? value of
when
n
u
for (3.1) attain
Along which curves starting from
u
decrease?
GNN
is not necessarily TI
does the
Here we study these questions assum-
1ng
(H1)
is simply connected and
R C R2
R,
f(R+)
c R,,
where
= ( 0 , +-I.
Yurt-her, we introduce the following Definition 2.
A
solution
(--
< P,(u)
u
of (3.1) is said to be
mild i f
{ v . (u))m 3 3 =1
where
< P,(u)
.<-
...
-t
+
-)
denotes
the set of eigenvalues of the differential operator -
A
tion
- f'(u) in o
I,
=
L2 ( ( 2 )
under the Dirichlet boundary cond-
o.
Most symmetric solutions mentioned in above examples are shown to be mild.
Mild symmetric solutions are the
Objects whose local profile we study in this section. Henceforth, res9ect to
R
xl-axis
c R2
is supposed to be symmetric with
without loss of generality.
there exists a Riemann mapping such that
g
:
D
= { 1 5 1 < l}
-
Then, R
Symmetric Domains and Elliptic Equations (3.4)
= g(5)
In terms of
g,
( 5 t
169
D).
we furthermore restrict the geometry of
That is.
G.
Im h( r ) > 0
(H2)
holds on
D+ = {ccDIIm
> O)r
5
where
(3.5)
This condition will be utilized instead of the ty.
GNN
proper-
Examples of such,domains with ( H 2 ) are performed by
Chen-Nakane-Suzuki ( 7 1 .
Some symmetric domains without
GNN
property satisfy (H2), and the converse is also true. We state the main result in this section. Theorem 2.
Under the assumptions (H1) and (H2), each
mild symmetric solution xl-axis.
of (3.1) attains its maximum on
In more details, let
Then, the value C
u
v
= urg
starting from a point
vector field
v(r) =
L
line
C
into
fie
=
&(
p
m(1+
c(t)) < 0
{r(t)lO 5 t < - 1
aD.
R2
.
decreases along each stream line on
L r\ D
and subject to the
52).
In other words, €or the flow
the relation
be the xl-axis in
5 =
(t 2 0 )
c(t)
defined by
holds.
is orthogonal to L
The stream and absorbed
Actually, it can be given explicitly in
terms of trigonometric functions.
We note that those stream
Takashi SUZUKI
170
D
lines on
are indegendent of the nonlinear term
also the Riemann mapping
f
and
g.
Theorem 2 does not hold without any assumgtions on the geometry of
other than its symmetry such as ( H 2 ) .
R
instance, take is mild.
a line R
:.
0
Let
T
f(t)
in (3.1), of which unique solution
= 2
c RL
o
be a symmetric domain with respect to
and composed of two circular discs
c,
a ? of
C 2 0 3 , cf. Payne C191).
its maximum on
T
we have
lirn U I -
In other words,
when
> 0
E
Proof of Theorem 2:
/uIT > 1
=
goVz
the solution = (u~goip~)
u
:
=
D
Recall that
u
does not attain
u(x)
g
:
D
For each
-
s1
Z E D,
is a we set
is another Riemann mapping.
12
A
(Sperb
is small in this case.
Riemann mapping satisfying (3.4).
gz
2 ~ . Then
a?
E L 0
We give the
Then;
of radius
ii,
connected by a channel of smallest width
for the outer
For
of (3.1), the function
vz
=
For
vz(c)
solves
(3.81
Henceforth we set
w
=
fi
and
v
=
vo.
The function
Symmetric Domains and Elliptic Equations
w = - va
I
satisfies
as s w s = o
L
w
=
171
(on a D ) .
0
By an elementary calculation, we have
so that
where
d,
=
of (H2), D,-
t
{ctD(Imh(<)P 0 1 because
2
=
By virtue
c d , and hence
(3.14.1) (-A-pof'(v))w~O (in D+),
w
f(v) 2- 0.
(on
0
(-A-Q
2
0f'(v))w,O
(in D-),
aD)
On the other hand,
where
is the vector field given in the statement of the theorem and
[
J
: C
assumption,
-----*.
u
R2
is the canonical isomorphism.
From the
is symmetric with respect to xl-axis and so
Takashi SUZUKI
172
is
v
e.
with respect to
thogonally to
= VV~CV31
WI
nD
1 crosses or-
=
0.
knD
Now, from the mildness of Lemma 2.
v
so that
.P
(3.11.2)
C
Furthermore,
u
The operator
follows S
-
=
A
2 - pof' (v) in
L~(D 2
under the Dirichlet condition is positive definite. Then we can aaply the strong maximum principle in (3.11) to conclude that
(3.12)
=
because
w
VV.[~I
(in D +- )
:0
c
0
(in
D+)
>
0
(in
D
implies
v ; , =~ 0, which contradicts to
v
-1
:
v > 0
I
(in
0
(in
D+)
D).
from
There-
fore, we have
for the flow
~ ( t ) defined in ( 3 . 6 ) .
5 =
To complete the proof, we give the Proof of Lemma 2: tor where
-
Csi,=
f'(u)
in
g(D+). -
L -1 ,' 0, where The domains
-
Let
LL(n+) -
6,
be the differential opera-
under the Dirichlet condition,
Then, the assertion is equivalent to t
p1
r?
5
denotes the first eigenvalue of
Jt
-
are symmetric with respect to xl-axis.
Making the odd extension of the first eigenfunction
+
9 1 > 0
173
Symmetric Domains and Elliptic Equations
A+,
of
we get an eigenfunction
-A-
tial operator
f I (u) in
L2 (
d, the differen-
of
$
under the Dirichlet
0 )
+
vl.
boundary condition with its eigenvalue function
has two nodal domains in
$
By virtue of the
R.
+
v 1 L A2(u),
nodal domain theorem of Courant, we have that
+
p1
from the mildness of
> 0
Remark 4 .
Riemann mapping
not unique.
Actually,
dition for
a c ( - 1 , 1).
tion
h = h(r)
ga
=
u.
g
gova
:
D
This eigen-
-
so
Similarly, P 1 > 0.
-
0 n
with ( 3 . 4 ) is
also satisfies this con-
With this modification, the func-
in (H2) is given as
Theorem 1 can be applied if (H2) is satisfied for this with some
a
t
Remark 5.
(-1, 1 ) .
g
a Riemann mapping
scz) =
for
h+(r) -
+(1
:
D
-
x1 $1
2
c2)
a g'(5) +
has two axile symme-
and x2
axes.
satisfying
We suppose that
g(Z).
=
n c R2
Suppose that
tries with respect to both
and
25.
R
Let
There exists
-g(-Z)
= g(2)
satisfies
u
=
u(x)
be a mild
solution of (3.1) symmetric with respect to both axes. u
ha
attains its maximum at
0,
the origin, by Theorem 2.
Then
Takashi SUZUKI
171 Set D1,
D1
= ( r, t D l R e
Ims
S,
t h e r e arise unique f l o w s
from
x1
fields
and
x2
2
)
By t h e s e stream l i n e s
tively.
and
Q
in
= r,-(t)
5-
subject to the vector
Q,
and
For e a c h p o i n t
c+(t)
=
S+
axes crossing
= a(l + 5
v+(c)
0 1.
>
w-(z)
= 1
-
c2 ,
respec-
i s divided i n t o four
D1
parts:
f
Let
and
I(Q)
for each
in
P
be t h e p a r t s i n d i c a t e d above.
II(Q)
The segmenrr c o n n e c t i n g
0
and
IvvlQ # 0
F u r t h e r , we have
vIp
we have
I(Q1,
Q
vIQ,
where
fci
(J
J [ u / I =.
< c
Dc
=
b y Theorem 2 .
ir, c D l v ( r ) < c l
c1
r
C
=
3nc
of
u
1s a
I(Q).
Therefore, under
is
u = u(x) star-shaped
In p a r t i c u l a r , each l e v e l set
L
x)
v = udg.
are c o n t a i n e d i n
t h e s e circumstances each m i l d symmetric s o l u t i o n has the property that
Then,
RC
i s simply connected, and its
c 1+6
Jordan curve for each
References
ClI
Amann, H . ,
M u l t i p l e p o s i t i v e f i x e d p o i n t s of a s y m p t o t i -
c a l l y l i n e a r maps, J . Func. A n a l . ,
17(1974)173-213.
Symmetric Domains and Elliptic Equations
c23
175
Ambrosetti, A , , On exact number of positive solutions
of convex nonlinear problems, Bollettino U.M.J.
(5)15-A
(1978)610-615. c3 3
Ambrosetti, A . , Hess, P., Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73(1980)411-422.
c4 3
Bandle, C., Isoperimetric Inequalities and Applications Pitman, Boston/London/Melbourne,
C53
1980.
Brezis, H., Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espace de Hilbert, North-Holland, Amsterdom/London/New York, 1973.
C63
Brezis, H., Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure A p p l . Math., 36(1983)437-477.
c73
Chen, Y.-G., Nakane, S., Suzuki, T., Elliptic equations on
2D
symmetric domains: local profile of mild solu-
tions, preprint.
C83
Coffman, C.V., A non-linear boundary value problem with many positive solutins, J. Diff. Eqs., 54(1984)429-437.
c91
Crandall, M.G., Rabinowitz, P.H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58(1975)207-218.
I101 Gel'fand, I.M., Some problems in the theory of quasilinear equations, Amer. Math. SOC. Transl., 1(2)29 (1963)295-381.
Takashi SUZUKI
176
C113 Gidas, B., Nil W.-M., Nirenberg, L., Symmetry and
related progerties via the maximum principle, Comm. Math. Phys., 68(1979)209-243. [121 Gustafsson, B., On the motion of a vortex in twodimensional flow of an ideal fluid in simply and multiply connected domains, Dep. Math., Royal Institute of Technology, Stockholm, Sweden (1979). [ 1 3 3 Joseph, D.D., Lundgren, T.S., Quasilinear Dirichlet
problems driven by positive sources, Arch. Rat. Mech. Anal., 49(1973)241-269. [141 Kawohl, B., Rearrangements and convexity of level sets in PDE, Lecture Notes in Math., #1150, Springer, Berlin/Heidelberg/New York/Tokyo, 1985, pp.95-97. [153 Kazdan, J.L., Warner, F.W., Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28(1975) 567-597. C 1 6 1 Moseley, J.L., Asymptotic solutions for a Dirichlet
problem with an exponential nonlinearity, SIAM J. Math. Anal., 14(1983)719-735. [173 Nakane, S . , private communication. C181 Nehari, Z., On a nonlinear differential equation aris-
ing in nuclear physics, Proc. Roy. Irish Acad., 62 (1963)117-135. l191 Payne, L.E., On two conjectures in the fixed membrane eigenvalue problem, ZAMP 24(1973)721-729. (201 Sperb, R.P., Extension of two theorems of Payne to some non-linear Dirichlet problems, ZAMP 26(1975)721-726.
Symmetric Domains and Elliptic Equations
177
C 2 1 1 Suzuki, T., Nagasaki, K., On the nonlinear eigenvalue
problem
Au+Ae'=O,
to appear in Trans.
AMS.
C 2 2 1 Suzuki, T., Nagasaki, K., Lifting of local subdifferen-
tiations and elliptic boundary value problems on symmetric domains, I, Proc. Japan Acad., Ser.
A 6411988)
1-4. c 2 3 1 Suzuki, T., Nagasaki, K., Lifting of local subdifferen-
tiations and elliptic boundary value problems on symmetric domains, 11, Proc. Japan Acad., Ser. A 6 4 ( 1 9 8 8 ) 29-32. C 2 4 3 Weston, V . H . ,
On asymptotic solution of a partial dif-
ferential equations with an exponential nonlinearity, SIAM J. Math., Anal., 9 ( 1 9 7 8 ) 1 0 3 0 - 1 0 5 3 .
This Page Intentionally Left Blank
Lecture Notes in Num. Appl. Anal., 10, 179-194 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
On the Cauchy Problem for the KPEquation Seiji Ukai Department of Applied Physics Osaka City University
Introduction
1.
To investiRate the transversal (y-directional) stability of one-dimensional KdV solitons, Kadomtsev and Petviashvili 1 1 3 proposed s o called KP equation which is a two-dimensional version o f KdV equation and given by (1.1)
(
where
II=U
position
lit
+
auuX
+
fluxxx I x
+
ruyy = 0,
t,x.y) is a scalar unknown fuction of t me t2O and x,y)ER2 while a . 6 , ~are real constants.
Kano C21
justified their intuitive derivation by establish ng Friedlichs’ exuansion for the Euler equation of water surface wave whose first order truncation leads to (1.1). l o c ~ l(in time) solutions to ( 1 . 1 1 ,
and also constructed
both f o r analytic initial
All his proof is based on the abstract Cauchy-Kowalevski
data. theorem,
Using the method of inverse scattering, Wickerhauser
161 proved the g obal existence assuming that y > O and initials
are small in the Sobolev space W2”(R2)n The aim of
WlSs(R2) with ~ 2 1 0 .
he present paper is to construct local
solutions for less smooth initials, namely, for those in W2*‘(Q) for sX3. with, however, a certain restriction stated later. 179
As
180
Seiji UKAI
for the domain Q. we deal with four cases
R2, RxT, TxR and T2
where T is a one-dimensional torus which implies the per odic boundary cond i t 1 on. In the seauel. HS denotes W 2 * s ( n ) while L Q ) ( I ; X ) ,
Co
I
;x)
and LlP(I;x) respectively the spaces of functions bounded, continuous, and Lipschitz continuous on an interval I with values in a space X.
Theorem (1.2)
s
as follows.
Let ~ 2 3 . For any uo satisfying
1.1.
uo
Our main resu t
wi h some V ~ ELOC' H ~ VVOEHS,
=
there i s a constant T > O and (1.1) has a unique solut on ~ ( t )E C 0(C-T,T3;HS)~Lip(t-T,Tl:Hs-3)
(1.3)
u has the form u=cpX with some c p € L ~ ( C - T . T l : HsL-o1c ) .
with u(O)=u 0 '
qyEL~(l-T.T3:HS-3).
Remark
1.2.
(i)
Any
U ~ € H ~ ~ , (2R) has a c p o € H ~ , , t R 2 )
such
t h a t uo=(pox: an easy choice i s ,
cp0(X,Y) = JoU0(X'Y)dX. X
The main restriction in (1.2) is, therefore, that
OY be n case Q is unbounded cp
nerindic i f s o is u 0 ' and in HS snfficient condition for this wi I b e discussed in 5 2 . ( 1 I )
Actually. we wi I 1 prove the theorem assuming
exlstrnce of
a V ~ E Hsatisfying ~
he
A
Cauchy Problem for the KP Equation (1.4)
181
uoY=vox*
which comes from ( 1 . 5 )
below.
And this is equivalent to (1.2).
see 5 2 , (1.2) has been assumed in 123 (see ( 2 . 1 4 )
(iii)
below),
whereas i t does not cover. nor is covered by, the assumution of 163.
(iv)
The uniqueness in the theorem s somewhat restrictive excent for the case R 2 . See Theorem 4 4 and Remark 4.5(i).
In the below, we will Rive an outline of the p r o o f o f Theorem 1.1.
The detail will be reported elsewhere.
Since (1.1)
is not a usual evolution equation, the general
local theory on quasilinear evolution equations (see e.g..131) cannot be applied directly, but will be used effectively in our Proof which relies on the technique of singular perturbation. First, in $2. (1.1) is seen to be equivalent in an L2 sense t o the system
ut
+ uuux + Buxxx +
(1.5)
v
X
rvy =
0,
- u =o. Y
with an auxiliary unknown v=v(t.x,y). implies the form u=oX as well as (1.4)-
The second eauation Note that the o r i ~ i n a l
KP equation given in 1 1 1 is (1.5). although only (1.1)
is quoted
in the recent literature. Next. (1.5) will be regarded as a reduced problem of the
Seiji UKAl
182
s i n ~ u l a rperturbation for an artificial evolution system U t + UUUX +RUXXX
!1.6) &Vt +
where
&
IS
v
YVY = 0 .
+
X
-
u
0,
Y
a small real parameter.
I t will be shown in s 3 that
i f & y > O , the general local theory mentioned above can auply.
~ l v i n a ;a unique solution (uE,vE) satisfying the initial condition (u,v)l t=O = (uo,vo). where u o must be the same as for c l . l ) , but vo may be chosen arbitrarily as long as (1.6) is concerned.
in order
to
prove the convergence of uE as
& -+O,
i t is
necessary t o show that uE exists on a time interval [ - T , T l common to
&
small and ( uE ) is compact in a strong topology.
This will be done by establishing uniform estimates for uE and IJ?
in
HS.
In particular. the uniform estimate for uy is found
t o exist only when the condition (1.4) is fulfilled.
we follow
t h e argument develoDed in [ 4 1 for the quasilinear symmetric
hyperbolic system, t o which, in fact, (1.6) reduces if the term f l u x x x is droDDed which vanishes when integrated by parts.
2.
Preliminaries The L2 equivalence of ( 1 . 1 )
and (1.5) as well as that of
( 1 . 2 ) a n d (1.4) come from an L2-version of the classical fact that
i f
f . g . f X , ~ y E0C ( R 2 ) and i f
183
Cauchy Problem for the KP Equation holds, then.
is in C 1 (R 2 ) and satisfies (2.2)
f = vy.
R = v X'
uniquely up to an additive constant.
EH; oc
We can prove the
Lemm 2 . 1 .
S U D D O S ~( 2 . 1 ) be fulfilled with f,g,fx,gY
( R 2 1.~20.
Then there exists
V€Hs,LA(R2)
satisfying ( 2 . 2 ) ,
uniquely UP to an additive constant.
Now ( 1 . 5 ) follows from ( 1 . 1 ) by setting f=ut+auux+buxxx ' g=-yu and ( 1 . 2 ) from ( 1 . 4 ) by f=vo, g=u0. Y' the converse is obvious.
For both cases,
I n case v€HS(R) is required, the situation differs according to the choice of R.
F o r example, v in the above is
not necessarily periodic even if f,g are, and similarly for the case L2 tR 2 ) .
Set, 2
p = ( l + X +y
2 1/2 )
Proposition 2 . 2 .
,
2 1/2
ql = (l+x 1
, q2
ql(y).
Let f , g and v be as in Lemma 2 . 1 .
Cnder the additional condition f.R,€HS with SLO. we have the followinp;. (i)
Let R = T 2 .
Then vEHS+l if and only if
Seiji UKAI
184
( i i )
Let R = R 2 .
I f we
Then, p K v € L 2 , VvEHS with K > 2 .
further assume
with some 6 > 1 / 2 . then vEHS+l (iii)
(2.5)
Let R = R x T tR=TxR).
UD
to an additive constant.
If
= 0 f o r 8.e.x
JTftx.y)dv
= 0 f o r a.e.y),
g(x.y)dx
then, a -1K v . ( q i K v ) E L 2 and Vv€HS with K > 1 .
If
in addition
(2.4)ta) ( ( b ) ) i s satisfied, then vEHS+l UP to an additive
cons tan t
ke
. shall now discriss the condition ( 1 . 2 ) .
Let
(Po
be given
a s in Remark 1 . 2 ( i ) .
Sunpose n o , u
I.emma 2 . 3 . ' i )
'2.6)
(11)
3
Let R=T'
or
TxR.
u0(x,y)dx = 0
Let R = R 2 o r RxT.
for a.e.y€T. Then
addition, q 8l u o € W 1 * s ( R ) and ( 2 . 6 ) k . then there
IS
a
@lCHS+l
EH'.
OY Then, w O E H s + l i. f uo satisfies
-K Q~ ( P ~ E H ' "
with K > 3 / 4 .
i s fulfilled with
such that uo=vIx.
If,
in
T replaced by
Cauchy Problem for the KP Equation 3.
185
Existence theorems for (1.6)
We first write ( 1 . 6 ) in the matrix form AOwt + A1(w)wx
(3.1)
+
A2wy
+
A3wxxx = 0 ,
where we have set w = t(U.v)
(column vector),
Note that the second equation in (1.6) is multiplied by y to make A
symmetric. 2 Throughout this paper we assme 8 Y > O so that . A
is positive
Thus, if A3=0, then (3.1) becomes a quasilinear
definite.
symmetric hyperbolic system.
Define
Then. we shall solve the Cauchy Problem ! + A(w)w dt
(3.3)
= 5,
w(0)
= wo,
t with wo= (uo,vo).
The Reneral local theory on the Cauchy problem o f the type (3.3) has been developed extensively, see e.g.133, which leads
to the
Theorem 3.1. (uO'vO)EHS.
Suupose & be such that & Y > O .
Let s23 and
Then. there is a T>O and (3.3) has a unisue
186
Seiji UKAI
s o 1 ut ion (3.4)
(uE(t).vE(t))
and the maD (uO.vO)
+
E Co(t-T.Tl;HS)
n C1(t-T,T1:Hs-3),
(uE,vE 1 is continuous in the class ( 3 . 4 ) .
To show that the life span T is independent of tiE
E
and that
converges to a solution of (1.1) as E + 0 , we shall establish
Since (3.1) is symmetric uniform estimates for u E and u t' hyperbolic i f B=O and since the term ux x x vanishes if intezrated by parts, we can follow the arzument in C41.
Let I I s denote
the norm o f Hs and define
Proceeding as in C41, we easily have the
Lemma 3 . 2 .
Let wE=(uE,vE) be as in Theorem 3.1.
Then,
holds with C20 independent of E .
Now the integral inequality which comes from (3.5) with lu~,SHwW, is to be comoared with the inteirral equation.
whose solution is b(t)=2{C(TO-ltl))-'
f o r (tl<2(CHw I )-'=To. 0 s
Cauchy Problem for the KP Equation As long as w 8 ( t ) exists in the interval
Lemma 3 . 3 . (-TO,TO),
187
i t satisfies
Set
which i s independent of E . so ( 3 . 6 )
b(t)ib(T)i(CT)-l. (3.8)
Hw"(t)Hs
If 18151, then T0/22T. and gives
luE(t)Is
+
< C.
EylvE(t)ls
For fixed lZ0. this provides a uniform estimate in t both for uE and v E , s o the well known continuation argument leads to the
Theorem 3 . 3 .
Theorem 3 . 1 is true with T given in (3.7)
independently of &*O such that E Y > O . E Now we shall evaluate wt.
ISl
A similar procedure could be
used i f ( 3 . 1 ) could be differentiated with respect to t , that E i s , if w t t were known to exist.
z(t)=(w'(t+h)-w
E (t))/h.
Actually, we shall evaluate
h > O , and then let h + O .
More
precicsely. we first note that A
~ + ZA ,~ ( u ~ ) z ~+ A
holds with f=zlu:(t+h),
~
+
ZA ~
~
-
t(fz , O ) ,~
~
~
Z = ~ ( Z ~ , Z ~ ) . Then we proceed just as in
Lemma 3.2, with L=s-3 in Place of s.
Instead of (3.41, we find
Seiji UKAI
188
whlch. together with (3.8) and by Cronwall's inequality, gives
01..
on
o n letting h + O .
[-T.TJ, with C > O indeDendent o f E. tElil. E
The initial value wt(0) i s , of course, to be specified throunh ( 3 . 1 ) or ( 1 . 6 ) .
Thus,
= Iauouox+8uxxx+~voyl~+
which is uniformly bounded f o r
E
:I uoy-vox
i f and only i f ( 1
2 1' 4)
holds.
This proves the
Lemma 3.4.
Under the situation of Theorem 3 . 3 .
further (1.4) be fulfilled. (3.9)
!ut(t)12 E
+
s-3
SUDDOS~
Then. with some constant C > O ,
EYlvt(t)l:-3 E
i
c
holds for l&lil and tE[-T.TJ.
A
corollary to ( 3 . 8 ) and (3.9) is the
Lemma 3.5.
{uE I ,
1El<1, i s compact in C0 (C-T,TJ;Hlo,) S-6
for
Cauchy Problem for the KP Equation
189
any b > O .
4.
Proof of Theorem 1 . 1 From the uniform estimates ( 3 . 8 ) and (3.9). i t also follows E
that there is a subsequence of uE, denoted arrain by u , such that uE(t)
+
u(t)
weakly* in Lm([-T,Tl;Hsl,
uE(t) t
+
u'(t)
weakly* in Lm(I-T,TI;HS-3).
(4.1)
as
E +O, with some limits u,u'.
Obviously u'=ut holds in the
distribution sense, so that u(t) E L ~ ~ I - T . T 3 : H S ~ ~ L i ~ ( C - T , T l : H s - 3 )
(4.2)
Lemma
The 1 mit u solves ( 1 . 1 ) in the distribution
4.1,
sense. Proof. (4.3)
According to Lemma 3.5, we may assume
uE(t)
+
strongly in C0(C-T,Tl:Hloc), 0-6
u(t)
E which then implies uE ux
hence, toaether with (4.4)
-yv; = fE
+
s - 2 - b ) , and uux strongly in C0([-T,TI;Htoc
(4.1),
=
ut+uu E E ux+Buxxx E E
weaklye in Lm([-T,TI;HS-3), (4.5)
t = U t + uuux
+
with Buxxx
*
+ f ,
IYO
Seiji UKAI E
On the other hand, (3.9) says a l s o that d G l v t l s - 3 i C which in turn Kives vE = uE X
Y
- EYvt E
+
u
Y
weakly* in Lw(C0,Tl;HS-3).
~ ,have Goinx t o the distributional limit in ( VE~ ) ~ = &( V ~ )we
-fX/Y=uyy
or (1.1).
Lemma 4 . 2 .
u
E
where C:
Cw([-T.TI:HS) 0
means the weak
continuity.
This can be Droven by modifvinR slightly the argument given
in
[4,
0.401 i n which the convergence (4.3) i s assumed to be
~ l o b a l ,that is. with Hfit replaced by HS-'.
Since (3.1) i s
t I me reversible, and by virtue of ( 3 . 5 ) and the above lemma, we
can
ollow t h e argument Riven in
Lemma 4 . 3 .
4.
p.441 t o conclude the
u ( t ) € C o ( I-TIT 1 ) HS).
Now the existence Dart o f Theorem 1.1 follows from ( 4 . 2 ) and Lemmas 4.1. 4.3.
To prove the uniqueness, we must look at
the equation ( 1 . 5 ) .
Theorem 4.4.
Associated with u o f Theorem 1 . 1 ,
there is a
v such that c4.6)
v ELrn( I-T.TI ; L 21 0 2 '
Vv E Lo( I-T.TI;HS-3),
and ' u , v ) solves ( 1 . 5 ) with u(0)=uo.
Further, writinK simply
Cauchy Problem for the KP Equation
191
Lm(L2) = Lm([-T,TI;L 2 1 , we have
And u is unique in the class (1.3),
and v in the class (4.6) and
(4.7) UD to an additive constant. Proof.
Write Lm(X),
Co(X) for Lm(I-T.Tl:X).
Co(I-T,TI:X)
Lemma 2.1 and Proposition 2.2, with f as in
respectively.
prove the existence of v satisfying (4.6). Y' In fact, f€Lm(HS-3) by (4.2) and REC'(H'-~) by Lemma 4.3. The ( 4 . 5 ) and g=-yu
latter implies f X ,
RY To prove (4.7)(i),
E cO(H'-~)
too, since f =
x RY' i t suffices to check (2.3).
Recall f E
of (4.4) to pat. by integration by parts, (4.8)
JT2P
E
&
dxdy = -YJT2vydxd~ = 0.
Owinq to (4.4), we can condition of (2.3).
KO
t o the limit and obtain the first
The second condition comes simply from the
definition ~ = u This proves (i). (4.7)Ciif is just P' Pronosition 2.2 (ii). and (2.5) for f comes just proceeding as
in (4.8) which proves (4.7)(iii). that ( 2 . 5 ) f o r p is fulfilled.
F o r (4.7)(iv),
we shall show
Integrate the second equation
of (1.6) in x and integrate by parts.
We Ket,
192
Seiji UKAI
In view of (3.9). (IhEilifi C w h e r e I D is the norm of Lm(L2(R>). Passinn to the limit E+O proves (2.5) f o r ~ = - y u Y' Let (uj,v.), The proof o f the uniaueness still remains. J
.i=1.2,
be two s o l u t i o n s of (1.5) satisfying (1.3).
(4.7). and set u = u -u 2 , v = v - v 1
U t + RUxxx
(4.9)
v
and u ( O ) = O .
+
X
YVY = - u
Y
2
*.
(4.6) and
and rL=-a(u u + U ~ ~ U ) . T h e n , 1 x
=o.
Let tl H be the norm of L
2
(R).
By intezration by
uarts, and by (1.3).
s o that we have from (4.9). p r o c e e d i n z as in (3.51,
\otict- that the terms which contain v cancel1 out b y inteRration b v usrts.
(4.11) Droves the uniaueness !u(t)l
Q=T2, d u e to (4.7)(i),
done for the c a s e
= 0 . so w e a r e
O t h e r e w i s e , however,
the intexration by parts is not lerqitimate because we do not know whether v E L 2
.
from (4.7), so ~t
But v W ' ( t e m o e r e d
distribution) a s seen
admits Fourier transform (series).
Let
u(f.n) denote the Fourier transform ( c o e f f i c i e n t ) o f u(x,y) and
x,(.E)
be such that
kritinp: u
b
x b = l for
= xd ( . E ) x 6 t r t ) u .
h
and = O otherwise.
etc.. we have from (4.91,
Cauchy Problem for the
KP Equation
193
u L t -it, 3 6 i i L + i n \ i L = $,
(4.12)
niis
- tGL
=
0.
Hence v s = n i ' L / e holds and is in L 2 (0) f o r each t,
being the
Substitution o f this into the first equation
R.
dual domain to
fl
of (4.12) yields
i-n; L m2 dt
-
ictee3-~n2/t,)iL,i,) =
H II being the norm of L 2 (R).
( $ L , i * ) i,i s t 0 ) = O ,
T a k i n g the real part and
i n t e m a t i n g , we have IIU,
G o i n K to the
imit b + O . usinn Parseva ' equality and (4.10). w e
get (4.11).
Thus, we a r e done.
Remark 4.5.
(i)
(4.6) is s a isfied automatically, but
(4.7) was established o n l y for those u constructed as limits of solutions uE o f (1.6). 1.1
Thus the unisunass statement in Theorem
is to be understood in the s e n s e of Theorem 4.4.
In this
rasuect. i t is desirable to prove the uniqueness only within the 2 class (4.6). This is the case for R=R ; we can s h o w that (4.6) imDlies (4.7). s e e [ 5 1 . T h e unioneness in Theorem 4.4 ensures the convergence
(ii)
1
of the whole sequence (u 1 . (i i i
)
On the other hand, vE does not necessari ly converKes;
(3.8) and (3.9) do not provide uniform estimates for vE in E.
O n e exceution is the case
R=T2 , see [51.
I94
Seiji UKAI
tteicrences [l]
Kadamtsev. B.B. and Petviashvili. V.1.: On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 1 5 (1970) 539-543.
C21
Kano, T.: L'6guation de Kadomtsev-Petviashvilli apurochant les ondes longues de surface de I'eau en ecoulment trois
dimensionnel,
Patterns and Waves,
Qualitative Analysis
o f \on1 inear Differential Equations. Studies in Mathematics
and I t s A~plications. pp431-444, K i n o k u n i y a / N o r t h - H o l l a n d , 1986.
r3!
Kato.
T.: The Cauchy Droblem for Quasilinear symmetric
hyverbollc systems, Arch. Ra ional Mech. Anal., 58(1975), 181-205. [4?
Yaida.
A.:
"Comuressible Flu d Flows and Studies o f
Conservation Laws in Several Space Variables". APPI
Math.
Sci.. vol.53. S ~ r i n ~ e r - V e r l aNew ~ . York, Berlin, He derberK Tokyo.
1984.
157
Lkai. S . : in prenaration.
C61
Wickerhauser. V.V.: Inverse scattering for the heat o ~ e r a t o rand evoliition in 2 + 1 variables. Commun Math Phys.. 108(1987). 67-87.
Hyperbolic Systems of Conservation Laws
195
Lecture Notes in Num. Appl. Anal., 10, 195-210 (198Y) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
Weak Asymptotic Solutions to Hyperbolic Systems of Conservation Laws By Atsushi Yoshikawa ' Department of Applied Science Faculty of hineering 36 Kyushu University Hakozaki, Fukuoka 612-Japan
1.
Introductlon and Motivation
Consider the initial value problem of finding an m-vector valued
...,urn),which solves the following system of partial
function u=(ul,
differential equations:
the initial data being prescribed as (1.2)
u =
8,
xeRn, t = 0.
Here Fk(u,x,t). k=l,...,n, and G(u.x,t) are smooth a-vector valued functions of (u,x,t)eRmxRnxR,which are bounded for large lul+lxl+itl together with all their derivatives with respect to u, x, t. Suppose that the syetem (1.1) is strictly hyperbolic in the sense that, for any f = ( f l ,
. . . .fn)~Rn\(0),
the martix
= Supported in part by Grant-in-Aid for Scientific Research, Ministry of Education, Science and Culture, No. 62460005.
Atsushi YOSHIKAWA
196
...., ~,(u,x,t,~). Here
has m distinct real eigenvalues ~~(u.x.t.f)
dFk(u,s.t;.) stand for the Frechet derivatives
for u,veRm. xtRn, teR. Let T>O. Let
u=0.1.2.., . . I II,.
Here H”( Rn) are the Sobolev space of order v with the norm
and CJ([O,T]:H’-j(Rn))
are the spaces of functions w(x.t) with
together with their derivatives with respect to t up to
values in H’-’(Rn)
the order j. The following result is then well-known f c f . Hajda [ 6 ] ) . Theorem 1 . l .
L e t v0=[+]+2
be the largest integer not. exceeding
(n/2)+2. Suppose
Then for a T>O depending on l/lgM
vO
, there is a unique solution
u(x.t)EX ‘ 0 (Rnx[O,T])” of the initial value problem (1.1) ( 1 . 2 1 in our present setting.
Remark.
For the validity of the above theorem, symmetric hyperbolicity
(or symmetrizability) is essential and the system need not be of conservation laws. The reason why we here assume the system be of
Hyperbolic Systems of Conservation Laws
197
conservation laws will become clear when we discuss weak solutions. Theorem 1.1 implies that even if the initial data g depend on a certain parameter, say
g=g(p).
the life spuns T ( p ) of the corresponding
solutions are bounded uniformly from below as far as g(p) remain bounded in H'0 ( Rn ) m
goEH
'0
.
Now consider the initial data of the form:
(Rn ) , ho(x) and gl(x,s) are smooth enough. We further assume that
gL(x,s) and all its derivatives with respect to x , s are bounded. rapidly
decreasing with respect to x.
so
Then
that only when r>uO the corresponding solutions are uniformly valid for
some T>O.
On the other hand, when r=l, (1.4)strongly resembles the oscillatory initial data in the geometrical optics approximation of solutions of linear hyperbolic system of partial differential equations.
In fact,
along this line, certain analogies to the geometrical optics approximation in nonlinear situations have been pushed forward by several authors. In most cases by substituting a presumed smooth solution depending on the parameter:
in the equation (1.11, and expanding the system with respect to first non-trivial term, to be picked up from u1 (x,t.p)as p-+-,
p.
the
is shown
to satisfy a certain scalar conservation law. In (1.6), uo(x,t) is the
108
Atsushi YOSHIKAWA
solution to (1.1) with data go. However, in view of the above observations on the life spuns of the solutions, the first non-trivial term just mentioned might not have any meaning even in the sense of asymptotics. Nevertheless, the idea is very important practically and theoretically. For its practical importance, see tlajda [ 6 ] and its references. We here pursue some of its theoretical importance. To begin with, we will suggest an approach to justify the geometrical optics approximation in a weak asymptotic form. Here it is essential that we have assumed the system (1.1) be of conservation laws
so
that the concept
of weak solutions makes sense. We can thus speak of asymptotic behaviors of weak solutions by coupling with properly chosen test functions which
contain the asymptotic parameter. Then as we shall show below. we can give a precise meaning to the above mentioned first non-trivial term of an
asymptotic weak solution and its equation of a conservations law.
In
fact, we shall in this way confirm previous observations by ChoquetBruhat [ Z ] . Hunter and Keller [ 4 ]and others. We have to admit, however, that our approach is less rigorous than that of DiPerna and Majda [ 3 ] in the sense that we can only speak of asymptotic weak solutions. From a theoretical point of view, there are more to note than the behavior of the first term. As we have observed, the initial data of the from (1.4)with r
When n=1, this can plausibly be explained
by the singularity occurring in the first non-trivial term of a weak asymptotic solution. However, other obstacles to uniform life expectancy may be present. As an example, we pick up an initial layer for the next order non-trivial term in a weak asymptotic solution.
Generally speaking,
when n22. it would be natural to expect in an asymptotic soluion a variety of layers of several levels, which, in total, share the original singularities of the initial data as expressed by undoundedness of their
Hyperbolic Systems of Conservation Laws
199
Sobolev norms ( 1 . 5 ) . It should also be noted that when riuO then the singularity structure of the solution is quite close to the one predicted by the linear theory
(see Bony [l]). 2.
Asymptotic classes and asymptotic weak solutions
Suppose u=u(x,t,p) is a weak solution of the initial value problem (1.1) (1.2) with g=g(x,p) given by (1.4) (r=l). Then for any P=P(x,t)c C;(R"X[O,T))~,
the integral
vanishes. Here P0=P0 (x)=P(x,O).
We choose test functions P in the form
or
where b,bleCG(R). a6C;(Rnx[0,T))m,
and ~EC~(R"X[O,T]). We require that
h(x.t) be a small perturbation of a linear function, i.e..
for some SERn\(0),
and that
Atsushi YOSHIKAWA
200
(2.5)
h(x,O) = h0 (x)
for h0(x) given in (1.4 .
(2.6)
Furthermore, we assume that for each te[O,T]
St(h) = {xcRn h(x,t)=O)
In particular, we may speak of be an embedded regular hypersurface in R". the standard density dSt(h) on the surface S t (h), which is defined by (2.7)
aJ
dSt ( h ) = (-1) n-1 a
dxl~.-*hdx,
when h( x,t )=O.
For b(s)ECG(R) and a0 Ix)ECG(R")',
Remark.
0 where a,(x) etc. denote the restrictions of a0( x ) etc. to the surface 1 1 0 S'th). We denote g,(*,sl by d g )(s.-:h ) and call it the spectral
c o e f f i c i e n t of g 1 along h0 .
Let
presumed weak solution u(x,t.p) be of the form (1.6). Instead of supposing smoothness of u1 (x,t.p), we pick up u1(x.t,p) from the our
asymptotic class WT which is specified by the following (O)-(V). 1 ( 0 ) u ( x . t . p ) i s bounded when ( x , t ) lies in any compact set and
p ~ p ~ > O .
( I ) For any test function of the form ( 2 . 2 ) , there is a generalized (m-vector valued) function ufu1 )(s.t..;h) on St(h) such that
Hyperbolic Systems of Conservation Laws
20 1
Here a,(.,t) is the restriction of a(x,t) to St(h). We call o(u 1 )(s.t..;h) the spectral c o s f f t c i e n t of u1(x,t,p)along the spectral function h(x,t).
1 For each te[O,T], u(u 1 )(s,t,.;h) has a local expression u*(s,t,y') in each (relatively compact) local coordinate patch Ut in St(h). We require 1 be smooth with respect to y'6Ut. bounded with respect to that u,(s,t.y') (1)
(s.y')eRxU1, and integrable in seR uniformly with respect to Y'EU t
.
Thus,
denoting rather symbolically
&(ht) =
(2.10)
L1(R;Cm(St(h) )nL"( R;C"( St(h) 1 ,
we can summarize these requirements as
for each te[O,T].
We require further that the mapping t--ru(u1 )(s,t,.:h)
be continuous, that is, with zT(h)=C([O,T];q(h)m),
(2.11)
1
U( u ) ( S, t , *
:h)eIT(.')h
Observe that ZT(h) is an algebra (over Cm(Rnx[O,T])) by the (almost everywhere (IU)
)
pointwise multiplication.
For any bilinear form Q( * ,
f o w (2.21,
)
on RmxRm, and any test function of the
202
Atsushi YOSHIKAWA
(2.12)
1 imp-pJoJ
1
b( ph)d x , t )Q(u1 ,ul)dxdt =
Remark .
ui(x,t.s) is bounded and integrable in
s
(uniformly with respect to x. t
on any compact set). 1 ( R ) When u ( u )(s.t,.:h)=O, there is a generalized (m-vector valued) function u'(u 1 )(s.t,.;h)on S t(h) such that
for any test function of the form ( 2 . 2 ) which, however, vanishes at t=O, a(x,O)=O. u'(u1 )(s.t,.;h) is called the outer subspectral coefficient along the spectral function h.
When u(u 1 )(s,t..;h)=O. there is another generalized (m-vector valued) function u"(u 1 )(s,.;h0) on S0(h) such that
(v)
(2.14)
1
-for any test function of the form (2.3). u"(u1 )(s,K,-;~) is called the inner subspectraL coefficient along the spectral function h. As
for the initial data. we assume
Hyperbolic Systems of Conservation Laws u ( g1 ) ( s ,
( 2.15)
3.
- :ho
)E&,(
ho )
203
.
Equations for the spectral coefficients h(x,t) is said to be a non-charactsristia spectral function if
when h(x,t)=O.
Then we have the following
Proposition 3.1.
For a non-characteristic spectral function h(x,t).
u(u 1 )(s,t.*:h) = 0.
(3.2)
For the outer subspectral coefficient, we have
Proposition 3.2.
For a non-characteristic spectral function h(x,t).
o'(ul)(s,t,-:h) is independent of t.
To compute the inner subspectral coefficient, let A,(0
a
)
be the
restriction of the matrix ht + M(uo,x,t;hx)
to the surface S0 (h) (Recall (1.3)). Let U,(r:h) be the equicontinuous ( i .e. Cob semigroup of continuous linear operators in q(ho)m generated
by -A=(o
a
.
Then we have
Proposition 3.9.
The inner subspectral coefficient is given by
(Recdl: ( 2 . 8 ) . (2.15)). t l ( x , t ) I S s a i d to be an L - t h charactertst~cspectraL functron l i
f o r t h e i - t h e i g e n v a l u e u t t.he matrix (1.3). Let. r - . ( w . x . t , f ) .:trid 3
fi.i(w,s.!
,:)
iua t ri s ?l( w ,x
be r e s p e c , t . i \ e l y be t-he 1-t.h r i w h t arid l e f t e i 8 e r i v e c t o r . s of ttie
.t . E
:
Let. h . ( s . t ) be t.lie i - t h c h a r a c t e r i s t i c f u n c t i o n w i t h h i ( x . O ) = h
. . . .UI.
Let.
j = l , , . . ,111.
11s write
Suppose T i s small enough that.
0
( A ) ,
i=l
Hyperbolic Systems of Conservation Laws when x&,
(3.5)
205
OrtrT. Then we can express u1(x,t,p)= ZjZl m uj(x.t.p)r.(x,t) 1
and speak of the spectral coefficients of u.(x,t,p)along hi(x,t). Xote 3
(3.6)
1
g (x,P)
=
~ y gj(x.p)r:(x,O), = ~ J
where g.(x.p)=k:(x,O).g 1(x,p). We can similarly speak of the spectral J J coefficients of g.(x.p). 3
Proposition 3.4.
For each i.j=l,
....m ,
The inner subspectral coefficients a"(u.)(s,r,.;hi)can readily be 3 computed. Let A 0i i , j , x ( * ) be the restriction of
to the surface S0 (hi)=S0 ( h0 ) .
Let U . *(r;hi) be the Co-semigroup of Jt
continuous linear operators in Zo(h0 ) generat-ed by h0i ,j , x ( have
Proposition 3.5.
For i#j,
(Recall (2.15). ( 3 . 6 ) ) .
a
.)-
as
.
Their we
Atsushi YOSHIKAWA
206
To compute the outer subspectral coefficients o'(u.)(s,t..;hi), J we need to know
a ( u1. )(s,t:;hi).
Let us compute a(ui)(s.t..;hi). Let
be the i-th characteristic vector field. Here A:~)(w.x.~,:)= azi(w,x,t,e)iatk. Let 9i(x.t)~Cm(Rnx[0.T])be such that
(3.12)
@'(X,O) = 1.
and
(3.3)
@'(x.t)
> 0, OitiT, xGR".
Here TLo is the linear partial differential operator:
on the right hand side standing for the matrix transpose. Let Jifx.t)~Cm(Rnx[O.T]) such that (3.15)
XiJi(x,t)= divXi-Ji(x,t),
(3.16)
J'(x,O) = 1
Then along the flow generated by X i , i.e., the i-th characteristic flow,
Hyperbolic Systems of Conservation Laws
207
we have
where
Ji('
,t) is the restriction of Ji(x.t) to the surface St(h) pulled-
back to S 0(h0) by the i-th characteristic flow. Note hi(x,O)=h0(x). As far as T is small enough, J'(x,t) and @'(x.t)
Reaark.
are
well-defined. Let
Then we have the following Proposition 3.6.
= o for any bcCz(R), c~Cz([0,T])and &Cz(Rn).
Here the integrand in the
second inner integral is evaluated as being pulled-back to S 0(h0 ) by the i-th characteristic flow. a,(.) is the restriction of a(x) to S 0 (h0 ) and J,(*,t) i are the restriction of vi(x,t), ai(x,t), Ji(x.t) v k , , ( * ) , @=(*,t), i
208
Atsushi YOSHIKAWA
to St(hi) , pulled-back to S0(h0 ) .
(3.19)means that o(ui)(s.t,-:hi)satisfies a scalar conservation law. It is known that i f olgi)(s,t..;h0 )c%(h 0 ) (recall (2.15)) and if o(ui)(s.t,.;hi)is an entropy solution of (3.19). then
as required (See Kruzhkov [5]. cf. Yoshikawa [ 7 ] ) . Finally, we compute the outer subspectral coefficients a ' (u . J
Let
(3.20)
;hi) , i#j.
s ,t,
3 .( x.t ) = t.( uo( x . t ) .x.t,hi 3
J
y .
j#i,
. ( x , t ) = a.(~,t).dr~(u~,",t,h~,~;ri), J
371
and
g = i , . . . ,m. j=1. . . . ,m. t#i, jti. A l s o recall (3.8). We further need the t' i elds
where r-! k)( w .x ,t ,{ )=ari( w ,x,t ,f
)
/atk.
"hen we have
Proposition 3 . 7 .
For j = l , . . . ,m. j#i a arid for any b( s)€C;IW)
P(~.~)Ec;(R~x(o.T)).
and
Hyperbolic Systems of Conservation Laws
209
2dSt(hi)dsdt (s,t,.;hi)
t(hi)dsdt + t,*;hi)dS
=
0.
Here p,t(.)
4.
etc. stand for the restriction of p(x,t) etc. to St(hi)
Concludlng remarks Proofs of these propositions and further discussions related to
generalized entropy conditions will be given elsewhere (See Yoshikawa [a]).
References
M. Bony, Calcul symbolique et propagation des singularites pour les gquations aux derivees partielles nonlineaires, Ann. Sci. Ecole Norm. Sup. ( 4 ) 14 (1981). 209-246.
Y. Choquet-Bruhat, Ondes asymptotiques et approch6es pour des systgmes d ' h t i o n s aux d6rivhes partielles non linkires, J. Math. Pures et Appl., 48 (19691, 117-158.
R. DiPerna and A . Majda, The validity of nonlinear geometrica opt cs for weak solutions of conservation laws, Commun. Math. Phys., 98
(1985) 313-347.
Atsushi YOSHIKAWA
210
[4] J. Hunter and J.B. Keller, Weakly nonlinear high frequency waves, h a m . Pure Appl. Math., 36 (1983)547-569. [5] S . N . Kruzhkov, First order quasilinear equations in several
independent variables, Rath. USSR Sbornik, 10 (1970) 217-243. [ 6 ] A . Hajda. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-V.. New York, 1984. [ 7 ] A . Yoshikawa. Introduction to systems of non-linear conservation laws
-a
theory of a conservation law-,
Sophia Lecture Notes in
ilathematics. Sophia University, Tokyo, 1985 (in Japanese).
181 A . Yoshikawa, An asymptotic theory for weak solutions of quasilinear hyperbolic systems of conservation laws (in preparation).
Lecture Notes in Num. Appl. Anal., 10, 211-243 (1989) Recent Topics in Nonlinear P D E IV, Kyoto, 1988
The Vanishing Viscosity Method in Two-Phase Stefan Problems with Nonlinear Flux Conditions
by Nobuyuki KENMOCHI
and
Department of Mathematics Faculty of Education
Systems Research Institute Polish Academy of Sciences
Chiba University, Chiba-shi Japan
1.
Irena PAWLOW
Newelska 6, 01-447 Warsaw Poland
Introduction
Two-phase Stefan problems with non-standard conditions imposed on the fixed boundary often create substantial difficulties at their analysis. Of such problems, those with maximal monotone (multivalued) operators in boundary conditions are especially interesting because of their relevance in continuum physics. In particular, problems with
time-dependent unilateral conditions of Signorini type fall into this category. Questions on the existence, and after all the uniqueness of s o l utions are quite difficult for this class of problems and require, as a rule, the use of rather advanced tools. Neither classical nor standard variational approaches turn out fruitful. One of the possible ways of solving the problems is based on introducing the so-called viscosity solutions and discussing their beh-
21 1
212
Nobuyuki KENMOCHI and Irena PAWLOW
avior as viscosity parameter vanishes. An approach of such a type, applicable to multi-dimensional problems, is developed in the present paper. Let R be a bounded domain in R N , N 2 2, with boundary of two disjoint, smooth and compact surfaces
r
and
rl.
aR consisting
For a fixed posit-
ive number T. denote
bie shall consider the following Problem
u
(1.1)
t
- AB(u) = 0
in Q ,
for r
’ p(r>
6(r)
[O,11
= ~
3: R =
-t
p(r)
(P):
< 0,
for r = 0,
+
1
for r
>
0;
k is an increasing and bi-Lipschitz continuous function, with p ( 0 )
0, that characterizes thermal properties of the material; the latent
heat of phase transition is normarized to the unity here; u r i a l datum on d ; g
and g
1
are given boundary data on C
0
0
is given ini-
and Z1,
respecti-
vely; - Y . ( * ) , i = 0, 1 , are maximal monotone graphs in R x R ; (a/an) denotes t h e outward normal derivative on aQ.
f’roblem ( P ) represents the enthalpy fixed-domain formulation of a
Two-Phase Stefan Problems with Nonlinear Flux Conditions
213
two-phase Stefan problem with unilateral boundary condition (1.3).
Free
boundary problems with conditions of type (1.3) arise in various physical situations. We mention here problems of nonsteady fluid flows through poro u s media [l], diffusion processes with membrane boundary conditions rela-
ted to piezodialysis or inverse osmosis phenomena [14],as well as those with Michaelis-Menten kinetics [ 201. Besides, conditions of the form (1.3) are applicable to describe temperature control action through the boundary [ 9 , 111 and heat flow subject to nonlinear cooling according to the
Stefan-Boltzmann radiation law [7]. In various applications, in particular, of special interest is Problem (P) with boundary conditions of Signorini type,described by the graphs
'
(1.5)
yo(r)
(resp. y,(r))
=
O (resp. 6)
, 6 (resp. 0) 0
<
0,
(resp. ( - , O ] )
[O.m)
In the case when the functions g
if r
if r
if r
>
=
0,
0.
and g l are independent of time t,
Problem (P) was studied by Magenes-Verdi-Visintin [22] who applied the 1 theory of nonlinear contraction semigroups in L ( a ) (cf., [2, 3, 81);
they showed that the solution is unique in the sense of Crandall-Liggett [ 6 ] . Besides, if the boundary condition is of the form
- - =an
y(t,x,B(u))
on
x = c0uzl,
with a smooth function y(t,x,r) on C x R , the existence and uniqueness of a solution to Problem (P) in the variational sense have been established by Niezgodka-Pawlow [24],Visintin 1271 and Niezgodka-Pawlow-Visintin [25]. We also refer to Cannon-DiBenedetto [5] and Nochetto [26] for some related results.
Nobuyuki KENMOCHI and Irena PAWLOW
214
In one-dimensional case, more general classes of two-phase Stefan problems have been considered by many authors. In particular we refer to Fasano-Primicerio [ 101, Hoffmann-Sprekels [ 12, 131, Yotsutani [ 2 8 ] , Knabrer [ 2 i ] and Kenmochi [17] in this respect.
However, to our knowledge, in multi-dimensional case there are no res u l t s con the existence and uniqueness of solutions to Problem
(P) with the
boundarv condition
w h c r e -y(t,x,*) is a general time-dependent maximal monotone graph in RxR. 111
the present paper we consider a particular form (1.3) of the bou-
ridarb condition (1.6). As already mentioned, the study of such a setting 1
s
:riterest in numerous applications.
ijf
T h e main result of the paper concerns the existence and uniqueness of
saliitioii to Problem
(P). The solution is constructed by applying the va-
nishing viscosity method. The underlying ideas related to the proof of ihis result have been already announced in the authors' note [ 1 9 ] .
Basic notations and convensions
-
l.ct H be a Hilbert space with scalar product (*,*), :>ct i l l bc a proper (i.e., (1.s.c.)
d
m
and $I
and norm
1.1
H'
$ a ) , lower semicontinuous
and convex function on H. The effective domain of iI is the set
and the subdifferential
a$
of 8 is a multivalued operator from H into H ,
3 c f 1 n e t l by
where
<;(a$)
= {
[z,w]
H
x
H; a@(z) *
@,
w E aJl(z)}
is the graph of
a@;
215
Two-Phase Stefan Problems with Nonlinear Flux Conditions D(W)
= {z
*
E H; a$(z)
range of the operator For E:
> 0 we
4) and R(a$)
=
uzEHaQ(z) are the domain and the
a@.
denote by Q, the standard regularization of Q defined by
As is well known, Q, is continuous and convex on H with D(QE) is singlevalued and Lipschitz continuous on
=
HI and aQE
H with Lipschitz constant
1/E.
Moreover,
a$,(z)
=
F1[ z
- (I+E~Q)-~~I, E H,
where I is the identity in H.
2. Main results 2.1. Basic assumptions concerning Problem (P) We discuss Problem (P) under the following assumptions (Al)
(Al)
- (A6).
B : R * R is a continuous, non-decreasing function which vani-
shes on the interval [0,1] and is bi-Lipschitz on
(-CO,O]
and
with
[l,m),
positive constants llc@ , cB, i.e., -1r 1 C@
1-r 2 I
2
lB(rl)-B(r2)l
2
for any ri E ( - , O ]
cBlrl-r21
o r r. € [1,m), i = 1, 2.
Let 8-l denote the inverse of 8 . (A2) yi, i
=
0, 1 , is a maximal monotone graph in R
x
R, i.e.,
the subdifferential of a proper 1.s.c. convex function y* on R, y . in
y . is
= ay?
R. (A3) gi G W1 92(0,T;H”2(ri))
A Lm(0,T;H3”(ri))
II Lm(Ci) f o r i=O,l.
Nobuyuki KENMOCHI and Irena PAWLOW
216
Lm(Q), meas{xe Q ; 0 <= uo(x)
5 1) = 0
1
and vo-B(uokH ( R ) .
(A4)
u
(Aj)
The initial and boundary data satisfy the following compatibi-
lity condition:
<
v~(vo(x)-gi(O,x))dr(x)
for i = 0, 1,
m
i'
where d r ( x ) stands for the usual surface element on
an.
l'he next assumption is related t o the requirement that the set {x€R; 0
2
u ( t , x ) 5 I ] (in the context of phase transitions usually referred to
an
as rriushv region) does not stick the boundary
over the time interval
!0,7'].
There are constants bi, ti, i = 0, 1, such that
(Ah)
(2.1)
bo 5 go
5
bl
a.e. on L o ,
co
5
g 1 5 c1
a.e. on E l ,
i and the solution V , i = 0, 1 , of the elliptic problems
- -3Vi
(EP)j
an Eyo(V
avi -an E y,(vi
i
- bi)
a.e. on
-
a.e. on
ci)
r rl, 0'
fulfil the following conditions (2.2) Y'
(2.3)
vo
2 v
5
-m
c v
(EP)i, i il &
=
0'
VO >,
ml
5
rl,
ml'
0, 1. has a unique solution V ic-
vi
a.e. on
In view of the results [ 4 ; Thms 1.10, 1.111, Problem
(0,l). Hence, due to (2.2)
(2.4)
R,
a.e. on To,
with some positive constants m Remark 2.1.
in
- m*
and ( 2 . 3 ) ,
in R ~ , ~ ,vi
2
my
~ ~ ( n c)""(6) n
with some
we may assert that in R 1 , 6 ,
i = 0, 1,
Two-Phase Stefan Problems with Nonlinear Flux Conditions t
217
t
for some positive constants m o , m l , 6, where
$,&=
(2.5)
2.2.
{x E R; dist(x,rk)
<
k = 0, 1.
61,
Definition of solutions to Problem (Pl
I n the sequel we shall use the following notations:
H
=
L2 ( Q ) ,
(v ,w)
X
=
H1 ( Q ) ,
a(v,w
X' is the dual space of X and X'
<*,a>:
x
X
-+
R
stands for the duality pairing between X' and X; (v,~),
i
DEFINITION 2.1.
=
jr v(x)w(x)dr(x),
i
=
0 , 1.
i
A function u : [ O , T ]
*H
is called a weak solution
(in variational sense) of Problem (P), if it satisfies the following conditions (Vl)
-
(vi)
(V3): w',~(o,T;x')
(V2) u ( 0 )
=
uo
n L~(o,T;H),
B ( ~ E) L~(o,T;x);
in the space H;
2 (V3) There are functions f i G L (ci), i
fi E yi(B(u)
=
0, 1 , such that
a.e. on zi,
- gi)
and (2.6)
t a(B(u(t)),z)
t
1 C (fi(t),z)r
=
i=O i for any z E X and a.e. t E [O,T].
0
Nobuyuki KENMOCHI and Irena PAWLOW
218
If the above definition contains the additional requir-
Remark 2.2. ement that
2 2 E L (0,T;H(nit,)) for some 6 > 0, i
B(u)
=
0, 1 ,
then bv virtue of Sobolev's trace theorem,
- ma ne
L2(0,T;H"2(ri)),
i
=
0, 1,
and ( v 3 ) is equivalent to the following system (see Lemma 3 . 1 ) : (2.7)
u
(2.8)
-
t
- Af3(u) = 0 in the sense of distributions in Q ,
B.W an
~y,(~(u)
- pi)
a.e. on C .
i
1 '
= 0 , 1.
Therefore, in such a case u is a weak solution of Problem (P) if and only if ( V l ) ,
( V 2 ) , ( 2 . 7 ) and ( 2 . 8 ) are satisfied. Moreover, we can take on C . as a function fi in condition ( V 3 ) .
- (h/an)s(u)
Now we introduce a specific class of weak solutions to Problem (P), constructed by the vanishing viscosity method. For a parameter w
> 0, let
8" be the approximation of 6. defined by B V ( r ) = B(r) We shall denote by (P)' 'u
=
i
rcplaced by
'6 and
respectively, Let uv denote the weak solution of (P)'
(@')-'(v0),
0, (P)'
r E R.
the Problem (P) with f3 and u
the sense of Definition 2.1. 'J
+ vr,
By virtue
in
of the results in [16], for each
has one and only one weak solution u V E W1'2(0,T;H) ( see
Proposition 3 . 3 ) .
DEFINlTION 2 . 2 .
A function u:
[O,T] + H is called a solution in the
vanishing viscosity sense of Problem (P) (a V-solution of (P)), if there k' exists a sequence { u .vk
-+
0 (as k
+ m)
V
1 of the weak solutions of (P)
such that
with wk
> 0 and
Two-Phase Stefan Problems with Nonlinear Flu Conditions UVk and as k +
c w1,2(o,T;H),
m,
Jk
+
u
weakly in W112(0,T;X') and weakly* in LOD(O,T;H), k'
(uVk)
+
@(u)
2
weakly in L (O,T;X),
fyk + fi weakly in L2 (Ii),
fi E yi(B(u) where :f
219
- pi) a.e.
i
=
on 1.
1'
0, 1, i
=
0, 1,
v
are functions in L2 (Ci) such that fik
yi(B
'k
k' (u )
- gi)
a.e.
on Ci, satisfying the variational identity ( 2 . 6 ) which corresponds to the
k' weak solution of (P)
.
It is easy to see that any V-solution of (P) is a weak solution in the sense of Definition 2 . 1 .
2.3.
Existence and uniqueness result for Problem (P)
The main result of this paper is stated in the following: THEOREM 2.1.
Under assumptions ( A l )
.. (A6),
only one V-solution u such that u E L"(Q),
B(~) c W~~'(O,T;H) n L~(o,T;x),
and for some positive constant 6,
B ( ~ > EL*(o,T;H~(sL,,)), where the sets Q.
196
i = 0 , 1,
are defined by (2.5).
Problem (P) has one and
Nobuyuki KENMOCHI and Irena PAWLOW
220
COROLLARY 2.1.
hold and let u v € W1”(0,T;H)
Assume that (Al)-,(A6)
be the unique weak solutlon of (P)’.
Then, as w + 0, u
V
converges to the
i-solution u of (P) in such a way that u‘ + u
6V (u“) <3sw e l l
p,’(u‘’)
-P
B(u)
weakly* in Lm(Q), weakly in W1’2(0,T;H) and weakly* in Lm(O,T;X),
as +
B(u)
Remark 2.3.
2 2 weakly In L (0,T;H( Q ) ) f o r some 6 1,6 If the functions g
> 0, 1
= 0,
1.
and g l are independent of time t,
C o r o l l a r y 2 . 1 is a special case in a result concerning the continuous
dependence of solutions of Problem (P) on f3 and y. (i
=
0, 1). due to
Bgnilan-Crandall-Sacks f 3 ] .
3. Some auxiliary results At.
first
we
recall here some results on the existence, uniqueness of
sulutions to Problem ( P ) and their monotone dependence upon the data in t h e case of smooth functions y.
1’
i
=
0, 1. The following two propositions
are derived directly from the results established in [ 2 4 , 2 5 , 271. Assume that ( A l ) ,
PKOPOSITION 3.1. +
K, i
(A3) and (A4) hold and let yi:
K
= 0 , 1 , be Lipschitz continuous and non-decreasing. Then, Problem
(P) h a s one and o n l y one weak solution which is a V-solution at the same time. Kemark 3.1. 111
By an extension of the arguments which have been used
[ 2 4 , 271, it can be inferred that for any u
unique weak solution ution u of ( P ) as v
u”E
+
W1”(O,T;H)
> 0,Problem
(P)’
has a
and { u v ) converges to the weak s o l -
0 in the sense of Definition 2 . 2 , when y. i 1’
= 0,1,
Two-Phase Stefan Problems with Nonlinear Flux Conditions
22 1
are Lipschitz contiunuous and non-decreasing on R. The next result is concerned with the monotone dependence of the solution of (P) with smooth yi, i = 0, l, upon the data.
PROPOSITION go, gl, uo) and
Assume that (Al) holds, and the data sets { y o , y l ,
3.2.
{yo, yl, g o , gl, ii0 }
satisfy the assumptions of Proposit-
ion 3 . 1 . Let u and ii be the weak solutions of Problem - . - . - . - .
to { B , yo, y l , go, gl, uol and IB,yo, y l , go, g19 Go) uo 5 ii
0
a.e. in R , gi
gi
4
a.e. on C 1' . i
=
(P) that correspond
,
respectively. If
0, 1,
and yi
2
Ti
on R,
i
=
0, 1,
then u s
a.e. in Q.
Now we are going to prove some results which characterize the weak solutions of Problem (P). LEMMA 3.1. Assume that (Al)
(A4) are satisfied. Let u be a weak
-.
solution of (P) such that 2
2
B ( u ) E L (0,T;H (R.
1,6
for some 6
))
> 0,
i = 0, 1.
Then ( 2 . 7 ) and ( 2 . 8 ) hold as well as
(3.1)
t a(B(u(t)),z)
=
for any Proof.
By taking
z =
z
E X and a.e. t
E
[O,T].
rlE %,(Q) in ( 2 . 6 ) we see that ( 2 . 7 ) holds. On n
AB(u)
E
the other hand, since u t
=
L (0,T;H1'2(r'.))
0, 1 , it follows from ( 2 . 7 ) by virtue of Green's
2
for i
=
L'((0,T)xQi,&)
and (a/an)B(u)
formula that the identity (3.1) is satisfied. Further, by combining ( 2 . 6 )
Nobuyuki KENMOCHI and Irena PAWLOW
222
with ( 3 . 1 ) w e c o n c l u d e t h a t
jTi
1 C
i=O
fi(t,x)z(x)dT(x)
+
j a Q w ( x ) d r ( x ) = 0
f o r a n y z & X a n d a.e. t
-
Therefore,
( a / a n ) B ( u ) = f . a.e. o n
LEWA 3 . 2 .
Assume t h a t ( A l )
-
ci,
E: [O,T].
i = 0, 1, and (2.8) h o l d s .
(A4) hold. L e t u and
0
b e t w o weak
s o l u t i o n s of ( P ) s u c h t h a t
(;.2)
i3.3) inen u
B(u), =
f,>r an) z
C
a.e.
t
in Q
6 > 0,
aitd t o r svme
rr.
a.e.
u
in
a(;)
1
t
6 [O,T]. T a k i n g
Further, n o t i c e t h a t a c c o I d i n g to (3.2),
z
const.
t h e sets
decomposed i n t o t h e f o l l o w i n g s u b s e t s f o r a . e .
r ,inJ
p
0, 1.
=
Q.
and a . e .
,Y
E L 2 (0,T;H 2 (Q1 , 6 ) I ,
= {XE
rl; t x u ( t , x ) ) <
=
r I'
1 i n (3.5), we g e t
i = 0, 1, can be
t E [O,T]:
accct,x)))
Two-Phase Stefan Problems with Nonlinear Flux Conditions
r;w
= I X E ri; B(u(t,x))
=
223
B(;(t,x))i.
On account of ( 3 . 4 ) and the monotonicity of yi, as(u(t,x)) an
(3.7) for a.e. x E r;(t),
2
as(;(
an
t ,x))
i = 0, 1. On the other hand, by virtue of ( 3 . 3 ) ,
( 3 . 7 ) is also true for a.e. x E ry(t). a.e. on C .
1'
0
i
=
0, 1 . Hence, by ( 3 . 6 ) we conclude that
- i'(t),
5
Therefore, inequality ( 3 . 7 ) holds
d 1> = z(u(t)
- ;(t),
1)
for a.e. t E [O,T].
Integrating both sides of this inequality over [ O , s ] , 0 < s 5 T, we get (u(s)
- C ( S ) , 1)
=
i,
(u(s,x)
- G(s,x))dx
2
This inequality together with (3.2) implies that u =
0.
i a.e.
on Q. V
Further on we shall examine the family of approximations (P) , v>O, to Problem (P). To this purpose, the time-dependent subdifferential operator techniques as developed by Kenmochi [15, 161 will be applied. For each t E [O,T] let us define a function
at: H + Ru{m}
by
n
otherwise,
m,
where j. i 1'
=
0, 1, is a proper
1.s.c.
and convex function on L2 ( r . ) ,
given by (3.9)
ji(w)
=
jr yq(w(x))dr(x),
w
L2 Ui), i
=
0, 1.
i
aL is L
Clearly, {zE
a proper 1.s.c. and convex function on H with D(6')
X; ji(z-gi(t))
<
03,
i = 0 , 1 ) . The subdifferential of
=
bt is charact-
erized by the following lemma. IJHtU 3.3.
Let ( A l )
- (A3) be satisfied. Then, for each t E [O,T],
224
Nobuyuki KENMOCHI and Irena PAWLOW
'
(3.10)
(3.121
wE
*
(w ,z--w)
+
a(w,w-z)
i
1
c
ji(w-gi(t))
an E yi(w-gi(t,*)) - -aw
Kaw E L2 (Ti)'
4
1
z
tr Z E X.
ji(z-gi(t)),
Ti, i
a.e. on
= 0,
1.
Proof. The equivalence between w* E abt(w) and (3.10) is immediate. Indeed, if w*E aQt(w), (3.13)
(w*,zl-w)
2
then by definition w 5' D(bt) and
at(z,) - @ t ( w )
Taking in (3.13) z1 = (1-a)w (3.10) by letting
CI
+ CIz
for any z1 C- D(bt).
with
E (0.1) and
0
z
t D(@
t
), we get
* 0 . Consequently (3.13) follows from (3.10), as it
can be easily seen. The regularity of the solution w of (3.10), i.e., w
E H2(R) (hence (a/an)w E H1'2(ri),
i n terms of system [(3.11),
i = 0, l), and its characterization
(3.12)) is due to [ 4 ; Chapter 11.
a
The next result is concerned with the existence and uniqueness of L:
solutions to Problem (P)
.
PROPOSITION 3.3. Let (Al)
- (AS) be satisfied. Then f o r any v > 0 V
there exists one and only one solution u
(3.14) AL t h e
(uV)'(t)
same time, u
+ a@t (6v (uv (t)))
=
0
W1l2(0,T;H) of the problem for a.e. t t I O , T ] ,
v is . the weak solution of Problem ( P ) V and
Two-Phase Stefan Problems with Nonlinear Flux Conditions (3.15)
Lm(O,T;X),
Bv(uv)
Proof.
ji(B
v
v
(u )
-
pi)
225
Lm(O,T), i = 0 , 1.
Let u s introduce a f u n c t i o n g = g ( t , x ) such t h a t
- Ag(t,*) (3.16)
i n Q,
= 0
a.e. on T i ,
= gi(t,*)
g(t,.)
i = 0 , 1.
By v i r t u e of a s s u m p t i o n ( A 3 ) , (3.17)
g
E W1 '2(0,T;H1(Q>) A Lm(0,T;H2(Q)) CI Loo(Q).
The mapping t -+ bt i s t h e n r e g u l a r i n t h e f o l l o w i n g s e n s e : f o r any s, t
E [O,T] and f o r any z
D($)
there e x i s t s a function
t D(@t)
such
that (3.18)
I
- 4s S
I n d e e d , g i v e n z E D(@ ), it is easy t o see t h a t ( 3 . 1 8 ) is s a t i s f i e d f o r
2 = z - g(s) t g ( t ) c D(I$~). Hence, by t h e r e s u l t s of [ 1 5 ] , f o r any v
>
0 problem (3.14) h a s t h e
u n i q u e s o l u t i o n uv E WlS2(O,T;H) w i t h r$(')(Bv(uv)) u s l y , on a c c o u n t of Lemma 3.3, s f i e s (3.15).
e
Lm(O,T). Simultaneo-
uv i s t h e weak s o l u t i o n of (P)'
t h a t sati-
a
L e t u s n o t e t h a t by v i r t u e o f Lemma 3 . 3 , any weak s o l u t i o n o f (P)", .
n
which b e l o n g s t o WlSL(O,T;H), s a t i s f i e s problem ( 3 . 1 4 ) . Now w e s h a l l e s t a b l i s h a comparison r e s u l t f o r s o l u t i o n s of (P)".
P r i o r t o t h i s , we recall a n o t i o n of o r d e r f o r convex f u n c t i o n s on R .
*
Given two p r o p e r , 1.s.c. and convex f u n c t i o n s on R , say T~ and the relation
*
+ * *
''T~6
*
T ~ "means t h a t
-rl(rlAr2) t T 2 ( r l v r 2 ) 5
*
+
*
-r2(r2)
f o r any r 1 , r2 E K,
T
*
~
,
Nobuyuki KENMOCHI and Irena PAWLOW
226
where r1Ar2 = mint rl,r2}, r1vr2 ;5
* *
=
max( r l,r2]. It is easy to see that
t
‘ I ~
T~ implies
-
(ri
where T
i
=
for any r i & D(ri) and any ri E ‘Ii(ri),i=l,2,
- r 2 )t 2 0
r;)(rl #.
ar. in R. We are ready to prove: Let v
PROPOSITION 3 . 4 .
> 0 and B
be given by (Al). Assume that the
-
-
-
-
.
“
data s e t s (yo, y l , g o , gl, uol and { y o , y I , go, g l ; uol satisfy assumpti-
5
go, gl,
v
uo 2 Go
gi *
(3.21)
*
-...
uo and 6, yo, y l ,
Ei
2
7,
t
a.e. on fi, a.e. on 1.
i
1’
=
ay., 7 .
0, 1,
-* ay.
and in R), i
=
0, 1 ,
For simplicity we shall u s e the notations: u
=
u v , u = u-w
yi
4
(where y .
=
1
1
=
s sw a.e. in Q.
Proof. = B”(u“),
B(t)
go, gl,
yl,
-
(3.20)
f3
W1’*(O,T;H) be the weak
u o , respectively. If
(3.19)
then u’
and Gv
(P) that correspond to B , yo,
s o l u t i o n s of 1
Let uw E W1”(0,T;H)
(A5).
ons ( A 2 )
B
=
1
,
BW(Gw). It follows from Lemma 3.3 and Remark 2 . 2 that
2
62 H ( 0 ) . B(t)
(3-
2
H (n), -(a/an)B(t,*)
and -(a/anlfi(t,*)E Ti(p(t,*)-gi(t,*)) i.t f o l l , ~ from s
a.e. on
r I’ . i
(2.6) thaL
‘ - 1
an(‘)
E yi(B(t,*)-gi(t,*))
-P
oo(r)
=
forr
0
for r
=
0,
. 1
for r
>
0,
=
a.e. on
r.
0, 1. Therefore,
Two-Phase Stefan Problems with Nonlinear Flux Conditions as n -+
m.
Let us take in ( 3 . 2 2 ) z =
a,([B-B]+).
221
Then, note that
(3.25) By virtue of ( 3 . 2 3 )
- ( 3 . 2 5 ) , after letting n
+ a,
we conclude from
( 3 . 2 2 ) that
Because of ( 3 . 1 9 1 , it follows from this inequality that [u i n Q , i.e.,uS;a.e.
4.
- ;I+
=
0 a.e.
inQ.
Approximations of Problem (P)
Throughout this section we shall assume that the assumptions ( A l ) ( A 6 ) hold. Let u s fix r i o E
R Cl if 0
0 E Yi(rio),
i
=
0, 1, so that
E R(yi)
and
r.
10
For
E
>
=
w
(resp.
0 and u
>
a),if
0 (i
= 0,
R(yi) C (-.0,0) (resp. C ( O l m > ) .
1) we define
.
228
Nobuyuki KENMOCHI and Irena PAWLOW for r
< ria,
for r = r.
10'
for r
>
r.
for r
<
rio,
10'
f o r r = r.
10'
for r
>
rio,
for r
<
r.
and 10'
for r = r.
10'
>
for r Let u s n o t e h e r e t h a t y .
= yi, a n d
1.00
(resp.
r.
=
0,
) is n o n - d e c r e a s i n g
10
a). Moreover,
rio.
Yi,€,,
remark t h a t for
E
=
>
~ ~ ) yi ,EO ( r e s p . Y ~ , if
0 and U
>
0,
Y. 1,EU
(i =
a n d L i p s c h i t z c o n t i n u o u s on R , a n d t h a t
(4. and
*
(4.2)
* *
s
'i,Eo
*
*
Q
where Y ,
1 ,Eo' it
and y.
'i, ou
l,E!J
Yi,Eo' ayi,ou = Yi,ou ~
(4.3)
9
~
'i ,EU
*
t
s
t
yi,ou*
i = 0, 1, it
a r e c o n v e x f u n c t i o n s o n R s u c h t h a t aY.
1,EO
aYi,Eu = ' i , E U
and
~ = ~~ q( , a~ ~~= ()Y aY ,~€ , ), ( ~ ~ =) Yi(ai)
,
fc;r some a . E D(y.) i n d e p e n d e n t o f 1
E
a n d U. L e t u s a l s o n o t e here t h a t
*
r h e s e are s t a n d a r d a p p r o x i m a t i o n s of y . . I n f a c t , we e a s i l y see t h a t
* (h.4)
as bcjr
C,
u
b 0,
(resp. E
*
Yi,ou) i n t h e s e n s e of Mosco ( c f . [ 2 3 ] )
* 0 ( r e s p . u * 0) f o r e a c h U
2
0 (resp.
l e t u s c o n s i d e r t h e e l l i p t i c problems
E t
0).
=
229
Two-Phase Stefan Problems with Nonlinear Flux Conditions
a vi
i
-aCiyO,Eu(VEII an
- bi)
a.e. on
r 0'
- ci>
a.e. on
rl,
a vi
--E
fl,Eu(~i
an
with constants bi, ci, i
V i for i
=
€11
=
i 0, 1, as in assumption (A6). Note that V 00
=
0, 1.
LEMMA 4.1. For any
E 2
unique solution Vi E H2(R) €11
(4.5)
vo s vo
(4.6)
voE O
vo€11
j
E!J
n C o S a ( 6 ) with v1
2
€0
i
0 and 11 L 0, problem (EP)
5
s V'
v1€11
some a E ( O , l ) ,
=
0 , 1) has a
such that
in R ,
ou
v
5
(i
in
OlJ
R.
Moreover,
(4.7)
'u:V
VZ0
C
uniformly on
6 as p
uniformly on
as
E
+ 0 for any fixed
0,
E ?.
+ 0 f o r any fixed 1-1 2 0.
Proof. The assertion follows by a direct application of the results in [4;Chapter 11. 0 By virtue of Lemma 4.1 and assumption ( A 6 ) , we may postulate that
(4.8)
vi€11
2
-
for i = 0, 1 and any
E
m
* in
vEiu
ao,&,
C- [ O , E ~ I ,
2
in R,,~,
m;
u E- [O,uol, where
*
*
u o , mo, ml, 6 are
E ~ ,
appropriately chosen positive constants. For v
> 0,E
replaced by yi Y
€11
t 0 , 11 2 0, let
(P)'
. Notice that (P)zo
section 3, Problem (P)'
EU
=
denote the problem (P)'
(P)'.
According to the results in V
ElJ
with y .
has a unique solution u
ElJ
E WlY2(O,T;H).
Our purpose now is to derive appropriate uniform estimates on the solutions of (P)'
€11
with respect to u,
E:
and
l ~ .
230
Nobuyuki KENMOCHI and Irena PAWLOW ESTIMATE (I).
For any v
vo
Proof.
ev
4
€0
>
v
0, 4
(UEJ
E
'
E [ O , E ~ I , u € [ O , u o l , we h a v e a.e. i n Q;
v OU
Let u s set
viv(t,x) EU
v -1
= (B )
i (vEu(x))
for ( t , x )
E Q,
i = 0 , 1.
C l e a r l y , "iv ( i = 0, 1) s a t i s f i e s t h e system EU
ED
t
-
Uiv(O,-) EU
I n view of ( 2 . 1 ) ,
abv(~iv)= EU
o
i n Q,
in
= (Bv)-'(VZU)
(2.2),
R,
( 4 . 2 ) , ( 4 . 5 ) and ( 4 . 6 ) , a n a p p l i c a t i o n of t h e
comparison r e s u l t ( P r o p o s i t i o n 3 . 4 ) y i e l d s t h e i n e q u a l i t y
u OU lV a . e .
uov €0
d uv
Vo
i 6 v (uEu) v 5
€0
EU
5
V1
OLJ
i n Q.
a . e . i n Q,
so t h a t (b.9) is also v a l i d . There e x i s t s a p o s i t i v e c o n s t a n t
ESTIMATE (11).
f o r any u
E
(0,1],
E
[ O , E ~ ] and
uE
[O,U,],
MI s u c h t h a t
where j .
19EU
is a p r o p e r , 1.
Two-Phase Stefan Problems with Nonlinear Flux Conditions S.C.
*
i'
2
and convex function on L
231
*
(Ti), defined by (3.9) with Y. replaced by
,ED'
Proof.
V
According to Proposition 3 . 3 , u
ElJ
is a unique solution of
the problem V
(4.10)
(uEu)'(t)
(4.11)
u
ji,E!J
t
a@,,,(B
v v
for a.e. t E [ O , T l ,
= 0
(uEu(t)))
V
V
where
t
EU
(0) = u
btQJ is a proper
0'
convex function on H, defined by (3.8) with
1.s.c.
in place of j. i = 0, 1. Further, for simplicity we shall use the 1'
V
notations: u = u
Ell'
B
=
V
t
B ,4
=
t bEIJ,, ji
=
ji,
u'
Let u s multiply both sides of equation (4.10) G(t) = vo
-
g(0)
+
g(t),
u
V
0
= u . 0
- G(t) with
by B(u(t))
where v = B(uo) and g is given by (3.16). 0
Then,
after integration over R , we get (u'(t),B(u(t))-G(t)) f o r a.e. t € [ O , T ] . (4.12)
t (abt(B(u(t))),B(u(t))-G(t))
Hence, it follows that
(u'(t),B(u(t))-G(t))
+
bt(B(u(t)))
By introducing the convex function
inequality (4.12)
to obtain
= 0
K(-)
S
t @ (G(t))
on H, given by
can be written i n the form
for a.e. t
E [O,T].
232
Nobuyuki KENMOCHI and Irena PAWLOW
ftlr somc positive constants p
(0,11,
r3
E \ O , E ~ I and u
k'
p;
(k = 1 , 2, 3 , 4) independent of v E
E [O,uol.
By virtue of (4.14), (4.15) and Estimate (I), we derive Estimate
( [ I ) from (4.13). ESTXMATE (111).
There exists a positive constant M
2
such that
and (4.17) f o r any v
E
(O,l],
E
[ O , E ~ ] and 1.1
E [O,uo]
with the positive constant 6
as i a Estimate (I).
In u r d e r to prove (4.16). (4.17) we formulate two lemmas. For a moment, let 6 be t h e same positive constant as in Estimate (I). Let u s consicier t h e iunction g& on Q l t d which is the solution of the problem
233
Two-Phase Stefan Problems with Nonlinear Flu Conditions for any t
E
[O,T]. Observe that g 6 k W’*2(0,T;H1(Q,,6))~Lm(0,T;H2(~l,6))
nLm(Q,,6). Further on we shall
X O f 6 = {zE X6;
use the notations:
z =
0 a.e. on
r‘l,6}.
Now, let u s define a proper 1.s.c. convex function @
Just as
a@,,,, t
t EU16
on H6 by
is a singlevalued mapping in H6, and regular in the
following sense: for any s,t E [O,T] and any z k‘ D(6Zu,6) there exists
LEMMA 4.2. w
Assume that v
t (0,1],
E
E
[O,E~
and
u E [O,u 1.
Let
W”2(0,T;H6) with w(0) k D(@ZDp6) and 3 I EU.6 $ ( * ) (w) E L2 (0,T;H6). Then
for any s , tt[O,T] with s 6 t.
Proof.
Inequality (4.21) can be derived from (4.20) by the same
arguments as in the proof of [18; Lemma 2.31 (or see 115; Corollary to Lemma 1.2.51).
0
Now let u s introduce a non-negative smooth function q on
a
such that
Nobuyuki KENMOCHI and Irena PAWLOW
234
where
B e s i d e s , s i n c e q = 1 i n a neighbourhood of
r 1'
Hence, on a c c o u n t of Lemma 3 . 3 , i t follows t h a t
Now w e a r e r e a d y t o p r o c e e d t o :
Proof of ESTIMATE (111).
@',
f for u
V
EU'
B,
t Bv, bEU,&, f g u , r e s p e c t i v e l y . Upon m u l t i p l y i n g b o t h s i d e s
of ( 4 . 2 2 ) by ( d / d T ) ( q B ( u ) ) ,
(4.23)
As p r e v i o u s l y , w e u s e t h e n o t a t i o n s u ,
(QU'(T)
we get
,~(B(u))'(T))~
t ( a @ T ( ~ 8 ( u ( W(qB(u) , )'(T))6
= (f(~).n(B(u))'(~)))~
for a.e.
T
C
[O,Tl.
Hence, a p p l y i n g i n e q u a l i t y ( 4 . 2 1 ) w i t h w = qB(u) and t a k i n g i n t o a c c o u n t ( 4 . 2 2 ) , we g e t
Two-Phase Stefan Problems with Nonlinear Flux Conditions
(4.24)
235
/
, -,t
for any t c [O,T]. Besides, observe that by virtue of assumption (Al) and because of ( 4 . 9 ) we have
Furthermore,
In view of ( 4 . 2 4 )
- (4.26),
it follows from ( 4 . 2 3 ) that
for any t E [O,T], where L1, L are some positive constants independent 2
of
vE
(0,1], E
E
[ O , f o ] and p & [O,po]. Inequality ( 4 . 2 7 ) , together with
Estimate (11), implies ( 4 . 1 6 ) , ( 4 . 1 7 ) ESTIMATE ( I V ) .
for some positive constant M2.
There exists a positive constant M
3
such that
and (4.29)
for any
ve
(0,1],
E
G
[O,Eo]
and p C i [O,pO].
Estimates ( 4 . 2 8 ) and ( 4 . 2 9 ) follow by the same arguments as in the
236
Nobuyuki KENMOCHI and Irena PAWLOW
proof of Estimate (111).
ESTIMATE
(Vl.
There exists a positive constant M4 such that
(4.30)
Proof. As
Let q be the function defined in the proof of Estimate (111).
previously, the same reduced notations are used for simplicity. Accor-
ding to the proof of Lemma 4 . 3 , we see that the function W:
nB(u)
satisfies the following system f o r a.e. t C [ O , T ] : ' -
AW(t,-) = f(t,*) - rlut(t,-)
W(t,*)
-
= 0
On
in R 1 , & ,
5,6*
W E yl(w(t,*)-gl(t,-))
a.e. on
rl.
Due tu the results of ( 4 ; Thms. 1.10, 1.111, we conclude that
where C is a constant independent of v,
E
and
u . By
virtue of ( 4 . 9 ) and
Estimates (11), (111), it follows from ( 4 . 3 1 ) that
with a constant M' independent of 4
imate h o l d s for IB(u)I
ESTIMATE ( V I ) .
2,
V,E
and
2 L [O,T;H ( Q o , 6 / 2 ) )
u . Clearly, the analoguos est-
. Hence ( 4 . 3 0 )
is obtained.
There exists a positive constant M5 such that
Two-Phase Stefan Problems with Nonlinear Flux Conditions for any v
E (0,111E E
[O,E~Iand
LJ
& [ O , u o l , where a' = R
is the same constant as in Estimate (I). Proof. Let u s introduce a smooth function q E on
n'. By
\
231
n6,2 and 6
aXn)such that q
= 1
applying the same reduced notations as previously, we have
5 If(t)lHln(B(u))'(t)lH
1
- xlrl(B(u))'(t)li
for a.e. t 6 [O,T].
B
Hence it f o l l o w s that for appropriately chosen positive constants L 3' L4' independent of v , E ,
d
x(V(rlB(U(t)))lH
u,
2
+ L31fl(B(U))'(t)lH
2
5
Lqlf(t)lH
2
for a.e. t
[O,T].
This inequality immediately implies Estimate (VI).
5. Convergence of approximate solutions Our purpose now is to prove Theorem 2.1. This will be done in a sequence of lemmas. Throughout this section assumptions (Al) ntained to be satisfied.
- (A6) are mai-
Nobuyuki KENMOCHI and Irena PAWLOW
238
Proof.
V
By v i r t u e of t h e u n i f o r m estimates on s o l u t i o n s u V
we c a n s e l e c t a s e q u e n c e ( u
- u
k =
uk + u
k
(5.2)
+
w i t h vk
0 (as k +
a)
such t h a t
weakly* i n L”(Q>,
z B(uk) +
(u,)
‘1
of (P)”,
vkuk +
5 weakly i n W’”(0,T;H)
a n d weakly
*
i n La(O,T;X),
as w e l l a s
k
(5.3) where
5 (u,)
Bk
=
V
5
+
2 2 weakly i n L (0,T;H
5
‘.
By v i r t u e of Aubin’s c o m p a c t n e s s theorem, ( 5 . 2 ) and (5.3) imply t h a t
B
k
2 2 1 i n L ( Q ) and L (0,T;H (RgI2)).
*5
(uk)
T h e r e f o r e , 5 = B ( u ) and
B k (u,) k k a5 (‘k) f . :-1 an
+ -
weakly i n L2 ( 0 , T ; H 1 ” ( r i ) ) ,
an
k k Hence, s i n c e f i E y i ( B (u,)
i = 0, 1,
i n L2 ( 0 , T ; H 1 ” ( r i ) ) ,
+ B(u)
-
9 . ) a.e. on Z .
1’
i = 0 , 1.
i = 0 , 1 , by s t a n d a r d mono-
t o n i c i t y a r g u m e n t s we c o n c l u d e t h a t
-
fi
u an . yi(B(u)
Now, a c c o r d i n g t o D e f i n i t i o n 2.2,
For any
5.2.
pi)
01.1
c.1 ’
E
E
( O , E ~ ] and
f (O,uo], l e t (P)Eo and ( P ) .EO
and yi 1
OD
ELI
t u
€0
2 i n L ( Q ) a s u + O
ou
, respectively.
r e s p e c t i v e l y have t h e u n i q u e V - s o l u t i o n s u
which s a t i s f y t h e s i m i l a r p r o p e r t y t o (5.1). Moreover, u
i = 0, 1.
w e see t h a t u i s a V - s o l u t i o n of ( P )
be t h e Problem ( P ) w i t h yi r e p l a c e d by yi Then (P)Eo and ( P )
a.e. on
0
and i t s a t i s f i e s ( 5 . 1 ) .
LLWA
-
€0
and u
ou
Two-Phase Stefan Problems with Nonlinear Flux Conditions
239
and u where u
ElJ
t u
ED
2
OD
in L (Q) as
E +
is a unique weak solution of (P)
with yi replaced by yi ,
,,,
i
=
0,
ED
((P)E,, denotes Problem (P)
0, 1).
Proof. We shall restrict the proof only to the case of (P)o,,.
Simi-
lar arguments can be applied to (P) €0’ Due to Proposition 3.1 and Remark 3 . 1 , (P)E,, has exactly one V-solution u
ElJ
which is the limit of the weak solutions u
V
ED
the sense of Definition 2.2. By virtue of Estimates (I) see that as
V +
- (VI) on
0,
Bv(u&)
in L2(Q),
B(u,,,)
+
weakly in W1’2(0,T;H) and
weakly* in Lm(O,T;X),
(5.4) v
0
v (UE,,)
B(U,,
-+
v v 38 (u,,> an
in L2(Z.), i = 0, 1,
)
1
weakly in L2( Z . ) ,
as
+ A
1
decreasing and bounded a.e. on Q as u Then, as
V
to (P)€,, as WO in
E +
u
* OU
(t,x) = lim u €4
0, EU
t u
B(u,,)
* 0l-I
+
E +
(t,x)
0 . Let us denote
for a.e. (t,x)E Q.
in L~(Q),
*
B(uO,)
in L2(Q),
weakly in W1’2(0,T;H) and
weakly* in Lm(O,T;X), O(uE,,)
t
i = 0, 1.
BLU~,,) 3
in L2(,Ii>,i = 0, 1,
V
uE,,,
we
230
Nobuyuki KENMOCHI and Irena PAWLOW
Since Y
e
*
~ * Yi,ou , ~ i n t~h e s e n s e o f Mosco a s
E
0 , f o r any f i x e d U
+
>
0,
i = 0 , 1 (see ( 4 . 4 ) ) , we i n f e r t h a t
Therefore, u
*
i s a weak s o l u t i o n of ( P )
ou
ou’
h a v i n g t h e similar p r o p e r t y
t o (5.1).
I n turn, let u
be any V - s o l u t i o n of ( P )
O’cc
estimates on a p p r o x i m a t e s o l u t i o n s t h a t u
u (2
ou’
weak s o l u t i o n s of problems ( P ) ( ’ J ~+
n o t e from t h e u n i f o r m
(O,uo], s a t i s f i e s t h e
V
u o i ) be a sequence of t h e
similar p r o p e r t y t o (5.1). F u r t h e r , l e t { u i u V
-
ou’
s u c h t h a t uk c o n v e r g e s t o u as k ou ou
OU’
+
0 ) i n t h e s e n s e of D e f i n i t i o n 2 . 2 . Then we c a n c o n c l u d e from Propou
s i t i o n 3.4 t h a t u& letting k *
cu
a.e. i n Q
2 uk
OD
f o r every
E
u
EU
5 u
ou
a.e. i n Q
f o r every
E
>
0.
* ou
s uou
a.e. i n Q.
An a p p l i c a t i o n of Lemma 3 . 2 t o t h e weak s o l u t i o n s u
ou
0. Hence,
* 0 yields that
E
u
Problem ( P )
>
we get
m,
Now, l e t t i n g
V
implies t h a t u
2
+
LEMMA 5.3.
u
ou
=
i n L (Q) a s
uoLl
and t h e V-solution
*
ou
u
ou E
t
ou
and u
ou
of
i n Q. C o n s e q u e n t l y ,
+
0
f o r any f i x e d IJ
of (P)ou i s u n i q u e .
>
0
a
Problem ( P ) h a s a u n i q u e V - s o l u t i o n u . T h i s s o l u t i o n c a n
be c o n s t r u c t e d a s t h e l i m i t
Two-Phase Stefan Problems with Nonlinear Flux Conditions u
where u
V
0l.l
V
u
2
<
If 0
I.I
OlJ
orm estimates on u
*
i' ,ou
+
it follows that u
>
u u
v
0 , so that u
ou
5 u
OG
a.e. in Q. Thus the
V OIJ'
*
+
OIJ
u
*
2
in L (Q) as
u
+
0. Hence,
0. By virtue of the unif-
and since
yi in the sense of Mosco as IJ
*
+
+
0, i
= 0, 1,
is a weak solution of (P) satisfying the similar prop-
On the other hand, for any V-solution u of (P) the inequa-
erty to (5.1). 9
9
then Yq,oIJ 5 Yi,oG. Hence,by Proposition 3 . 4 ,
1 is bounded and non-increasing in L2 (Q) as
there exists the limit u
5
9
< 4,
a.e. in Q for any v
OD
sequence { u
lity u
2
in L (Q) as LI + 0,
is the V-soluti-on of (P)ou.
OIJ
Proof. u
+u
OIJ
24 1
holds a.e. in Q, because v ou
s u
a.e. in Q for any v
Again, by Lemma 3.2, u = u
*
0 and
u > 0.
in Q.
This shows the assertion of the lemma. In view of Lemmas 5.1
>
p
- 5.3, Theorem 2.1 follows immediately. REFERENCES
1.
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2.
-
228.
Ph. Bgnilan, Equations d'Evolution dans un Espace de Banach Quelconque et Applications, Publications Math. Orsay, Univ. Paris Sud, Orsay, vol. 25,1972.
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6.
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N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time-dependent constraints, Nonlinear Anal. T. M. A.
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N. Kenmochi and I. Pawlow, The vanishing viscosity method and a two-
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