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Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equation
Physica D 50 (1991) 135-151 North-Holland
Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equation X i a o s o n g Liu Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14214-3093, USA
Received 24 May 1990 Revised manuscript received 8 November 1990 Accepted 12 November 1990 Communicated by J. Guckenheimer
We present some results on the Kuramoto-Sivashinsky equation, which models pattern formation on unstable flame fronts and thin hydrodynamic films. We show Gevrey class regularity of the solutions and decay of the orbits to small neighborhoods (exponentially small) of some simple manifolds (approximate inertial manifolds).
1. Introduction and notations Inspired by the work of Foias and T e m a m [12] and Jolly, Kevrekidis and Titi [14], we present results concerning Gevrey class regularity and long-term behavior of the solutions of the Kuramoto-Sivashinsky equation. Because there is no forcing term in the Kuramoto-Sivashinsky equation, by looking at solutions for longer times we are able to get stronger results than those in ref. [14] under the same assumptions. We apply the techniques of refs. [6, 12, 14] to prove our results. The same ideas can be applied to other dissipative partial differential equations. An inertial manifold is a finite-dimensional, positively invariant, Lipschitz manifold which attracts all orbits exponentially, and so it contains the global attractor. If it exists, it enables us to reduce the long-time behavior of an infinite-dimensional dynamical system (PDE) to that of a finite-dimensional dynamical system (ODE). If the P D E is restricted to an inertial manifold, the P D E is equivalent to an O D E called an inertial form. Because of its explicit form, approximate inertial manifolds are easier to compute numerically than the inertial manifolds themselves, and the results give interesting, relatively simple representations of the dynamics of the system considered; see Foias et al. [4], Jauberteau et al. [13] and Jolly et al. [14, 15]. The Kuramoto-Sivashinsky equation models pattern formations in different physical contexts, and it is a paradigm of low-dimensional behavior. The existence of inertial manifolds for the K u r a m o t o Sivashinsky equation has been established in refs. [2, 3, 6, 9]. It does not provide the inertial form in an explicit form, so an approximate inertial manifold is necessary. To do this, a number of approximation schemes have been introduced in literature (see ref. [14] and references there). Following the approach of ref. [14], because we consider the Gevrey class regularity, we are able to improve the error estimates in ref. [14]. We also have a similar result regarding the Navier-Stokes equations [16]. But we did not consider Gevrey class regularity there; if one considers Gevrey class regularity, then one can show [21] the decay of solutions of the Navier-Stokes equations to exponentially small neighborhoods of some simple approximate inertial manifolds.
0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
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136
We consider the normalized Kuramoto-Sivashinsky equation: 0b/
04U
02U
0 - - 7 +0X - - +4 - - + 0X z
0/A U"~ =0
u(x,O) = u o
i n R X R +, in R,
with the periodic boundary condition
u(x,t) =u(x + L,t). In this paper, we only consider odd initial conditions and odd solutions. The Kuramoto-Sivashinsky equation can be written as [6, 18] 0u
O--t-+ A u - A 1 / 2 u + B ( u , u ) = 0 Vu ~ H,
(1)
where H consists of those odd functions u such that U=
~_, i u j e i(2"~jx/L),
ujisreal,
uj=-u_/,
(2)
j = --cx~
~1uil2
(3) has an orthonormal basis {wj = sin(2rrjx/L)}, which are eigenfunctions of A eigenvalues At = ( 2 ~ j / L ) 4, j = 1, 2 . . . . . We denote by (.,.), 1.1 the scalar product respectively. We define P =Pm to be the orthogonal projection on the span of Q = Qm = I - Pm" l e t u(t) = p ( t ) + q(t) (where p(t) = Pu(t), q(t) = eu(t)) be the (1) can be written as
dp +Ap -A1/2p + P B ( p + q , p +q) = O, dt
(4)
dq + A q - A l / 2 q + Q B ( p + q dt
(5)
p+q)=O.
H e r e B is the bilinear operator 0u
B ( u , v ) =U o x . For s, z, a >_ 0, the domain of A ' e "A" is the set of functions u satisfying (2) and
L ~ lujlZl2nrj/Z 18, e 2"(2"J/L)"~ < ~.
X. Liu / Kuramoto -Sivashinsky equation
137
We recall Agmon's inequality in the one-dimensional case, (7)
l u l ~ ~ v ~ - l u [ 1/2 [A1/4u[1/2,
and also that by integration by parts
(B(u,u),u)=O
VuED(AI/4).
(8)
In the following we denote by (.,.)~, 1.1, the scalar product and the norm in D(e'A"), that is, [ul, = le'A~ul,
( u , v ) , = (e'A°u,e'A°v).
Let the initial data for (1) be u 0 ~ H, and let u(t) be the corresponding odd solution of (1). Then we derive a bound for M'u(t)l for s >_ 0 (where 4s is an integer) (theorem 1 of section 2), and we show A~erA~/4u(t) (where s and r are appropriate constants) is analytic for t large enough (corollary 1 of section 3). We also prove all the solutions approach some exponentially small neighborhood of some simple manifolds (corollary 3 and theorem 5 of section 4).
2. Some estimates
First we give an estimate of the nonlinear term B by following ref. [12] closely.
Lemma 1. For fixed s, ~', ot where s, ~-> 0 and 0 < a _< 1/4, there exists a constant c, > 0 only depending on s, such that , ,1/2 [A1/au~1/2 IA,+ 1/4U[- r f o r u ~ D(AS+I/4e~A=). [A~B(u,u)l~<_CslUl~
(9)
Proof. Let u,v ~ D(AS+l/ae~A"),w ~ D(e~A=), and set u = ~ i u j e i(2~ix/L),
u 7 = e "r(2~rj/L)4~ lujl.
J
We use similar notations for v and w. We have
B ( u , v) = - i
~, ujvl(2'rrl/L ) e i(2"rrkx/L).
(10)
j+l=k
Since 0 < a < 1/4, we have
I( ASB( u, v ) , w )~l = L j+/+~k=oUjVt--ff-[----ff-J2"rrl [ 2 ~ k "~4S e2,r(2.r,.k/L)4,,Wk <_~t
, ,2'rrl
,[2avk'I 4s
E uj u I ~ - - - w k ~-----~] j+l+k=O
< c ' ( I + II),
e ~'[(2~rk/L)'-(2~rj/L)4a-(2wl/L)4a]
(11)
138
X. Liu / Kuramoto-Sivashinsky equation
where c s is a constant such that
Ij + l] 4s < cs(lji, . 4, + ill4,), and
I=L
U* [ 2'rrj /4s/~ • ( ~_( }
E
' t L )
'
w~= f , < ( x ) q , l ( x ) O , ( x ) d x ,
j+l+k=O
II=L
E uTv~( j+l+k=O
2~_l )4s + 1
w~=
fg~
~2(x)llt2(x)O2(x)dx,
where ~l(X) = ]~._,u?(2,rrj/L) 4s e i(2"rrjx/L), O,( x) = ~_~c'[~(2~rl/L )e i(2~'~/'~), l J 01(x ) = ~ w ~ e i(2~kx/L), k and ~:2(x) = ~_~u* e i(2~j~/l~), J
0 2 ( x ) = ~_~v?(2.rrl/L)4"+'ei(2~t~/L),
l
o~(x) = E w ~ e ~(2~kx/~).
k
By (7) and H61der's inequality, we get
I < I~:,1 Iqql~10,1 -< f2lA'ul= IA1/2cI'/2 IA1/4c1'/2 Iwl.-.
(12)
II < Is~2[= ]q,z1 ]021 < v/2[ull~/2 IAI/aulIT/2 [A'~+I/4uI~ Iwl~.
(13)
Suppose s > 1/4. Then, because
IASu I, ]AI/ztA rl/2 __< Ill (1/4)/(,~+r I/4)1 AS+ 1/4/,/Is/( s+ 1/4) ]l/ (s/2-r l/8)/(s+ I/4) IA,.+ /4.u 1(l/4)/(s+.r 1/4) <_ lu]l,/2lA'+l/4ul~ , from (11), (12), and (13) we find
MSB(u, u)l, < ~2c'[ASu I, IAl/2u II=/2 IA1/4N lit/2 + V~C;lU I1,/2 IAl/4u I,l-/2 IN F+ 1/4u [~ <_2~[5c;lu. ,1/2 iAl/4.ul~ll/2 IAS+l/4Ulr.
(14)
If 0 _< s < 1/4, since I _< Is~,l~ 1011 Io~l _< ~lA'ull~/2 IA~+l/4u[I,/2 IA1/4uI, Iwl,,
(is)
X. Liu / Kuramoto-Sivashinsky equation
139
and
IA'u I, IA1/4u I¢ < lu I~/4)/(s + 1/4)IA,+ 1/4 u i~/(, + 1/4)[u i~/(s + 1/4) iAS+ 1/4 u lo/4)/(s+ 1/4) _< lul,
[AS+l/4u[z,
from (11), (13) and (14) we see that
IASB(u, u)l, _< 2v~c'Iull,/2 [A1/4u 1/2 iAs+l/4ul,.
(16)
Now from (14) and (16), we see that l e m m a 1 is true for all s >_ 0 with c, = 2V~-c'. So we have proved l e m m a 1.
[]
Remarks. T h e best estimate in l e m m a 1 is achieved when a = 1 / 4 . L e m m a 1 is also true in the complexified case. Now we give a b o u n d for the orbits of the K u r a m o t o - S i v a s h i n s k y equation.
Theorem 1. Let L > 1, s > 0, and let 4s be an integer. T h e n for any u 0 ~ H there exists a constant to, only d e p e n d i n g on luo[, such that if t > t, = t o + 4s, then IASu(t)[ < Ps = Ms L4s+5/2,
(17)
where M s > 0 only d e p e n d s on s. T h e p r o o f below is formal. It can be m a d e rigorous by using Galerkin approximation and then passing to the limit using the a p p r o p r i a t e c o m p a c t n e s s t h e o r e m s and the vector valued Vitali t h e o r e m (see e.g. refs. [1, 10, 12]). T h e s a m e consideration applies to the proofs of t h e o r e m s 2 and 3 below.
Proof. We use induction to prove t h e o r e m 1. For s = 0, 1 / 4 , by refs. [6, 17] we know t h e o r e m 1 is true. Taking the scalar product of (1) with get
AZSu and using H61der's inequalities and l e m m a 1 (take ~- = 0), we
1 dIASul 2 2 d-----T-+ IAs+l/2ul2 IAs+l/4ul 2 -{- Cslul I/2 IA1/4ull/2 IAs+l/4ul IASu[
<_ [A s- 1/4U[2/3 [As+ 1/Zu]4/3 + c s [u[ 1/2 IAl/4u]l/2 [As+ 1/4u[ [ASu[ 2 s + l / 2 u [2 +cslu [1/2 IAl/4u i1/2 iAs+l/4 u I lASul, <½lAS-1/4ul2+3lA SO dlASul 2
dt
+ 2IA'+ 1/2U[2 - 2 As-l~4,, 2 - 2cslul 1/2 IAi/4u[ 1/2 [AS+1/4ul IASul <_0.
(18)
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140
Now let s >_ 1 / 4 and assume t h e o r e m 1 is true for r, where r < s, and 4r is a nonnegative integer. Next we let 2 s- 1/4u[2-2c, /3(t) = xIA 2 s+ j / Z u l 2 _ 51A
lu[ 1/2 IAl/4ul
1/2 [AS+ l/4u]
IASul .
(19)
Because
Ms+a/4ul < IAs-l/4u[ 1/3 IAS+I/2ul:/3,
(20)
IA'u[ ~ [As-I/4u[ 2/3 [As+l/2ul 1/3
(21)
we get
/3(t) >
IAs+l/4ul IA'ul IA'+I/eul 3 IA'-W4ul - 2c, [ul 1/2 Ml/4ul 1/2 [A ~+ 1/4ul IA'ul + llA,+l/2ul
>
2
2 xls-l/4tll 2
IAs+l/4ul M'ul 3lAS_l/4u I /31(t ) + / 3 2 ( t ) ,
(22)
where /31(t ) =
]A'+l/2ul - 6CslU] 1/2 IA1/4u] 1/2 ]A" 1/4ul,
~ 2 ( t ) = 51A 1 s+ 1 / 2 u l 2 _ 51A 2 , - 1/4Ul2.
From(20)
]ASul3 - 6c, lul 1/2 [Al/4ul 1/2 [A~,_l/4ul [3l(t ) > iAS-1/4ul2 1 (iA~u[ 3 _ >--iAS_l/4ul2
6CslUll/2 M1/aull/2 iAS_l/4ul3),
(23)
and
IASul 6 B2(t) > 3lA,-1/4u]4
~IA'-1/4U12 >
.
1
-- 3lAs_l/4ul4(M'u
By the induction hypothesis, we know for t > t s 1/4 that lu(t)l-
IA~/4u(t)l <-<-P,/4,
and
IAS-l/4u(t)l <--Ps-1/4"
16 _
21A,_1/4u16).
(24)
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141
N o w we let o-~ b e a c o n s t a n t such t h a t trs > p" = m a x { ( 6 c ~ )
1/3(
PoPl/4)
1/6
Ps-1/4,21/6ps-1/4} "
(25)
T h e n from (18) a n d ( 2 2 ) - ( 2 5 ) we see that if t h e r e exists a t ' >_ t s_ n/4 such t h a t
IASu(t')l < o's,
(26)
[ASu( t )l < ~r~ for t > t'.
(27)
then
If s = 1 / 2 , we t a k e the scalar p r o d u c t o f (1) with u. T h e n by (8) a n d H61der's i n e q u a l i t y we get
f----[lu[2 + IA1/2ul2 <_ lul 2 T h e r e f o r e if t > t o, we have
tt+llA1/2ul
ds < 2p 2,
a n d h e n c e t h e r e exists a t ' ~ [t, t + 1] such that
IA1/2u( t')l < V~Po.
(28)
N o w we t a k e tl/2 = t o + 2 a n d
Pl /2 = max { P~/2 , x/2 po } . T h e n f r o m ( 2 6 ) - ( 2 9 ) we see that
]A1/Zu(t)l ~ P l / 2
for t >__tl/2,
and
t91/2 -~ MI/2 L9/2. So t h e o r e m 1 is true for s = 1 / 2 . If s __ 3 / 4 , we s u b s t i t u t e s - 1 / 2 for s in (18), o b t a i n i n g
dlAS-l/2ul 2 dt + 2]Asul2 - 2 l A S - 3 / 4 u l 2 - 2Cs_l/2lull/2lA1/4ujl/2 lAS-1/2ul [AS-1/4u] <_0. F r o m the a b o v e i n e q u a l i t y we get for t _> L - 1/4
ft t+llAsul2 ds -
(29)
X. Liu / Kuramoto-Sit:ashinsky equation
142
Hence there exists a t' ~ [t, t + 1] such that IAS( t')l <_ r/~,
(30)
where Tls = Ps-3/4 + (3Cs 1/2)xl/2;tPOPl/2)xl/4/tPs I/2Ps-l/4)xl/2
q'-
( 3 / 2 ) 1/2 Ps-l/2.
(31)
Now we take t, = t s_ ~/4 + 1 = t o + 4s and p, = max{p~, r/~}. From (26), (27), (30) and (31) for t > t~ = t 0 + 4s we get MSu(t)l
and Ps = Ms L4s +5/2.
[]
We have proved theorem 1.
3. Gevrey class regularity of the solutions
Now we show the Gevrey class regularity of the solutions. We define a function T , ( r ) : [0, ~) ~ E by
T,(r)
=
( 16X/~C2 2 ~
+ 12~/2
r
)-I
(32)
T h e o r e m 2. Let 0 _< a _< 1/4, s >_ 0 and assume u o e D(A'), then u(t): t--+ e ra`" A ' u ( r e i°) is analytic on (0, T , ([A*uol)) (in the region A([A'uoq) , where A(IA'uo[ ) is given by (48) below), and IA s e rA" u( r ei°)] 2 _< 2[A~uo[ 2,
(33)
for r e i° E A([ASuo[). If, moreover, there exists a constant M, such that IA~u(t)l<_M
Vt>O,
(34)
then u ( t ) : t ~ A" eT*(M)A" u ( r e i°) is analytic on ( T , ( M ) , o o )
(in the region A(M), where A(M) is given by (50) below), and
le 7*(M)A'' ASu(05)l <_ ~ / 2 M ,
for 05 ~ A(M).
(35)
143
X. Liu / Kuramoto-Sivashinsky equation
Proof. As in refs. [11, 12] we consider complexified spaces and functions. Then (1) can be written as du
d---( + Au - A1/2u + B( u, u) = 0.
(36)
For ~ = r e i°, taking the scalar product of (36) with A2Su(r e i°) in D(CA"), multiplying it by e i°, and taking the real part of the result, we have
-~+Au-A1/4u+B(u,u),A2"u
=0. r
Now
R e[e [ i°[du [-~-A2Su)~]=Re '
( erA°A~dU -d-r' CA°ASu ( re i° )) = R e ( d ( C A " A s u ) dr
_12 dr INs e rAau 12 - (erA'~ AS+au,erA" ASu) > "~
e rA" A S + a u , erA '~ A S u )
[A s +aU ]rlASulr "
Therefore 1 d
2 dr
IASul2r - IA'+~ul~lASulr + cos OIA'+l/2ul 2 < Re[ei°(Al/2u,A2"u)r ] - Re[eiO(B(u,u),A2,U)r].
(37)
Since 0 < a < 1/4 we get
iA.+~ul~lA.ulr<_ iAs+l/2u 2,~-. r ~Zl lg 2-2a r <
(4o)o ,1o,
IA'+l/2UlZr+(1-a ) ~
[A'ul 2,
(38)
and we have
Re[ei°( A'/2u,A2"u)r] < [A'+l/4ul2 ~ IA'ulrlAS+l/2u[r ~_~ -~[As+l/2U[2r
-~- co~ASul
2.
(39)
By lemma 1, we find that
I( B(u,u),A2Su),]
<-~Cs A~/2.
< ~IASul2
--
1
iA.+l/2ulr < cos 0 As+I/2,,]2 --
8
. . . .
2c 2
"t- A2s+l/4cosoMSUl 4.
(40)
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144
We put (38)-(40) into (37) to obtain:
d l A ' u 12r+ cos 0 IA ' + 1/2u 12r <2 (1-a)~]
{ as ]~/('-")
2 ]
2
4c2,
+ -cosO ]lASulr + /~21s+l/4cosO
IA'ul 4.
(41)
Now let y ( r ) = IASu(rei°)12r= [A~erA"u(rei°)12 F r o m (41) we have
dy < b y ( y + c ) dr
(42)
where
b-
~.2s+ 1/4 COS 0 '
c=2
1-c~
~
+
co~s-
]
A2S+I/4 COS 0
4c~
We try to find a bound T . on r such that y(r)<_2yl,
for r e [ 0 , ~ r . l .
(43)
We solve (42) and get the inequality
y(r) Yo eh,,~. y(r) +c - Yo +C Since z / ( z + c) increases as z increases in (0, ~), we only need YO ebcr < 2Yo Yo + c - 2y o + c "
Thus, 1
r _< ~
log
2y o + c
because for z ~ (0, 1), log(l+z)
> z/2.
'
(44)
145
X. Liu / Kuramoto-Sivashinsky equation
Hence we only need 1
c
1
r < bc 2(2y o + c)
(45)
2b(2y o + c)"
We restrict to cos 0 _> f 2 / 2 . Then 2[{ 1 ]"/('-~) 2 ] /~21S+1/4 COS 0 _ 2 [ ( 1 ) 1/3 2 c < [~ c--o-if-O] + co---70] 4c 2 < ~ + co--7
A s+1/4cOS 0
4c 2
< 2[(COS O) 2/3 + 2]h'2s+l/._.....~ 4 3A2s+1/4 -4C 2 < C - 2- - ' ~
(46)
Also 4c 2 4Vt2c 2 < - b = /~2s+1/4COS 0 -~21s+1/4 "
(47)
From (45)-(47), we get
(
[ 8v~-c 2 312,+ ,/4 ]]-1 r< ~ 2Yo+ ~-7Cs2 ]] So if 0 < r < T , ( l A ' u o l ) and cos 0 > V~-/2, then (43) is true, where
r , OA'uol) = [( 16v/2c2lA , .o 2 + 12v~-)-' That is, for r e i° in the region A(IA'uol): A(IA'uo I) = {r ei°: r < T , (IA'u ol), cos 0 > v~-/2}
(48)
we have IA' e rA" u( r ei°)12 < 21A'uol 2. If (34) is true, then we repeat the argument above around each t o > 0 to get [erA"ASu(re i° + to)] < v~M,
for r e i° E A(lA'u(to)l). In particular, we have ler*(M)A" A ' u ( cb)l < v ~ M ,
(49)
146
)d Liu / Kuramoto-Sivashinsky equation
where
2
[ 16~2c2
T,(M)=I~M
+12v/2
) 1
,
and (b ~ A ( M ) which consists of the strip Re~b>__T,(M),
IIm~bl<(v~/Z)T,(M).
(50) []
We have proved theorem 2. Now from theorem 1 and theorem 2, we easily prove
Corollary 1. Let 0 < ~ < 1/4, s >_ 0 and 4s be an integer, and assume u 0 • H. Then u ( t ) - ~ A s e r*(o')A'~ u( r e i°)
is analytic on ( T ,
(Ps) + t~, oo), and
[ASel*(°')AC'u(4))[ < v~p,
V4~A(ps)
+ts,
(51)
where A(O,) + t, = {4~: Re4~ >- T , ( p s ) + 6 , [Im~b[ < T , ( p , ) / v ~ } .
Ps, t, are the same as in theorem 1, and T,(p,) is given by (32).
4. Approximate inertial manifolds Now we consider approximate inertial manifolds. We show all t h e ' trajectories approach some exponentially small neighborhood of some simple manifolds (cf. refs. [5, 14, 20]).
Theorem 3. Let 0 < a < 1/4, s > 0, and ]ASu(t)] < M
Vt>0.
(52)
Then we have
IASu(qb)lT, < v~M
V& • A ( M ) ,
(53t
X. Liu / Kuramoto-Sivashinsky equation
147
and if t > (1 + 1 / ~ / 2 ) T , , then
A , dU(t) 2M dt T. < T . '
(54)
IAS+lu(t)lr. <_K,( M ) ,
(55)
As+ldU(t) dt
T, -<
16K1(M) T,
(56)
where T , = T , ( M ) , A(M) are the same as before and 8M
KI( M ) = ~
3c4/3M7/3
+2M+
1~4s/3_1/6
.
Pro@ Eq. (53) follows from theorem 2. Applying Cauchy's integral formula to the circle F centered at any point on the real axis t > (1 + 1 / ~ - ) T , with radius T , / v ~ , we have u(t) =
1__1__[u(~b) drk. 2rri Jrgb - t
From this and (53) and using theorem 1, we get for t > (1 + 1 / v ~ - ) T ,
A s ~ t t)
< 2M . r, - T ,
Thus, (54) is true. Applying lemma 1, (52), (53) and (54) we establish (55) and (56).
[]
From theorem 3 and theorem 1, we get
Corollary 2. Let 0 _ 0, and let 4s be an integer. Assume u o ~ H. Then we have IA'u(4~)IT.(,,)--< v~-p, V4~ ~ A ( p , ) + ts, and if t > (1 + 1 / v ~ ) T , ( p s )
A~ dUd(~)
+ t,, then
2p~
r.(.~) < T , ( p s ) ,
IA~+lu( t)IT.(p,) < Kl(Ps), 16Kl(ps)
As+l d U ( t ) T.(p,)
dt
<
(57)
r,(pA
(58) (59) (60)
X. Liu / Kuramoto-Sit,ashinsky equation
148
A(p,), p,, t~, T,(p,)are the same as before and
where
4/3 7/3
8p~
gl(Ps)
3c,
T,(p,~ + 2 p , +
Ps
t~4s/3-1/6"
F r o m corollary 2, we easily get
Corollary 3. U n d e r the same assumptions as corollary 2, then there exists a t~ = t o + 4s such that if 1/v/2)T.(p~)+ t,, we have
t > (1 +
[q(t)l_<
K'(Ps) a , + ~ e - T*{p')~';'+'
(61)
"'m + 1
~ t t)
<-
e 7.~o,,~';,,+,
16K'(Ps)
(62)
Remark. T h e case s = 0 or 1 / 4 , T.(ps) = 0 was considered in ref. [14]. We see that every solution of the K u r a m o t o - S i v a s h i n s k y equation a p p r o a c h e s some exponentially small n e i g h b o r h o o d of the linear manifold PmH. If the solution u(t) is on the universal attractor, then (61), (62) are true for all t ~ N. Next, as in refs. [14, 20] we analyze the steady approximate inertial manifold stationary solutions of (1)), where ~ ' = graph(q~); q~: /3 ~ Q H such that
A~(p)-A1/2ff)(p)+QB(p+q~(p),p+clg(p))
=0,
p~/3.
de" (which contains all the
(63)
For fixed s > 1 / 4 , r, a > 0, we define /3 = {p ~ P H : /3 ± = {q ~ Q H :
[A'pI,<_ V~Ps}, [ASql, <_f2p,}.
Now we show the existence of q~.
Theorem 4. (cf. refs. [10, 14, 20]) If m satisfies Am+l >- - max{4r2, r 21 /. 2
2/
(64)
P,],
where
rl
=
~-1/4~-s+1/8 2 V~ps+8Cs-I/4A'm+IAI Ps,
and r2
=
1 + 4V~C,_ l / 4 "~m- -+l /l4") tl
--s
+ 1/8Ps,
then there exists a unique m a p p i n g @: /3 ~ Q H satisfying (63). Its graph ~/" is a C-analytic manifold.
X. Liu / Kuramoto- Sivashinsky equation
149
Moreover IWq:'( p)l~- < Am 1/2rl Vp
Proof. (See also refs. [10, 14, 20].) For p ~/3 define Tp(q): Tp(q)=A-1/2q-A-1QB(p+q,p+q)
Vq~/3±
By lemma 1, we find that
IA'Zp( q)[, < Aml+/2rl,
(65)
and
A, OT~q)~7
(66)
+1
We see Tp is a contraction map if (64) is true. Now let the fixed point of Tp be @(p), then we have [] proved theorem 4 and the graph of q~ is an analytic manifold (see refs. [7, 10]). Now we define • 0 ( p ) = 0,
qbk(P)=Tp(qgk_l(p) ) Vp~fl
and
k=l,2 .....
From theorem 4, we get ]A'(@(p) - @k(P))]~ -< 2(rzA~nl+/z)klASdPl(P)]~,
(67)
since q~l(P) =
-A-1QB(p,P) •
So by lemma 1 for s > 1/4, [ASqbl( P)]~ -<
]As- 1QB( P, P )lr < Am3+/14lZS-1~4an( p, p)]~
-<" 'Jtm-3/4r, + l ~s-l/41/1~ 1/2 ~"
- 3 / 4 A -1' + l / S l h ~ p l Z ~ . IAI/4pIIfE IASpl, ~-~Cs--1/4 A rn--1
(68)
Combining (67) and (68), we have IA'(@(P) -
@k(P))[~ -< 2Cs-1/4AlS+l/8Am3/14(r2Am
1/2+1) klA ~(p)[,,2
(69)
X. Liu / Kuramoto-Sivashinsky equation
150
and from the above inequality and corollary 1, we see that if t > T , ( p s ) + t,, then
IAS(q)(P) -q)k(P))lr.(z) <4c,
l-s+l/8)L-3/4[r
1/4"1
)~
I/2 ~k r'2
" m + l k ' 2 " ' m + l J P's"
(70)
N o w we consider the distance between u(t) and . ~ = graph(q)).
Theorem 5. Let 0 < a < 1 / 4 , s > 1 / 4 , u 0 ~ H, and assume m to be large e n o u g h such that (64) and the following are true l at- 2 V ~ P , / 4 A m 7 / 8 + J' "~P -0 n l / 2 ~P1/4"'m+l l / 2 )' - 3/4 ~ 1 / 2 . A m +1/2
If t > ( 1 + 1 / v ~ ) T , ( p s ) + t
Iq(t)-q)k(P(t))
,, then 32 K ~( ps ) -,-2 -T,(paA';,+l r , ( p s ) Am+ ' e
Iq(t)--q)(p(t))l_<
I<-
(71)
- - s - 3 / 4 g[r2Am+l - - - l / 2 X) k P52~) e-T*(°')a';"+ 3 T2 K s 2 +4e,.-,/4A 1- s + l / 8 -Am+l , ( Ip(sP)s ) Am+l ',
(72)
(73)
where Ps, ts, etc. are the same as before.
Proof. By ref. [14] we know if (71) is true and t > tl/4, then IA(q(t) - q)(p(t)))l < 2 ~
.
F r o m this and corollary 3, we see (72) is true. N o w combine (72) and (70) we get (73) is true.
[]
Remark. T h e case s = 1 / 4 , T , ( p , ) = 0 was considered in ref. [14]. F r o m t h e o r e m 5 and corollary 3, we see the distance of orbits to ~," is closer than it is to Pm H by a factor of Am +1"~ A n d graph(q) 3) achieves the same estimates as graph(q)). N o t e that for large s, Ps is very large, and then there might not be any gain by taking large s (see (']2)). T h e real gain in this case is the exponential term and even here T , d e p e n d s on p,. So, in real c o m p u t a t i o n s one should probably take s relatively small, say s = 0 or s = 1 / 4 .
Acknowledgements T h e a u t h o r thanks his advisor Professor Nicholas D. Kazarinoff for valuable advice and suggestions. T h e a u t h o r also thanks the a n o n y m o u s referee for helpful suggestions.
References [1] P. Constantin and C. Foias, Navier-Stokes Equations (The University of Chicago Press, Chicago, 1988). [2] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Mathematics Sciences, No. 70 (Springer, Berlin, 1989). [3] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynam. Diff. Eqs. 1 (1989) 45-73.
X. Liu / Kuramoto -Siuashinsky equation
151
[4] C. Foias, M.S. Jolly, I.G. Kevrekidis, G.R. Sell and E.S. Titi, On the computation of inertial manifolds, Phys. Lett. A 131 (1988) 433-436. [5] F. Foias, O. Manley and R. Temam, Modelling of the interaction of small and large eddies in two dimensional turbulent flows, Math. Modelling Numerical Anal. 22 (1988) 93-118. [6] C. Foias, B. Nicolaenko, G.R. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl. 67 (1988) 197-226. [7] C. Foias and J.-C. Saut, Remarques sur les equations de Navier-Stokes stationnaires, Ann. Scuola Norm. Sup. Pisa Ser. 4, 10 (1983) 169-177. [8] C. Foias, G.R. Sell and E.S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dynam. Diff. Eqs. 1 (1989) 199-224. [9] C. Foias, G.R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Diff. Eqs. 73 (1988) 309-353. [10] C. Foias and R. Temam, Remarques sur les 6quations de Navier-Stokes stationnaires et les ph6nom~nes successifs de bifurcation, Ann. Scuola Norm. Sup. Pisa Ser. 4, 5 (1978) 29-63. [11] C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl. 58 (1979) 339-368. [12] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989) 359-369. [13] F. Jauberteau, C. Rosier and R. Temam, The nonlinear Galerkin method in computational fluid dynamics, Appl. Num. Math., to appear. [14] M.S. Jolly, I.G. Kevrekidis and E.S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations, Physica D 44 (1990) 38-60. [15] M.S. Jolly, I.G. Kevrekidis and E.S. Titi, Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation, J. Dynam. Diff. Eqs. (1990), in press. [16] X. Liu, On approximate inertial manifolds for two and three dimensional turbulent flows, J. Math. Anal. Appl., to appear. [17] B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equation: Nonlinear stability and attractors, Physica D 16 (1985) 155-183. [18] R. Temam, Infinite Dimensional Dynamics Systems in Mechanics and Physics (Springer, Berlin, 1988). [19] E.S. Titi, Une vari6t6 approximante de l'attracteau universel des 6quations de Navier-Stokes, non lin6aire, de dimension finie, Compt. Rend. Acad. Sci. Paris I 307 (1988) 383-385. [20] E.S. Titi, On approximate inertial manifolds to the Navier-Stokes equations, MSI Preprint 88-119, J. Math. Anal. Appl., to appear. [21] E.S. Titi, private communication.
Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equation X i a o s o n g Liu Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14214-3093, USA
Received 24 May 1990 Revised manuscript received 8 November 1990 Accepted 12 November 1990 Communicated by J. Guckenheimer
We present some results on the Kuramoto-Sivashinsky equation, which models pattern formation on unstable flame fronts and thin hydrodynamic films. We show Gevrey class regularity of the solutions and decay of the orbits to small neighborhoods (exponentially small) of some simple manifolds (approximate inertial manifolds).
1. Introduction and notations Inspired by the work of Foias and T e m a m [12] and Jolly, Kevrekidis and Titi [14], we present results concerning Gevrey class regularity and long-term behavior of the solutions of the Kuramoto-Sivashinsky equation. Because there is no forcing term in the Kuramoto-Sivashinsky equation, by looking at solutions for longer times we are able to get stronger results than those in ref. [14] under the same assumptions. We apply the techniques of refs. [6, 12, 14] to prove our results. The same ideas can be applied to other dissipative partial differential equations. An inertial manifold is a finite-dimensional, positively invariant, Lipschitz manifold which attracts all orbits exponentially, and so it contains the global attractor. If it exists, it enables us to reduce the long-time behavior of an infinite-dimensional dynamical system (PDE) to that of a finite-dimensional dynamical system (ODE). If the P D E is restricted to an inertial manifold, the P D E is equivalent to an O D E called an inertial form. Because of its explicit form, approximate inertial manifolds are easier to compute numerically than the inertial manifolds themselves, and the results give interesting, relatively simple representations of the dynamics of the system considered; see Foias et al. [4], Jauberteau et al. [13] and Jolly et al. [14, 15]. The Kuramoto-Sivashinsky equation models pattern formations in different physical contexts, and it is a paradigm of low-dimensional behavior. The existence of inertial manifolds for the K u r a m o t o Sivashinsky equation has been established in refs. [2, 3, 6, 9]. It does not provide the inertial form in an explicit form, so an approximate inertial manifold is necessary. To do this, a number of approximation schemes have been introduced in literature (see ref. [14] and references there). Following the approach of ref. [14], because we consider the Gevrey class regularity, we are able to improve the error estimates in ref. [14]. We also have a similar result regarding the Navier-Stokes equations [16]. But we did not consider Gevrey class regularity there; if one considers Gevrey class regularity, then one can show [21] the decay of solutions of the Navier-Stokes equations to exponentially small neighborhoods of some simple approximate inertial manifolds.
0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
X. Liu / Kuramoto-Sicashinsky equation
136
We consider the normalized Kuramoto-Sivashinsky equation: 0b/
04U
02U
0 - - 7 +0X - - +4 - - + 0X z
0/A U"~ =0
u(x,O) = u o
i n R X R +, in R,
with the periodic boundary condition
u(x,t) =u(x + L,t). In this paper, we only consider odd initial conditions and odd solutions. The Kuramoto-Sivashinsky equation can be written as [6, 18] 0u
O--t-+ A u - A 1 / 2 u + B ( u , u ) = 0 Vu ~ H,
(1)
where H consists of those odd functions u such that U=
~_, i u j e i(2"~jx/L),
ujisreal,
uj=-u_/,
(2)
j = --cx~
~1uil2
(3) has an orthonormal basis {wj = sin(2rrjx/L)}, which are eigenfunctions of A eigenvalues At = ( 2 ~ j / L ) 4, j = 1, 2 . . . . . We denote by (.,.), 1.1 the scalar product respectively. We define P =Pm to be the orthogonal projection on the span of Q = Qm = I - Pm" l e t u(t) = p ( t ) + q(t) (where p(t) = Pu(t), q(t) = eu(t)) be the (1) can be written as
dp +Ap -A1/2p + P B ( p + q , p +q) = O, dt
(4)
dq + A q - A l / 2 q + Q B ( p + q dt
(5)
p+q)=O.
H e r e B is the bilinear operator 0u
B ( u , v ) =U o x . For s, z, a >_ 0, the domain of A ' e "A" is the set of functions u satisfying (2) and
L ~ lujlZl2nrj/Z 18, e 2"(2"J/L)"~ < ~.
X. Liu / Kuramoto -Sivashinsky equation
137
We recall Agmon's inequality in the one-dimensional case, (7)
l u l ~ ~ v ~ - l u [ 1/2 [A1/4u[1/2,
and also that by integration by parts
(B(u,u),u)=O
VuED(AI/4).
(8)
In the following we denote by (.,.)~, 1.1, the scalar product and the norm in D(e'A"), that is, [ul, = le'A~ul,
( u , v ) , = (e'A°u,e'A°v).
Let the initial data for (1) be u 0 ~ H, and let u(t) be the corresponding odd solution of (1). Then we derive a bound for M'u(t)l for s >_ 0 (where 4s is an integer) (theorem 1 of section 2), and we show A~erA~/4u(t) (where s and r are appropriate constants) is analytic for t large enough (corollary 1 of section 3). We also prove all the solutions approach some exponentially small neighborhood of some simple manifolds (corollary 3 and theorem 5 of section 4).
2. Some estimates
First we give an estimate of the nonlinear term B by following ref. [12] closely.
Lemma 1. For fixed s, ~', ot where s, ~-> 0 and 0 < a _< 1/4, there exists a constant c, > 0 only depending on s, such that , ,1/2 [A1/au~1/2 IA,+ 1/4U[- r f o r u ~ D(AS+I/4e~A=). [A~B(u,u)l~<_CslUl~
(9)
Proof. Let u,v ~ D(AS+l/ae~A"),w ~ D(e~A=), and set u = ~ i u j e i(2~ix/L),
u 7 = e "r(2~rj/L)4~ lujl.
J
We use similar notations for v and w. We have
B ( u , v) = - i
~, ujvl(2'rrl/L ) e i(2"rrkx/L).
(10)
j+l=k
Since 0 < a < 1/4, we have
I( ASB( u, v ) , w )~l = L j+/+~k=oUjVt--ff-[----ff-J2"rrl [ 2 ~ k "~4S e2,r(2.r,.k/L)4,,Wk <_~t
, ,2'rrl
,[2avk'I 4s
E uj u I ~ - - - w k ~-----~] j+l+k=O
< c ' ( I + II),
e ~'[(2~rk/L)'-(2~rj/L)4a-(2wl/L)4a]
(11)
138
X. Liu / Kuramoto-Sivashinsky equation
where c s is a constant such that
Ij + l] 4s < cs(lji, . 4, + ill4,), and
I=L
U* [ 2'rrj /4s/~ • ( ~_( }
E
' t L )
'
w~= f , < ( x ) q , l ( x ) O , ( x ) d x ,
j+l+k=O
II=L
E uTv~( j+l+k=O
2~_l )4s + 1
w~=
fg~
~2(x)llt2(x)O2(x)dx,
where ~l(X) = ]~._,u?(2,rrj/L) 4s e i(2"rrjx/L), O,( x) = ~_~c'[~(2~rl/L )e i(2~'~/'~), l J 01(x ) = ~ w ~ e i(2~kx/L), k and ~:2(x) = ~_~u* e i(2~j~/l~), J
0 2 ( x ) = ~_~v?(2.rrl/L)4"+'ei(2~t~/L),
l
o~(x) = E w ~ e ~(2~kx/~).
k
By (7) and H61der's inequality, we get
I < I~:,1 Iqql~10,1 -< f2lA'ul= IA1/2cI'/2 IA1/4c1'/2 Iwl.-.
(12)
II < Is~2[= ]q,z1 ]021 < v/2[ull~/2 IAI/aulIT/2 [A'~+I/4uI~ Iwl~.
(13)
Suppose s > 1/4. Then, because
IASu I, ]AI/ztA rl/2 __< Ill (1/4)/(,~+r I/4)1 AS+ 1/4/,/Is/( s+ 1/4) ]l/ (s/2-r l/8)/(s+ I/4) IA,.+ /4.u 1(l/4)/(s+.r 1/4) <_ lu]l,/2lA'+l/4ul~ , from (11), (12), and (13) we find
MSB(u, u)l, < ~2c'[ASu I, IAl/2u II=/2 IA1/4N lit/2 + V~C;lU I1,/2 IAl/4u I,l-/2 IN F+ 1/4u [~ <_2~[5c;lu. ,1/2 iAl/4.ul~ll/2 IAS+l/4Ulr.
(14)
If 0 _< s < 1/4, since I _< Is~,l~ 1011 Io~l _< ~lA'ull~/2 IA~+l/4u[I,/2 IA1/4uI, Iwl,,
(is)
X. Liu / Kuramoto-Sivashinsky equation
139
and
IA'u I, IA1/4u I¢ < lu I~/4)/(s + 1/4)IA,+ 1/4 u i~/(, + 1/4)[u i~/(s + 1/4) iAS+ 1/4 u lo/4)/(s+ 1/4) _< lul,
[AS+l/4u[z,
from (11), (13) and (14) we see that
IASB(u, u)l, _< 2v~c'Iull,/2 [A1/4u 1/2 iAs+l/4ul,.
(16)
Now from (14) and (16), we see that l e m m a 1 is true for all s >_ 0 with c, = 2V~-c'. So we have proved l e m m a 1.
[]
Remarks. T h e best estimate in l e m m a 1 is achieved when a = 1 / 4 . L e m m a 1 is also true in the complexified case. Now we give a b o u n d for the orbits of the K u r a m o t o - S i v a s h i n s k y equation.
Theorem 1. Let L > 1, s > 0, and let 4s be an integer. T h e n for any u 0 ~ H there exists a constant to, only d e p e n d i n g on luo[, such that if t > t, = t o + 4s, then IASu(t)[ < Ps = Ms L4s+5/2,
(17)
where M s > 0 only d e p e n d s on s. T h e p r o o f below is formal. It can be m a d e rigorous by using Galerkin approximation and then passing to the limit using the a p p r o p r i a t e c o m p a c t n e s s t h e o r e m s and the vector valued Vitali t h e o r e m (see e.g. refs. [1, 10, 12]). T h e s a m e consideration applies to the proofs of t h e o r e m s 2 and 3 below.
Proof. We use induction to prove t h e o r e m 1. For s = 0, 1 / 4 , by refs. [6, 17] we know t h e o r e m 1 is true. Taking the scalar product of (1) with get
AZSu and using H61der's inequalities and l e m m a 1 (take ~- = 0), we
1 dIASul 2 2 d-----T-+ IAs+l/2ul2 IAs+l/4ul 2 -{- Cslul I/2 IA1/4ull/2 IAs+l/4ul IASu[
<_ [A s- 1/4U[2/3 [As+ 1/Zu]4/3 + c s [u[ 1/2 IAl/4u]l/2 [As+ 1/4u[ [ASu[ 2 s + l / 2 u [2 +cslu [1/2 IAl/4u i1/2 iAs+l/4 u I lASul, <½lAS-1/4ul2+3lA SO dlASul 2
dt
+ 2IA'+ 1/2U[2 - 2 As-l~4,, 2 - 2cslul 1/2 IAi/4u[ 1/2 [AS+1/4ul IASul <_0.
(18)
X. Liu / Kuramoto-Siuashinsky equation
140
Now let s >_ 1 / 4 and assume t h e o r e m 1 is true for r, where r < s, and 4r is a nonnegative integer. Next we let 2 s- 1/4u[2-2c, /3(t) = xIA 2 s+ j / Z u l 2 _ 51A
lu[ 1/2 IAl/4ul
1/2 [AS+ l/4u]
IASul .
(19)
Because
Ms+a/4ul < IAs-l/4u[ 1/3 IAS+I/2ul:/3,
(20)
IA'u[ ~ [As-I/4u[ 2/3 [As+l/2ul 1/3
(21)
we get
/3(t) >
IAs+l/4ul IA'ul IA'+I/eul 3 IA'-W4ul - 2c, [ul 1/2 Ml/4ul 1/2 [A ~+ 1/4ul IA'ul + llA,+l/2ul
>
2
2 xls-l/4tll 2
IAs+l/4ul M'ul 3lAS_l/4u I /31(t ) + / 3 2 ( t ) ,
(22)
where /31(t ) =
]A'+l/2ul - 6CslU] 1/2 IA1/4u] 1/2 ]A" 1/4ul,
~ 2 ( t ) = 51A 1 s+ 1 / 2 u l 2 _ 51A 2 , - 1/4Ul2.
From(20)
]ASul3 - 6c, lul 1/2 [Al/4ul 1/2 [A~,_l/4ul [3l(t ) > iAS-1/4ul2 1 (iA~u[ 3 _ >--iAS_l/4ul2
6CslUll/2 M1/aull/2 iAS_l/4ul3),
(23)
and
IASul 6 B2(t) > 3lA,-1/4u]4
~IA'-1/4U12 >
.
1
-- 3lAs_l/4ul4(M'u
By the induction hypothesis, we know for t > t s 1/4 that lu(t)l-
IA~/4u(t)l <-<-P,/4,
and
IAS-l/4u(t)l <--Ps-1/4"
16 _
21A,_1/4u16).
(24)
X. Liu / Kuramoto- Sivashinsky equation
141
N o w we let o-~ b e a c o n s t a n t such t h a t trs > p" = m a x { ( 6 c ~ )
1/3(
PoPl/4)
1/6
Ps-1/4,21/6ps-1/4} "
(25)
T h e n from (18) a n d ( 2 2 ) - ( 2 5 ) we see that if t h e r e exists a t ' >_ t s_ n/4 such t h a t
IASu(t')l < o's,
(26)
[ASu( t )l < ~r~ for t > t'.
(27)
then
If s = 1 / 2 , we t a k e the scalar p r o d u c t o f (1) with u. T h e n by (8) a n d H61der's i n e q u a l i t y we get
f----[lu[2 + IA1/2ul2 <_ lul 2 T h e r e f o r e if t > t o, we have
tt+llA1/2ul
ds < 2p 2,
a n d h e n c e t h e r e exists a t ' ~ [t, t + 1] such that
IA1/2u( t')l < V~Po.
(28)
N o w we t a k e tl/2 = t o + 2 a n d
Pl /2 = max { P~/2 , x/2 po } . T h e n f r o m ( 2 6 ) - ( 2 9 ) we see that
]A1/Zu(t)l ~ P l / 2
for t >__tl/2,
and
t91/2 -~ MI/2 L9/2. So t h e o r e m 1 is true for s = 1 / 2 . If s __ 3 / 4 , we s u b s t i t u t e s - 1 / 2 for s in (18), o b t a i n i n g
dlAS-l/2ul 2 dt + 2]Asul2 - 2 l A S - 3 / 4 u l 2 - 2Cs_l/2lull/2lA1/4ujl/2 lAS-1/2ul [AS-1/4u] <_0. F r o m the a b o v e i n e q u a l i t y we get for t _> L - 1/4
ft t+llAsul2 ds -
(29)
X. Liu / Kuramoto-Sit:ashinsky equation
142
Hence there exists a t' ~ [t, t + 1] such that IAS( t')l <_ r/~,
(30)
where Tls = Ps-3/4 + (3Cs 1/2)xl/2;tPOPl/2)xl/4/tPs I/2Ps-l/4)xl/2
q'-
( 3 / 2 ) 1/2 Ps-l/2.
(31)
Now we take t, = t s_ ~/4 + 1 = t o + 4s and p, = max{p~, r/~}. From (26), (27), (30) and (31) for t > t~ = t 0 + 4s we get MSu(t)l
and Ps = Ms L4s +5/2.
[]
We have proved theorem 1.
3. Gevrey class regularity of the solutions
Now we show the Gevrey class regularity of the solutions. We define a function T , ( r ) : [0, ~) ~ E by
T,(r)
=
( 16X/~C2 2 ~
+ 12~/2
r
)-I
(32)
T h e o r e m 2. Let 0 _< a _< 1/4, s >_ 0 and assume u o e D(A'), then u(t): t--+ e ra`" A ' u ( r e i°) is analytic on (0, T , ([A*uol)) (in the region A([A'uoq) , where A(IA'uo[ ) is given by (48) below), and IA s e rA" u( r ei°)] 2 _< 2[A~uo[ 2,
(33)
for r e i° E A([ASuo[). If, moreover, there exists a constant M, such that IA~u(t)l<_M
Vt>O,
(34)
then u ( t ) : t ~ A" eT*(M)A" u ( r e i°) is analytic on ( T , ( M ) , o o )
(in the region A(M), where A(M) is given by (50) below), and
le 7*(M)A'' ASu(05)l <_ ~ / 2 M ,
for 05 ~ A(M).
(35)
143
X. Liu / Kuramoto-Sivashinsky equation
Proof. As in refs. [11, 12] we consider complexified spaces and functions. Then (1) can be written as du
d---( + Au - A1/2u + B( u, u) = 0.
(36)
For ~ = r e i°, taking the scalar product of (36) with A2Su(r e i°) in D(CA"), multiplying it by e i°, and taking the real part of the result, we have
-~+Au-A1/4u+B(u,u),A2"u
=0. r
Now
R e[e [ i°[du [-~-A2Su)~]=Re '
( erA°A~dU -d-r' CA°ASu ( re i° )) = R e ( d ( C A " A s u ) dr
_12 dr INs e rAau 12 - (erA'~ AS+au,erA" ASu) > "~
e rA" A S + a u , erA '~ A S u )
[A s +aU ]rlASulr "
Therefore 1 d
2 dr
IASul2r - IA'+~ul~lASulr + cos OIA'+l/2ul 2 < Re[ei°(Al/2u,A2"u)r ] - Re[eiO(B(u,u),A2,U)r].
(37)
Since 0 < a < 1/4 we get
iA.+~ul~lA.ulr<_ iAs+l/2u 2,~-. r ~Zl lg 2-2a r <
(4o)o ,1o,
IA'+l/2UlZr+(1-a ) ~
[A'ul 2,
(38)
and we have
Re[ei°( A'/2u,A2"u)r] < [A'+l/4ul2 ~ IA'ulrlAS+l/2u[r ~_~ -~[As+l/2U[2r
-~- co~ASul
2.
(39)
By lemma 1, we find that
I( B(u,u),A2Su),]
<-~Cs A~/2.
< ~IASul2
--
1
iA.+l/2ulr < cos 0 As+I/2,,]2 --
8
. . . .
2c 2
"t- A2s+l/4cosoMSUl 4.
(40)
X. Liu /Kuramoto-Sit,ashinsky equation
144
We put (38)-(40) into (37) to obtain:
d l A ' u 12r+ cos 0 IA ' + 1/2u 12r <2 (1-a)~]
{ as ]~/('-")
2 ]
2
4c2,
+ -cosO ]lASulr + /~21s+l/4cosO
IA'ul 4.
(41)
Now let y ( r ) = IASu(rei°)12r= [A~erA"u(rei°)12 F r o m (41) we have
dy < b y ( y + c ) dr
(42)
where
b-
~.2s+ 1/4 COS 0 '
c=2
1-c~
~
+
co~s-
]
A2S+I/4 COS 0
4c~
We try to find a bound T . on r such that y(r)<_2yl,
for r e [ 0 , ~ r . l .
(43)
We solve (42) and get the inequality
y(r) Yo eh,,~. y(r) +c - Yo +C Since z / ( z + c) increases as z increases in (0, ~), we only need YO ebcr < 2Yo Yo + c - 2y o + c "
Thus, 1
r _< ~
log
2y o + c
because for z ~ (0, 1), log(l+z)
> z/2.
'
(44)
145
X. Liu / Kuramoto-Sivashinsky equation
Hence we only need 1
c
1
r < bc 2(2y o + c)
(45)
2b(2y o + c)"
We restrict to cos 0 _> f 2 / 2 . Then 2[{ 1 ]"/('-~) 2 ] /~21S+1/4 COS 0 _ 2 [ ( 1 ) 1/3 2 c < [~ c--o-if-O] + co---70] 4c 2 < ~ + co--7
A s+1/4cOS 0
4c 2
< 2[(COS O) 2/3 + 2]h'2s+l/._.....~ 4 3A2s+1/4 -4C 2 < C - 2- - ' ~
(46)
Also 4c 2 4Vt2c 2 < - b = /~2s+1/4COS 0 -~21s+1/4 "
(47)
From (45)-(47), we get
(
[ 8v~-c 2 312,+ ,/4 ]]-1 r< ~ 2Yo+ ~-7Cs2 ]] So if 0 < r < T , ( l A ' u o l ) and cos 0 > V~-/2, then (43) is true, where
r , OA'uol) = [( 16v/2c2lA , .o 2 + 12v~-)-' That is, for r e i° in the region A(IA'uol): A(IA'uo I) = {r ei°: r < T , (IA'u ol), cos 0 > v~-/2}
(48)
we have IA' e rA" u( r ei°)12 < 21A'uol 2. If (34) is true, then we repeat the argument above around each t o > 0 to get [erA"ASu(re i° + to)] < v~M,
for r e i° E A(lA'u(to)l). In particular, we have ler*(M)A" A ' u ( cb)l < v ~ M ,
(49)
146
)d Liu / Kuramoto-Sivashinsky equation
where
2
[ 16~2c2
T,(M)=I~M
+12v/2
) 1
,
and (b ~ A ( M ) which consists of the strip Re~b>__T,(M),
IIm~bl<(v~/Z)T,(M).
(50) []
We have proved theorem 2. Now from theorem 1 and theorem 2, we easily prove
Corollary 1. Let 0 < ~ < 1/4, s >_ 0 and 4s be an integer, and assume u 0 • H. Then u ( t ) - ~ A s e r*(o')A'~ u( r e i°)
is analytic on ( T ,
(Ps) + t~, oo), and
[ASel*(°')AC'u(4))[ < v~p,
V4~A(ps)
+ts,
(51)
where A(O,) + t, = {4~: Re4~ >- T , ( p s ) + 6 , [Im~b[ < T , ( p , ) / v ~ } .
Ps, t, are the same as in theorem 1, and T,(p,) is given by (32).
4. Approximate inertial manifolds Now we consider approximate inertial manifolds. We show all t h e ' trajectories approach some exponentially small neighborhood of some simple manifolds (cf. refs. [5, 14, 20]).
Theorem 3. Let 0 < a < 1/4, s > 0, and ]ASu(t)] < M
Vt>0.
(52)
Then we have
IASu(qb)lT, < v~M
V& • A ( M ) ,
(53t
X. Liu / Kuramoto-Sivashinsky equation
147
and if t > (1 + 1 / ~ / 2 ) T , , then
A , dU(t) 2M dt T. < T . '
(54)
IAS+lu(t)lr. <_K,( M ) ,
(55)
As+ldU(t) dt
T, -<
16K1(M) T,
(56)
where T , = T , ( M ) , A(M) are the same as before and 8M
KI( M ) = ~
3c4/3M7/3
+2M+
1~4s/3_1/6
.
Pro@ Eq. (53) follows from theorem 2. Applying Cauchy's integral formula to the circle F centered at any point on the real axis t > (1 + 1 / ~ - ) T , with radius T , / v ~ , we have u(t) =
1__1__[u(~b) drk. 2rri Jrgb - t
From this and (53) and using theorem 1, we get for t > (1 + 1 / v ~ - ) T ,
A s ~ t t)
< 2M . r, - T ,
Thus, (54) is true. Applying lemma 1, (52), (53) and (54) we establish (55) and (56).
[]
From theorem 3 and theorem 1, we get
Corollary 2. Let 0 _ 0, and let 4s be an integer. Assume u o ~ H. Then we have IA'u(4~)IT.(,,)--< v~-p, V4~ ~ A ( p , ) + ts, and if t > (1 + 1 / v ~ ) T , ( p s )
A~ dUd(~)
+ t,, then
2p~
r.(.~) < T , ( p s ) ,
IA~+lu( t)IT.(p,) < Kl(Ps), 16Kl(ps)
As+l d U ( t ) T.(p,)
dt
<
(57)
r,(pA
(58) (59) (60)
X. Liu / Kuramoto-Sit,ashinsky equation
148
A(p,), p,, t~, T,(p,)are the same as before and
where
4/3 7/3
8p~
gl(Ps)
3c,
T,(p,~ + 2 p , +
Ps
t~4s/3-1/6"
F r o m corollary 2, we easily get
Corollary 3. U n d e r the same assumptions as corollary 2, then there exists a t~ = t o + 4s such that if 1/v/2)T.(p~)+ t,, we have
t > (1 +
[q(t)l_<
K'(Ps) a , + ~ e - T*{p')~';'+'
(61)
"'m + 1
~ t t)
<-
e 7.~o,,~';,,+,
16K'(Ps)
(62)
Remark. T h e case s = 0 or 1 / 4 , T.(ps) = 0 was considered in ref. [14]. We see that every solution of the K u r a m o t o - S i v a s h i n s k y equation a p p r o a c h e s some exponentially small n e i g h b o r h o o d of the linear manifold PmH. If the solution u(t) is on the universal attractor, then (61), (62) are true for all t ~ N. Next, as in refs. [14, 20] we analyze the steady approximate inertial manifold stationary solutions of (1)), where ~ ' = graph(q~); q~: /3 ~ Q H such that
A~(p)-A1/2ff)(p)+QB(p+q~(p),p+clg(p))
=0,
p~/3.
de" (which contains all the
(63)
For fixed s > 1 / 4 , r, a > 0, we define /3 = {p ~ P H : /3 ± = {q ~ Q H :
[A'pI,<_ V~Ps}, [ASql, <_f2p,}.
Now we show the existence of q~.
Theorem 4. (cf. refs. [10, 14, 20]) If m satisfies Am+l >- - max{4r2, r 21 /. 2
2/
(64)
P,],
where
rl
=
~-1/4~-s+1/8 2 V~ps+8Cs-I/4A'm+IAI Ps,
and r2
=
1 + 4V~C,_ l / 4 "~m- -+l /l4") tl
--s
+ 1/8Ps,
then there exists a unique m a p p i n g @: /3 ~ Q H satisfying (63). Its graph ~/" is a C-analytic manifold.
X. Liu / Kuramoto- Sivashinsky equation
149
Moreover IWq:'( p)l~- < Am 1/2rl Vp
Proof. (See also refs. [10, 14, 20].) For p ~/3 define Tp(q): Tp(q)=A-1/2q-A-1QB(p+q,p+q)
Vq~/3±
By lemma 1, we find that
IA'Zp( q)[, < Aml+/2rl,
(65)
and
A, OT~q)~7
(66)
+1
We see Tp is a contraction map if (64) is true. Now let the fixed point of Tp be @(p), then we have [] proved theorem 4 and the graph of q~ is an analytic manifold (see refs. [7, 10]). Now we define • 0 ( p ) = 0,
qbk(P)=Tp(qgk_l(p) ) Vp~fl
and
k=l,2 .....
From theorem 4, we get ]A'(@(p) - @k(P))]~ -< 2(rzA~nl+/z)klASdPl(P)]~,
(67)
since q~l(P) =
-A-1QB(p,P) •
So by lemma 1 for s > 1/4, [ASqbl( P)]~ -<
]As- 1QB( P, P )lr < Am3+/14lZS-1~4an( p, p)]~
-<" 'Jtm-3/4r, + l ~s-l/41/1~ 1/2 ~"
- 3 / 4 A -1' + l / S l h ~ p l Z ~ . IAI/4pIIfE IASpl, ~-~Cs--1/4 A rn--1
(68)
Combining (67) and (68), we have IA'(@(P) -
@k(P))[~ -< 2Cs-1/4AlS+l/8Am3/14(r2Am
1/2+1) klA ~(p)[,,2
(69)
X. Liu / Kuramoto-Sivashinsky equation
150
and from the above inequality and corollary 1, we see that if t > T , ( p s ) + t,, then
IAS(q)(P) -q)k(P))lr.(z) <4c,
l-s+l/8)L-3/4[r
1/4"1
)~
I/2 ~k r'2
" m + l k ' 2 " ' m + l J P's"
(70)
N o w we consider the distance between u(t) and . ~ = graph(q)).
Theorem 5. Let 0 < a < 1 / 4 , s > 1 / 4 , u 0 ~ H, and assume m to be large e n o u g h such that (64) and the following are true l at- 2 V ~ P , / 4 A m 7 / 8 + J' "~P -0 n l / 2 ~P1/4"'m+l l / 2 )' - 3/4 ~ 1 / 2 . A m +1/2
If t > ( 1 + 1 / v ~ ) T , ( p s ) + t
Iq(t)-q)k(P(t))
,, then 32 K ~( ps ) -,-2 -T,(paA';,+l r , ( p s ) Am+ ' e
Iq(t)--q)(p(t))l_<
I<-
(71)
- - s - 3 / 4 g[r2Am+l - - - l / 2 X) k P52~) e-T*(°')a';"+ 3 T2 K s 2 +4e,.-,/4A 1- s + l / 8 -Am+l , ( Ip(sP)s ) Am+l ',
(72)
(73)
where Ps, ts, etc. are the same as before.
Proof. By ref. [14] we know if (71) is true and t > tl/4, then IA(q(t) - q)(p(t)))l < 2 ~
.
F r o m this and corollary 3, we see (72) is true. N o w combine (72) and (70) we get (73) is true.
[]
Remark. T h e case s = 1 / 4 , T , ( p , ) = 0 was considered in ref. [14]. F r o m t h e o r e m 5 and corollary 3, we see the distance of orbits to ~," is closer than it is to Pm H by a factor of Am +1"~ A n d graph(q) 3) achieves the same estimates as graph(q)). N o t e that for large s, Ps is very large, and then there might not be any gain by taking large s (see (']2)). T h e real gain in this case is the exponential term and even here T , d e p e n d s on p,. So, in real c o m p u t a t i o n s one should probably take s relatively small, say s = 0 or s = 1 / 4 .
Acknowledgements T h e a u t h o r thanks his advisor Professor Nicholas D. Kazarinoff for valuable advice and suggestions. T h e a u t h o r also thanks the a n o n y m o u s referee for helpful suggestions.
References [1] P. Constantin and C. Foias, Navier-Stokes Equations (The University of Chicago Press, Chicago, 1988). [2] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Applied Mathematics Sciences, No. 70 (Springer, Berlin, 1989). [3] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynam. Diff. Eqs. 1 (1989) 45-73.
X. Liu / Kuramoto -Siuashinsky equation
151
[4] C. Foias, M.S. Jolly, I.G. Kevrekidis, G.R. Sell and E.S. Titi, On the computation of inertial manifolds, Phys. Lett. A 131 (1988) 433-436. [5] F. Foias, O. Manley and R. Temam, Modelling of the interaction of small and large eddies in two dimensional turbulent flows, Math. Modelling Numerical Anal. 22 (1988) 93-118. [6] C. Foias, B. Nicolaenko, G.R. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl. 67 (1988) 197-226. [7] C. Foias and J.-C. Saut, Remarques sur les equations de Navier-Stokes stationnaires, Ann. Scuola Norm. Sup. Pisa Ser. 4, 10 (1983) 169-177. [8] C. Foias, G.R. Sell and E.S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dynam. Diff. Eqs. 1 (1989) 199-224. [9] C. Foias, G.R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Diff. Eqs. 73 (1988) 309-353. [10] C. Foias and R. Temam, Remarques sur les 6quations de Navier-Stokes stationnaires et les ph6nom~nes successifs de bifurcation, Ann. Scuola Norm. Sup. Pisa Ser. 4, 5 (1978) 29-63. [11] C. Foias and R. Temam, Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl. 58 (1979) 339-368. [12] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal. 87 (1989) 359-369. [13] F. Jauberteau, C. Rosier and R. Temam, The nonlinear Galerkin method in computational fluid dynamics, Appl. Num. Math., to appear. [14] M.S. Jolly, I.G. Kevrekidis and E.S. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations, Physica D 44 (1990) 38-60. [15] M.S. Jolly, I.G. Kevrekidis and E.S. Titi, Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation, J. Dynam. Diff. Eqs. (1990), in press. [16] X. Liu, On approximate inertial manifolds for two and three dimensional turbulent flows, J. Math. Anal. Appl., to appear. [17] B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equation: Nonlinear stability and attractors, Physica D 16 (1985) 155-183. [18] R. Temam, Infinite Dimensional Dynamics Systems in Mechanics and Physics (Springer, Berlin, 1988). [19] E.S. Titi, Une vari6t6 approximante de l'attracteau universel des 6quations de Navier-Stokes, non lin6aire, de dimension finie, Compt. Rend. Acad. Sci. Paris I 307 (1988) 383-385. [20] E.S. Titi, On approximate inertial manifolds to the Navier-Stokes equations, MSI Preprint 88-119, J. Math. Anal. Appl., to appear. [21] E.S. Titi, private communication.