Time analyticity and gevrey regularity for solutions of a class of dissipative partial differential equations

Nonlinear Andysir. Theory, Printed in Great Britain.

Methods

& Applicotiom,

Vol.

16, No.

II, pp. 959-980,

1’991. 0

0362-546X/91 $3.00+ .W 1991 Pergamon Press plc

TIME ANALYTICITY AND GEVREY REGULARITY FOR SOL,UTIONS OF A CLASS OF DISSIPATIVE PARTIAL DIFFERENTIAL EQUATIONS KEITH PROMISLOW The Institute

for Applied

Mathematics

and Scientific Computing, 618 East Third Street, Bloomington, IN 47405, U.S.A.

Indiana

Univeristy,

(Received 9 March 1990; received in revised form 1 October 1990; received for publication 6 November 1990) Key words and phrases: Time analaticity, Ginzburg-Landau,

Navier-Stokes,

inertial manifolds, dissipative Cahn-Hilliard, Gevrey regularity.

PDEs,

reaction

diffusion,

INTRODUCTION THIS PAPER

is concerned with the time analyticity for the solutions of a class of dissipative partial differential equations, extending previous results obtained for the Navier-Stokes equations to include a general abstract framework (see Foias and Temam [6,7], and Foias et al. [9]). Moreover, it is shown that these solutions have values in the domain of the principle linear operator, and for a slightly more restrictive abstract setting, in the Gevrey class of functions, which is a subset of em. In particular, it is shown that the following dynamical systems are included in this abstract setting: the Ginzburg-Landau equation in space dimensions 1, 2, and 3, the Cahn-Hilliard equation in space dimensions 1 through 5, and a general reactiondiffusion framework with a polynomial nonlinearity, which is shown to contain the HodgkinHuxley equations for nerve impulse transmission, an equation modeling superconductivity in fluids, the Belousov-Zhabontinsky reaction in chemical dynamics, and an equation from combustion theory. Of course we also recover the previous conclusions for the Navier-Stokes equations in space dimensions 2 and 3. The main result, theorem 1.1, states that if the nonlinear term of the equation under consideration satisfies certain bounds in terms of the dissipative linear operator and if the solution is approximately bounded at a given time t, then it can be extended as a time analytic function in a complex domain of the form

where 0, and T,depend on the solution only through its norm at time t. Hence, if the solution is known to be uniformly bounded on the positive real axis, then it has a time analytic extension to a pencil-shaped domain in the complex plane with width d > 0,centered around the positive real axis (see Fig. 2). On this domain Cauchy’s Formula may be used to obtain estimates on the time derivatives of the solution in terms of the bounds on the solution and the width d of the strip. Such estimates are very useful in establishing the existence and refining the order of approximate inertial manifolds. An additional result, theorem 3.1, strengthens theorem 1.1 in a slightly more restrictive setting, showing that these solutions are time analytic with values in the Gevrey class of functions. Moreover, we may deduce from this the exponential decay of the Fourier coefficients of the solution (for an application of this see [S]). 959

960

K. PROMISLOW

The foundation of the proof is based upon that in [6]. We complexify the spaces on which the equations act, and impliment a complex-time Galerkin approximation. These approximate equations are ODES, and hence we have an analyticity result in domains where the solution is bounded. Using uniform a priori bounds in the various norms, we obtain a common domain of analyticity and are able to pass to the limit, preserving the time analyticity with values in these norms. The paper is divided into three parts, the first presents the abstract equation and the proof of the analyticity result, the second contains applications to the equations mentioned above, and the third contains the strenthening of the time analyticity result to include the Gevrey class regularity. 1. TIME

1.1.

ANALYTICITY

IN THE

GENERAL

CASE

The equation and its environment

Let H be an infinite dimensional real Hilbert Space with scalar product (. , *) and norm (- I. Further we are given an unbounded, positive, self-adjoint, closed linear operator A with operator domain D(A) C H and an inverse A-’ compact in H; also we are given a nonlinear R : D(A) + H which is analytic on finite dimensional subspaces of D(A) and has analytic extensions to the complexifications of these subspaces. Due to the compactness of A-’ in H, there exists a basis {wj] of H such that the Wjs are eigenvectors of A. That is, there exists an orthonormal basis of H consisting of the eigenvectors ( Wj ~ of A with eigenvalues nj E IR, such that AWj = IWjl

=

~jWj

1

and the Aj + l tooas j -+ 00. Also, we define AS, the power of A, for any s E [R; AS maps D(A”) onto H; IA”* 1is a Hilbert norm on D(A’). In particular we define V = D(A”‘) and denote its norm and inner product by 11.(1 and (( *, a)) respectively. We will denote by 1. I 1 the norm (11. II2 + 1. 12)1’2. For k, a positive integer, we define H = @:H, V = @f V, and D(A) = @f D(A) and give them the usual Hilbert norms and inner products for a direct sum. For a vector u E H we define Au = (Au,, . . . , Au,). We consider the following abstract initial value problem involving the vector function u: IR, -+ D(A), u’(t) + DAu(t) + R(u(t)) = f(t), te R+, (1.1) u(O) = %I,

t = 0,

(1.2)

where f: R, + H is analytic and D = (dii)kxk is a real, positive definite matrix. Let Hc, Vc, and D(A)c denote the complexifications of H, V, and D(A) respectively. Recall that H, is a subset of H @ H over C with a typical element being h, + ih2, multiplication by a complex constant performed in the natural manner, A(hl + ih2) = Ah, + iAh,, and with inner product given by (4

+ ih,, g, + &I = (h, a) + (h2, a) + Ub,

gl) - (4, gdl.

Once again A-’ is compact in Hc and in fact, using the vectors ( wjj that span H, it is easy to construct an orthonormal basis (wj) of He: consisting of eigenvectors of A.

Time analyticity and Gevrey regularity

961

1.2. Time analyticity We assume that there exist 1 < y < co, K > 0, and a function E > 0, R satisfies

C E e(O, 00: R,) such that for all

I(R(u), Au + u)l % E[Au~~+ C(&)(uI:y + K, and for every compact

4 R(u)

weakly in L2(X; H)

when U, --f u uniformly from X into D(A) endowed exists A4 > 0 such that f satisfies Ilf(0ll* theorem

(1.3a)

set X c C, R(U

The following

vu E D(A),,

contains

with the weak topology,

vte

5 MY

(1.3b) and that there

I?,.

(1.4)

the main result of this paper.

THEOREM 1 .I. Let u be a solution of (1.1) and (1.2) above, then there exists 19~satisfying l6,( I 7r/4 and a function T, E C(R+, R,) such that if lu(O)], is finite then u has a D(A), valued analytic extension to a complex region of the form A(lu(O)(,) (N.B. see theorem uniformly bounded

= {z = seie I/O/ I 0, and 0 I s 5 T,(Ju(O)),)).

2.1, theorem 2.2, theorem 2.3, and theorem 2.4.) Moreover if lu(t)[, by B on an interval (a, b) then this region can be extended to include A =

For all compact

K contained

sup I(d54/dzk)(z) ZEK

u t + A(B). tE@,b)

inside this region

the following

I 1 I 2”2(2/d)k(k!)(

sup [Au(z)/ I

ZEK

r,(K)

inequalities

1 + lU,l~>““,

hold,

d = dist(K, aA)

< 00,

(1.5) (1.6)

sup jA(dku/dzk)(z)l I 2”(k!)[d(K, aA(u,))]-kT2(K’),

(1.7)

ZEK

where K’ = (z E A(uO) 1d(z, aA(

is

2 id(K, aA(u

Proof. Consider the Galerkin approximation in complexified time; i.e. the complex differential system in H, = (Cw, + .. - + Cw,). Let P,,, be the projection onto H, . We look for a solution u,(z)

= E g;(z)w;;

g,:C

+ C

i=l

of

(au,/az + DAU, + ~(2.4,) -f(t),~) U,(O) = P*u,.

= 0,

VVEH,,,,

(1.8) (1.9)

Since A has a particularly simple form on H, , i.e. AU,(Z) = C ligi(z)wi, (1.8)-(1.9) reduce to a system of m ordinary differential equations, and so by Cauchy-Kovalevski’s theorem (see Folland [lo] for example) it possesses a unique analytic solution U, defined in a complex neighborhood, r, of the origin. Also, U, reduces to the real time Galerkin approximation to (1 . l)-( 1.2) when restricted to the real axis.

962

K. PROM~SLOW

a priori estimates,

Now, to obtain

(au/az,

we take u = Au,

Au) + (&4/&,

u) + (DAU, Au) + (DA,

+ (R(u), Au + 2.4)- (f(t), where,

for convenience,

we have dropped

+ u, in (1.8), yielding u)

Au + 2.4)= 0,

(1.10)

m on u. Integration

the subscript

by parts produces

((&4/8z, u)) + (&4/&z, u) + (DAU, Au) + ((Du, 2.4)) + (R(u), Au + 24) - (f(t),

Au + u) = 0.

(1.11)

Now, write z = s eis, viewing 0 as fixed between -n/2 and n/2, multiply equation (1.11) by eie and take the real part. Since D is a real, positive definite matrix it follows that there exists an cyO2 0 such that Re(Dz, z) L CY~~Z[~ for all z E C=“. Using this fact we obtain t(d/~)(ll#

+ 1~1~) + a0

cos(W142 + 11412) 5 IsW)l 11~11,(1~42 + 11412) + I(W),Au + 4 + Iu-w,~~ + @I.

Using

(1.3a) with E = CY~ cos(8)/8

+(d/&)l&

and (1.4) on the last two terms we obtain

+ (.Y,cos(~)(/Au~~ + Ilul12)5 I(DII,Isin(0)J(IAu12

+ llul12) + (a,/8)c~@)lAu)~

+ K + c(e)(@ Now,

further

restrict

+ M(IAuJ + lul).

(1.12)

n/4)

(1.13)

0 so that (Yec0s(e) L JID ll,Isin(B)).

For instance 161 I Min(ltan-‘(cr,/4110(l,)I, Then,

using Young’s

inequality

+(d/ds)lul:. 5

+ a,c0s(e)()Au(2

(a,/4)

cos(e)[Ih@

+ 4M2/(cXo which simplifies

on the last term in (1.12), we obtain

+

cos(e)) +

of the form

llul12)

)l2#1 +

(a,/8)

c0s(e)\A@

+

K + c(e)l@

(cy0/8) cos(8)~Au12 + MlU)2,

to

(d/&)lul:

+ a,cos(e)()Au12

With B restricted as in (1.13) we obtain that is (1.14) is of the form (d/d.s)lu(se’e)l: Momentarily a differential

+

an equation

dropping inequality

+ ll?.#) I c(e)\@

bounds

independent

This clearly indicates

c,). By integrating

this equation

- (y - l)c,.s)“(‘-~)

the existence

K.

of f3 on the coefficients

the term c,(lAu12 + Ilulj2), and writing of the form

0 < y(s) I (y(o)‘-y

+

+ c,(lAu12 + l)ul12) I c2 + c3)u(sei”)lfY.

Y’(S) 4 C‘!YY@) where c, = Maxlc,,

+ 4hz2/(a,c0s(e))

of a constant

(1.14) in (1.14);

(1.15)

y(s) = 1 + Iti(s eie)l: we obtain

for s 1 0 we find for 0 5 s < ~(o)‘-~[(y

Tr = ~(0)‘-~(1

- (t)Y-‘)/((y

- l)cJ-i. - 1)~) depending

Time analyticity

Fig. 1. The local domain

and Gevrey

of analyticity

regularity

of a solution

963

of

(1.1) and (1.2).

only on [u,,l1 and independent of m, such that I~,(se”)j,

for all 0 5 s 5 T,(lu,J,)

5 2(1 + Iu,(O)(:)“~ 5 2(1 + lu,l~)“’

for all 0 satisfying (1.13).

(1.16)

Here we have reintroduced the subscript m from (1.8). Hence the theory of existence of solutions of ODES implies that U, extends to an analytic solution of (1.8) in a region containing A(lu(O)l,) = (z = seie) 0 < s < ~r(lu~lr) and B satisfies (1.13)). By design

SUPl%ml, 5 20 + l%lf>“‘,

z E Nld,).

Moreover, by Cauchy’s formula (dkU,/dZk)(Z) = (k!)/(27ri) i’Iz--gt= d/2 where d = d(z, aA(

in the above. Therefore kdkGdzk)Ii

In particular,

u,(l;l)(rl - z)-(~+‘) drl

5 (2/d)%!)

s”d;lp, ,b,Azh. %1

for K a compact subset of A(q,), we have sup [(dku,/dzk)(z)I ZEK

1 5 2”2(2/d)k(k!)(l

(1.17)

+ l~~l:)“~

where d = d(K, aA(u Now, to obtain apriori bounds on U, in D(A) we observe that for any compact subset K of A(u,) and for any z = seie E K, we have I(d/ds)(u,(seie)lfl

= (2((du,/dz,

u,)) + 2(du,/dz,

u,)l I 21du,/dzl,(u,(, I 4(2/d)(l

+ lu&)

(1.18)

where d = d(K, aA(luOl r)). Inserting this into (1.14) and dropping the I/u,))~ term yields c11A~,J2 I c, + c,(2(1 + lu,l:))

+ 4(2/d)(l

+ lz&.

(1.19)

Thus for any compact set Kc A(lu,l r) we have U, uniformly bounded (i.e. independent of m)

964

K.

PROMISLOW

in L”(K, D(A)), sup \&,&)I

LEK

T, = T,(K).

5 T, < ~0,

By Cauchy’s formula we obtain, for any compact K such that d(K, ?IA(~u,\,)) 1 d > 0, A(d$,Jd?)(z)

= (k!)/(2ni)

5 Iz-_?I=d/2

Nn(r)/(~

for all z E K.

- z)~+’ drl,

Denoting by K’ the compact set {z E A(u,)[d(z, ~A(~u,~,)) 2 *d(K, aA((u,I,))) > K, we obtain INd%Jdz%)l

5 2”(k!)V(K

~NOl-k

y,

sup IA(dku,/dzk)(z)I

s 2”(k!)[d(K,

~A(u,))]-~T~(K’).

Anne (1.20)

ZEK

We now proceed with the passage to the limit as m + 43. Since the functions U, : C -+ D(A) are uniformly bounded on the set A( Iq,( J and are analytic there we can apply the vector version of the classical Monte1 theorem to the sequence (u,) and extract a subsequence which converges uniformly in the D(A) norm on every compact subset of A(@ to a function u* which is D(A) analytic on A(uO). Moreover, since u,) co,r) is the corresponding real Galerkin approximation and u,I(,,~ + U, the solution of (1 .l) and (1.2), we must have that U* is just an analytic extension of u to a domain containing A(uJ. Thus U* is the unique analytic extension of u to d(u,), and hence we will denote u* by U. So any converging subsequence of (u,) converges to U, and hence the whole sequence converges to U. Clearly we have that SUP k4z)l

ZEK

5

T,(K),

for any compact K c A(u,).

Similarly (1.17) and (1.20) imply that dku,/dzk -+ dku/dzk,

in D(A) uniformly on compact subsets of A(uJ, and that u also satisfies (1.5)-(1.7). These uniform convergence results, together with (1.3b), easily yield that u is a solution of (1.1) and (1.2).

Fig. 2. The domain

of analyticity

of a solution

bounded

in V on R,

Time analyticity

and Gevrey

regularity

965

If (u] 1 is bounded by R on the real interval ((Y,fl), then the arguments made for t = 0 can be repeated for any point t E (a, p), yielding u: C -+ D(A), is a D(A) valued time analytic function on the open set tvO” + A(lu(t)],)] 3 It;8jlt

+ A(R)].

Remarks. 1. Under the same hypotheses as theorem 1.1, for 0 < t I ($)T’( (u,,]1), we have the following relations: @ku/dtk)(t))l

IA(d%/dtk)(t)]

I 2kk!c-k(2i’2)(1

+ lu,lt)““,

5 2kk!c-k(c2 + c,(2(1 + lu,lf>)’ + 4(4/c)(l

(1.21) + ~u,~:)]“~/c,,

(1.22)

where here c = (tm/(l + m)2)1’2 is the distance of t to the aA, m = (r,,/41(D)l, is the slope of the line that forms the boundary, and c 1, c2, and y are as in equation (1.9). These estimates are obtained by taking the compact set K in (1.5) and (1.7) to be the subset [E, ($)T’(lu,],)] of the real line and explicitly calculating d = dist(k, aA) and T,(K’). 2. In the case that u is not smooth at t = 0, the derivatives of u tend to infinity in the H-norm as t -+ 0. Equations (1.21) and (1.22) give bounds on this blow-up. Similar results have been obtained using different methods by Iooss and Brezis (see [15] and [2]). 2. APPLICATIONS

We now consider applications of theorem 1.1 to various dissipative partial differential equations. In particular we consider reaction-diffusion, Ginzburg-Landau, and Cahn-Hilliard equations, with several specific examples from reaction-diffusion; for the sake of completeness we include results for the Navier-Stokes problem which were previously obtained in [6,7, 17,291. The main obstacle to applying theorem 1.1 is proving the estimate (1.3a) on the nonlinear term R. Since, in the following examples, R restricted to &D(A) is a polynomial map C” --t C”; it is clearly analytic. The verification of (1.3b) imposes weaker conditions on R than (1.3a) does, and is omitted. 2.1. Reaction-diffusion

equations

As our first example we consider the reaction-diffusion equations with a polynomial nonlinearity; these equations govern a variety of phenomena, including transmission of messages by nerve endings, modeling of superfluids, and some combustion problems, as well as Belousov-Zhabotinsky reactions in chemical dynamics. Given Q, an open bounded subset of R”, our model reaction-diffusion equation reads:

adat - D AU + g(u) = 0 u(0) = zig

(2.1) (2.2)

where u = (u,, . . . , uk) is a vector-valued function definded on Q x IR, , D is a positive diagonal matrix with diagonal entries d,, . . . , dk, and g: Rk + Rk is a function on R” whose ith component is a polynomial of degree r 2 2 of the form gj(x) =

c c;xp1 *. * xp. lol(dT

(2.3)

966

K. PROMISLOW

We supplement this equation with one of the following boundary conditions: U(X,t) = 0,

for all x on %2, and all t > 0,

(2.4a)

u( *, t) is C&periodic, for each t 2 0, where Q = (0, L)“, (&/&r)(x, t) = 0,

(2.4b)

for all x on %2, and all t > 0.

(2.4~)

First observe that in the case of boundary conditions (2.4a, b, or c) we have, respectively, H = JP(sz)

L2(Q)

L2(Q)

V = H&2)

H;,,(Q)

H’(n)

H&,(Q)

1~ E ~~(a)

D(A) = H2(a)

r-~H&2)

1au/an=

0).

To put (2.1) in the form of (1.1) we have only to write A = -A and R(u) = g(u). For certain polynomials g (see Temam [28] and Marion [16]) the regularity of this problem is well known, as is the fact that A is a positive, self-adjoint, unbounded linear operator with a compact inverse. It remains to show that R: D(A) --t H and that (1.3) and (1.4) are satisfied for this equation. Let u E D(A) and examine (R(u)1

But we can find constants ci and c2, independent c I& /#a 10115r

of i, such that

5 c,J@’ + c,.

So we conclude that (R(u)) 5 C(Iz&‘)r + clszl. Now we must bound the rth power of the L2’ norm of u using the HZ norm. This argument is carried out below in the proof that R satisfies (1.3) and will not be repeated here. At this point we observe that (1.4) is trivially verified with A4 = 0 sincef = 0 in this equation. To verify (1.3a) we write u = (Ui, . . . , uk) and notice that

Ium),Au + u)l

5

IS IOL FCY1~~~$“~’

.a* Ugk)(-AUi + ui) dx m

Now we bring the absolute values inside the integrals and bound (uilai with (uIui to obtain

But we can find constants ci and c2, independent

of i, such that

c lc;llz4’a’ 5 CiJUlr + c2. /aIsr

Time analyticity and Gevrey regularity

967

So we conclude that l(Nu),Au

+ 41 5

c Isirk

+

c~(u~~/AL+[+ c,[AuJ dx 1

5 Cl(lUl~~~)‘IL4Ul + c,Inl”+lul

(c~($+~ + c4)cLy s

+ C&4(L’+‘)r+1 + c&l.

(2.5)

Now we must bound the rth power of the L” norm of u and the (r + l)th power of the L’+’ norm of u using the H2 norm. To simplify these calculations we consider different cases, corresponding to different values of the space dimension n. Case n = 1 or 2. In this case we have that H’ is continuously This gives us an inequality of the form I&

5

embedded in Lp for any p 2 1.

+&‘.

(2.6)

Using (2.6) in (2.5), and then Young’s inequality in the next line, we obtain I(N~),~~

+ 4

5 c&,,l41a44

+ c21w2L4~l

+

C3(C,+I14J+1 + cml

5 &(AU)2 + c(1/&)((Iu~,)2’ + PO + c(14,)r+1

+ c&d.

Thus we may conclude that in space dimension 1 or 2, (1.3) holds with y = r where r may be any positive integer. Case n 2 3. In this case we prove the following lemma. LEMMA 2.1. There exist 0 and t satisfying 1 > T, o L 0 and rg < 1 and (r + 1)r < 2, such that the following inequalities are valid for all u E D(A) in space dimension n 2 3,

(Iz~I~Z~)~~ c(r, n, Cl)(JAuJ2+ Iu(~)~~‘~()UI~)~(‘-~)

if 1 I r < (n + 2)/(n - 2)

(2.7)

(Iu/~~+I)~+~ 5 c(r, n, Cl)(JAu12+ (uJ~)(~+~)~‘~(~uI~)(‘+~)(‘-‘) if 1 I r < (n + 6)/(n - 2). (2.8) Proof. First we attempt to embed H1+LT into L2’; the Sobolev embedding theorems show that this is possible when 1/(2r) 2 l/2 - (1 + o)/n (2.9) 2r 5 2n/(n - 2(1 +a)). We now observe that H1+uis the interpolate of H’ and Hz, and that the norm (1~~1~ + /u(‘)~‘~ is equivalent to the usual norm of H2. So we obtain I&lf~

I c(Jf4~j&0((uJH~)1-~

(I&‘+=)r I C(JAUJ2+ (U)2)ro’2((U(H’)r(l-u). So to have the exponent of /~u( to be less than one, we must have ro < 1. Using this in (2.9) we obtain 2r < 2n/(n - 2(1 + l/r)) r < (n + 2)/(n - 2). Hence (2.7) is verified since the H1+O norm dominates the L2’ norm for such r.

K. PROMISLOW

968

For (2.8) we attempt to embed H1+’ into L’+‘; the Sobolev embedding theorems show that this is possible when l/(r + 1) 1 l/2 - (1 + r)/n (2.10)

r + 1 I 2n/(n - 2(1 + r)).

We now observe that H’+‘is the interpolate of H’ and Hz, and that the norm (IAu)~ + 1~1~)~‘~ is equivalent to the usual norm on H2. So we obtain ((UIH’+)r+i 5 C(jAUJ2 + IU12)(r+l)r’2(lUIN,)(r+l)(l-T). Here we want the exponent of \Aul to be less than 2, so we take r satisfying (r + 1)~ < 2, and insert this into (2.10) to obtain r + 1 < 2n/(n - 2(1 + 2/(r + 1))) r < (n + 6)/(n - 2). So for such r we have the inequality in (2.8), owing to the embedding of H1+’ into L’+l. Using this lemma in (2.5) yields 1(R(u),Au

+ u)l 5 c(lAu12 + )u)~)‘~‘~(~u(~)~(‘-~)~Au~ + c,~Q\“~)Au( + c(lAu12 + ~~1~)(‘+‘)“~((~l~)(‘+l)(~-~) + c&J(

I(R(u),Au

+ u)( 5 c/2((AulrS+’ + j~(‘~)(I~jr)~(~-~)+ c,\Ql”“lA~\ + c(lAul(‘+‘)’ + (uI(‘+‘)‘)((uI~)(‘+‘)(‘-~)+ c,lCIl.

Since the power of each IAul term is less than 2, we may apply Young’s inequality to the terms containing (Au1 to obtain an inequality of the form (1.3). We summarize these results in the following theorem. THEOREM2.1. In space dimension 1 or 2 there exists a &, satisfying (@,I s n/4 and a function TI E C?(R+, I?,), depending on g and the space dimension, such that if for any t, (u(t)( 1 is finite, then u has a D(A), valued analytic extension to a complex region of the form A = t + A(lu(t)l,) and (1.5)-( 1.7) hold on A. In space dimension 3 the result holds so long as T, the degree of g, is equal to 1,2, 3, or 4. In space dimension 4 or 5, it is sufficient to restrict r to be 1 or 2 in order to obtain this result. We now describe some specific examples of reaction-diffusion

equations.

Example 2.1. Hodgkin-Huxley equation. This system, as proposed by Hodgkin and Huxley [13], models the nerve impulse transmission. Here n = 1 or 2 and Q = (0, L)“, the equation reads: c?u,/& = d, Au, - g(ui, u2, u3, u,), ch,/at

= d2 Au, - k,(u,)(h,(u,)

&,/&

= d3 Au, - k2(ul)(h2(ul) - u,),

c%,/at = d4 Au, - k,(u,)(h,(u,)

- u2),

- u,),

Time analyticity

and Gevrey

regularity

969

where g(u) = -y1u~u3(6, - ul) - yzui(62 - ui) - ~~(6~ - u,) and 6, > a3 > 0 > &. The functions ki > 0 and the functions 1 > hi > 0 for i = 1, 2, 3 are usually assumed in c”(Q) but we may take them to be polynomials without loss of generality. Also, in the original model, d2 = dJ = d4 = 0; we use the model proposed by Smoller [26] and assume di > 0. In this model u1 represents the electric potential in the nerve, while u2, u3, u4 represent chemical concentrations that may range between 0 and 1. The existence of an absorbing set in V = N’ yields the existence of a t > 0 such that ]u(t)l 1 I A4 for all t > 7, so by theorem 1.l u is a D(A)-valued time analytic function on a domain of the form A = (z = seis 1Re(z) >

7

and (Im(z)I < d)

for some d > 0.

Example 2.2. This equation arises in the study of super-conducting bounded domain in P, n 2 1 and u = (u,, . . . , u,) a solution of

liquids. We have Q a

au/at - DAu = (1 - lu12)u, where D is a positive diagonal matrix. It is proved in [28] that this equation possesses an absorbing set in I/ c H', hence u can be uniformly bounded in the H’ norm on IR, . Consequently, theorem 1.1 shows that for n = 1, 2, or 3 this solution has a time analytic extension to a strip of uniform thickness in the complex plane which contains the positive real axis. Example 2.3. This equation models the Belousov-Zhabotinsky reactions in chemical dynamics (see Howard and Kopell [14], Hastings and Murray [12]). Here k = 3, Q c R" is open and bounded, and u = (ui, . . , , u,) satisfies au,/& &,/at

- d, Au, - CY(U~ - uIu2 + u, - pu;) = 0, - d2 Au, - (l/cy)(yu, - uluz - u2) = 0,

au,/at - d3 Au, - 6(u, - 2.43)= 0, where 01, p, y, and 6 are positive constants and the Ui denote chemical concentrations. The existence of an absorbing set in H’ (see Temam [28] and Marion [16]) allows us to apply theorem 2.1 on the positive real axis to obtain a complex, D(A)-valued, time analytic extension of u to a strip of uniform thickness about the positive real axis. Example 2.4. This last example of a reaction diffusion equation is borrowed from combustion theory. Let Sz be an open bounded subset of IR”, n 2 1. We consider the following problem

au,/at - dl Au, - (U,)Ph(U,) = 0, a24,iat - d2 Au, - (u2)p&.4J = 0, where h(s) = (s(~exp(-cu/s) and p, 01, y are positive constants. It can be shown (see Marion [16]) that if u,(O) 2 0 a.e. then ul(t) 2 0 a.e. and similarly for u2. So the nonlinear term R has an analytic extension to the complexification of H, V, and D(A). Also, the term exp( -a/s) and its derivatives of order up to two can easily be bounded by a constant on {z E G ) Re(z) > 0), so it does not effect the calculation of a priori estimates. Using the existence of absorbing sets in H’, we conclude that when p and y are positive integers, theorem 2.1 applies in space dimension 1 or 2 and also in space dimensions 3, 4, and 5 but with restrictions on the sum Y + I?.

970

K.

2.2. Ginzburg-Landau

PROMISLOW

equation

We now consider the Ginzburg-Landau equation, a Schrodinger equation with a nonlinear term, which governs the finite amplitude evolution of instability waves in a large variety of instability phenomena which are close to criticality. Various forms of the Ginzburg-Landau equation arise in hydrodynamic instability theory (see Blennerhassett [3], Moon et al. [ 18, 191, Newell and Whitehead [21], Stuart and Di Prima [27]). Given a, an open bounded subset of R”, n = 1, 2, 3; the Ginzburg-Landau equation reads: adat

-

(V +

iol)Au +

(K

+

i/l)lu12u - ~2.4= 0

(2.11)

u(0) = 2.40

(2.12)

where u is a complex-valued function defined on Q x R,, and the parameters v, (Y, K, p, y are real numbers satisfying v > 0 and K > 0. We supplement this equation with one of the following boundary conditions: u(x, t) = 0

for all x on XJ, and all t > 0

(2.13a)

U(‘, 0

is Q-periodic for each t 2 0, where Q = (0, L)”

(2.13b)

for all x on aQ and all t > 0.

(2.13~)

(au/an)(x,

t) = 0

To put (2.11) in the form of (1.1) we write u = U, + iu,, take A = -A and define the 2 matrix D and the nonlinear term R = (R,, R2) as follows D=

v _a [ CY V 1

R,(u) = (u: + U;)(‘+

- /I%) - YU,

R2(u) = (uf +

+ &)

U;)(KU,

-

x

2

yu,.

We observe that in the case of boundary conditions (2.13a, b, or c) we have, respectively, H = L2(sz>

L2(Q)

v = HA(Q)

H&,(Q)

or

H’(Q)

H&(Q)

or

(V E H2(sZ) I au/an

D(A) = H2(sZ) rl H;(Q)

L2(Q

= 01.

It is clear that A = -A is a positive, self-adjoint, closed linear operator with a compact inverse. Now it remains to show that R maps D(A) into H and that (1.3a) and (1.4) are satisfied for this equation. That R: D(A) -+ H is a consequence of the subsection 2.1; since R is a polynomial of degree 3, lemma 2.1 implies the result in space dimensions 1, 2, and 3. To verify (1.3) we observe that

I& + $11 Vdd2u+ (114+ lb4 0(Vu)d.~ Ii 5 4431v& + dvl + l)l1412.

I@(u), Au + u)l 5

Now we use the fact that H3’4 is continuously embedded in L4 for space dimension 1, 2 or 3 and that H3’4 1s . the interpolate of L2 and H’ to conclude that [z&4 5 c~z4~1’4(~U~1)?

(2.14)

Time analyticity and Gevrey regularity

971

We may conclude, using the fact that ((AU]* + )u]*)~‘*is a norm on N* equivalent to the usual one, that ]vz& I c]]ul11’4(]/lu]2 + ]ul2)3’8. (2.15) Using (2.14) and (2.15) in our estimate on /(R(U), Au + u)] we find that [(R(u),/424 + u)] I cIu11’2(Iu11)3’2(~uIJ1’2(111~2 + (Au]2)3’4 + (Iy] + l)((ul12

4aZ + .44* + ~~~-3~1~1211~112~1~1~~6 + (lrl + m412 I &4u12+ C(E)-~(IU[,)‘~ + K. 5

Where C = max((4e))‘cl, (1~1 + 1)). We have thus found that (1.3a) is satisfied with y = 5. A very similar argument shows that R maps D(A) into H. Equation (1.4) is trivially satisfied since f = 0. Thus theorem 1.l holds and we find that the solution to the Ginzburg-Landau equation is a D(A) analytic function in each component although u itself is not expected to be analytic. Owing to the existence of an absorbing set in H’, the domain of analyticity contains a set of the form A = (z E UZI Re(z) > r and ]Im(z)] < d] and (1.5) through (1.7) hold on this set, in particular for all t > ‘5. Because of the existence of absorbing sets in V in space dimension 1 and 2 (see Promislow [25] or Ghidaglia and Heron [ll]) we can conclude: 2.2. In space dimensions 1,2 there exist d, T > 0 such that the real and imaginary components of the solution to the Ginzburg-Landau equation (2.11) and (2.12) have analytic extensions to a domain of the form

THEOREM

Ad = {z E C ( Re(z) > t and 1Im(z)I < d] and (1.5)-(1.7) hold on Ad, in particular for all t > t. In space dimension 3, at each time t such that [u(t)/ 1 is finite we obtain analytic extensions of the real and imaginary parts of the solution to a domain containing A( [u(t)] 1). 2.3. Navier-Stokes

equations in 20 and 30

For complete details of the time and space analyticity of solutions of the Navier-Stokes equation see Foias and Temam [6, 71; the following is included for the sake of completeness. The Navier-Stokes equations arise in the study of the flow of viscous incompressible fluids. Let Q be a region with boundary r in IR”with n = 2 or 3, then the quantity u = (u,, . . . , un), u: I?” -P I?“, is the velocity of the fluid at each point x in Sz. These equations read:

adat

- vAu +

f

(2.16)

div u = 0

(2.17)

u-Vu

=

u(0) = z&J

(2.18)

where f(t) E L*(Q)” and is uniformly bounded independent of t E IR. To these equations we add a boundary condition 24(x,I) = 0

for all x on aa, and all t > 0

(2.19a)

U(‘, 0

is a-periodic

(2.19b)

for each t 2 0, where Q = (0, L)“.

912

K. PROMISLOW

We observe

that in the case of boundary

conditions

(2.19a or b) we have, respectively,

H = (u E L’(Q)” 1div u = 0, u * n = 0 on I)

or

(U E L’(Q)” 1div u = 0, uilr; = -uiIq+,, V = (u E Ha”

) div u = O] {u E Hi,,(n)”

D(A) = ZY2(L’)”f~ V

or H&,(Q)”

Here we define the operator A = -VA the regularity of this equation is well closed linear operator with a compact (1.3a) and (1.4) are satisfied. First we

I(R(u), u)l 5 c]~]~‘~\\uI] lAul”21~I we quickly

or 1div u = 0 and S u d_x = 0) fl V.

and the nonlinear term R by R(u) = u - VU. Once again known and it is clear that A is a positive, self-adjoint, inverse. It remains to show that R: D(A) -+ H and that recall two facts (see Temam [28] for example):

(R(u), u) = 0

With these inequalities

1 5 i 5 n)

for all u E D(A)

(2.20)

for all u E D(A) and v E H.

(2.21)

see that

IR(u)I 5 c(u[~‘~\[u(\IAuI~‘~ I(R(u),Au So Young’s

+ u)l = I(R(u),A(u))

inequality

immediately

+ (R(u), u)l = I(R(u),Au))I

(- ~lul~‘~bll lAu13’2.

yields

I(R(u),Au + u)(I EIAU(~ +

(c/E)(u(:.

That is, R: D(A) + H and (1.3) is verified in space dimensions 2 and 3. The condition (1.4) is amply verified by assuming that f(t) is bounded in L2 independent of t. We summarize our results in the following theorem. 2.3. In space dimension 2 or 3 there exists a 0,, satisfying ]&,I 5 7c/4 and a function T, E (?(I?+, R,), depending on v and the space dimension, such that if for any t, ]u(t)ll is finite then U, the solution to the Navier-Stokes equation (2.16)-(2.18), has a D(A)c valued analytic extension to a complex region of the form

THEOREM

A = t + (]u(t)],) and (1.5) E (1.7) hold on A. 2.4.

Cahn-Hilliard

equation

The nonlinear Cahn-Hilliard equation [5] is a continuous model for the description of the dynamics of pattern formation in a phase transition. For a description of the physical aspects of this equation the reader is referred to Novick-Cohen and Segel [24], for further mathematical properties see Novick-Cohen and Segel [24], Nicolaenko and Scheurer [22], Nicolaenko et al. [23], Constantin et al. [4]. The equation is fourth order with a negative viscosity term. Let fi be an open, bounded subset of K?”with a smooth boundary I, n = 1, 2, or 3. We look

Time analyticity and Gevrey regularity

973

for a function U: R + L2(SJ) such that

adat

- AK(u) = 0,

(2.22) (2.23)

u(O) = uo, where here K(U) = -v Au + f(u), v > 0 and f is a polynomial of order p f(X) = i

p 1 1.

CljXj,

(2.24)

j=l

The equation is supplemented with boundary conditions of one of the following two types: U(‘, 0

(adan)(x,

t) = 0

is Q-periodic for each t I 0, where &2= (0, L)”

(2.25a)

for all x on an, and all t > 0.

(2.25b)

Observe that in the case of boundary conditions (2.25a or b) we have, respectively: H = L2(sz)

L2(Q)

v = &?&)

1~ E H2(n) I au/an 10 E Hi

W-4) = H&(Q)

) adan

= 0 on r] = a Adan

= 0).

To put (2.22) into the form of (1.1) we write A = vA2 and R(u) = -Af(u)

= -f’(u) Au +

f "(u)IVU(~. It is easy to see that A is a positive, self-adjoint, unbounded, closed linear operator.

Since (1.4) is immediate and the proof of R : D(A) -+ H is similar to that for the verification of (1.3), we will only prove that (1.3) holds for R. To this effect we note the following inequalities: I(&2 I c((A”%(2 + ],12]i’2 = c(llU//2 + (L4)2)r’25 CIU(i.

(2.26)

By Sobolev embeddings H”‘4 c L4, so JvuIIp 5 C~IU(Iff~f”/4 % C~VU~1-n’4(AU~n’4 I CIUy-n’8

IUy+y

(2.27)

and, by Agmon’s inequalities, interpolation inequalities, and the equivalence, for every q > 0, of the norms {(AURA+ ~Iu~~]“~ with the H2 norm on Y and (lA2~(2 + ~]u)~]~‘~with the H4 on D(A) (see [20]), we have

I&- 5 cJvu( 5 c~u~1’2~u~:‘2

n = 1,

(2.28)

I&- 5 cI&*+C 5 CIU(1’2-r’2lUI:‘2+r’2

n = 2,

(2.29)

I&- 5 C(VU(“2((&2)“2 I c(u(i’4(ul:‘4

n = 3,

(2.30)

I&- 5 CJVU~“2()&3)“2 I c(uJ”‘+(:‘2(Au11’4

n = 4,

(2.31)

I&- 5 C(lUIH,)1’2(lUIH,)“2 5 clul:‘4~AU11’4

n = 5.

(2.32)

y< 00.

(2.33)

Now, to bound IR(u),Au

+ u)l we

begin with

](W), Au

+ u)l 5 ]R(u)](]Aul + lul),

so it is sufficient to show that kR(u)l 5 ck$‘I~k,

whereOIa<

land01

974

K. PROMISLOW

But by the definition of R(u) we have IR(u)l 5 \f’(&-lA~I

+ ~f”(~)kk&,

and there exist kl, k2, k3, and k4 such that

so we

If’(s)l I klsP-’

+ k2,

If”@)I

+ k4,

4 k3F2

may conclude that IR(u)( I c(~uIL-)p-1~z4(I+ C((UIL=)P-21U11-n’41UI:+n’4 + C(UII.

(2.34)

In the cases n = 1,2, or 3 we are done since we may bound the L” norms without using a power of IAul. In the cases n = 4 or 5 we require that (p - 1)/4 < 1, P <

5,

or equivalently p 5 4.

Since in the cases of space dimension 1 or 2 we have an absorbing set in T/ (see [28]), we may summarize our result as follows. THEOREM2.4. In space dimensions 1, 2 there exist d, T > 0 such that the solution to the Cahn-Hilliard equations (2.22) and (2.23) has an analytic extension to a domain of the form A, = (z E CcI Re(z) > r and )Im(z)I < dJ and (1.5)-( that lu(t)ll A((u(t)) I). we restrict

1.7) hold on Ad, in particular for all t > r. In space dimension 3, at each time t such is finite we obtain an analytic extension of the solution to a domain containing In space dimension 4 or 5 we obtain the same conclusion as in space dimension 3 if the degree of the polynomial to be less than or equal to 4. 3. GEVREY CLASS REGULARITY

In this section we enhance the strength of the previous regularity results, showing that the solutions are analytic functions with values not only in D(A) but in a Gevrey class of functions on Sz, and hence in e”(a). However, the scope of the general equation must be narrowed; we restrict the nonlinear term R to be a differential polynomial and consider only the periodic boundary condition. Additionally, we require that the conditions from Section 2 on the degree of the polynomial and the order of the differentials in the nonlinear term R hold. We are given a positive, linear, self-adjoint, unbounded, homogeneous elliptic differential operator A of order 2p, with domain D(A), and a compact inverse A -I acting on the Hilbert space H c L’(Q). Thus A has the form

Thus, in the case where Q = (cyl, . . . . (II,) is a multi-index with loll = cyl + ... +a,. 0 = [0, 2n]“, with periodic boundary conditions, A’s Fourier transform, A^, is a positive, homogeneous polynomial of degree 2p, which is contained in a cone defined by the 2pth power of the Euclidean norm on IR”(see Agmon [l]) c-‘l~l2P I A(r) 5 cl
(3.1)

915

Time analyticity and Gevrey regularity

Observing that the eigenvectors of A are the exponential functions (ei”x)j,Hn, whose eigenvalues (Aj = A(j)JjE Ln are as in Section 1, we identify H with a subspace of the space of functions u = C Ujeij’“, uj = (uj’, . . . . Ui”)E 6” (3.2) C IUj12< 00,

JU12s (&r

where the sums, as in the following, are overj E Z”, unless specified otherwise. We assume for convenience that u(x, t) dx = 0,

for all t L 0 and all solutions u.

(3.3)

n

Note that (3.3) is equivalent to the condition u,, = 0 in (3.2). Now, the norm on D(A) can be written, C lAj121Uj12*

IAU12= (&y Defining the differential

operator B = (-l)“AP; IBu)~ =

~

then we see that B(t) = l~/~p, and hence, ’ C lil”“lUjl’.

0

Thus, from (3.1) with r = j, we immediately obtain, on D(A), c-‘IBu(~

5 [Au12 I clBu12.

Now, for t > 0, we consider the Gevrey class of functions, D(exp(rB”2P)); of the form (3.2)-(3.3) such that, [lu/f = lexp(rB”2p)u12

=

&

(3.4) the set of functions

n C e2rli’Iuj)2 < CO.

0

(3.5)

At this point we are ready to reconsider the abstract initial value problem (1 .l)-(1.2) with periodic boundary conditions and the further constraint that the nonlinear term R be a polynomial in (D” Jloll5 d. Since the Fourier transform is an isometry on L2, there is a function F of the Fourier coefficients, Uj, of u, and the multi-indices, CY~, of R, such that (R(U), AU) = F(Uj,j*‘).

We require that for the y, K, and C given by (1.3), for all E > 0, F satisfies, IF(lUjl, Ij”il)l I &IAU12 + C(&)llul12’ + K.

(3.6)

That is, for a given u in H, after replacing its Fourier coefficients with their magnitude and the powers of the indices with their magnitudes, then F evaluated in this manner still satisfies (1.3a) for the original choice of u. Note that this is a very mild additional restriction on R since we retain the H norm and relations of the form (1.3a) are typically attained through the use of Sobolev estimates which exhibit behaviour of the type found in (3.6). Also, in the place of (1.4) we assume that there exist constants A4, rs > 0 such that f satisfies

IfwI, 5 M

for all t E IR, .

(3.7)

With these assumptions on equations (1.1) and (1.2), we state the main result of this section.

916

K. PROMISLOW

THEOREM 3.1.Given a differential operator A as described above, an initial condition u,, in V, f satisfying (3.7), and a differential polynomial R satisfying (3.6) and (1.3b); then, there exists T* ,depending only on the data and on u,, through llu,,ll, an d a re g’ion A 3 (0, T,), in the complex

plane, such that the following holds: (i) There exists a unique regular solution u of (1.1) and (1.2) such that the mapping t + [Al’2 ewG#4W”2pMf) is analytic on (0, T*) C A with values in H, 4(t) = min(t, o, T*). (ii) If the solution u of (1.1) and (1.2) exists and is uniformly bounded in V for all positive times, then u is analytic on (0, CQ),with values in D(A1’2 exp(aB”2P)) and A contains (0, 43). Remark.

The nature of the region A in the complex plane is made explicit in the course of the

proof. We proceed the proof of theorem 3.1 with a lemma. LEMMA 3.1.Let u be given in D(A 1’2eTB”2P),for some t > 0. Then if R satisfies (3.6), we have for the same y, K, and C as in (1.3a), for all E > 0, I(R(u),Au),l Proof.

s E(Au[: + C(~)[A”~u)fy

+ K.

(3.8)

We set

and define u* = C uj*eij’, = exp(sB1’2Pu) where @ = eTljIuj, A general notation for R, the differential we will examine only a single term, R(u) =

for j E Z”.

polynomial, c fi terms k =

is cumbersome to write down, hence

D%‘k 1

where CY,,. . . . CY~are arbitrary multi-indices and the i,, . . . , id are integers between 1 and n, denoting the component of u = (u’, . . . , u”). Thus we have, (R(u), Au), = (R(u), exp(2rB1’2P)Au), (RW,

c

Au), =

&

~~~~joe2Me-ij0~x

j. E h”

>

this integral is zero, by the nature of the exponential function, unless -jO + ji + j2 + ’ .’ + jk = 0, in which case the integral has value (2n)“. Hence,

I(R(z.4, Au),! 5 (270” c

C

tems jI+...+jd=jO

Iu~oI

. . .

Iu$dl

IjpIl

. . .

Ij~dl;lioe~~jOl-ljlt-"'-Ijd')

Time analyticity

but Ij,l = Ij, +j, obtain,

+ . ..+j.l

Iuw,~~)*l

and Gevrey

regularity

I (j,l + ... + Ij,l, so exp(r(lj,l

911

- Ij,l - e.. - Ij,J)) 5 1. We

5 (277)”teImS c j,+,,,~jd=jolu:,“I *** h;~l IPI .** l.Lm.jo.

Using the above notation for u* = exp(rB1’2P)u, we have established, I(Z?(u,Au),l I IE(luj*l, Ijail) s EIAu*[~ + C(E)IA”~~*I~~ + K. This completes the proof of lemma 3.1. Proof of theorem 3.1. The framework of this proof is identical to that of theorem 1.1: we prove a priori estimates on the solution in the )0IT norm, in complex time. We establish a domain A in the complex plane where the solutions are uniformly bounded, then we pass to the limit in m, the dimension of the projected space. However, since the proof differs from that for theorem 1.1 only in the derivation of the a priori estimates, we will only sketch the description of the domain A and will omit the passage to the limit. Furthermore, to simplify the notation, we will only consider the solution u of the full problem; thus all the calculations are formal. To begin, we complexify the equation (1.1) as before, write r#~(t)= min(t, o), z = seie, s > 0, cos(8) > 0, and Es = exp(+(s cos B)B”2p); viewing 0 as a fixed element of (-n/2, n/2). Take the scalar product of (1.1) with Au(se”) in D(E,), multiply by eie and take the real part. Here we may drop the u from (1.10) since we are in the space-periodic case with zero integral average. This yields, Re e”((E, &/&&se”), E,Au(se”)) + (EJlAu, E,Au) + (E,R(u), E,Au)) = Re eie(Esf, E,Au).

(3.9)

But, using the relation, d/ds = ePie d/dz, we obtain Re e”[(E, &./&(.seie), EsAu(seie)))

= Re(A1’2 d/ds(E,u(.se”) - @(s cos B)(cos B)EsAu(seie), E,A1’2u(se’e)) = (i) d/~IA1’2u(Seie)1~(,,,, - (~0s Ob’(~co~

0)

e)(Aul~(,,,,e)IA1’2UI~(scose)

2 (9) d/dr;lA1’2u(seie)l~(,,,,e)- (ao~0sW4IAuI&,,,,e) - (~0s~/dA1’2d&,se~.

(3.10)

Also, as before, we have Ree’e(ES~Au,ESAu)

2 cos~oOIAul&,cose) - (sin 01 II~II*IA~I&sco~e~;

restricting 0 as in (1.13), we obtain Re e”(E,DAu,

E,Au) 2 ($>cos 0 (Y~IAu(&,,

ej.

(3.11)

Inserting (3.10) and (3.11) into (3.9) and writing I$ in place of 4(s cos 0) yields (+)d/d+ll;

+ (c~,cos 0)/21A4;

5 I(f,Au),I

+ I(M4&),l.

(3.12)

K. PROMISLOW

978

Treating the I(f, Au),1 term as before and using lemma 3.1, we arrive at an inequality of the form of (1.15), (d/d.s)llu(.se’e)ll~ + c,jAu(seie)I~ I c2 + c,lu(seie)l~, (3.13) where the constants depend only on the data and are independent Continuing as in Section 1, we write

of 13satisfying (1.13).

J$S) = 1 + I&41’2&eie)12,

(3.14) (3.15)

Y’(S) 5 c‘0W to conclude that lE+41’22((.seie)125 2 + 21A1’2~,12

(3.16)

in the region A(llz+,ll) given by, 0 I s 5 7i(lluJ> = y(o)‘-Y(1 - (+)‘-‘)/((y

- 1)cq) (3.17)

I@ 5 Min(ltan-‘((y,/411011*)),

n/4).

Thus, even if u0 is only in D(A”2), then U(se”) belongs to D(E,A1’2) in the angular region ~(llu,ll) of C given by (3.17) (see Fig. 1). If )Iu 1)is uniformly bounded by M on II?, , then we see that A(llu,,jl) may be extended to include A = Ut,o(t + A(M)), which is similar to Fig. 2. Additionally, arguing exactly as in Section 1, we can show that for all compact K contained inside A, the following inequalities hold, sup ~~(dkz.4/dzk)(seie)~~~ 5 21’2(2/d)k(k!)(l ZEK

+ ~~~~~~~~~~~~d = dist(K, aA)

sup IAu(seie)I, 5 T,(K) < 00,

(3.18) (3.19)

ZEK

sup IA(dku/dzk)(seie)l,

5 2k(k!)[d(K,

aA(u,))]-kT2(K’),

(3.20)

ZEK

where K’ E (z E A@,,) I d(z, aA(

2 +d(K, aA(u

1. Under the same hypotheses as theorem 3.1, for 0 < t 5 (+)~,(llu,ll>, we have the following relations:

Remarks.

Il(dku/dtk)(t)(&, IA(dku/dtk)(t)(+

I 2kk!c-k(21’2)(1 + IIz+,~~~)~‘~, I 2kk!c-k/cl(c2

+ c,(2(1 + Ib~,lj~))~ + 4(4/c)(l

(3.21) + /u,,[\~)]~‘~,

(3.22)

where here c = tm/(l + m2)1’2 is the distance oft to the aA., m = CQ/~~~D II* is the slope of the line that forms the boundary, and cl, c2, and y are as in equation (3.13). 2. In the case that the solution u is uniformly bounded by M in the V norm for all positive time, we see from equation (3.16) above that &4”2u(t)~2

in particular,

5 2 + 2M2,

(3.23)

from the definition of E,, we obtain luj(t)12 5 (2 + 2M2)LJ:1e-2Vl+to.

This is the exponential decay of the Fourier coefficients

luj]j ELn.

(3.24)

Time analyticity

and Gevrey

regularity

979

3. Observe that, for (Ya given multi-index, and summing over j E Z”,

But there exists a constant M(a, r) > 0, such that lj”l 5 ertjl,

for all j E Z” such that Ijl 2 M,

so we can easily show that if 1~1, < m for some t > 0 then ID%] < 00. Thus the Gevrey class of functions is contained inside of C!“(Q). 4. It can easily be verified that all the equations considered in Section 2 trivially satisfy the conditions on the linear operator A as well as (3.6), thus theorems 2.1-2.4 can be strengthened, in the space periodic case, to state that the solutions are time analytic with values in the Gevrey class of functions. Acknowledgements-These studies were partially supported by the National Science Foundation under Grant No. NSF-DMS-8802596 and by the U.S. Air Force Office of Science Research under Grant No. AFOSR-88-103, while the author was an Office of Naval Research Fellow at the Institute for Applied Mathematics and Scientific Computing of Indiana University, Bloomington. REFERENCES Princeton, NJ (1965). 1. AGMON S.. Lectures on Elliptic Boundary Value Problems. Van Nostrand Company, de semi-groups-non-lineaires, These, Universite de Paris. 2. BR~ZIS D.; Interpolation of waves bv wind. Phil. Trans. R. Sot., Lond. (Ser. A), 298, 451-494 3. BLENNERHA~SETTP. J.. On the generation (1980). 4. CONSTANTIN P., FO~AS C., NICOLAENKO B. & TEMAM R., Integral and Inertial Mantfolds for Dissipative Partial Differential Equations, Vol. 70. Springer, New York (1988). system 1: interfacial free energy, J. Chem. Phys. 28, 5. CAHN J. W. & HILLIARD J. E., Free energy of a non-uniform 258 (1958). equations, J. funct. Analysis 6. FOIAS C. & TEMAM R., Gevrey class regularity for the solutions of the Navier-Stokes (to appear). statistical solutions of the Navier-Stokes equations and turbulence, 7. FOIAS C. & TEMAM R., On the stationary Publ. Math. d’Orsay (1975). and Stokes eigenfunctions and the far-dissipative turbulent 8. FOIAS C., MANLEY 0. & SIROVICH L., Empirical spectrum, Phys. Fluids (A)2 (1990). des petits et grands tourbillons dans des Ccoulements 9. Foms C., MANLEY 0. & TEMAN R., Sur I’interaction turbulents, Cr. Acad. SC. Paris (Ser. I) 305, 497-500 (1987); Modelisation of the interaction of small and large eddies in turbulent flows, Math. Mod. numer. Analysis (1988). 10. FOLLAND G., Introduction to Partial Differential Equations. Princeton University Press, Princeton, NJ (1976). partial differential 11. GHIDAGLIA J. M. & HBRON B., Dimension of the attractors associated to the Ginzburg-Landau equation, Physica 28D, 282-304 (1987). 12. HASTINGS S. & MURRAY J., The existance of oscillatory solutions in the Field-Noyes model for the Belousov-Zhabotinski reaction, SIAM J. Appt. Math. 28, 678-688 (1975). to conduction and 13. HODGKW A. & HUXLEY A., A qualitative description of membrane current and its application excitation in nerves, J. Physiol. 117, 500-544 (1952). 14. HOWARD L. & KOPEL N., Plane wave solutions to reaction-diffusion equations, Stud. appl. Math. 52, 291-328 (1973). 15. Iooss G., Bifurcation of Maps and Applications. North-Holland, Amsterdam (1979). 16. MARION M., Attractors for reaction-diffusion equations: existence and estimate of their dimension, Applic. Analysis 25, 101-147 (1987). 17. MASUDA K., On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equations, Proc. Japan Acad. 43, 827-832 (1967).

980

K PROMISLOW

18. MOON H. T., HUERRE P. & REDEKOPPL. G., Three frequency motion and chaos in the Ginzburg-Landau equation, Phys. Rev. Lett. 49, 458-460 (1982). 19. MOONH. T., HUERREP. & REDEKOPPL. G., Transitions to Chaos in Ginzburg-Landau equation, Physica 7D, 135-150 (1983). 20. NIRENBERGL., On elliptic partial differential equations, Annali. Scu. norm. sup. Piss 13, 116-162 (1959). 21. NEWELLA. C. & WHITEHEADJ. A., Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279-304 (1969). 22. NICOLAENKO B. & SCHEURERB., Low-dimension behavior of the pattern formation Cahn-Hilliard equation, in Trends in the Theory and Practice of Nonlinear Analysis. 23. NICOLAENKO B., SCHEURERB. & T&AM R., Some global dynamical properties of a class of pattern formation equations, Communs partial diff. Eqns 14, 245-297 (1989). 24. NOVICK-COHEN A. & SEGELL. A., Nonlinear aspects of the Cahn-Hilliard equation, Physica lOD, 277-298 (1984). 25. PROMISLOW K. S., Localization and approximation of attractors for the Ginzburg-Landau partial differential equation, J. Dynamics diff. Eqns (to appear). 26. SMOLLER J., Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983). 27. STUARTJ. T. & DI PRIMAR. C., The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. R. Sot. Land. (Ser. A) 362, 27-41 (1978). 28. TEMAMR., Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988). 29. TEMAMR., Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Reg.Conf. Ser. Appf. Math., SIAM, Philadelphia, PA (1983).