Pergamon

Nonlinear Analysis, Theory, Methods & Applications, Vol. 32, No. 1, pp. 71-85, 1998 © 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0362-546X/98 $19.00+0.00

PII: S0362-546X(97)00453-7

T H E G L O B A L A T T R A C T O R F O R T H E 2D N A V I E R - S T O K E S F L O W O N S O M E U N B O U N D E D D O M A I N S

RICARDO ROSA The Institute for Scientific Comput ing & Applied Mathematics , Indiana University, Bloomington, U.S.A.; and

Departmento de Matem~ltica Aplicada, Rio de Janeiro, Brazil

(Received 14 February 1996; received for publication 18 February 1997)

Key words and phrases: Navier-Stokes equations, global attractor, attractor dimension, weak dissipation, unbounded domains.

1. I N T R O D U C T I O N

The global attractor for the 2D Navier-Stokes equations was first obtained for bounded domains in the works of Ladyzhenskaya [1] and Foias and Temam [2], with the latter work showing also the finite dimensionality of the attractor in the sense of the Hausdorff dimension (see also [3-5]). Later on, the unbounded domain case was treated by Abergel [6] and Babin [7], but the forcing term was required to lie in some weighted space. However the dimension estimate of the attractor in this case was independent of the weighted norm of the forcing term and it was natural to expect the existence of the global attractor for more general forces.

It is the purpose of this note to show that indeed the global attractor exists for more general forces than can even lie in the natural dual space V' (see below) of the theory of the Navier-Stokes equations. The finite dimensionality of the global attractor is also obtained in this case and an estimate of its dimension is given in terms of suitable Reynolds and Grashof numbers defined for forces in V'.

Another condition required in the works of Abergel [6, 8], and Babin [7] is the smoothness of the boundary of the spatial domain in order to obtain the appropriate regularity of the domain of the Stokes operator. This regularity is no longer required here and the spatial domain can be an arbitrary open set as long as the Poincar6 inequality holds on it. This includes the channel flow, the multi-channel flow and other variations. It should be remarked that for the bounded case, the assumption of the smoothness of the domain was removed by Ladyzhenskaya in [9].

The existence of the global attractor for dissipative evolution equations has always relied on some kind of compactness of the semigroup generated by such equations. Usually, this compactness is obtained through some regularization property of such equations together with the compact imbedding of the relevant Sobolev spaces (see [5] for instance). This approach is suitable only for bounded domains since the Sobolev imbeddings are no longer compact otherwise.

For unbounded domains, the remedy was to consider weighted spaces (see Abergel [6, 8], Babin [7], Babin and Vishik [10] and Feireisl et al. [11]), but with the drawback that the forcing term and in some cases even the initial condition had to be restricted to the weighted spaces.

71

72 R. ROSA

Our aim here is to avoid weighted spaces by exploiting the energy equation valid for the 2D Navier-Stokes equation in order to obtain the so-called asymptotic compactness of the semigroup. The concept of asymptotic compactness was already used by Abergel [6, 8] and by Ladyzhenskaya [12], and is implicit in [5], Theorem 1.1.1, (1.1.3)] (see also the concept of asymptotic smoothing in the works of Hale [13] and Haraux [14]).

The idea of using the energy equation to obtain the existence of the global attractor was successfully applied to some weakly damped hyperbolic equations by Ball [15], and then by Ghidaglia [16] and Wang [17]. Although the equations were all considered on bounded domains, the proofs did not make essential use of the compactness of the Sobolev imbeddings, being then natural to extend this idea to equations on unbounded domains. We illustrate this point here for the case of the 2D Navier-Stokes equation, while other equations will be studied elsewhere.

This paper is organized as follows. In Section 2 we recall the mathematical setting of the problem and the existence and regularity results, and establish some weak continuity results of the corresponding semigroup. In Section 3 we prove the asymptotic compactness property and deduce the existence of the global attractor. Finally in Section 4 we prove the finite dimensionality result.

2. PRELIMINARIES

We consider the flow of an incompressible viscous fluid of constant density enclosed in a region ~ C R E with rigid boundary 0~) and governed by the Navier-Stokes equations. We denote by u(x, t) e B E and p(x, t) ~ JR, respectively, the velocity and the pressure of the fluid at the point x e ~ and at time t _> 0, which are then determined by the following initial-boundary value problem:

I Ou v Au + (u" V)u + Vp = f in ~, Ot V • u = 0 in ~), (2.1)

u = 0 on 0£2,

~, u(., 0) = u0 in ~,

where v > 0 is the kinematic viscosity of the fluid and f = f(x) ~ ~2 is the external body force (assumed to be time independent).

The domain fl can be an arbitrary bounded or unbounded open set in ~2 without any regularity assumption on its boundary 3~) and with the only assumption that the Poincar4 inequality holds on it. More precisely, we assume only the following:

There exists 2~ > 0 such that

t <_ 1

The mathematical framework of (2.1) is now classical (see [18], for instance): first let tl_z(~) = (L2(~)) 2 and IHol(~) --- (Hol(~)) z endowed, respectively, with the inner products

(u ,v )= f u . v d r , u ,v~E2(~) ,

Global attractor for 2D Navier-Stokes flow 73

and

C 2_ ((u, v)) = | [ , v u j . Vv~dx , u = (Ul, u2), v = (Vl, v2) e IHol(fl),

J f 2 j = l

and norms I" I = ( ' , .)1/2, I1" II = ( ( ' , .))1/2. Note that thanks to (2.2) the norm I1" I] is equivalent to the usual one in IH~(fl). Set now

= {v e (5)(g2))2; V • v = 0 in £2],

V = closure o f '-q in IHol(~),

H = closure of ~2 in ~2(~),

with H and V endowed with the inner product and norm of, respectively, tL2(~) and IHi(~). It follows f rom (2.2) that

1 [ul~ ~ T, Ilull~' v u e v. (2.3)

We then consider the following weak formula t ion o f (2.1): to find

u e L*°(0, T; H) N L2(0, T; V), ¥ T > 0, (2.4)

such that

d ~ t (u , v) + v((u, v)) + b(u, u, v) = ( f , v), v v ~ V, ¥ t > 0, (2.5)

and

u(0) = Uo, (2.6)

where b: V × V x V -* [~ is given by

2 l Ov~ b(u, v, w) = i,j= ~ ui ~ixi W) dr , (2.7)

and ( . , • ) is the duali ty product between V' and V when we identify H with its dual, and we assumed for simplicity that f ~ V'. The weak formula t ion (2.5) is equivalent to the funct ional equat ion

u ' + v A u + B(u) = f , in V', for t > 0, (2.8)

where u ' = d u / d t , A : V ~ V ' is the Stokes opera tor defined by

( A u , v) = ((u, v)), v u , v ~ V, (2.9)

and B(u) = B(u, u) is a bilinear opera tor B: V × V ~ V' defined by

(B(u, v), w) = b(u, v, w), v u, v, w ~ V.

The Stokes opera tor is an isomorphism from V into V', while B satisfies the following inequality (see [18, Lemma III.3.4]):

IIB(u)[I , -< 21/2lulllul[, v u 6 v. (2.10)

We have the following reesult [18].

74 R. ROSA

THEOREM 2.1. G i v e n f ~ V' and u0 ~ H, there exists a unique u ~ L®(~+; H) O LZ(0, T; V), v T > 0, such that (2.5) (hence (2.8)) and (2.6) hold. Moreover, u ' ~ L2(0, T; V'), v T > 0 and u ~ C(~+; H).

Now, let u = u(t), t >_ O, be a solution of (2.5) given by Theorem 2.1. Since u ~ L2(O, T; V) and u' ~ L2(O, T; V ' ) , we have

l d 2 dt lu12 = (U' , U), (2.11)

so that from (2.8),

l d - - - l u l 2 = ( f - v A u - B(u), u) 2 dt

= <f , u> - vllul[ z - b ( u , u, u) .

Hence, from the well-known orthogonality property

b(u, v, v) = O, q u , v ~ V, (2.12)

we deduce that

d lu12 + 2vllul12 2(f , u), Vt > O, (2.13) dt

in the distribution sense on R +. From (2.13) and using (2.3) one can easily deduce the classical estimates

1 [u(t)l 2 - luol z e -~x't + ~ Ilfll~,,, v t _> o, (2.14)

and

I S t 1 1 Ilu(s)ll2ds < - - l u o l 2 + Ilfll~,,, v t > o. (2.15)

7 o - t v -~

The energy equation (2.13) will be further used in Section 3. For the moment, note that thanks to Theorem 2.1, we can define a continuous semigroup {S(t)lt~ o in H by

S(t)Uo = u(t), t >_ O,

where u is the solution of (2.5) with u(0) = Uo ~ H. It is not difficult to see that the map S(t): H ~ H, for t >_ 0, is Lipschitz continuous on bounded subsets of H. Moreover, from (2.14) it follows that the set

63 = v e H; Ivl <- P0 -- - II/llv' (2.16) v

is absorbing in H for the semigroup. We will also need in Section 3 the following weak continuity of the semigroup [S(t)}t ~ o.

G l o b a l a t t r a c t o r fo r 2D N a v i e r - S t o k e s f low 75

LEMMA 2.1. Let [Uo,] . be a sequence in H converging weakly in H to an element Uo e H. Then

S(t)Uo, ~ S ( t ) u o weakly in H, v t _> 0, (2.17)

and

S(.)Uo, ~ S(')Uo weakly in L2(0, T; V), v T > 0. (2.18)

P r o o f . Let u . ( t ) = S(t)Uo. and u(t) = S(t)Uo, for t _> 0. From (2.14) and (2.15) we find that

{u. 1. is bounded in L=(~ +, H) A L2(0, T; V), v T > 0. (2.19)

Hence I T-a I T-a

[u , ( t + a) - u,(t)12 d t <_ c r a 1/2 IluAt + a) - u.(t)[I dt. (2.22) 0 0

Using the Cauchy-Schwarz inequality and (2.19) we find from (2.22) that

i t - . + a) - _<

[u.(t Un(t) 12 dt (T al/2, 0

for another positive constant 6r independent of n. Therefore

i 'T-a

lim sup IluAt + a) - u.(t)ll~2<ar)dt = 0, (2.23) a ~ O n 0

for all r > 0, where f~r = f2 ~ Ix e ~2; Ixl < r / . Moreover, from (2.19),

{u.l~rl. is bounded in L2(0, T; IHl(~"~r)) f') t~(0, T; [L2(~2r)), (2.24)

u" = f - v A u n - B(un) ,

and since A is a bounded linear operator from Vinto V' and B satisfies (2.10), it follows that

{u']. is bounded in Lz(O, T; V'), ¥ t > 0. (2.20)

Then, for a l l v e V a n d 0 _ < t _ < t + a _ < T, with T > 0 ,

l t+a

( u . ( t + a) - u . ( t ) , v) = (u~,(s), v> ds t

<-H ~J2 , vlla I lu . l l~ (o ,T;v , )

<_ cr[lvlla 1/2, (2 .21)

where Cr is positive, independent of n. Then, for v = u . ( t + a) - u . ( t ) , which belongs to V for almost every t, we find from (2.21) that

[u . ( t + a) - u . ( t ) [ z <_ c r a l / Z l l u . ( t + a) - u.(t)ll.

Hence, since

76 R. R O S A

for all r > 0. Consider now a truncation function r e C~(~ ÷) with r(s) = 1 for s e [0, l] and r(s) = 0 for s ~ [2, +o0). For each r > 0, define Vn,r(x) = 7 : ( I x I 2 / r 2 ) U n ( X ) for x ~ t)2r. Then, from (2.23), we find that

I T - a

lim sup Ilv,,r(t + a) 2 - v,,At)ll~2(m~) dt = 0, w T > 0, Wr > 0, (2.25) a ~ O n 0

while from (2.24) we find that

[Vn,r] n is bounded in L2(0, T; IH~(~2r)) n L°~(0, T; ~2(~2r)), V T > 0, Vr > 0. (2.26)

Thus, by a compactness theorem ([19, Theorem 16.3] with X = U_2(~2r), Y = IHI(fl2A and p = 2; see also [20, Theorem 13.3]),

{v~.A, is relatively compact in L2(0, T; I12(~2r)), v T > 0, v r > 0. (2.27)

It follows then from (2.27) that

lu~ I nr}~ is relatively compact in LZ(0, T; tl-Z(f~r)), V T > 0, V r > 0. (2.28)

Then from (2.19) and (2.28), and by a diagonal process, we can extract a subsequence [u~,l,, such that

u~, --, ~ weak-star in L~(IR÷; H) ,

weakly in L2oc(~+; F), (2.29)

strongly in L2oc(~÷; II 2(f~r)), Vr > 0,

for some

a n) n Hoc( +; v). (2.30)

The convergence (2.29) allows us to pass to the limit in the equation for un, to find that t~ is a solution of (2.5) with ti(0) = Uo. By the uniqueness of the solutions we must have ~ = u. Then by a contradiction argument we deduce that the whole sequence [U,}n converges to u in the sense of (2.29). This proves (2.18).

Now, from the strong convergence in (2.29) we also have that u~(t) converges strongly in 112(~r) to u(t) for almost every t _> 0 and all r > 0. Hence for all v e ~ ,

(u , ( t ) , v) ~ (u(t) , v), for a.e. t e ~+.

Morever, from (2.19) and (2.21), we see that [(u~(t), v)l, is equibounded and equicon- tinuous on [0, T], for all T > 0. Therefore

(un(t) , v) ~ (u(t) , v), V t e [R +, V v ~ ~ . (2.31)

Finally, (2.17) follows from (2.31) by taking into account (2.19) and the fact that ~ is dense in H. •

Global attractor for 2D Navier-Stokes flow 77

3. T H E G L O B A L A T T R A C T O R

The existence of the global attractor will follow from an abstract result as soon as we prove the asymptot ic compactness of the semigroup { S ( t ) } t ~ o. Such a semigroup is said to be asymptotically compac t in a given metric space if

IS(t ,)u,} is precompact (3.1)

whenever

[u,}, is bounded and t, --, ~o. (3.2)

In order to show that IS(t)]t> o is asymptotically compact in H, we use the energy equation (2.13).

First define [., "]: V x V--, ~ by

[u, v] = v((u, v)) - v ~ (u, v), ¥ u, v ~ V. (3.3)

Clearly, [ . , . ] is bilinear and symmetric. Moreover, from (2.3),

Hence

[u] = ~ [u, u] = vlluff = - v - ~ lui =

V 12 vllull2 - 5 Ilul12 = 2 Ilul12.

Ilulf 2 ~ In] 2 ~ vlJull 2, v u e V. (3.4)

Thus [ ' , "] defines an inner product in Vwith norm [.] = [ . , -1 1/2 equivalent to I[" il. N o w , add and subtract v;t~[ul 2 from the energy equation (2,1 3) to find

d lul2 + v2,lul 2 + 2[ul z =.2G¢, u), (3.5) dt

for any solution u = u(t) = S(t)u o, u o ~ H. Then, by the variation of constants formula,

l' lu( t ) l z = lUof z e -~x~' + 2 e-"Xl~t-s)((f, u(s)) - [u(s)] 2) ds, , 0

which can be written as

f' fs( t)Uof 2 = luol 2 e -~x ' ' + 2 e - ~ X ' < ' - ' ( ( f , S(s )uo) - [S(s)uol 2) ds , (3 .6) o

for all u o ~ H, and t ___ 0. We are now in position to show the asymptotic compactness of the semigroup. For that

purpose, let B C H be bounded and consider lunl, C B and [tn},, t, _> 0, t, ~ co. Since the set 63 defined in (2.16) is absorbing, there exists a time T(B) > 0 such that

S(t)B C 63, v t >_ T(B),

so that for t, large enough (t, >__ T(B)),

S ( t , )u , ~ 63. (3.7)

78 R. ROSA

Thus [S(tn)un} ~ is weakly precompact in H and

S(t.,)un, ~ w weakly in H, (3.8)

for some subsequence n ' and w e CB (since (B is closed and convex). Similarly for each T > 0, we also have

S(t~ - T)u~ e (B, (3.9)

for t . >_ T + T(B). Thus [ S ( t ~ - T)U~}n is weakly precompact in H, and by using a diagonal process and passing to a further subsequence if necessary we can assume that

SUn, - T)un, ~ wr weakly in H, v T ~ N, (3.10)

with wT ~ (B. Note then by the weak continuity of S(t) established in Lemma 2.1 that

w = limHw S(t . , )u . , = limHw S(T)S(tn, - T )u . , n I n ~

= S ( T ) lim/4w S(t~, - T )u , , = S ( T ) w r, n ~

where limH~ denotes the limit taken in the weak topology of H. Thus

w = S ( T ) w r , v T ~ N. (3.11)

Now, from (3.8), we find

[wl - lim inf[S(tn,)Un,[, (3.12) n '

and we shall now show that

l imsup lS ( t . , )u . , I <_ Iwl. n ~

For T ~ N and t. > T we have by (3.6)

IS(tn)U.lZ= [ S ( T ) S ( t . - T)u . I 2

= I S ( t n - T)Unl2e -~xlr

+ 2 e-~Xlcr-s)I(f, S(s)S(t. - r ) u . > - [S(s)S(t. - r ) u . 1 2 1 d s . (3.13) 0

From (3.9) we find

lira sup (e-~X:[s(t~, - T)Un,[2) _< p~ e-,X:. (3.14)

Also, by the weak continuity (2.18) we deduce from (3.10) that

S( . )S( t . , (3.15)

Then since

we find

lim

- T)u . , ~ S ( ' ) w r weakly in L2(0, T; V).

s ~ e-"×lcr-s)f c LZ(0, T; V'),

l T e-VX'<r-s)(f, S(s)S(t. , - T ) u . , ) ds = e-V×'tr-s)(f, S(s)wr) ds. 0

(3.16)

Global attractor for 2D Navier-Stokes flow

Moreover, since [.] is a norm in V equivalent to I1" [I and

0 < e -~xlr _< e -'x~(r-s) _< 1, Vs c [0, T],

we see that

79

Hence

I T

lim sup - 2 e-"X~(r-s)[S(s)S(t., - T)Un,] 2 ds rt' 0

S = - lim inf 2 e-~X~(r-s)[S(s)S(t., - T )u . , ] 2 ds n ' 0

<- - 2 e-~Xl(r-s)[S(s)wrl 2 ds. (3.18) 0

We can now pass to the lira sup as n goes to infinity in (3.13), taking (3.14), (3.16) and (3.18) into account to obtain

lira sup IS(t.,)u.,I a <_ pg e -~×'r + 2 e-~Xl(r-s)I(f, S(s)wr> - [S(s)wrl21 ds. (3.19) " ' 0

On the other hand, we obtain from (3.6) applied to w = S ( T ) w r that

J w l 2 = Is(r)wTI 2

= e-~X'rlWrl2 + 2 e-'X1(r-s)[(f, S(s)wr> - [S(s)wrl 2] ds. (3.20) 0

From (3.19) and (3.20) we then find

lim sup [S(t.,)u., 12 _< i wl 2 ÷ (p~ _ [w~ 12) e - ~ : . t

_< Iwl 2 + p2e-"X:, q T e N. (3.21)

Let T go to infinity in (3.21) to obtain

lim sup Is(t . , )u. , 12 -< I wl 2, (3.22)

as claimed. Since H is a Hilbert space, (3.22) together with (3.8) imply

S ( t . , ) u . , ~ w strongly in H. (3.23)

( i f e-~Xl(r-~)[. ]2 ds t t/2

is a norm in L2(0, T; V) equivalent to the usual norm. Hence from (3.15) we deduce that

l l e-~Xl(r-s)[S(s)wr] 2 ds <_ l iminf e-~Xltr-s)[S(s)S(t . - T)u.] 2 ds. (3.17) 0 0

80 R. ROSA

This shows that [S(t , )u,] , is precompact in H, and hence that IS(t)] t > o is asymptotically compact in H. Since [S(t)]t ~ o has also a bounded absorbing set 63 in H, the existence of the global attractor follows from a general result which we state here for completeness.

THEOREM 3.1. Let E be a complete metric space and let [S(t)lt>0 be a semigroup of continuous (nonlinear) operators in E. If (and only if) [S(t)lt :. 0 possesses an absorbing set 63 bounded in E and is asymptotically compact in E then [S(t)]t > o possesses a (compact) global attractor ~2 = 09(63). Moreover, if t ~ S(t)Uo is continuous from ~+ into E and 63 is connected in E, then 6t is connected in E.

The proof of Theorem 3.1 is essentially that of [5, Theorem I.l .1]; see [8, 12]. Using Theorem 3.1, we prove our main result.

THEOREM 3.2. Let £2 be an open set satisfying (2.2). Assume v > 0 and f e V'. Then, the dynamical system {S(t)}t>0 associated to the evolution equation (2.5) possesses a global attractor in H, i.e. a compact invariant set (~ in H which attracts all bounded sets in H. Moreover, ~ is connected in H and is maximal for the inclusion relation among all the functional invariant sets bounded in H.

Remark 3.1. The global attractor 6t obtained in the Theorem 3.2 is actually included and bounded in V. This follows from an a priori estimate obtained by Ladyzhenskaya [9]. More precisley, it follows from estimate (7~-) of Theorem 2 in [9], which implies the following more explicit estimate:

t l[s(t)uoll 2 <_ ec(l+luol'+tzllfll4v ,) , v u 0 ~ H, v t >__ 0, (3.24)

where c is a constant depending only on v. Estimate (3.24) does not rely on either the Poincar6 inequality or the smoothness of the domain and can be shown to hold for arbitrary unbounded domains in ~z.

4. D I M E N S I O N OF T H E A T T R A C T O R

For the analysis of the dimension of the global attractor 6t obtained in Section 3 we use the general theory developed by Constantin et al. [21], in the compact case, and by Ghidaglia and Temam [22], in the noncompact case, i.e. when the solution operator for the linearized flow is not compact, which is the case here. We follow the presentation given in [5], to which the reader is referred for more details.

Let Uo ~ H and set u(t) = S(t)Uo, for t _ 0. From (2.8) we see that the linearized flow around u is given by the equation

I U' + v A U + B(u, U) + B(U, u) = 0, in V', (4.1)

u ( 0 ) = 4.

As for the nonlinear problem, one can show the following:

given ~ ~ H, there exists a unique U ¢ L=(0, T; H) fq L2(0, T; V), v T > 0, (4.2)

satisfying (4.1). Moreover, U' ¢ L2(0, T; V') and U ~ e([0, T]; H), v T > 0.

Global attractor for 2D Navier-Stokes flow 81

We can then define a linear map L(t; Uo): H --, H by setting L( t ; Uo)~ = U(t). It can also be proven that L(t; Uo) is bounded and that {S(t)Jt~o is uniformly differentiable on (~, i.e.

IS ( t )Vo - S ( t )Uo - L( t ; Uo)" (Vo - Uo)[ lim sup = 0. (4.3) ~--.o ~o,~o~ IVo - Uol

o< luo-vol <~

Write (4.1) as

U' = F ' ( u ) U - - v A U - B(u, U) - B(U, u), (4.4)

and define numbers qm, m • IN, by

qm limsup sup sup 1 i t = Tr(F'(S(r)Uo) ° Qm(r)) dr, (4.5) t--*~ U o ~ g;iEH -t dO

[(il < I i= l , . . . ,m

where Qm(r) = Q,,(r; Uo, ~1 . . . . . ~m) is the orthogonal projector in H onto the space spanned by L(t; Uo)~l, . . . , L(t; Uo)~m. The trace (denote Tr) of F'(S(t)Uo) o Qm(t) in (4.5) is defined at least almost everywhere in t and will be made precise below.

From the general result in [5, Section V.3.41, we have that if qm < 0 for some m • N then the global attractor has finite Hausdorff and fractal dimensions estimated respectively as

dimH(~) --< m, (4.6)

dimF(~2)_ m(1 + max (qJ)+~ 1 <j<_m ~ / " (4.7)

In order to estimate the numbers qm, let Uo e (t and (t . . . . . (,~ e H. Set u(t) = S(t)uo and Uj(t) = L(t; Uo)(j, t > O. Let (a~(t) . . . . . (am(t), t >_ O, be an orthonormal basis in H for SpanlUl(t) . . . . . U,,(t)}. Since U i ( t ) e V (at least almost everywhere in t) we can assume (aj(t) • V (by the Gram-Schmidt orthogonalization process). Then we have

Tr F'(u(r)) o Qm(r) = ~ (F'(u(r))(aj, (aj) j = l

= ~ ( - vA(a j - B(u, (aj) - B((aj, u), (aft j = l

= ~ (-vll(ajll 2 - b(u , (aj, (aj) - b((aj, u, (aj)) j = l

= (using (2.12))

= ~ (-vl[~o/ll 2 - b((aj, u, (aj)). j = 1 ( 4 . 8 )

82 R. ROSA

Now

j~=l b(~j'u'~j) = j~=l Ill k,~l=l ~jk ~-~k(~jldX

k , I = ~ L Oxk \ . i = 1

, , = -

< (t'12 ~ (0U/'~2 dx')l/2 ( C ~(J ~=l~jk~jl)2dX)l/2 - k , t = l \ O x k / i \ j r 2 k , t = l

where

-< I1.11 o5 ~, k = ] j = ] I ] j = l

\ 2 "~ 1/2

= Ilull lplL2¢.), (4.9)

rn p(x) = ~ I~p~(x)l 2.

j = l

Since the ~0j s are or thonormal in H, hence in 112(f~), and belong to V C IH~(f~), we have the Lieb-Thirring inequality (see Ghidaglia et al. [231, Corollary 4.3 with n = 2, p = 2, m = 1)

IPi[2(n) = in p(x)2 dx

j = l ~ k = l

= k ~ [l~jll z, (4.10) j = l

for an absolute constant k. Insert (4.10) into (4.9) to find

j=~ b(¢j, u, Cj) -< Ilull I1~112 1

,~llull 2 + v ~ i1~112. 2v 2 j = l

Global attractor for 2D Navier-Stokes flow 83

Hence (4.8) gives

v m TrF ' (u ( r ) ) o Qm(r) -< -2j~,.= II~jllZ + ~v Ilullg

_< (using (2.3))

_< _ - - kojl 2 + ~ Ilull 2 2 j = l

= (since I~0jl = 1)

1/21 /( - ~ m + ~v Ilullz"

Define the energy dissipation flux

1 f l e v21 lim sup sup / = IIs(r)Uo[I z dr, t~oo u0~0~ 7 J0

which is finite thanks to (2.15). Then, from (4.11), we find

< ~ z , m q m - 2 + 2V-~l e'

Therefore, if m ' • N is defined by

v m e N .

m ' - 1 <_ v- -~ e < m ' ,

then q,,, < 0, so that from (4.6),

ke dimn(6t) < m ' < 1 + v322.

Moreover, if m" • N is defined by

2k m " - 1 < "~v32---~e <_ m " ,

then (see [5, Lemma VI.2.2])

qm" < 0 and (qJ)+ < 1 I q . . , , J - '

so that from (4.7),

¥ j = 1 . . . . . m",

417 d i m e(a ) -< 2m" _ 2 + iv32--~ e.

Using (2.15) we can estimate the energy dissipation flux e by

/~1 e -~ --Ilfll~,, .

V

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

(4.18)

84 R. ROSA

For forces in V', if we take ).]/2 for a characteristic length for the problem (it has the Ilftlv, &l as a character- dimension of length as one can see from (2.3)), we can regard 1/2 1/4

istic velocity and define the Reynolds number

R e - Ilfll~(2 v)- ~/4 • (4.19)

We can also define the generalized Grashof number

IIfIJv, G = v2)-~/2 (= Re2). (4.20)

Therefore, using (4.18) we prove the following.

THEOREM 4.1. The global attractor 6t obtained in Theorem 3.2 has finite Hausdorff and fractal dimensions, which can be estimated in terms of the Reynolds number (4.19) and the Grashof number (4.20) by

dimn((~) < 1 + kG 2 = 1 + k R e 4 (4.21)

and

dimF(6~) < 2(1 + 2~G 2) = 2(1 + 2k Re4), (4.22)

for an absolute constant k.

Remark 4.1. In case the domain ~ is bounded and of class t32, the Hausdorff and fractal dimensions of the global attractor were estimated by Temam [24] (see also Constantin and Foias [3]) to be bounded from above by cG, for some constant c depending on the shape of the domain. This estimate is based on the asymptotic distribution of the eigenvalues [)-/Ij*°- - 1 of the 2D Stokes operator in such domains, which is 2j -~- C)-lj (see M&ivier [25]). The same bound cG can also be obtained for bounded domains less regular than e 2, as long as )-j --- c)-lj. This is the case, for instance, for Lipschitz bounded domains [25].

Acknowledgements--This work was partly supported by the CNPq-Brasflia, Brazil, under the Grant 200308/920; by the National Science Foundation under the Grant NSF-DMS-9400615; and by the Research Fund of the Indiana University. I would also like to thank Professor Roger Temam for valuable suggestions pertaining to this work.

R E F E R E N C E S

1. Ladyzhenskaya, O., On the dynamical system generated by the Navier-Stokes equations. Zapiskii ofnauch- nish seminarovs LOMI, 1972, 27, 91-114; English translation in J. o f Soviet Math., 1975, 3(4).

2. Foias, C. and Temam, R., Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations. J. Math. Pures et Appl., 1979, 58, 334-368.

3. Constantin, P. and Foias, C., Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractor for the 2D Navier-Stokes equations. Comm. Pure Appl. Math., 1985, XXXVlII, 1-27.

4. Constantin, P., Foias, C., Manley, O. and Temam, R., Determining modes and fractal dimension of turbulent flows. J. Fluid Mech., 1988, 150, 427-440.

5. Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, Vol. 68 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988.

6. Abergel, F., Attractors for a Navier-Stokes flow in an unbounded domain. Math. Mod. and Num. Anal., 1989, 23(3), 359-370.

Global attractor for 2D Navier-Stokes flow 85

7. Babin, A. V., The attractor of a Navier-Stokes system in an unbounded channel-like domain. J. Dynamics and Diff. Eqs., 1992, 4(4), 555-584.

8. Abergel, F., Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains. J. Diff. Equations, 1990, 83(1), 85-108.

9. Ladyzhenskaya, O., First boundary value problem for the Navier-Stokes equations in domains with nonsmooth boundaries. C. R. Acad. Sci. Paris, Serie I, 1992, 314, 253-258.

10. Babin, A. V. and Vishik, M. I., Attractors of partial differential equations in an unbounded domain. Proc. Roy. Soc. Edinburgh, 1990, lI6A, 221-243.

11. Feireisl, E., Laurenqot, P., Simondon, F. and Tour6, H., Compact attractors for reaction-diffusion equations in ~N. C. R. Acad. Sci. Paris, S~rie 1, 1994, 319, 147-151.

12. Ladyzhenskaya, O., Attractors for Semigroups and Evolution Equations. Lezioni Lincei, Cambridge University Press, 1991

13. Hale, J. K., Asymptotic behavior and dynamics in infinite dimensions. In Research Notes in Mathematics, eds. J. K. Hale and P. Martlnez-Amores, Vol. 132. Pitman, New York, 1985, pp. 1-420.

14. Haraux, A., Two remarks on hyperbolic dissipative problems. In Nonlinear Partial Differential Equations and Their Applications (College de France Seminar VoL VII), eds. H. Brezis and J. L. Lions. Pitman, 1985, pp. 161-179.

15. Ball, J. M., A proof of the existence of global attractors for damped semilinear wave equations. (To appear.) Cited in [16].

16. Ghidaglia, J. M., A note on the strong convergence towards the attractors for damped forced KdV equations. J. Diff. Equations, 1994, 110, 356-359.

17. Wang, X., An energy equation for the weakly damped driven nonlinear Schr6dinger equations and its appli- cations to their attractors. Physica D, 1995, 88, 167-175.

18. Temam, R., Navier-Stokes Equations. North-Holland, Amsterdam, 1979. 19. Marion, M. and Temam, R., Navier-Stokes equation-theory and approximation. Handbook of Numerical

Analysis., Vol. 6, Elsevier, North-Holland (to appear). 20. Temam, R., Navier-Stokes equations and nonlinear functional analysis. In CBMS-NSF Regional

Conference Series in Applied Mathematics, Vol. 66. SIAM, Philadelphia, 1983 (2nd edition, 1995). 21. Constantin, P., Foias, C. and Temam, R., Attractors representing turbulent flows. Mere. Amer. Math.

Soc., 1985, 53(314). 22. Ghidaglia, J. M. and Temam, R., Attractors for damped nonlinear hyperbolic equations. J. Math. Pures

Appl., 1987, 66, 273-319. 23. Ghidaglia, J. M., Marion, M. and Temam, R., Generalizations of the Sobolev-Lieb-Thirring inequalities

and application to the dimension of the attractor. Differential and Integral Equations, 1988, 1, 1-21. 24. l 'emam, R., Infinite dimensional dynamical systems in fluid mechanics. In Nonlinear Functional Analysis

and its Applications, ed. F. Browder, Vol. 45 of Proceedings o f Symposia in Pure Mathematics. American Mathematical Society, 1986, pp. 413-445.

25. M6tivier, G., Valeurs propres d'op6rateaurs d6finis par la restriction de syst6mes variationelles a des sous-espaces. J. Math. Pures Appl., 1978, 57, 133-156.