Research Article The Finite-Dimensional Uniform Attractors for the Nonautonomous g-navier-stokes Equations

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1 Hindawi Publishin Corporation Journal of Applied Mathematics Volume 2009, Article ID , 17 paes doi: /2009/ Research Article The Finite-Dimensional Uniform Attractors for the Nonautonomous -Navier-Stokes Equations Delin Wu Collee of Science, China Jilian University, Hanzhou , China Correspondence should be addressed to Delin Wu, Received 3 July 2008; Accepted 17 January 2009 Recommended by Alberto Tesi We consider the uniform attractors for the two dimensional nonautonomous -Navier-Stokes equations in bounded domain Ω. Assumin f f x, t L 2, we establish the existence of the loc uniform attractor in L 2 Ω and D A 1/2. The fractal dimension is estimated for the kernel sections of the uniform attractors obtained. Copyriht q 2009 Delin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the oriinal work is properly cited. 1. Introduction In this paper, we study the behavior of solutions of the nonautonomous 2D -Navier-Stokes equations. These equations are a variation of the standard Navier-Stokes equations, and they assume the form, u νδu u u p f in Ω, t u u u 0inΩ, where x 1,x 2 is a suitable smooth real-valued function defined on x 1,x 2 Ω and Ω is a suitable bounded domain in R 2.Noticethatif x 1,x 2 1, then 1.1 reduce to the standard Navier-Stokes equations. In addition, we assume that the function f,t : f t L 2 R; E is translation loc bounded, where E L 2 Ω or H 1 Ω. This property implies that f 2 L 2 b f 2 sup L 2 R;E b t R t 1 t f s 2 E ds <. 1.2

2 2 Journal of Applied Mathematics We consider this equation in an appropriate Hilbert space and show that there is an attractor A which all solutions approach as t. The basic idea of our construction, which is motivated by the works of 1, 2. In 1, 2 the author established the lobal reularity of solutions of the -Navier- Stokes equations. For the boundary conditions, we will consider the periodic boundary conditions, while same results can be ot for the Dirichlet boundary conditions on the smooth bounded domain. For many years, the Navier-Stokes equations were investiated by many authors and the existence of the attractors for 2D Navier-Stokes equations was first proved by Ladyzhenskaya 3, 4 and independently by Foias and Temam 5. The finite-dimensional property of the lobal attractor for eneral dissipative equations was first proved by Mallet- Paret 6. For the analysis on the Navier-Stokes equations, one can refer to 7 and specially 8 for the periodic boundary conditions. The book in 9 considers some special classes of such systems and studies systematically the notion of uniform attractor parallellin to that of lobal attractor for autonomous systems. Later on, 10 presents a eneral approach that is well suited to study equations arisin in mathematical physics. In this approach, to construct the uniform or trajectory attractors, instead of the associated process {U σ t, τ, t τ, τ R} one should consider a family of processes {U σ t, τ }, σ Σ in some Banach space E, where the functional parameter σ 0 s, s R is called the symbol and Σ is the symbol space includin σ 0 s. The approach preserves the leadin concept of invariance which implies the structure of uniform attractor described by the representation as a union of sections of all kernels of the family of processes. The kernel is the set of all complete trajectories of a process. In the paper, we study the existence of compact uniform attractor for the nonautonomous the two dimensional -Navier-Stokes equations in the periodic boundary conditions Ω. We apply measure of noncompactness method to nonautonomous -Navier- Stokes equations equation with external forces f x, t in L 2 R; E which is normal function loc see Definition 4.2. Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained. 2. Functional Settin Let Ω 0, 1 0, 1 and we assume that the function x x 1,x 2 satisfies the followin properties: 1 x C per Ω and 2 there exist constants m 0 m 0 and M 0 M 0 such that, for all x Ω, 0 < m 0 x M 0. Note that the constant function 1 satisfies these conditions. We denote by L 2 Ω, the space with the scalar product and the norm iven by as well as H 1 Ω, with the norm where u/ x i D i u. u, v u v dx, u 2 u, u, 2.1 Ω [ u H 1 Ω, u, u 2 ( Di u, D i u ) ] 1/2, 2.2 i 1

3 Journal of Applied Mathematics 3 Then for the functional settin of 1.1, we use the followin functional spaces H Cl L 2 per Ω, V { u C per Ω : u 0, } udx 0, Ω } { u Hper Ω, 1 : u 0, udx 0 Ω, 2.3 where H is endowed with the scalar product and the norm in L 2 Ω,, andv is the spaces with the scalar product and the norm iven by u, v u v dx, u u, u. 2.4 Ω Also, we define the orthoonal projection P as P : L 2 per Ω, H 2.5 and we have that Q H, where Q Cl L 2 per Ω, { φ : φ C 1 ( Ω, R )}. 2.6 Then, we define the -Laplacian operator Δ u 1 u Δu 1 u 2.7 to have the linear operator A u P [ 1 u ]. 2.8 For the linear operator A, the followin hold see 1, 2 : 1 A is a positive, self-adjoint operator with compact inverse, where the domain of A,D A V H 2 Ω,. 2 There exist countable eienvalues of A satisfyin 0 <λ λ 1 λ 2 λ 3, 2.9 where λ 4π 2 m/m and λ 1 is the smallest eienvalue of A. In addition, there exists the correspondin collection of eienfunctions {e 1,e 2,e 3,...} which forms an orthonormal basis for H.

4 4 Journal of Applied Mathematics Next, we denote the bilinear operator B u, v P u v and the trilinear form b u, v, w n ( ) u i Di v j wj dx ( P u v, w ), Ω i,j where u, v, w lie in appropriate subspaces of L 2 Ω,. Then, the form b satisfies b u, v, w b u, w, v for u, v, w H We denote a linear operator R on V by Ru P [ 1 u ] for u V, 2.12 and have R as a continuous linear operator from V into H such that Ru, u u u u m 0 m 0 λ 1/2 for u V We now rewrite 1.1 as abstract evolution equations, du dt νa u B u νru P f, u τ u τ Hereafter c will denote a eneric scale invariant positive constant, which is independent of the physical parameters in the equation and may be different from line to line and even in the same line. 3. Abstract Results Let E be a Banach space, and let a two-parameter family of mappins {U t, τ } {U t, τ t τ, τ R} act on E: U t, τ : E E, t τ, τ R. 3.1 Definition 3.1. A two-parameter family of mappins {U t, τ } is said to be a process in E if U t, s U s, τ U t, τ, t s τ, τ R, U τ, τ Id, τ R. 3.2 By B E we denote the collection of the bounded sets of E. We consider a family of processes {U σ t, τ } dependin on a parameter σ Σ. The parameter σ is said to be the

5 Journal of Applied Mathematics 5 symbol of the process {U σ t, τ } and the set Σ is said to be the symbol space. In the sequel Σ is assumed to be a complete metric space. A family of processes {U σ t, τ }, σ Σ is said to be uniformly with respect to w.r.t. σ Σ bounded if for any B B E the set U σ t, τ B B E. σ Σ τ R t τ 3.3 AsetB 0 E is said to be uniformly w.r.t. σ Σ absorbin for the family of processes {U σ t, τ }, σ Σ if for any τ R and every B B E there exists t 0 t 0 τ, B τ such that σ Σ U σ t, τ B B 0 for all t t 0. AsetP E is said to be uniformly w.r.t. σ Σ attractin for the family of processes {U σ t, τ }, σ Σ if for an arbitrary fixed τ R, ( ( lim sup dist E Uσ t, τ B, P )) 0. t σ Σ 3.4 A family of processes possessin a compact uniformly absorbin set is called uniformly compact and a family of processes possessin a compact uniformly attractin set is called uniformly asymptotically compact. Definition 3.2. AclosedsetA Σ E is said to be the uniform w.r.t. σ Σ attractor of the family of processes {U σ t, τ }, σ Σ if it is uniformly w.r.t. σ Σ attractin and it is contained in any closed uniformly w.r.t. σ Σ attractin set A of the family of processes {U σ t, τ }, σ Σ : A Σ A. A family of processes {U σ t, τ }, σ Σ actin in E is said to be E Σ,E -continuous, if for all fixed t and τ, t τ, τ R the mappin u, σ U σ t, τ u is continuous from E Σ into E. A curve u s, s R is said to be a complete trajectory of the process {U t, τ } if U t, τ u τ u t, t τ, τ R. 3.5 The kernel K of the process {U t, τ } consists of all bounded complete trajectories of the process {U t, τ }: K { u u satisfies 3.6, u s E M u for s R }. 3.6 The set K s {u s u K} E 3.7 is said to be the kernel section at time t s, s R.

6 6 Journal of Applied Mathematics For convenience, let B t σ Σ s t U σ s, t B, the closure B of the set B and R τ {t R t τ}. Define the uniform w.r.t. σ Σ ω-limit set ω τ,σ B of B by ω τ,σ B t τ B t which can be characterized, analoously to that for semiroups, the followin: y ω τ,σ B there are sequences { x n } B, { σn } Σ, { tn } Rτ such that t n, U σn t n,τ x n y n. 3.8 We recall characterize the existence of the uniform attractor for a family of processes satisfyin 3.8 in term of the concept of measure of noncompactness that was put forward first by Kuratowski see 11, 12. Let B B E. Its Kuratowski measure of noncompactness κ B is defined by κ B inf{δ >0 B admits a finite coverin by sets of diameter δ}. 3.9 Definition 3.3. A family of processes {U σ t, τ }, σ Σ is said to be uniformly w.r.t. σ Σ ω-limit compact if for any τ R and B B E the set B t is bounded for every t and lim t κ B t 0. We present now a method to verify the uniform w.r.t. σ Σ ω-limit compactness see 13, 14. Definition 3.4. A family of processes {U σ t, τ }, σ Σ is said to satisfy uniformly w.r.t. σ Σ Condition C if for any fixed τ R, B B E and ε>0, there exist t 0 t τ, B, ε τ and a finite-dimensional subspace E 1 of E such that i P σ Σ t t 0 U σ t, τ B is bounded; and ii I P σ Σ t t 0 U σ t, τ x ε, x B, where P : E E 1 is a bounded projector. Therefore we have the followin results. Theorem 3.5. Let Σ be a metric space and let {T t } be a continuous invariant semiroup T t Σ Σ on Σ. A family of processes {U σ t, τ }, σ Σ actin in E is E Σ,E -(weakly) continuous and possesses the compact uniform w.r.t. σ Σ attractor A Σ satisfyin ( ) ( ) A Σ ω 0,Σ B0 ωτ,σ B0 K σ 0, τ R, 3.10 σ Σ if it i has a bounded uniformly w.r.t. σ Σ absorbin set B 0, and ii satisfies uniformly w.r.t. σ Σ Condition C Moreover, if E is a uniformly convex Banach space then the converse is true.

7 Journal of Applied Mathematics 7 4. Uniform Attractor of Nonautonomous -Navier-Stokes Equations This section deals with the existence of the attractor for the two-dimensional nonautonomous -Navier-Stokes equations with periodic boundary condition see 1, 2. It is similar to autonomous case that we can establist the existence of solution of 2.14 by the standard Faedo-Galerkin method. In 1, 2, the authors have shown that the semiroup S t : H H t 0 associated with the autonomous systems 2.14 possesses a lobal attractor. The main objective of this section is to prove that the nonautonomous system 2.14 has uniform attractors in H and V. To this end, we first state some the followin results of existence and uniqueness of solutions of Proposition 4.1. Let f V be iven. Then for every u τ H there exists a unique solution u u t on 0, of 2.14, satisfyin u τ u τ. Moreover,one has u t C [ 0,T; H ) L 2 ( 0,T; V ), T > Finally, if u τ V,then u t C [ 0,T; V ) L 2 ( 0,T; D ( A )), T > Proof. The Proof of Proposition 4.1 is similar to autonomous in 1, 15. Now we will write 2.14 in the operator form t u A σ t u, u t τ u τ, 4.3 where σ s f x, s is the symbol of 4.3.Thus,ifu τ H, then problem 4.3 has a unique solution u t C 0,T ; H L 2 0,T ; V. This implies that the process {U σ t, τ } iven by the formula U σ t, τ u τ u t is defined in H. Now recall the followin facts that can be found in 13. Definition 4.2. A function ϕ L 2 R; E is said to be normal if for any ε>0, there exists η>0 loc such that sup t R t η t ϕ s 2 Eds ε. 4.4 Remark 4.3. Obviously, L 2 n R; E L 2 b R; E. Denote by L2 c R; E the class of translation compact functions f s, s R, whose family of H f is precompact in L 2 R; E. It is proved loc in 13 that L 2 n R; E and L 2 c R; E are closed subspaces of L 2 R; E, but the latter is a proper b subset of the former for further details see 13. We now define the symbol space H σ 0 for 4.3. Let a fixed symbol σ 0 s f 0 s f 0,s be normal functions in L 2 loc R; E ; that is, the family of translation {f 0 s h,h R}

8 8 Journal of Applied Mathematics forms a normal function set in L 2 loc T 1,T 2 ; E, where T 1,T 2 is an arbitrary interval of the time axis R. Therefore H ( σ 0 ) H ( f0 ) [ f0 x, s h h R ] L 2 loc R;E. 4.5 Now, for any f x, t H f 0, the problem 4.3 with f instead of f 0 possesses a correspondin process {U f t, τ } actin on V. As is proved in 10, the family {U f t, τ f H f 0 } of processes is V H f 0 ; V -continuous. Let K f { u f x, t for t R u f x, t is solution of 4.3 satisfyin uf,t H M f t R } 4.6 be the so-called kernel of the process {U f t, τ }. Proposition 4.4. The process {U f t, τ } : H H V associated with the 4.3 possesses absorbin sets B 0 { } u H u ρ 0, B1 { } u V u ρ which absorb all bounded sets of H. Moreover B 0 and B 1 absorb all bounded sets of H and V in the norms of H and V, respectively. Proof. The proof of Proposition 4.4 is similar to autonomous -Navier-Stokes equation. We can obtain absorbin sets in H and V the followin from 1 and the proof of the main results as follow. The main results in this section are as follows. Theorem 4.5. If f 0 x, s is normal function in L 2 loc R; V, then the processes {U f0 t, τ } correspondin to problem 2.14 possess compact uniform w.r.t. τ R attractor A 0 in H which coincides with the uniform w.r.t. f H f 0 attractor A H f0 of the family of processes {U f t, τ }, f H f 0 : ( ) A 0 A H f0 ω 0,H f0 B0 f H f 0 K f 0, 4.8 where B 0 is the uniformly w.r.t. f H f 0 absorbin set in H and K f is the kernel of the process {U f t, τ }. Furthermore, the kernel K f is nonempty for all f H f 0. Proof. As in the previous section, for fixed N,letH 1 be the subspace spanned by w 1 ;...; w N, and H 2 the orthoonal complement of H 1 in H. We write u u 1 u 2 ; u 1 H 1,u 2 H 2 for any u H. 4.9 Now, we only have to verify Condition C. Namely, we need to estimate u 2 t 2, where u t u 1 t u 2 t is a solution of 2.14 iven in Proposition 4.1.

9 Journal of Applied Mathematics 9 Multiplyin 2.14 by u 2, we have ( ) du dt,u 2 ( ) νa u, u 2 ( ) B u, u,u 2 ( ) f, u 2 ( Ru, u 2 ) It follows that 1 d u2 2 2 dt ν u 2 ( ( ) ( B u, u,u 2 ) f, u2 Ru, u2 ) Since b satisfies the followin inequality see 15 : b u, v, w c u 1/2 u 1/2 v w 1/2 w 1/2, u, v, w V, 4.12 thus, ( B u, u,u 2 ) c u 1/2 u 3/2 u2 1/2 u2 1/2 c u 1/2 u 3/2 u2 λ m 1 ν u2 2 6 cρ 0ρ Next, the Cauchy inequality, ( Ru, u 2 ) ( ) ν u, u 2 ν m 0 u u ν u νρ m 2 0 λ. λ m 1 Finally, we have ( f, u 2 ) f V u2 ν u ν f 2 V Puttin toether, there exist constant M 1 M 1 m 0,,ρ 0,ρ 1 such that 1 d u2 2 2 dt 1 2 ν u2 2 3 f V M 1. 2ν 4.16

10 10 Journal of Applied Mathematics Therefore, we deduce that d u2 2 dt νλ m 1 u M 1 3 ν f 2 V Here M 1 depends on λ m 1, is not increasin as λ m 1 increasin. By the Gronwall inequality, the above inequality implies u 2 t 2 ( u 2 t0 1 ) 2 2 e νλ m 1 t t 0 1 2M 1 3 t e νλm 1 t s f 2 νλ m 1 ν V ds. t Applyin 4.4 for any ε 3 t e νλm 1 t s f 2 ν V ds < ε t Usin 2.9 and let t 1 t 0 1 1/νλ m 1 ln 3ρ 2 0 /ε, then t t 1 implies 2M < ε νλ m 1 3 ; u 2 t e νλ m 1 t t 0 1 ρ 2 0 e νλ m 1 t t 0 1 /2 < ε Therefore, we deduce from 4.18 that u2 2 2 ε, t t 1,f H ( f 0 ), 4.21 which indicates {U f t, τ }, f H f 0 satisfyin uniform w.t.r. f H f 0 Condition C in H. Applyin Theorem 3.5 the proof is complete. Theorem 4.6. If f 0 x, s is normal function in L 2 loc R; H, then the processes {U f0 t, τ } correspondin to problem 2.14 possesses compact uniform w.r.t. τ R attractor A 1 in V which coincides with the uniform w.r.t. f H f 0 attractor A H f0 of the family of processes {U f t, τ }, f H f 0 : ( ) A 1 A H f0 ω 0,H f0 B1 K f 0, f H f where B 1 is the uniformly w.r.t. f H f 0 absorbin set in V and K f is the kernel of the process {U f t, τ }. Furthermore, the kernel K f is nonempty for all f H f 0. Proof. Usin Proposition 4.4, we have the family of processes {U f t, τ }, f H f 0 correspondin to 4.3 possesses the uniformly w.r.t. f H f 0 absorbin set in V. Now we prove the existence of compact uniform w.r.t. f H f 0 attractor in V by applyin the method established in Section 3, that is, we testify that the family of processes

11 Journal of Applied Mathematics 11 {U f t, τ }, f H f 0 correspondin to 4.3 satisfies uniform w.r.t. f H f 0 Condition C. Multiplyin 2.14 by A u 2 t, we have ( ) dv dt,a u 2 ( ) ( ) νa u, A u 2 B u, u,a u 2 ( ) ( f, A u 2 Ru, A u 2 ) It follows that 1 d u dt ν A u 2 2 ( B u, u,a u 2 ) ( ) f, A u 2 ( ) Ru, A u To estimate B u, u,au 2, we recall some inequalities 16 : for every u, v D A : B u, v c u 1/2 u 1/2 v 1/2 u 1/2 A u 1/2 A v 1/2, v 4.25 see 16 w L Ω 2 ( A c w w ) 1/2 1 lo 4.26 λ w 2 from which we deduce that B u, v c u L Ω v u v L Ω, 4.27 and usin, 4.26 ( u v B u, v c 1 lo A ) u 2 1/2, λ w 2 u A v A 3/2 v (1 2 lo λ A v 2 )1/ Expandin and usin Youn s inequality, toether with the first one of 4.28 and the second one of 4.25, we have ( ) ( ( ) ) ( ( ) ) B u, u,a u 2 B u1,u 1 u 2,A u 2 B u2,u 1 u 2,A u 2 cl 1/2 ( u1 A u u1 2 ) u2 cu2 1/2 A u 2 3/2 ν A u c ν ρ4 1 L c ν 3 ρ2 0 ρ4 1, t t 0 1, 4.29

12 12 Journal of Applied Mathematics where we use A u 1 2 λ m u and set L 1 lo λ m 1 λ Next, usin the Cauchy inequality, ( Ru, A u 2 ) ( ν u, A u 2 ) ν m 0 u A u ν A u ν 2 2 ρ 2 1. Finally, we estimate f, A u 2 by ( ) f, A u 2 f A u 2 2 ν A u ν f Puttin toether, there exists a constant M 2 such that d u 2 2 dt νλ m 1 u ν f M Here M 2 M 2 ρ 0,ρ 1,L,ν, depends on λ m 1, is not increasin as λ m 1 increasin. Therefore, by the Gronwall inequality, the above inequality implies u 2 2 ( u 2 t0 1 ) 2 e νλ m 1 t t 0 1 2M 2 3 t e νλm 1 t s f 2 νλ m 1 ν ds. t Applyin 4.4 for any ε 3 t e νλm 1 t s f 2 ν ds < ε t

13 Journal of Applied Mathematics 13 Usin 2.9 and let t 1 t 0 1 1/νλ m 1 ln 3ρ 2 1 /ε, then t t 1 implies 2M 2 < ε νλ m 1 3 ; ( u2 t0 1 ) 2 e νλ m 1 t t 0 1 ρ 2 1 e νλ m 1 t t 0 1 < ε Therefore, we deduce from 4.35 that u2 2 ε, t t 1, f H ( f 0 ), 4.38 which indicates {U f t, τ }, f H f 0 satisfyin uniform w.t.r. f H f 0 Condition C in V. 5. Dimension of the Uniform Attractor In this section we estimate the fractal dimension for definition see, e.., 2, 10, 15 of the kernel sections of the uniform attractors obtained in Section 4 by applyin the methods in 17. Process {U t, τ } is said to be uniformly quasidifferentiable on {K s } τ R, if there is a family of bounded linear operators {L t, τ; u u K s, t τ, τ R}, L t, τ; u : E E such that lim sup ε 0 τ R U t, τ v U t, τ u L t, τ; u v u sup 0. u,v K s v u 0< u v ε 5.1 We want to estimate the fractal dimension of the kernel sections K s of the process {U t, τ } enerated by the abstract evolutionary Assume that {L t, τ; u } is enerated by the variational equation correspondin to 2.14 t w F u, t w, w t τ w τ E, t τ, τ R, 5.2 that is, L t, τ; u τ w τ w t is the solution of 5.2, andu t U t, τ u τ is the solution of 2.14 with initial value u τ K τ. For natural number j N, weset ( 1 q j lim sup sup T T τ R u τ K τ τ T τ Tr ( F u s,s ) ) ds, 5.3 where Tr is trace of the operator. We will need the followin Theorem VIII.3.1 in 10 and 2.

14 14 Journal of Applied Mathematics Theorem 5.1. Under the assumptions above, let us suppose that U τ R K τ is relatively compact in E, and there exists q j,j 1, 2,..., such that q j q j, for any j 1, q n0 0, q n0 1 < 0, for some n 0 1, q j q n0 ( q n0 q n0 1)( n0 j ), j 1, 2, Then, q n0 d F K τ d 0 : n 0, τ R. 5.5 q n0 q n0 1 We now consider 2.14 with f L 2 n R; V. The equations possess a compact uniform w.r.t. f H f attractor A H f and τ R K f τ A H f.by 2, 10, 15, we know that the associated process {U f t, τ } is uniformly quasidifferentiable on {K f τ } τ R and the quasidifferential is Hölder-continuous with respect to u τ K f τ. The correspondin variational equation is t w νa u B u νru P f F u t,t w, w t τ w τ E, τ R. 5.6 We have the main results in this section. Theorem 5.2. Suppose that f t satisfies the assumptions of Theorem 4.5. Then, if γ 1 2 / m 0 λ 1/2 > 0, the Uniform attractor A 0 defined by 4.8 satisfies d F ( A0 ) β α, 5.7 where α c 2νm 0 λ 1 γ 2M 0, β c 1d 1 2ν 3 m 0 γ sup ϕ j H ϕ j 1 j 1,2,...,m 1 T τ T τ f s 2 V ds, 5.8 the constant c 1,c 2 of (3.29) and (3.32) of Chapter VI in [15] and [2], λ 1 is the first eienvalue of the Stokes operator and d 1 2 /4m 0 M 0. Proof. With Theorem 4.5 at our disposal we may apply the abstract framework in 2, 10, 15, 17. For ξ 1,ξ 2,...,ξ m H,letv j t L t, u τ ξ j, where u τ H.Let{ϕ j s ; j 1, 2,...,m} be an orthonormal basis for span {v j ; j 1, 2,...,m}. Since v j V almost everywhere s τ, we can also assume that ϕ j s V almost everywhere s τ. Then, similar to the Proof

15 Journal of Applied Mathematics 15 process of Theorems 4.5 and 4.6, we may obtain m ( m m F U s, τ u τ,s ϕ i,ϕ i ) ν ϕ j 2 ( ) b ϕj,u s, τ u τ,ϕ j i 1 i 1 i 1 i 1 m ( ) ν ϕ j,ϕ j, 5.9 almost everywhere s τ. From this equality, and in particular usin the Schwarz and Lieb- Thirrin inequality see 2, 10, 15, 17, oneobtains m ϕ 2 λ 1 λ m m 0 ( λ 1 M ) m 0 λ m c 2 λ 1 i 1 0 M m2, 0 ( Tr j F U s, τ u τ,s ) ( ν 1 ) m ϕ m 0 λ 1/2 j 2 ( c1 U s, τ u d 1 m τ ϕ j 2 m 1 i 1 0 i 1 ν ( 1 2 ) m ϕj 2 2 m 0 λ 1/2 c 1d 1 U s, τ uτ 2 2νm 1 i 1 0 νm ( ) c 2M 0 m 0 λ 1/2 2 λ 1 m2 c 1d 1 U s, τ uτ 2 2νm, 0 1 ) 1/ on the other hand, we can deduce 2.14 that d U s, τ u τ 2 dt ν f 2 U s, τ u τ 2 V ν 2ν m 0 λ 1/2 U s, τ u τ for λ 4π 2 m 0 /M 0, and then t τ ( U s, τ uτ 2 1 t ds f s 2 ν 2 V ds u τ 2 ) ( 1 2 ) 1, t τ τ ν m 0 λ 1/2 Now we define q m sup ϕ j H ϕ j 1 j 1,2,...,m ( 1 T τ T τ Tr j ( F ( U s, τ u τ,s ) ds )), 5.13

16 16 Journal of Applied Mathematics Usin Theorem 5.1, we have q m νm 0 2M 0 ( 1 2 m 0 λ 1/2 1 ) νm ( ) c 2M 0 m 0 λ 1/2 2 λ 1 m2 1 c ( 1d νm sup 0 ν 2 T ϕ j H ϕ j 1 j 1,2,...,m c 2 λ 1 m2 c 1d 1 2νm 0 τ T τ sup ϕ j H ϕ j 1 j 1,2,...,m ( 1 T τ T τ ) U s, τ uτ 2 ds f s 2 V ds u τ 2 ) ( 1 2 ) 1 νt m 0 λ 1/2, 5.14 q m lim sup q m αm 2 β, T Hence dim F A 0 τ β α Acknowledment The author would like to thank the reviewers and the editor for their valuable suestions and comments. References 1 J. Roh, Dynamics of the -Navier-Stokes equations, Journal of Differential Equations, vol. 211, no. 2, pp , M. Kwak, H. Kwean, and J. Roh, The dimension of attractor of the 2D -Navier-Stokes equations, Journal of Mathematical Analysis and Applications, vol. 315, no. 2, pp , O. A. Ladyzhenskaya, Über ein dynamisches System, das durch die Navier-Stokesschen Gleichunen erzeut wird, Zapiskii of Nauchnich, Seminarov LOMI, vol. 27, pp , O. A. Ladyzhenskaya, A dynamical system enerated by the Navier-Stokes equations, Journal of Mathematical Sciences, vol. 3, no. 4, pp , C. Foiaş and R. Temam, Finite parameter approximative structure of actual flows, in Nonlinear Problems: Present and Future, A. R. Bishop, D. K. Campbell, and B. Nicolaenko, Eds., vol. 61 of North- Holland Mathematics Studies, pp , North-Holland, Amsterdam, The Netherlands, J. Mallet-Paret, Neatively invariant sets of compact maps and an extension of a theorem of Cartwriht, Journal of Differential Equations, vol. 22, no. 2, pp , P. Constantin and C. Foiaş, Navier-Stokes Equations, Chicao Lectures in Mathematics, University of Chicao Press, Chicao, Ill, USA, R. Temam, Navier-Stokes Equations, vol. 2 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 3rd edition, A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, vol. 17 of Recherches en Mathématiques Appliquées, Masson, Paris, France, 1991.

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