The Finite-Dimensional Uniform Attractors for the Nonautonomous g-Navier-Stokes Equations

Volume 2009, Article ID 150420,17pages doi:10.1155/2009/150420

Research Article

The Finite-Dimensional Uniform Attractors for the Nonautonomous g-Navier-Stokes Equations

Delin Wu

College of Science, China Jiliang University, Hangzhou 310018, China

Correspondence should be addressed to Delin Wu,[email protected] Received 3 July 2008; Accepted 17 January 2009

Recommended by Alberto Tesi

We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domainΩ. Assumingf fx, t ∈ L²_loc, we establish the existence of the uniform attractor inL²ΩandDA^1/2. The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

Copyrightq2009 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 original work is properly cited.

1. Introduction

In this paper, we study the behavior of solutions of the nonautonomous 2D g-Navier-Stokes equations. These equations are a variation of the standard Navier-Stokes equations, and they assume the form,

∂u

∂t −νΔu u· ∇u∇pf inΩ, 1

g∇ ·gu ∇g

g ·u∇ ·u0 inΩ, 1.1

whereg gx1, x₂ is a suitable smooth real-valued function defined onx1, x₂ ∈ Ωand Ωis a suitable bounded domain inR². Notice that ifgx1, x₂ 1, then1.1reduce to the standard Navier-Stokes equations.

In addition, we assume that the function f·, t : ft ∈ L²_locR;E is translation bounded, whereEL²ΩorH⁻¹Ω. This property implies that

f²_L2

b f²_L2

bR;Esup

t∈R

_t1

t

fs²_Eds <∞. 1.2

We consider this equation in an appropriate Hilbert space and show that there is an attractorAwhich all solutions approach ast → ∞. The basic idea of our construction, which is motivated by the works of1,2 .

In 1, 2 the author established the global regularity of solutions of the g-Navier- Stokes equations. For the boundary conditions, we will consider the periodic boundary conditions, while same results can be got for the Dirichlet boundary conditions on the smooth bounded domain. For many years, the Navier-Stokes equations were investigated by many authors and the existence of the attractors for 2D Navier-Stokes equations was first proved by Ladyzhenskaya3,4 and independently by Foias and Temam5 . The finite-dimensional property of the global attractor for general dissipative equations was first proved by Mallet- Paret6 . For the analysis on the Navier-Stokes equations, one can refer to7 and specially 8 for the periodic boundary conditions.

The book in 9 considers some special classes of such systems and studies sy- stematically the notion of uniform attractor parallelling to that of global attractor for autonomous systems. Later on,10 presents a general approach that is well suited to study equations arising in mathematical physics. In this approach, to construct the uniform or trajectoryattractors, instead of the associated process{Uσt, τ, t ≥ τ, τ ∈ R}one should consider a family of processes{Uσt, τ},σ∈Σin some Banach spaceE, where the functional parameterσ₀s, s ∈Ris called the symbol andΣis the symbol space includingσ₀s. The approach preserves the leading 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 non- autonomous the two dimensional g-Navier-Stokes equations in the periodic boundary conditionsΩ. We apply measure of noncompactness method to nonautonomous g-Navier- Stokes equations equation with external forcesfx, tinL²_locR;Ewhich is normal function see Definition 4.2. Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

2. Functional Setting

LetΩ 0,1×0,1and we assume that the functiongx gx1, x₂satisfies the following properties:

1gx∈C^∞_perΩand

2there exist constantsm₀ m₀gandM₀ M₀gsuch that, for allx ∈ Ω,0 <

m0≤gx≤M0. Note that the constant functiong≡1 satisfies these conditions.

We denote byL²Ω, gthe space with the scalar product and the norm given by u, v_g

Ωu·vg dx, |u|²_g u, u_g, 2.1 as well asH¹Ω, gwith the norm

uH¹Ω,g

u, u_g²

i1

D_iu, D_iu

g

1/2

, 2.2

where∂u/∂x_iD_iu.

Then for the functional setting of1.1, we use the following functional spaces

HgCl_L²_perΩ,g

u∈C^∞_perΩ:∇ ·gu0,

Ωu dx0 , V_g

u∈H_per¹ Ω, g:∇ ·gu0,

Ωu dx0 ,

2.3

whereH_gis endowed with the scalar product and the norm inL²Ω, g, andV_gis the spaces with the scalar product and the norm given by

u, v_g

Ω∇u· ∇vg dx, ug u, u_g. 2.4

Also, we define the orthogonal projectionPgas

Pg :L²_perΩ, g−→Hg 2.5

and we have thatQ⊆H_g^⊥, where

QCl_L²_perΩ,g

∇φ:φ∈C¹ Ω,R

. 2.6

Then, we define theg-Laplacian operator

−Δgu≡ 1

g∇ ·g∇u−Δu− 1

g∇g· ∇u 2.7

to have the linear operator

A_guP_g

− 1

g∇ ·g∇u

. 2.8

For the linear operatorAg, the following holdsee1,2 :

1Agis a positive, self-adjoint operator with compact inverse, where the domain of A_g, DAg V_g∩H²Ω, g.

2There exist countable eigenvalues ofAgsatisfying

0< λg≤λ1≤λ2≤λ3 ≤ · · ·, 2.9 whereλ_g 4π²m/M andλ₁ is the smallest eigenvalue ofA_g. In addition, there exists the corresponding collection of eigenfunctions{e1, e2, e3, . . .}which forms an orthonormal basis forH_g.

Next, we denote the bilinear operatorBgu, v Pgu· ∇vand the trilinear form

b_gu, v, w ⁿ

i,j1

Ωu_i D_iv_j

w_jg dx

P_gu· ∇v, w

g, 2.10

whereu, v, wlie in appropriate subspaces ofL²Ω, g. Then, the formb_gsatisfies

bgu, v, w −bgu, w, v foru, v, w∈Hg. 2.11 We denote a linear operatorRonVgby

RuPg

1

g∇g· ∇u

foru∈Vg, 2.12

and haveRas a continuous linear operator fromV_gintoH_gsuch that

|Ru, u| ≤ |∇g|_∞

m0 ug|u|g≤ |∇g|_∞

m₀λ^1/2_g ug foru∈V_g. 2.13

We now rewrite1.1as abstract evolution equations, du

dt νAguBguνRuPgf, uτ u_τ. 2.14 Hereafter c will denote a generic scale invariant positive constant, which is ind- ependent of the physical parameters in the equation and may be different from line to line and even in the same line.

3. Abstract Results

LetEbe a Banach space, and let a two-parameter family of mappings{Ut, τ} {Ut, τ| t≥τ, τ ∈R}act onE:

Ut, τ:E−→E, t≥τ, τ∈R. 3.1

Definition 3.1. A two-parameter family of mappings{Ut, τ}is said to be a process inEif Ut, sUs, τ Ut, τ, ∀t≥s≥τ, τ ∈R,

Uτ, τ Id, τ∈R. 3.2

By BEwe denote the collection of the bounded sets of E. We consider a family of processes{Uσt, τ} depending on a parameter σ ∈ Σ. The parameter σ is said to be the

symbol of the process{Uσt, τ}and the setΣis said to be the symbol space. In the sequelΣis assumed to be a complete metric space.

A family of processes {Uσt, τ}, σ ∈ Σ is said to be uniformly with respect to w.r.t.σ∈Σbounded if for anyB∈ BEthe set

σ∈Σ

τ∈R

t≥τ

U_σt, τB∈ BE. 3.3

A setB₀⊂Eis said to be uniformlyw.r.t. σ ∈Σabsorbing for the family of processes {Uσt, τ},σ ∈Σif for anyτ ∈Rand everyB ∈ BEthere existst0 t0τ, B≥τ such that

σ∈ΣU_σt, τB⊆B₀for allt≥t₀.

A setP ⊂ Eis said to be uniformlyw.r.t. σ ∈Σattracting for the family of processes {Uσt, τ}, σ∈Σif for an arbitrary fixedτ∈R,

tlim→∞

sup

σ∈Σ dist_E

U_σt, τB, P

0. 3.4

A family of processes possessing a compact uniformly absorbing set is called uniformly compact and a family of processes possessing a compact uniformly attracting set is called uniformly asymptotically compact.

Definition 3.2. A closed setA_Σ ⊂ Eis said to be the uniformw.r.t. σ ∈ Σattractor of the family of processes {Uσt, τ}, σ ∈ Σ if it is uniformly w.r.t. σ ∈ Σattracting and it is contained in any closed uniformlyw.r.t. σ ∈Σattracting setAof the family of processes {Uσt, τ}, σ∈Σ: A_Σ⊆ A.

A family of processes{Uσt, τ}, σ ∈Σacting inEis said to beE×Σ, E-continuous, if for all fixedtandτ, t≥τ, τ ∈Rthe mappingu, σ→Uσt, τuis continuous fromE×Σ intoE.

A curveus, s∈Ris said to be a complete trajectory of the process{Ut, τ}if

Ut, τuτ ut, ∀t≥τ, τ∈R. 3.5

The kernelKof the process{Ut, τ}consists of all bounded complete trajectories of the process{Ut, τ}:

K

u·|u·satisfies3.6, usE≤M_u fors∈R

. 3.6

The set

Ks {us|u·∈ K} ⊆E 3.7

is said to be the kernel section at timets, s∈R.

For convenience, letBt

σ∈Σ

s≥tUσs, tB, the closureBof the setBandRτ {t∈ R | t ≥ τ}. Define the uniformw.r.t. σ ∈ Σω-limit setω_τ,ΣBofBbyω_τ,ΣB

t≥τB_t which can be characterized, analogously to that for semigroups, the following:

y∈ω_τ,ΣB⇐⇒there are sequences xn

⊂B, σn

⊂Σ, tn

⊂Rτ

such thattn−→∞, Uσntn, τxn−→yn−→ ∞. 3.8 We recall characterize the existence of the uniform attractor for a family of processes satisfying3.8in term of the concept of measure of noncompactness that was put forward first by Kuratowskisee11,12 .

LetB∈ BE. Its Kuratowski measure of noncompactnessκBis defined by

κB inf{δ >0|Badmits a finite covering by sets of diameter≤δ}. 3.9

Definition 3.3. A family of processes {Uσt, τ}, σ ∈ Σ is said to be uniformlyw.r.t. σ ∈ Σω-limit compact if for any τ ∈ Rand B ∈ BE the setB_t is bounded for every tand lim_{t→ ∞}κBt 0.

We present now a method to verify the uniformw.r.t. σ ∈ Σω-limit compactness see13,14 .

Definition 3.4. A family of processes{Uσt, τ}, σ ∈Σis said to satisfy uniformlyw.r.t. σ ∈ ΣConditionCif for any fixedτ ∈R, B ∈ BEandε >0, there existt0 tτ, B, ε≥τand a finite-dimensional subspaceE₁ofEsuch that

iP

σ∈Σ

t≥t0U_σt, τBis bounded; and iiI−P

σ∈Σ

t≥t0U_σt, τx ≤ε,∀x∈B, whereP:E → E₁is a bounded projector.

Therefore we have the following results.

Theorem 3.5. LetΣbe a metric space and let{Tt}be a continuous invariant semigroupTtΣ Σ onΣ. A family of processes{Uσt, τ}, σ ∈ Σacting in EisE×Σ, E-(weakly) continuous and possesses the compact uniformw.r.t. σ ∈ΣattractorA_Σsatisfying

A_Σω_0,Σ B0

ω_τ,Σ B0

σ∈Σ

Kσ0, ∀τ∈R, 3.10

if it

ihas a bounded uniformlyw.r.t. σ∈Σabsorbing setB₀, and iisatisfies uniformlyw.r.t. σ∈ΣConditionC

Moreover, ifEis a uniformly convex Banach space then the converse is true.

4. Uniform Attractor of Nonautonomous g-Navier-Stokes Equations

This section deals with the existence of the attractor for the two-dimensional nonautonomous g-Navier-Stokes equations with periodic boundary conditionsee1,2 .

It is similar to autonomous case that we can establist the existence of solution of2.14 by the standard Faedo-Galerkin method.

In1,2 , the authors have shown that the semigroupSt:H_g → H_gt≥0associated with the autonomous systems2.14possesses a global attractor. The main objective of this section is to prove that the nonautonomous system 2.14 has uniform attractors in Hg

andV_g.

To this end, we first state some the following results of existence and uniqueness of solutions of2.14.

Proposition 4.1. Letf ∈Vbe given. Then for everyu_τ ∈H_gthere exists a unique solutionuut on0,∞of 2.14, satisfyinguτ u_τ. Moreover,one has

ut∈C

0, T;H_g

∩L²

0, T;V_g

, ∀T >0. 4.1

Finally, ifu_τ ∈V_g, then

ut∈C 0, T;Vg

∩L² 0, T;D

Ag

, ∀T >0. 4.2

Proof. The Proof ofProposition 4.1is similar to autonomous in1,15 . Now we will write2.14in the operator form

∂tuA_σtu, u|tτ uτ, 4.3

whereσs fx, sis the symbol of4.3. Thus, ifuτ ∈Hg, then problem4.3has a unique solutionut∈C0, T ;H_g∩L²0, T ;V_g. This implies that the process{Uσt, τ}given by the formulaU_σt, τuτutis defined inH_g.

Now recall the following facts that can be found in13 .

Definition 4.2. A functionϕ∈L²_locR;Eis said to be normal if for anyε >0, there existsη >0 such that

sup

t∈R

_tη

t

ϕs²_Eds≤ε. 4.4

Remark 4.3. Obviously, L²_nR;E ⊂ L²_bR;E. Denote by L²_cR;E the class of translation compact functionsfs, s∈R, whose family ofHfis precompact inL²_locR;E. It is proved in13 thatL²_nR;EandL²_cR;Eare closed subspaces ofL²_bR;E, but the latter is a proper subset of the formerfor further details see13 .

We now define the symbol spaceHσ0for 4.3. Let a fixed symbolσ0s f0s f₀·, sbe normal functions inL²_locR;E; that is, the family of translation{f0sh, h ∈ R}

forms a normal function set inL²_locT1, T2 ;E, whereT1, T2 is an arbitrary interval of the time axisR. Therefore

H σ0

H f0

f0x, sh|h∈R

L²_locR;E. 4.5

Now, for any fx, t ∈ Hf0, the problem 4.3 with f instead of f0 possesses a corresponding process{Uft, τ}acting on V_g. As is proved in 10 , the family{Uft, τ | f∈ Hf0}of processes isVg× Hf0;V_g-continuous.

Let Kf

u_fx, tfort∈R|u_fx, tis solution of4.3satisfying u_f·, t

H≤M_f ∀t∈R 4.6

be the so-called kernel of the process{Uft, τ}.

Proposition 4.4. The process {Uft, τ} : H_g → H_gVg associated with the 4.3 possesses absorbing sets

B0

u∈H_g| |u|g≤ρ₀

, B1

u∈V_g| ug ≤ρ₁

4.7 which absorb all bounded sets ofH_g. MoreoverB0andB1absorb all bounded sets ofH_gandV_gin the norms ofHgandVg, respectively.

Proof. The proof of Proposition 4.4is similar to autonomous g-Navier-Stokes equation. We can obtain absorbing sets in H_g and V_g the following from1 and the proof of the main results as follow.

The main results in this section are as follows.

Theorem 4.5. If f₀x, s is normal function in L²_locR;V_g, then the processes {Uf0t, τ} corresponding to problem 2.14 possess compact uniform w.r.t. τ ∈ R attractor A0 in H_g which coincides with the uniform w.r.t. f ∈ Hf0 attractor AHf0 of the family of processes {Uft, τ}, f ∈ Hf0:

A0 A_Hf0ω_0,Hf0 B0

f∈Hf0

Kf0, 4.8

whereB0is the uniformlyw.r.t. f ∈ Hgf0absorbing set inHgandKfis the kernel of the process {Uft, τ}. Furthermore, the kernelKf is nonempty for allf ∈ Hf0.

Proof. As in the previous section, for fixedN, letH₁be the subspace spanned byw₁;. . .;w_N, andH2the orthogonal complement ofH1inHg. We write

uu₁u₂; u₁∈H₁, u₂∈H₂ for any u∈H_g. 4.9 Now, we only have to verify Condition C. Namely, we need to estimate |u2t|₂, whereut u₁t u₂tis a solution of2.14given inProposition 4.1.

Multiplying2.14byu2, we have du

dt, u₂

g

νA_gu, u₂

g

Bu, u, u2

g

f, u₂

g− Ru, u₂

g. 4.10

It follows that 1 2

d dtu2²

gνug²

g≤Bu, u, u2

gf, u2

g Ru, u2

g. 4.11

Sinceb_gsatisfies the following inequalitysee15 :

b_gu, v, w≤c|u|^1/2_g u^1/2_g vg|w|^1/2_g w^1/2_g , ∀u, v, w∈V_g, 4.12

thus,

Bu, u, u2

g≤c|u|^1/2_g u^3/2_g u₂^1/2

g u₂^1/2

g

≤ c

λ_m1|u|^1/2_g u^3/2_g u2

g

≤ ν 6u₂²

gcρ₀ρ₁³.

4.13

Next, the Cauchy inequality,

Ru, u₂

g

ν

g∇g· ∇u, u2

g

≤ ν

m₀|∇g|∞ugu₂

g

≤ ν 6u₂²

g 3νρ²₁|∇g|²_∞ 2m²₀λ_gλ_m1.

4.14

Finally, we have

f, u2

g≤ |f|Vgu2≤ ν 6u2²

g 3 2ν|f|²_V

g. 4.15

Putting4.13–4.15together, there exist constantM1 M1m0,|∇g|_∞, ρ0, ρ1such that 1

2 d dtu2²

g1 2νu2²

g≤ 3|f|Vg

2ν M1. 4.16

Therefore, we deduce that d dtu₂²

gνλ_m1u₂²

2≤2M₁ 3 ν|f|²_V

g. 4.17

HereM₁depends onλ_m1, is not increasing asλ_m1increasing.

By the Gronwall inequality, the above inequality implies u2t²

g ≤u2

t01²

2e^{−νλm1t−t01} 2M1

νλ_m1 3 ν

_t

t01e^{−νλm1t−s}|f|²_Vds. 4.18

Applying4.4for anyε

3 ν

_t

t01e^{−νλm1t−s}|f|²_V gds < ε

3. 4.19

Using2.9and lett₁t₀1 1/νλ_m1ln3ρ₀²/ε, thent≥t₁implies 2M

νλ_m1 < ε 3; u₂t01²

2e^{−νλm1t−t01}≤ρ²₀e^{−νλm1t−t01/2}< ε 3.

4.20

Therefore, we deduce from4.18that u2²

2≤ε, ∀t≥t1, f∈ H f0

, 4.21

which indicates{Uft, τ}, f ∈ Hf0satisfying uniformw.t.r. f ∈ Hf0ConditionC inH_g. ApplyingTheorem 3.5the proof is complete.

Theorem 4.6. If f0x, s is normal function in L²_locR;Hg, then the processes {Uf0t, τ} corresponding to problem 2.14 possesses compact uniform w.r.t. τ ∈ R attractor A1 in Vg

which coincides with the uniform w.r.t. f ∈ Hf0 attractor A_Hf0 of the family of processes {Uft, τ}, f ∈ Hf0:

A1 AHf0ω_0,Hf0 B1

f∈Hf0

Kf0, 4.22

whereB1is the uniformlyw.r.t. f ∈ Hf0absorbing set inV_gandKf is the kernel of the process {Uft, τ}. Furthermore, the kernelKf is nonempty for allf ∈ Hf0.

Proof. UsingProposition 4.4, we have the family of processes{Uft, τ},f ∈ Hf0corresp- onding to4.3possesses the uniformlyw.r.t. f ∈ Hf0absorbing set inV_g.

Now we prove the existence of compact uniformw.r.t. f ∈ Hf0attractor inVgby applying the method established inSection 3, that is, we testify that the family of processes

{Uft, τ}, f ∈ Hf0corresponding to4.3satisfies uniformw.r.t. f ∈ Hf0Condition C.

Multiplying2.14byA_gu₂t, we have dv

dt, Agu2

νAgu, Agu2

Bgu, u, Agu2

g

f, Agu2

−

Ru, Agu2

g. 4.23

It follows that 1 2

d dtu₂²

gνA_gu₂²

g≤B_gu, u, Agu₂

gf, A_gu₂

gRu, A_gu₂

g. 4.24

To estimateBgu, u, Au2_g, we recall some inequalities16 : for everyu, v∈DAg:

B_gu, v≤c

⎧⎨

⎩

|u|^1/2g u^1/2g v^1/2g A_gv^1/2

g ,

|u|^1/2g A_gu^1/2

g vg

4.25

see16

|w|_L∞Ω² ≤cwg

1log Agw λgw²_g

_1/2

4.26

from which we deduce that

Bgu, v≤c|u|L^∞Ω|∇v|g|u|g|∇v|_L^∞Ω, 4.27 and using,4.26

B_gu, v≤c

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩

ugvg

1log |Agu|² λgw²g

_1/2 ,

|u|gA_gv

g

1log A^3/2_g v² λ_gAgv²g

1/2

.

4.28

Expanding and using Young’s inequality, together with the first one of4.28and the second one of4.25, we have

Bgu, u, Agu2≤Bg

u1, u1u2

, Agu2Bg

u2, u1u2

, Agu2

≤cL^1/2u₁

gA_gu₂

gu₁

gu₂

g

cu₂^1/2

g A_gu₂^3/2

g

≤ ν

6Agu2²

g c

νρ⁴₁L c

ν³ρ²₀ρ⁴₁, t≥t01,

4.29

where we use

Agu1²

g ≤λmu1²

g 4.30

and set

L1logλ_m1 λg

. 4.31

Next, using the Cauchy inequality, Ru, Agu2

g|

ν

g∇g· ∇u, Agu2

g

≤ ν

m₀|∇g|∞ugA_gu₂

g

≤ ν

6Agu2²

g 3ν

2 |∇g|²_∞ρ²₁.

4.32

Finally, we estimate|f, Agu₂|by

f, Agu2≤ |f|gAgu2

2≤ ν

6Agu2²

g 3

2ν|f|²_g. 4.33

Putting4.29–4.33together, there exists a constantM₂such that d

dtu₂²

gνλ_m1u₂²

g≤ 3

ν|f|gM₂. 4.34

Here M₂ M₂ρ0, ρ₁, L, ν,|∇g| depends on λ_m1, is not increasing as λ_m1 increasing.

Therefore, by the Gronwall inequality, the above inequality implies u2²

g≤u2

t01²

ge^{−νλm1t−t01} 2M₂ νλ_m1 3

ν _t

t01e^{−νλm1t−s}|f|²_gds. 4.35

Applying4.4for anyε

3 ν

_t

t01e^{−νλm1t−s}|f|²_gds < ε

3. 4.36

Using2.9and lett1t01 1/νλm1ln3ρ₁²/ε, thent≥t1implies 2M2

νλ_m1 < ε 3; u₂

t₀1²

ge^{−νλm1t−t01}≤ρ²₁e^{−νλm1t−t01}< ε 3.

4.37

Therefore, we deduce from4.35that u2²

g≤ε, ∀t≥t1, f ∈ H f0

, 4.38

which indicates{Uft, τ}, f ∈ Hf0satisfying uniformw.t.r. f ∈ Hf0ConditionC inVg.

5. Dimension of the Uniform Attractor

In this section we estimate the fractal dimensionfor definition see, e.g.,2,10,15 of the kernel sections of the uniform attractors obtained in Section 4 by applying the methods in17 .

Process{Ut, τ}is said to be uniformly quasidifferentiable on{Ks}_τ∈R, if there is a family of bounded linear operators{Lt, τ;u |u ∈ Ks, t ≥τ, τ ∈ R}, Lt, τ;u : E → E such that

lim sup

ε→0τ∈R sup

u,v∈Ks 0<|u−v|≤ε

|Ut, τv−Ut, τu−Lt, τ;uv−u|

|v−u| 0. 5.1

We want to estimate the fractal dimension of the kernel sectionsKsof the process {Ut, τ}generated by the abstract evolutionary2.14. Assume that{Lt, τ;u}is generated by the variational equation corresponding to2.14

∂twFu, tw, w|tτwτ ∈E, t≥τ, τ ∈R, 5.2 that is,Lt, τ;u_τwτ wtis the solution of 5.2, and ut Ut, τuτ is the solution of 2.14with initial valueuτ ∈ Kτ. For natural numberj∈N, we set

qj lim

T→∞sup

τ∈R sup

uτ∈Kτ

1 T

_τT

τ

Tr

Fus, s ds

, 5.3

where Tr is trace of the operator.

We will need the following Theorem VIII.3.1 in10 and2 .

Theorem 5.1. Under the assumptions above, let us suppose thatU_τ∈RKτis relatively compact in E, and there existsq_j, j1,2, . . ., such that

q_j≤q_j, for anyj≥1, qn0≥0, qn01<0, for somen0≥1, qj≤qn0

qn0−qn01 n0−j

, ∀j1,2, . . . .

5.4

Then,

d_FKτ≤d₀ :n₀ qn0

qn0−qn01, ∀τ∈R. 5.5 We now consider2.14withf ∈L²_nR;V_g. The equations possess a compact uniform w.r.t. f ∈ Hf attractor AHf and

τ∈RKfτ ⊂ AHf. By 2, 10, 15 , we know that the associated process {Uft, τ} is uniformly quasidifferentiable on {Kfτ}_τ∈R and the quasidifferential is H ¨older-continuous with respect to u_τ ∈ Kfτ. The corresponding variational equation is

∂tw−νAgu−Bgu−νRuPgf≡Fut, tw, w|tτ wτ ∈E, τ ∈R. 5.6

We have the main results in this section.

Theorem 5.2. Suppose that ft satisfies the assumptions of Theorem 4.5. Then, if γ 1 − 2|∇g|_∞/m0λ^1/2_g >0, the Uniform attractorA0defined by4.8satisfies

dF

A0

≤

β

α, 5.7

where

α c2νm0λ₁γ 2M0

,

β c₁d₁

2ν³m₀γ sup

ϕj∈Hg|ϕj|≤1 j1,2,...,m

1 T

_τT

τ

fs²_V gds,

5.8

the constantc₁, c₂of (3.29) and (3.32) of ChapterV Iin [15] and [2],λ₁is the first eigenvalue of the Stokes operator andd₁ |∇g|²_∞/4m₀|∇g|∞M₀.

Proof. WithTheorem 4.5at our disposal we may apply the abstract framework in2,10,15, 17 .

Forξ₁, ξ₂, . . . , ξ_m∈H_g, letv_jt Lt, uτ·ξ_j, whereu_τ ∈H_g. Let{ϕjs;j1,2, . . . , m}

be an orthonormal basis for span{vj;j1,2, . . . , m}. Sincevj ∈Vgalmost everywheres≥τ, we can also assume that ϕ_js ∈ V_g almost everywheres ≥ τ. Then, similar to the Proof

process of Theorems4.5and4.6, we may obtain

m i1

FUs, τuτ, sϕi, ϕ_i

g−ν^m

i1

ϕj²_g−^m

i1

b_g

ϕ_j, Us, τuτ, ϕ_j

−^m

i1

ν

g∇g· ∇ϕj, ϕj

g

,

5.9

almost everywheres ≥τ. From this equality, and in particular using the Schwarz and Lieb- Thirring inequalitysee2,10,15,17 , one obtains

m i1

ϕ²_g≥λ₁· · ·λ_m≥ m₀ M0

λ₁· · ·λ_m

≥ m₀ M0

c₂λ₁m²,

Tr_j

FUs, τuτ, s

g≤ −ν

1− |∇g|_∞ m0λ^1/2₁

_m

i1

ϕ_j²

gUs, τuτ

g

c₁d₁ m₀

m i1

ϕ_j²

g

_1/2

≤ −ν 2

1−2|∇g|_∞ m₀λ^1/2₁

_m

i1

ϕ_j²

g c₁d₁ 2νm0

Us, τuτ²

g

≤ −νm0

2M0

1−2|∇g|∞

m₀λ^1/2₁

c2λ₁m² c1d1

2νm0

Us, τuτ²

g,

5.10

on the other hand, we can deduce2.14that

d

dtUs, τuτ²

gνUs, τuτ²

g≤ f²_V g

ν 2ν

m₀λ^1/2_g |∇g|∞Us, τuτ²

g 5.11

forλ_g4π²m₀/M₀, and then _t

τ

Us, τuτ²

gds≤ 1

ν² _t

τ

fs²_V

gds |uτ|² ν

1−2|∇g|∞

m₀λ^1/2_{g −1}

, t≥τ. 5.12

Now we define

qm sup

ϕj∈Hg|ϕj|≤1 j1,2,...,m

1 T

_τT

τ

Trj

F

Us, τuτ, s ds

g

, 5.13

UsingTheorem 5.1, we have

qm≤ −νm0

2M0

1−2|∇g|∞

m₀λ^1/2₁

c2λ₁m² c1d1

2νm0

⎛

⎜⎜

⎝ sup

ϕj∈Hg|ϕj|≤1 j1,2,...,m

1 T

_τT

τ

Us, τuτ²

gds ⎞

⎟⎟

⎠

≤ −νm₀ 2M₀

1−2|∇g|_∞ m0λ^1/2₁

c₂λ₁m²

c1d1

2νm₀

⎛

⎜⎜

⎝1 ν² sup

ϕj∈Hg|ϕj|≤1 j1,2,...,m

1 T

_τT

τ

fs²_V

gds |uτ|² νT

1−2|∇g|∞

m0λ^1/2g

₋₁⎞

⎟⎟

⎠,

qmlim sup

T→ ∞ qm≤ −αm²β,

5.14

Hence

dim_FA0τ≤

β

α. 5.15

Acknowledgment

The author would like to thank the reviewers and the editor for their valuable suggestions and comments.

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