Abstract

We consider the asymptotic behaviour of nonautonomous 2D g-Navier-Stokes equations in bounded domain Ω. Assuming that 𝑓𝐿2loc, which is translation bounded, the existence of the pullback attractor is proved in 𝐿2(Ω) and 𝐻1(Ω). It is proved that the fractal dimension of the pullback attractor is finite.

1. Introduction

In this paper, we study the behavior of solutions of the nonautonomous g-Navier-Stokes equations in spatial dimension 2. These equations are a variation of the standard Navier-Stokes equations, and they assume the form 𝜕𝑢1𝜕𝑡𝜈Δ𝑢+(𝑢)𝑢+𝑝=𝑓inΩ,𝑔(𝑔𝑢)=𝑔𝑔𝑢+𝑢=0inΩ,(1.1) where 𝑔=𝑔(𝑥1,𝑥2) is a suitable smooth real-valued function defined on (𝑥1,𝑥2)Ω and Ω is a suitable bounded domain in 2. Notice that if 𝑔(𝑥1,𝑥2)=1, then (1.1) reduce to the standard Navier-Stokes equations.

In addition, we assume that the function 𝑓(,𝑡)=𝑓(𝑡)𝐿2loc(;𝐸) is translation bounded, where 𝐸=𝐿2(Ω) or 𝐻1(Ω). This property implies that𝑓2𝐿2𝑏=𝑓2𝐿2𝑏(;𝐸)=sup𝑡𝑡𝑡+1𝑓(𝑠)2𝐸𝑑𝑠<.(1.2)

We consider this equation in an appropriate Hilbert space and show that there is a pullback attractor 𝔄. This is the basic idea of our construction, which is motivated by the works of [1].

Let Ω=(0,1)×(0,1). We assume that the function 𝑔(𝑥)=𝑔(𝑥1,𝑥2) satisfies the following properties:(1)𝑔(𝑥)𝐶per(Ω),(2)there exist constants 𝑚0=𝑚0(𝑔) and 𝑀0=𝑀0(𝑔) such that, for all 𝑥Ω, 0<𝑚0𝑔(𝑥)𝑀0. Note that the constant function 𝑔1 satisfies these conditions.

We denote by 𝐿2(Ω,𝑔) the space with the scalar product and the norm given by(𝑢,𝑣)𝑔=Ω(𝑢𝑣)𝑔𝑑𝑥,|𝑢|2𝑔=(𝑢,𝑢)𝑔,(1.3) as well as 𝐻1(Ω,𝑔) with the norm𝑢𝐻1(Ω,𝑔)=(𝑢,𝑢)𝑔+2𝑖=1𝐷𝑖𝑢,𝐷𝑖𝑢𝑔1/2,(1.4) where 𝜕𝑢/𝜕𝑥𝑖=𝐷𝑖𝑢.

Then for the functional setting of the problems (1.1), we use the following functional spaces:𝐻𝑔=𝐶𝑙𝐿2per(Ω,𝑔)𝑢𝐶per(Ω)𝑔𝑢=0,Ω,𝑉𝑢𝑑𝑥=0𝑔=𝑢𝐻1per(Ω,𝑔)𝑔𝑢=0,Ω,𝑢𝑑𝑥=0(1.5) where 𝐻𝑔 is endowed with the scalar product and the norm in 𝐿2(Ω,𝑔) and 𝑉𝑔 is the spaces with the scalar product and the norm given by((𝑢,𝑣))𝑔=Ω(𝑢𝑣)𝑔𝑑𝑥,𝑢𝑔=((𝑢,𝑢))𝑔.(1.6) Also, we define the orthogonal projection 𝑃𝑔 as𝑃𝑔𝐿2per(Ω,𝑔)𝐻𝑔,(1.7) and we have that 𝑄𝐻𝑔, where𝑄=𝐶𝑙𝐿2per(Ω,𝑔)𝜙𝜙𝐶1Ω,.(1.8) Then, we define the 𝑔-Laplacian operatorΔ𝑔1𝑢𝑔1(𝑔)𝑢=Δ𝑢𝑔(𝑔)𝑢(1.9) to have the linear operator𝐴𝑔𝑢=𝑃𝑔1𝑔((𝑔𝑢)).(1.10) For the linear operator 𝐴𝑔, the following hold (see [1]).

(1) 𝐴𝑔 is a positive, self-adjoint operator with compact inverse, where the domain of 𝐴𝑔 is 𝐷(𝐴𝑔)=𝑉𝑔𝐻2(Ω,𝑔).

(2) There exist countable eigenvalues of 𝐴𝑔 satisfying0<𝜆𝑔𝜆1𝜆2𝜆3,(1.11) where 𝜆𝑔=4𝜋2𝑚0/𝑀0 and 𝜆1 is the smallest eigenvalue of 𝐴𝑔. In addition, there exists the corresponding collection of eigenfunctions {𝑒1,𝑒2,𝑒3,} which forms an orthonormal basis for 𝐻𝑔.

Next, we denote the bilinear operator 𝐵𝑔(𝑢,𝑣)=𝑃𝑔(𝑢)𝑣 and the trilinear form𝑏𝑔(𝑢,𝑣,𝑤)=2𝑖,𝑗=1Ω𝑢𝑖𝐷𝑖𝑣𝑗𝑤𝑗𝑃𝑔𝑑𝑥=𝑔(𝑢)𝑣,𝑤𝑔,(1.12) where 𝑢, 𝑣, and 𝑤 lie in appropriate subspaces of 𝐿2(Ω,𝑔). Then, the form 𝑏𝑔 satisfies𝑏𝑔(𝑢,𝑣,𝑤)=𝑏𝑔(𝑢,𝑤,𝑣)for𝑢,𝑣,𝑤𝐻𝑔.(1.13) We denote a linear operator 𝑅 on 𝑉𝑔 by𝑅𝑢=𝑃𝑔1𝑔(𝑔)𝑢for𝑢𝑉𝑔(1.14) and have 𝑅 as a continuous linear operator from 𝑉𝑔 into 𝐻𝑔 such that||||||||(𝑅𝑢,𝑢)𝑔𝑚0𝑢𝑔|𝑢|𝑔||||𝑔𝑚0𝜆𝑔1/2𝑢𝑔for𝑢𝑉𝑔.(1.15)

We now rewrite (1.1) as abstract evolution equations: 𝑑𝑢𝑑𝑡+𝜈𝐴𝑔𝑢+𝐵𝑔𝑢+𝜈𝑅𝑢=𝑃𝑔𝑓,𝑢(𝜏)=𝑢𝜏.(1.16) In [1] the author established the global regularity of solutions of the g-Navier-Stokes equations. The Navier-Stokes equations were investigated by many authors, and the existence of the attractors for 2D Navier-Stokes equations was first proved in [2] and independently in [3]. The finite-dimensional property of the global attractor for general dissipative equations was first proved in [4]. For the analysis of the Navier-Stokes equations, one can refer to [5], specially [6] for the periodic boundary conditions.

The theory of pullback (or cocycle) attractors has been developed for both the nonautonomous and random dynamical systems (see [713]) and has shown to be very useful in the understanding of the dynamics of nonautonomous dynamical systems.

The understanding of the asymptotic behaviour of dynamical systems is one of the most important problems of modern mathematical physics. One way to treat this problem for a system having some dissipativity properties is to analyse the existence and structure of its global attractor, which, in the autonomous case, is an invariant compact set which attracts all the trajectories of the system, uniformly on bounded sets. This set has, in general, a very complicated geometry which reflects the complexity of the long-time behaviour of the system (see [1417] and the references therein). However, nonautonomous systems are also of great importance and interest as they appear in many applications to natural sciences. In this situation, there are various options to deal with the problem of attractors for nonautonomous systems (kernel sections [18], skew-product formalism [16, 19], etc.); for our particular situation we have preferred to choose that of pullback attractor (see [9, 10, 13, 20]) which has also proved extremely fruitful, particularly in the case of random dynamical systems (see [11, 13]).

In this paper, we study the existence of compact pullback attractor for the nonautonomous g-Navier-Stokes equations in bounded domain Ω with periodic boundary condition. It is proved that the fractal dimension of the pullback attractor is finite.

Hereafter 𝑐 will denote a generic 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.

2. Abstract Results

We now recall the preliminary results of pullback attractors, as developed in [810, 13].

The semigroup 𝑆(𝑡) property is replaced by the process 𝑈(𝑡,𝜏) composition property𝑈(𝑡,𝜏)𝑈(𝜏,𝑠)=𝑈(𝑡,𝑠)𝑡𝜏𝑠,(2.1) and, obviously, the initial condition implies that 𝑈(𝜏,𝜏)=Id. As with the semigroup composition 𝑆(𝑡)𝑆(𝜏)=𝑆(𝑡+𝜏), this just expresses the uniqueness of solutions.

It is also possible to present the theory within the more general framework of cocycle dynamical systems. In this case the second component of 𝑈 is viewed as an element of some parameter space 𝐽, so that the solution can be written as 𝑈(𝑡,𝑝)𝜚, and a shift map 𝜃𝑡𝐽𝐽 is defined so that the process composition becomes the cocycle property 𝑈(𝑡+𝜏,𝑝)=𝑈𝑡,𝜃𝜏𝑝𝑈(𝜏,𝑝).(2.2) However, when one tries to develop a theory under a unified abstract formulation, the context of cocycle (or skew-product flows) may not be the most appropriate to deal with the problem. In this paper, we apply a process 𝑈(𝑡,𝜏) to (1.16) by using the concept of measure of noncompactness to obtain pullback attractors.

By (𝐸) we denote the collection of the bounded sets of 𝐸.

Definition 2.1. Let 𝑈 be a process on a complete metric space 𝐸. A family of compact sets {𝒜(𝑡)}𝑡 is said to be a pullback attractor for 𝑈 if, for all 𝜏, it satisfies(i)𝑈(𝑡,𝜏)𝒜(𝜏)=𝒜(𝑡) for all 𝑡𝜏,(ii)lim𝑠dist(𝑈(𝑡,𝑡𝑠)𝐷,𝒜(𝑡))=0, for 𝐷(𝐸).The pullback attractor is said to be uniform if the attraction property is uniform in time, that is, lim𝑠sup𝑡dist(𝑈(𝑡,𝑡𝑠)𝐷,𝒜(𝑡))=0,for𝐷(𝐸).(2.3)

Definition 2.2. A family of compact sets {𝒜(𝑡)}𝑡 is said to be a forward attractor for 𝑈 if, for all 𝜏, it satisfies(i)𝑈(𝑡,𝜏)𝒜(𝜏)=𝒜(𝑡) for all 𝑡𝜏,(ii)lim𝑡dist(𝑈(𝑡,𝜏)𝐷,𝒜(𝑡))=0, for 𝐷(𝐸).The forward attractor is said to be uniform if the attraction property is uniform in time, that is, lim𝑡sup𝜏dist(𝑈(𝑡+𝜏,𝜏)𝐷,𝒜(𝑡+𝜏))=0,for𝐷(𝐸).(2.4) In the definition, dist(𝐴,𝐵) is the Hausdorff semidistance between 𝐴 and 𝐵, defined as dist(𝐴,𝐵)=sup𝑎𝐴inf𝑏𝐵𝑑(𝑎,𝑏),for𝐴,𝐵𝐸.(2.5) Property (i) is a generalization of the invariance property for autonomous dynamical systems. The pullback attracting property (ii) considers the state of the system at time 𝑡 when the initial time 𝑡𝑠 goes to .

The notion of an attractor is closely related to that of an absorbing set.

Definition 2.3. The family {𝐵(𝑡)}𝑡 is said to be (pullback) absorbing with respect to the process 𝑈 if, for all 𝑡 and 𝐷(𝐸), there exists 𝑆(𝐷,𝑡)>0 such that for all 𝑠𝑆(𝐷,𝑡)𝑈(𝑡,𝑡𝑠)𝐷𝐵(𝑡).(2.6) The absorption is said to be uniform if 𝑆(𝐷,𝑡) does not depend on the time variable 𝑡.

Now we recall the abstract results in [21].

Definition 2.4. The family of processes {𝑈(𝑡,𝑡𝑠)} is said to be satisfying pullback Condition (C) if, for any fixed 𝐵(𝐸) and 𝜀>0, there exist 𝑠0=𝑠(𝐵,𝑡,𝜀)0 and a finite dimensional subspace 𝐸1 of 𝐸 such that(i){𝑃(𝑠𝑠0𝑈(𝑡,𝑡𝑠)𝐵)𝐸} is bounded,(ii)(𝐼𝑃)(𝑠𝑠0𝑈(𝑡,𝑡𝑠)𝐵)𝐸𝜀, where 𝑃𝐸𝐸1 is a bounded projector.

Theorem 2.5. Let the family of processes {𝑈(𝑡,𝜏)} acting in 𝐸 be continuous and possess compact pullback attractor 𝒜(𝑡) satisfying 𝒜(𝑡)=𝐵𝜔(𝐵,𝑡),for𝑡,(2.7) if it (i)has a bounded (pullback) absorbing set 𝐵,(ii)satisfies pullback Condition (C).Moreover if 𝐸 is a uniformly convex Banach space, then the converse is true.

3. Pullback 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 in a bounded domain Ω with periodic boundary condition.

In [1], the author has shown that the semigroup 𝑆(𝑡)𝐻𝑔𝐻𝑔(𝑡0) associated with the autonomous systems (1.16) possesses a global attractor in 𝐻𝑔 and 𝑉𝑔. The main objective of this section is to prove that the nonautonomous system (1.16) has uniform attractors in 𝐻𝑔 and 𝑉𝑔.

To this end, we first state the following results of existence and uniqueness of solutions of (1.16).

Proposition 3.1. Let 𝑓𝑉 be given. Then for every 𝑢𝜏𝐻𝑔 there exists a unique solution 𝑢=𝑢(𝑡) on [0,) of (1.16), satisfying 𝑢(𝜏)=𝑢𝜏. Moreover, one has 𝑢(𝑡)𝐶𝜏,𝑇;𝐻𝑔𝐿2𝜏,𝑇;𝑉𝑔,𝑇>𝜏.(3.1) Finally, if 𝑢𝜏𝑉𝑔, then 𝑢(𝑡)𝐶𝜏,𝑇;𝑉𝑔𝐿2𝐴𝜏,𝑇;𝐷𝑔,𝑇>𝜏.(3.2)

Proof. The Proof of Proposition 3.1 is similar to autonomous case in [1, 17].

Proposition 3.2. The process {𝑈(𝑡,𝑡𝑠)}𝑉𝑔𝑉𝑔 associated with the system (1.16) possesses (pullback) absorbing sets, that is, there exists a family {𝐵(𝑡)}𝑡𝑅 of bounded (pullback) absorbing sets in 𝐻𝑔 and 𝑉𝑔 for the process 𝑈, which is given by 0=𝐵(𝑡)=𝑢𝐻𝑔|𝑢|𝑔𝜌0,1=𝐵(𝑡)=𝑢𝑉𝑔𝑢𝑔𝜌1,(3.3) which absorb all bounded sets of 𝐻𝑔. Moreover 0 and 1 absorb all bounded sets of 𝐻𝑔 and 𝑉𝑔 in the norms of 𝐻𝑔 and 𝑉𝑔, respectively.

Proof. The proof of Proposition 3.2 is similar to autonomous g-Navier-Stokes equation. We can obtain absorbing sets in 𝐻𝑔 and 𝑉𝑔 from [1].

The main results in this section are as follows.

Theorem 3.3. If 𝑓(𝑥,𝑡)𝐿2𝑏(𝑅;𝑉) and 𝑢𝜏𝐻𝑔, then the processes {𝑈(𝑡,𝑡𝑠)} corresponding to problem (1.16) possess compact pullback attractor 𝒜0(𝑡) in 𝐻𝑔 which coincides with the pullback attractor: 𝒜0(𝑡)=0𝜔(0,𝑡),(3.4) where 0 is the (pullback) absorbing set in 𝐻𝑔.

Proof. As in the previous section, for fixed 𝑁, let 𝐻1 be the subspace spanned by 𝑤1,,𝑤𝑁, and 𝐻2 the orthogonal complement of 𝐻1 in 𝐻𝑔. We write 𝑢=𝑢1+𝑢2,𝑢1𝐻1,𝑢2𝐻2forany𝑢𝐻𝑔.(3.5) Now, we only have to verify Condition (C). Namely, we need to estimate |𝑢2(𝑡)|𝑔, where 𝑢(𝑡)=𝑢1(𝑡)+𝑢2(𝑡) is a solution of (1.16) given in Proposition 3.1.
Multiplying (1.16) by 𝑢2, we have 𝑑𝑢𝑑𝑡,𝑢2𝑔+𝜈𝐴𝑔𝑢,𝑢2𝑔+𝐵𝑔(𝑢,𝑢),𝑢2𝑔=𝑓,𝑢2𝑔𝜈𝑅𝑢,𝑢2𝑔.(3.6) It follows that 12𝑑||𝑢𝑑𝑡2||2𝑔𝑢+𝜈𝑔2𝑔|||𝐵(𝑢,𝑢),𝑢2𝑔|||+|||𝑓,𝑢2𝑔|||+𝑅𝑢,𝑢2𝑔.(3.7) Since 𝑏𝑔 satisfies the following inequality (see [6]): ||𝑏𝑔||(𝑢,𝑣,𝑤)𝑐|𝑢|𝑔1/2𝑢𝑔1/2𝑣𝑔|𝑤|𝑔1/2𝑤𝑔1/2,𝑢,𝑣,𝑤𝑉𝑔,(3.8) thus, |||𝐵(𝑢,𝑢),𝑢2𝑔|||𝑐|𝑢|𝑔1/2𝑢𝑔3/2||𝑢2||𝑔1/2𝑢2𝑔1/2𝑐𝜆𝑚+1|𝑢|𝑔1/2𝑢𝑔3/2𝑢2𝑔𝜈6𝑢22𝑔+𝑐𝜌0𝜌31.(3.9) Next, using the Cauchy inequality, |||𝜈𝑅𝑢,𝑢2𝑔|||=||||𝜈𝑔(𝑔)𝑢,𝑢2𝑔||||𝜈𝑚0||||𝑔𝑢𝑔||𝑢2||𝑔𝜈6𝑢22𝑔+3𝜈𝜌21||||𝑔22𝑚20𝜆𝑔𝜆𝑚+1.(3.10) Finally, we have |||𝑓,𝑢2𝑔|||||𝑓||𝑉𝑔𝑢2𝜈6𝑢22𝑔+3||𝑓||2𝜈2𝑉𝑔.(3.11) Putting (3.9)–(3.11) together, there exists constant 𝑀1=𝑀1(𝑚0,|𝑔|,𝜌0,𝜌1) such that 12𝑑||𝑢𝑑𝑡2||2𝑔+12𝜈𝑢22𝑔3||𝑓||𝑉𝑔2𝜈+𝑀1.(3.12) Therefore, we deduce that 𝑑||𝑢𝑑𝑡2||2𝑔+𝜈𝜆𝑚+1||𝑢2||2𝑔2𝑀1+3𝜈||𝑓||2𝑉𝑔.(3.13)Here, 𝑀1 depends on 𝜆𝑚+1, is not increasing as 𝜆𝑚+1 increasing.
By the Gronwall inequality, the above inequality implies that ||𝑢2||(𝑡)2𝑔||𝑢2||(𝜏)2𝑔𝑒𝜈𝜆𝑚+1(𝑡𝜏)+2𝑀1𝜈𝜆𝑚+1+3𝜈𝑡𝜏𝑒𝜈𝜆𝑚+1(𝑡𝑠)||𝑓||2𝑉𝑑𝑠.(3.14) If we consider the time 𝑡𝑠 instead of 𝜏 (so that we can use more easily the definition of pullback attractors), we have 3𝜈𝑡𝜏𝑒𝜈𝜆𝑚+1(𝑡𝜎)||||𝑓(𝜎)2𝑉3𝑑𝜎=𝜈𝑡𝑡𝑠𝑒𝜈𝜆𝑚+1(𝑡𝜎)||||𝑓(𝜎)2𝑉𝑑𝜎.(3.15) Applying continuous integral and Lemma II  1.3 in [18] for any 𝜀, there exists 𝜂=𝜂(𝜀)>0 such that 𝑡𝑡𝜂||||𝑓(𝜎)2𝑉𝑑𝜎<𝜈𝜀;18(3.16) thus, we have 3𝜈𝑡𝑡𝜂𝑒𝜈𝜆𝑚+1(𝑡𝜎)||||𝑓(𝜎)2𝑉𝜀𝑑𝜎6,3(3.17)𝜈𝑡𝜂𝑡𝑠𝑒𝜈𝜆𝑚+1(𝑡𝜎)||||𝑓(𝜎)2𝑉3𝑑𝜎𝜈𝑡𝜂𝑡𝜂1𝑒𝜈𝜆𝑚+1(𝑡𝜎)||||𝑓(𝜎)2𝑉+3𝑑𝜎𝜈𝑡𝜂1𝑡𝜂2𝑒𝜈𝜆𝑚+1(𝑡𝜎)||||𝑓(𝜎)2𝑉3𝑑𝜎+𝜈𝑒𝜈𝜆𝑚+1𝜂𝑡𝜂𝑡𝜂1||||𝑓(𝜎)2𝑉𝑑𝜎+𝑒𝜈𝜆𝑚+1𝑡𝜂1𝑡𝜂2||||𝑓(𝜎)2𝑉3𝑑𝜎+𝜈𝑒𝜈𝜆𝑚+1𝜂1+𝑒𝜈𝜆𝑚+1+sup𝑠𝑅𝑠𝑠1||||𝑓(𝜎)2𝑉𝑑𝜎(3/𝜈)𝑒𝜈𝜆𝑚+1𝜂1𝑒𝜈𝜆𝑚+1𝑓2𝐿2𝑏.(3.18) Using (1.11) and letting 𝑠1=(1/𝜈𝜆𝑚+1)ln(3𝜌20/𝜀), then 𝑠𝑠1 implies that 3𝜈𝑡𝜂𝑡𝑠𝑒𝜈𝜆𝑚+1(𝑡𝜎)||||𝑓(𝜎)2𝑉𝑑𝜎(3/𝜈)𝑒𝜈𝜆𝑚+1𝜂1𝑒𝜈𝜆𝑚+1𝑓2𝐿2𝑏(𝑅;𝑉)𝜀6,(3.19)2𝑀1𝜈𝜆𝑚+1𝜀3,||𝑢2||(𝜏)2𝑔𝑒𝜈𝜆𝑚+1(𝑡𝜏)𝜌20𝑒𝜈𝜆𝑚+1𝑠1𝜀3.(3.20) Therefore, we deduce from (3.14) that ||𝑢2||2𝑔𝜀,𝑠𝑠1,(3.21) which indicates {𝑈(𝑡,𝜏)} satisfying pullback Condition (C) in 𝐻𝑔. Applying Theorem 2.5, the proof is complete.

According to Propositions 3.1-3.2, we can now regard that the families of processes {𝑈(𝑡,𝜏)} are defined in 𝑉𝑔 and 1 is a pullback absorbing set in 𝑉𝑔.

Theorem 3.4. If 𝑓(𝑥,𝑡)𝐿2𝑏(𝑅;𝐻𝑔), then the processes {𝑈(𝑡,𝜏)} corresponding to problem (1.16) possess compact pullback attractor 𝒜1(𝑡) in 𝑉𝑔: 𝒜1(𝑡)=1𝜔(1,𝑡),(3.22) where 1 is the absorbing set in 𝑉𝑔.

Proof. Using Proposition 3.2, we have that the family of processes {𝑈(𝑡,𝜏)} corresponding to (1.16) possess the pullback absorbing set in 𝑉𝑔.
Now we testify that the family of processes {𝑈(𝑡,𝜏)} corresponding to (1.16) satisfies pullback Condition (C).
Multiplying (1.16) by 𝐴𝑔𝑢2(𝑡), we have 𝑑𝑣𝑑𝑡,𝐴𝑔𝑢2+𝜈𝐴𝑔𝑢,𝐴𝑔𝑢2+𝐵𝑔(𝑢,𝑢),𝐴𝑔𝑢2𝑔=𝑓,𝐴𝑔𝑢2𝜈𝑅𝑢,𝐴𝑔𝑢2𝑔.(3.23) It follows that 12𝑑𝑑𝑡𝑢22𝑔||𝐴+𝜈𝑔𝑢2||2𝑔|||𝐵𝑔(𝑢,𝑢),𝐴𝑔𝑢2𝑔|||+|||𝑓,𝐴𝑔𝑢2𝑔|||+|||𝜈𝑅𝑢,𝐴𝑔𝑢2𝑔|||.(3.24) To estimate (𝐵𝑔(𝑢,𝑢),𝐴𝑢2)𝑔, we recall some inequalities (see [22]), for every 𝑢,𝑣𝐷(𝐴𝑔), ||𝐵𝑔||(𝑢,𝑣)𝑐|𝑢|𝑔1/2𝑢𝑔1/2𝑣𝑔1/2||𝐴𝑔𝑣||𝑔1/2,|𝑢|𝑔1/2||𝐴𝑔𝑢||𝑔1/2𝑣𝑔,(3.25)|𝑤|𝐿(Ω)2𝑐𝑤𝑔||𝐴1+log𝑔𝑤||𝜆𝑔𝑤2𝑔1/2,(3.26) from which we deduce that ||𝐵𝑔||(𝑢,𝑣)𝑐|𝑢|𝐿(Ω)||||𝑣𝑔|𝑢|𝑔||||𝑣𝐿(Ω),(3.27) and using (3.26), ||𝐵𝑔||(𝑢,𝑣)𝑐𝑢𝑔𝑣𝑔1+log|𝐴𝑔𝑢|2𝜆𝑔𝑢2𝑔1/2,||𝑢|𝑔||𝐴𝑔𝑣|𝑔1+log|𝐴𝑔3/2𝑣|2𝜆𝑔𝐴𝑔𝑣2𝑔1/2.(3.28) Expanding and using Young's inequality, together with the first one of (3.28) and the second one of (3.25), we have ||𝐵𝑔(𝑢,𝑢),𝐴𝑔𝑢2||||𝐵𝑔𝑢1,𝑢1+𝑢2,𝐴𝑔𝑢2||+||𝐵𝑔𝑢2,𝑢1+𝑢2,𝐴𝑔𝑢2||𝑐𝐿1/2𝑢1𝑔||𝐴𝑔𝑢2||𝑔𝑢1𝑔+𝑢2𝑔||𝑢+𝑐2||𝑔1/2||𝐴𝑔𝑢2||𝑔3/2𝜈6||𝐴𝑔𝑢2||2𝑔+𝑐𝜈𝜌41𝑐𝐿+𝜈3𝜌20𝜌41,𝑡𝑡0+1,(3.29) where we use ||𝐴𝑔𝑢1||2𝑔𝜆𝑚𝑢12𝑔(3.30) and set 𝜆𝐿=1+log𝑚+1𝜆𝑔.(3.31) Next, using the Cauchy inequality, |||𝑅𝑢,𝐴𝑔𝑢2𝑔|||=||||𝜈𝑔(𝑔)𝑢,𝐴𝑔𝑢2𝑔||||𝜈𝑚0||||𝑔𝑢𝑔||𝐴𝑔𝑢2||𝑔𝜈6||𝐴𝑔𝑢2||2𝑔+3𝜈2||||𝑔2𝜌21.(3.32) Finally, we estimate |(𝑓,𝐴𝑔𝑢2)| by ||𝑓,𝐴𝑔𝑢2||||𝑓||𝑔||𝐴𝑔𝑢2||2𝜈6||𝐴𝑔𝑢2||2𝑔+3||𝑓||2𝜈2𝑔.(3.33) Putting (3.29)–(3.33) together, there exists a constant 𝑀2 such that 𝑑𝑑𝑡𝑢22𝑔+𝜈𝜆𝑚+1𝑢22𝑔3𝜈||𝑓||𝑔+𝑀2.(3.34) Here, 𝑀2=𝑀2(𝜌0,𝜌1,𝐿,𝜈,|𝑔|) depends on 𝜆𝑚+1, is not increasing as 𝜆𝑚+1 increasing. Therefore, by the Gronwall inequality, the above inequality implies that 𝑢22𝑔𝑢2(𝜏)2𝑔𝑒𝜈𝜆𝑚+1(𝑡𝜏)+2𝑀2𝜈𝜆𝑚+1+3𝜈𝑡𝜏𝑒𝜈𝜆𝑚+1(𝑡𝑠)||𝑓||2𝑔𝑑𝑠.(3.35) We consider the time 𝑡𝑠 instead of 𝜏. The following result is similar to (3.17)–(3.19), for any 𝜀: 2𝑐𝜈𝑡𝜏𝑒𝜈𝜆𝑚+1(𝑡𝜎)/2||𝑓||2𝑔𝜀𝑑𝜎3.(3.36) Using (1.11) and letting 𝑠2=(2/𝜈𝜆𝑚+1)ln(3𝜌21/𝜀), then 𝑠𝑠2 implies that 2𝑀2𝜈𝜆𝑚+1𝜀3,𝑢2(𝜏)2𝑔𝑒𝜈𝜆𝑚+1(𝑡𝜏)𝜌21𝑒𝜈𝜆𝑚+1𝑠<𝜀3.(3.37) Therefore, we deduce from (3.35) that 𝑢22𝑔𝜀,𝑠𝑠1,(3.38) which indicates {𝑈(𝑡,𝜏)} satisfying pullback Condition (C) in 𝑉𝑔.

4. The Dimension of the Pullback Attractor

To estimate the dimension of the pullback attractor 𝒜0(𝑡), we will apply the abstract machinery in [18, 23]. Let 𝐹𝑉𝑔×𝑉𝑔 be a given family of nonlinear operators such that, for all 𝜏 and any 𝑢𝜏𝐻𝑔, there exists a unique function 𝑢(𝑡)=𝑢(𝑡;𝜏,𝑢0) satisfying𝑢𝐿2𝜏,𝑇;𝑉𝑔𝐶𝜏,𝑇;𝐻𝑔,𝐹(𝑢(𝑡),𝑡)𝐿1𝜏,𝑇;𝑉𝑔,𝑇>𝜏,𝑑𝑢𝑑𝑡=𝐹(𝑢(𝑡),𝑡),𝑡>𝜏,𝑢(𝜏)=𝑢𝜏,(4.1) where 𝐹(𝑢)=𝜈𝐴𝑔𝑢𝐵𝑔𝑢𝜈𝑅𝑢+𝑃𝑔𝑓.

Using the standard methods (see [17, 18]), we can show that {𝑈(𝑡,𝜏)} is uniformly quasidifferentiable on {𝐵(𝑡)}𝑡. Then, for all 𝜏𝑇 and any 𝑢𝜏,𝑣𝜏𝐻𝑔, there exists a unique 𝑣(𝑡)=𝑣(𝑡;𝜏,𝑢𝜏,𝑣𝜏), which is a solution of𝑣𝐿2𝜏,𝑇;𝑉𝑔𝐶𝜏,𝑇;𝐻𝑔,𝑑𝑣𝑑𝑡=𝐹𝑈(𝑡,𝜏)𝑢𝜏,𝑡𝑣,𝑣(𝜏)=𝑣𝜏.(4.2) For all 𝜏<𝑇, we define the linear operator 𝐿(𝑡,𝑢𝜏)𝐻𝑔𝐻𝑔 by𝑣𝑡;𝜏,𝑢𝜏,𝑣𝜏=𝐿𝑡,𝜏,𝑢𝜏𝑣𝜏.(4.3)

Theorem 4.1. Suppose that 𝑓(𝑡) satisfies the assumptions of Theorem 3.3. Then, if 𝛾=1(2|𝑔|/𝑚0𝜆𝑔1/2)>0, the Pullback attractor (uniformly in the past) 𝒜0 defined by (3.4) satisfies 𝑑𝐹𝒜0𝛽𝛼,(4.4) where 𝑐𝛼=2𝜈𝑚0𝜆1𝛾2𝑀0,𝑐𝛽=1𝑑12𝜈3𝑚0𝛾sup𝜑𝑗𝐻𝑔,|𝜑𝑗|1𝑗=1,2,,𝑚1𝑇𝜏𝜏𝑇𝑓(𝑠)2𝑉𝑔𝑑𝑠,(4.5) with the constant 𝑐1,𝑐2 of (3.29) and (3.32) of Chapter VI in [17], 𝜆1 is the first eigenvalue of the Stokes operator and 𝑑1=|𝑔|2/4𝑚0+|𝑔|+𝑀0.

Proof. With Theorem 3.3 at our disposal we may apply the abstract framework in [17, 18, 23, 24].
For 𝜉1,𝜉2,,𝜉𝑚𝐻𝑔, let 𝑣𝑗(𝑡)=𝐿(𝑡,𝑢𝜏)𝜉𝑗, where 𝑢𝜏𝐻𝑔. Let {𝜑𝑗(𝑠);𝑗=1,2,,𝑚} be an orthonormal basis for span {𝑣𝑗;𝑗=1,2,,𝑚}. Since 𝑣(𝑠;𝜏,𝑢𝜏,𝑣𝑗𝜏)𝑉𝑔 almost everywhere 𝑠𝜏, we can also assume that 𝜑𝑗(𝑠)𝑉𝑔 almost everywhere 𝑠𝜏. Then, similar to the proof process of Theorems 3.3 and 3.4, we may obtain 𝑚𝑖=1𝐹𝑈(𝑠,𝜏)𝑢𝜏𝜑,𝑠𝑖,𝜑𝑖=𝜈𝑚𝑖=1𝜑𝑗2𝑔𝑚𝑖=1𝑏𝑔𝜑𝑗,𝑈(𝑠,𝜏)𝑢𝜏,𝜑𝑗𝑚𝑖=1𝜈𝑔(𝑔)𝜑𝑗,𝜑𝑗𝑔,(4.6) almost everywhere 𝑠𝜏. From this equality, and in particular using the Schwarz and Lieb-Thirring inequality (see [17, 18, 23, 24]), one obtains 𝑚𝑖=1𝜑2𝑔𝜆1++𝜆𝑚𝑚0𝑀0𝜆1++𝜆𝑚𝑚0𝑀0𝑐2𝜆1𝑚2,𝑇𝑟𝑗(𝐹𝑈(𝑠,𝜏)𝑢𝜏||||,𝑠𝜈1𝑔𝑚0𝜆11/2𝑚𝑖=1𝜑𝑗2𝑔+𝑈(𝑠,𝜏)𝑢𝜏𝑐1𝑑1𝑚0𝑚𝑖=1𝜑𝑗2𝑔1/2𝜈22||||1𝑔𝑚0𝜆11/2𝑚𝑖=1𝜑𝑗2𝑔+𝑐1𝑑12𝜈𝑚0𝑈(𝑠,𝜏)𝑢𝜏2𝑔𝜈𝑚02𝑀02||||1𝑔𝑚0𝜆11/2𝑐2𝜆1𝑚2+𝑐1𝑑12𝜈𝑚0𝑈(𝑠,𝜏)𝑢𝜏2𝑔.(4.7) On the other hand, we can deduce that 𝑑||𝑑𝑡𝑈(𝑠,𝜏)𝑢𝜏||2𝑔+𝜈𝑈(𝑠,𝜏)𝑢𝜏2𝑔𝑓2𝑉𝑔𝜈+2𝜈𝑚0𝜆𝑔1/2|𝑔|𝑈(𝑠,𝜏)𝑢𝜏2𝑔(4.8) for 𝜆𝑔=4𝜋2𝑚0/𝑀0, and then 𝑡𝜏𝑈(𝑠,𝜏)𝑢𝜏2𝑔1𝑑𝑠𝜈2𝑡𝜏𝑓(𝑠)2𝑉𝑔||𝑢𝑑𝑠+𝜏||2𝜈2||||1𝑔𝑚0𝜆𝑔1/21,𝑡𝜏.(4.9)
Now we define 𝑞𝑚=sup𝜑𝑗𝐻𝑔,|𝜑𝑗|1𝑗=1,2,,𝑚1𝑇𝜏𝜏𝑇𝑇𝑟𝑗𝐹𝑈(𝑠,𝜏)𝑢𝜏,,𝑠𝑑𝑠̃𝑞𝑚𝜈𝑚02𝑀02||||1𝑔𝑚0𝜆11/2𝑐2𝜆1𝑚2+𝑐1𝑑12𝜈𝑚0sup𝜑𝑗𝐻𝑔,|𝜑𝑗|1𝑗=1,2,,𝑚1𝑇𝜏𝜏𝑇𝑈(𝑠,𝜏)𝑢𝜏2𝑔𝑑𝑠𝜈𝑚02𝑀02||||1𝑔𝑚0𝜆11/2𝑐2𝜆1𝑚2+𝑐1𝑑12𝜈𝑚01𝜈2sup𝜑𝑗𝐻𝑔,|𝜑𝑗|1𝑗=1,2,,𝑚1𝑇𝜏𝜏𝑇𝑓(𝑠)2𝑉𝑔||𝑢𝑑𝑠+𝜏||22||||𝜈𝑇1𝑔𝑚0𝜆𝑔1/21𝑞),𝑚=limsup𝑇̃𝑞𝑚𝛼𝑚2+𝛽.(4.10) Hence, dim𝐹𝒜0(𝜏)𝛽𝛼.(4.11)

Acknowledgments

The author would like to thank the reviewers and the editor for their valuable suggestions and comments. This work was supported by the National Science Foundation of China (Grant no. 10901147).