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Discrete Dynamics in Nature and Society
VolumeΒ 2010Β (2010), Article IDΒ 893240, 16 pages
http://dx.doi.org/10.1155/2010/893240
Research Article

On the Dimension of the Pullback Attractors for g-Navier-Stokes Equations

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

Received 3 June 2010; Accepted 14 November 2010

Academic Editor: BinggenΒ Zhang

Copyright Β© 2010 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.

Abstract

We consider the asymptotic behaviour of nonautonomous 2D g-Navier-Stokes equations in bounded domain Ξ©. Assuming that π‘“βˆˆπΏ2loc, which is translation bounded, the existence of the pullback attractor is proved in 𝐿2(Ξ©) and 𝐻1(Ξ©). 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 πœ•π‘’1πœ•π‘‘βˆ’πœˆΞ”π‘’+(π‘’β‹…βˆ‡)𝑒+βˆ‡π‘=𝑓inΞ©,𝑔(βˆ‡β‹…π‘”π‘’)=βˆ‡π‘”π‘”β‹…π‘’+βˆ‡β‹…π‘’=0inΞ©,(1.1) where 𝑔=𝑔(π‘₯1,π‘₯2) is a suitable smooth real-valued function defined on (π‘₯1,π‘₯2)∈Ω and Ξ© is a suitable bounded domain in ℝ2. Notice that if 𝑔(π‘₯1,π‘₯2)=1, then (1.1) reduce to the standard Navier-Stokes equations.

In addition, we assume that the function 𝑓(β‹…,𝑑)=βˆΆπ‘“(𝑑)∈𝐿2loc(ℝ;𝐸) is translation bounded, where 𝐸=𝐿2(Ξ©) or π»βˆ’1(Ξ©). This property implies that‖𝑓‖2𝐿2𝑏=‖𝑓‖2𝐿2𝑏(ℝ;𝐸)=supπ‘‘βˆˆβ„ξ€œπ‘‘π‘‘+1‖𝑓(𝑠)β€–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 Ξ©=(0,1)Γ—(0,1). We assume that the function 𝑔(π‘₯)=𝑔(π‘₯1,π‘₯2) satisfies the following properties:(1)𝑔(π‘₯)∈𝐢∞per(Ξ©),(2)there exist constants π‘š0=π‘š0(𝑔) and 𝑀0=𝑀0(𝑔) such that, for all π‘₯∈Ω, 0<π‘š0≀𝑔(π‘₯)≀𝑀0. Note that the constant function 𝑔≑1 satisfies these conditions.

We denote by 𝐿2(Ξ©,𝑔) the space with the scalar product and the norm given by(𝑒,𝑣)𝑔=ξ€œΞ©(𝑒⋅𝑣)𝑔𝑑π‘₯,|𝑒|2𝑔=(𝑒,𝑒)𝑔,(1.3) as well as 𝐻1(Ξ©,𝑔) with the norm‖𝑒‖𝐻1(Ξ©,𝑔)=(𝑒,𝑒)𝑔+2𝑖=1𝐷𝑖𝑒,𝐷𝑖𝑒𝑔1/2,(1.4) where πœ•π‘’/πœ•π‘₯𝑖=𝐷𝑖𝑒.

Then for the functional setting of the problems (1.1), we use the following functional spaces:𝐻𝑔=𝐢𝑙𝐿2per(Ξ©,𝑔)ξ‚»π‘’βˆˆπΆβˆžperξ€œ(Ξ©)βˆΆβˆ‡β‹…π‘”π‘’=0,Ξ©ξ‚Ό,𝑉𝑒𝑑π‘₯=0𝑔=ξ‚»π‘’βˆˆπ»1perξ€œ(Ξ©,𝑔)βˆΆβˆ‡β‹…π‘”π‘’=0,Ξ©ξ‚Ό,𝑒𝑑π‘₯=0(1.5) where 𝐻𝑔 is endowed with the scalar product and the norm in 𝐿2(Ξ©,𝑔) and 𝑉𝑔 is the spaces with the scalar product and the norm given by((𝑒,𝑣))𝑔=ξ€œΞ©(βˆ‡π‘’β‹…βˆ‡π‘£)𝑔𝑑π‘₯,‖𝑒‖𝑔=((𝑒,𝑒))𝑔.(1.6) Also, we define the orthogonal projection 𝑃𝑔 asπ‘ƒπ‘”βˆΆπΏ2per(Ξ©,𝑔)βŸΆπ»π‘”,(1.7) and we have that π‘„βŠ†π»βŸ‚π‘”, where𝑄=𝐢𝑙𝐿2per(Ξ©,𝑔)ξ‚†βˆ‡πœ™βˆΆπœ™βˆˆπΆ1ξ‚€Ξ©,ℝ.(1.8) Then, we define the 𝑔-Laplacian operatorβˆ’Ξ”π‘”1𝑒≑𝑔1(βˆ‡β‹…π‘”βˆ‡)𝑒=βˆ’Ξ”π‘’βˆ’π‘”(βˆ‡π‘”β‹…βˆ‡)𝑒(1.9) to have the linear operator𝐴𝑔𝑒=π‘ƒπ‘”ξ‚Έβˆ’1𝑔(βˆ‡β‹…(π‘”βˆ‡π‘’)).(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)=π‘‰π‘”βˆ©π»2(Ξ©,𝑔).

(2) There exist countable eigenvalues of 𝐴𝑔 satisfying0<πœ†π‘”β‰€πœ†1β‰€πœ†2β‰€πœ†3≀⋯,(1.11) where πœ†π‘”=4πœ‹2π‘š0/𝑀0 and πœ†1 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 𝐡𝑔(𝑒,𝑣)=𝑃𝑔(π‘’β‹…βˆ‡)𝑣 and the trilinear form𝑏𝑔(𝑒,𝑣,𝑀)=2𝑖,𝑗=1ξ€œΞ©π‘’π‘–ξ€·π·π‘–π‘£π‘—ξ€Έπ‘€π‘—ξ€·π‘ƒπ‘”π‘‘π‘₯=𝑔(π‘’β‹…βˆ‡)𝑣,𝑀𝑔,(1.12) where 𝑒, 𝑣, and 𝑀 lie in appropriate subspaces of 𝐿2(Ξ©,𝑔). Then, the form 𝑏𝑔 satisfies𝑏𝑔(𝑒,𝑣,𝑀)=βˆ’π‘π‘”(𝑒,𝑀,𝑣)for𝑒,𝑣,π‘€βˆˆπ»π‘”.(1.13) We denote a linear operator 𝑅 on 𝑉𝑔 by𝑅𝑒=𝑃𝑔1𝑔(βˆ‡π‘”β‹…βˆ‡)𝑒forπ‘’βˆˆπ‘‰π‘”(1.14) and have 𝑅 as a continuous linear operator from 𝑉𝑔 into 𝐻𝑔 such that||||≀||||(𝑅𝑒,𝑒)βˆ‡π‘”βˆžπ‘š0‖𝑒‖𝑔|𝑒|𝑔≀||||βˆ‡π‘”βˆžπ‘š0πœ†π‘”1/2‖𝑒‖𝑔forπ‘’βˆˆπ‘‰π‘”.(1.15)

We now rewrite (1.1) as abstract evolution equations: 𝑑𝑒𝑑𝑑+πœˆπ΄π‘”π‘’+𝐡𝑔𝑒+πœˆπ‘…π‘’=𝑃𝑔𝑓,𝑒(𝜏)=π‘’πœ.(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 [7–13]) 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 [14–17] 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 Ξ© 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 [8–10, 13].

The semigroup 𝑆(𝑑) property is replaced by the process π‘ˆ(𝑑,𝜏) composition propertyπ‘ˆ(𝑑,𝜏)π‘ˆ(𝜏,𝑠)=π‘ˆ(𝑑,𝑠)βˆ€π‘‘β‰₯𝜏β‰₯𝑠,(2.1) and, obviously, the initial condition implies that π‘ˆ(𝜏,𝜏)=Id. As with the semigroup composition 𝑆(𝑑)𝑆(𝜏)=𝑆(𝑑+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 π‘ˆ is viewed as an element of some parameter space 𝐽, so that the solution can be written as π‘ˆ(𝑑,𝑝)𝜚, and a shift map πœƒπ‘‘βˆΆπ½β†’π½ is defined so that the process composition becomes the cocycle property π‘ˆξ€·(𝑑+𝜏,𝑝)=π‘ˆπ‘‘,πœƒπœπ‘ξ€Έπ‘ˆ(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 π‘ˆ(𝑑,𝜏) to (1.16) by using the concept of measure of noncompactness to obtain pullback attractors.

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

Definition 2.1. Let π‘ˆ be a process on a complete metric space 𝐸. A family of compact sets {π’œ(𝑑)}π‘‘βˆˆβ„ is said to be a pullback attractor for π‘ˆ if, for all πœβˆˆβ„, it satisfies(i)π‘ˆ(𝑑,𝜏)π’œ(𝜏)=π’œ(𝑑) for all 𝑑β‰₯𝜏,(ii)limπ‘ β†’βˆždist(π‘ˆ(𝑑,π‘‘βˆ’π‘ )𝐷,π’œ(𝑑))=0, for π·βˆˆβ„¬(𝐸).The pullback attractor is said to be uniform if the attraction property is uniform in time, that is, limπ‘ β†’βˆžsupπ‘‘βˆˆβ„dist(π‘ˆ(𝑑,π‘‘βˆ’π‘ )𝐷,π’œ(𝑑))=0,forπ·βˆˆβ„¬(𝐸).(2.3)

Definition 2.2. A family of compact sets {π’œ(𝑑)}π‘‘βˆˆβ„ is said to be a forward attractor for π‘ˆ if, for all πœβˆˆβ„, it satisfies(i)π‘ˆ(𝑑,𝜏)π’œ(𝜏)=π’œ(𝑑) for all 𝑑β‰₯𝜏,(ii)limπ‘‘β†’βˆždist(π‘ˆ(𝑑,𝜏)𝐷,π’œ(𝑑))=0, for π·βˆˆβ„¬(𝐸).The forward attractor is said to be uniform if the attraction property is uniform in time, that is, limπ‘‘β†’βˆžsupπœβˆˆβ„dist(π‘ˆ(𝑑+𝜏,𝜏)𝐷,π’œ(𝑑+𝜏))=0,forπ·βˆˆβ„¬(𝐸).(2.4) In the definition, dist(𝐴,𝐡) is the Hausdorff semidistance between 𝐴 and 𝐡, defined as dist(𝐴,𝐡)=supπ‘Žβˆˆπ΄infπ‘βˆˆπ΅π‘‘(π‘Ž,𝑏),for𝐴,π΅βŠ†πΈ.(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 𝑑 when the initial time π‘‘βˆ’π‘  goes to βˆ’βˆž.

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

Definition 2.3. The family {𝐡(𝑑)}π‘‘βˆˆβ„ is said to be (pullback) absorbing with respect to the process π‘ˆ if, for all π‘‘βˆˆβ„ and π·βˆˆβ„¬(𝐸), there exists 𝑆(𝐷,𝑑)>0 such that for all 𝑠β‰₯𝑆(𝐷,𝑑)π‘ˆ(𝑑,π‘‘βˆ’π‘ )π·βŠ‚π΅(𝑑).(2.6) The absorption is said to be uniform if 𝑆(𝐷,𝑑) does not depend on the time variable 𝑑.

Now we recall the abstract results in [21].

Definition 2.4. The family of processes {π‘ˆ(𝑑,π‘‘βˆ’π‘ )} is said to be satisfying pullback Condition (C) if, for any fixed π΅βˆˆβ„¬(𝐸) and πœ€>0, there exist 𝑠0=𝑠(𝐡,𝑑,πœ€)β‰₯0 and a finite dimensional subspace 𝐸1 of 𝐸 such that(i)⋃{‖𝑃(𝑠β‰₯𝑠0π‘ˆ(𝑑,π‘‘βˆ’π‘ )𝐡)‖𝐸} is bounded,(ii)⋃‖(πΌβˆ’π‘ƒ)(𝑠β‰₯𝑠0π‘ˆ(𝑑,π‘‘βˆ’π‘ )𝐡)β€–πΈβ‰€πœ€, where π‘ƒβˆΆπΈβ†’πΈ1 is a bounded projector.

Theorem 2.5. Let the family of processes {π‘ˆ(𝑑,𝜏)} acting in 𝐸 be continuous and possess compact pullback attractor π’œ(𝑑) satisfying π’œ(𝑑)=ξšπ΅βˆˆβ„¬πœ”(𝐡,𝑑),forπ‘‘βˆˆβ„,(2.7) if it (i)has a bounded (pullback) absorbing set 𝐡,(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 Ξ© with periodic boundary condition.

In [1], the author has shown that the semigroup 𝑆(𝑑)βˆΆπ»π‘”β†’π»π‘”(𝑑β‰₯0) 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 π‘“βˆˆπ‘‰β€² be given. Then for every π‘’πœβˆˆπ»π‘” there exists a unique solution 𝑒=𝑒(𝑑) on [0,∞) of (1.16), satisfying 𝑒(𝜏)=π‘’πœ. Moreover, one has 𝑒(𝑑)∈𝐢𝜏,𝑇;π»π‘”ξ€Έβˆ©πΏ2ξ€·πœ,𝑇;𝑉𝑔,βˆ€π‘‡>𝜏.(3.1) Finally, if π‘’πœβˆˆπ‘‰π‘”, then 𝑒(𝑑)∈𝐢𝜏,𝑇;π‘‰π‘”ξ€Έβˆ©πΏ2ξ€·ξ€·π΄πœ,𝑇;𝐷𝑔,βˆ€π‘‡>𝜏.(3.2)

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

Proposition 3.2. The process {π‘ˆ(𝑑,π‘‘βˆ’π‘ )}βˆΆπ‘‰π‘”β†’π‘‰π‘” associated with the system (1.16) possesses (pullback) absorbing sets, that is, there exists a family {𝐡(𝑑)}π‘‘βˆˆπ‘… of bounded (pullback) absorbing sets in 𝐻𝑔 and 𝑉𝑔 for the process π‘ˆ, which is given by ℬ0ξ€½=𝐡(𝑑)=π‘’βˆˆπ»π‘”βˆ£|𝑒|π‘”β‰€πœŒ0ξ€Ύ,ℬ1ξ€½=𝐡(𝑑)=π‘’βˆˆπ‘‰π‘”β€–π‘’β€–π‘”β‰€πœŒ1ξ€Ύ,(3.3) which absorb all bounded sets of 𝐻𝑔. Moreover ℬ0 and ℬ1 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(π‘₯,𝑑)∈𝐿2𝑏(𝑅;𝑉′) and π‘’πœβˆˆπ»π‘”, then the processes {π‘ˆ(𝑑,π‘‘βˆ’π‘ )} corresponding to problem (1.16) possess compact pullback attractor π’œ0(𝑑) in 𝐻𝑔 which coincides with the pullback attractor: π’œ0(𝑑)=ξšβ„¬0βˆˆβ„¬πœ”(ℬ0,𝑑),(3.4) where ℬ0 is the (pullback) absorbing set in 𝐻𝑔.

Proof. As in the previous section, for fixed 𝑁, let 𝐻1 be the subspace spanned by 𝑀1,…,𝑀𝑁, and 𝐻2 the orthogonal complement of 𝐻1 in 𝐻𝑔. We write 𝑒=𝑒1+𝑒2,𝑒1∈𝐻1,𝑒2∈𝐻2foranyπ‘’βˆˆπ»π‘”.(3.5) Now, we only have to verify Condition (C). Namely, we need to estimate |𝑒2(𝑑)|𝑔, where 𝑒(𝑑)=𝑒1(𝑑)+𝑒2(𝑑) is a solution of (1.16) given in Proposition 3.1.
Multiplying (1.16) by 𝑒2, we have 𝑑𝑒𝑑𝑑,𝑒2𝑔+ξ€·πœˆπ΄π‘”π‘’,𝑒2𝑔+𝐡𝑔(𝑒,𝑒),𝑒2𝑔=𝑓,𝑒2ξ€Έπ‘”βˆ’ξ€·πœˆπ‘…π‘’,𝑒2𝑔.(3.6) It follows that 12𝑑||𝑒𝑑𝑑2||2𝑔‖‖𝑒+πœˆπ‘”β€–β€–2𝑔≀|||𝐡(𝑒,𝑒),𝑒2𝑔|||+|||𝑓,𝑒2𝑔|||+𝑅𝑒,𝑒2𝑔.(3.7) Since 𝑏𝑔 satisfies the following inequality (see [6]): ||𝑏𝑔||(𝑒,𝑣,𝑀)≀𝑐|𝑒|𝑔1/2‖𝑒‖𝑔1/2‖𝑣‖𝑔|𝑀|𝑔1/2‖𝑀‖𝑔1/2,βˆ€π‘’,𝑣,π‘€βˆˆπ‘‰π‘”,(3.8) thus, |||𝐡(𝑒,𝑒),𝑒2𝑔|||≀𝑐|𝑒|𝑔1/2‖𝑒‖𝑔3/2||𝑒2||𝑔1/2‖‖𝑒2‖‖𝑔1/2β‰€π‘πœ†π‘š+1|𝑒|𝑔1/2‖𝑒‖𝑔3/2‖‖𝑒2β€–β€–π‘”β‰€πœˆ6‖‖𝑒2β€–β€–2𝑔+π‘πœŒ0𝜌31.(3.9) Next, using the Cauchy inequality, |||ξ€·πœˆπ‘…π‘’,𝑒2𝑔|||=||||ξ‚΅πœˆπ‘”(βˆ‡π‘”β‹…βˆ‡)𝑒,𝑒2𝑔||||β‰€πœˆπ‘š0||||βˆ‡π‘”βˆžβ€–π‘’β€–π‘”||𝑒2||π‘”β‰€πœˆ6‖𝑒2β€–2𝑔+3𝜈𝜌21||||βˆ‡π‘”2∞2π‘š20πœ†π‘”πœ†π‘š+1.(3.10) Finally, we have |||𝑓,𝑒2𝑔|||≀||𝑓||𝑉′𝑔‖𝑒2πœˆβ€–β‰€6‖𝑒2β€–2𝑔+3||𝑓||2𝜈2𝑉′𝑔.(3.11) Putting (3.9)–(3.11) together, there exists constant 𝑀1=𝑀1(π‘š0,|βˆ‡π‘”|∞,𝜌0,𝜌1) such that 12𝑑||𝑒𝑑𝑑2||2𝑔+12πœˆβ€–π‘’2β€–2𝑔≀3||𝑓||𝑉′𝑔2𝜈+𝑀1.(3.12) Therefore, we deduce that 𝑑||𝑒𝑑𝑑2||2𝑔+πœˆπœ†π‘š+1||𝑒2||2𝑔≀2𝑀1+3𝜈||𝑓||2𝑉′𝑔.(3.13)Here, 𝑀1 depends on πœ†π‘š+1, is not increasing as πœ†π‘š+1 increasing.
By the Gronwall inequality, the above inequality implies that ||𝑒2||(𝑑)2𝑔≀||𝑒2||(𝜏)2π‘”π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœ)+2𝑀1πœˆπœ†π‘š+1+3πœˆξ€œπ‘‘πœπ‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’π‘ )||𝑓||2𝑉′𝑑𝑠.(3.14) If we consider the time π‘‘βˆ’π‘  instead of 𝜏 (so that we can use more easily the definition of pullback attractors), we have 3πœˆξ€œπ‘‘πœπ‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)||||𝑓(𝜎)2𝑉′3π‘‘πœŽ=πœˆξ€œπ‘‘π‘‘βˆ’π‘ π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)||||𝑓(𝜎)2π‘‰β€²π‘‘πœŽ.(3.15) Applying continuous integral and Lemma II  1.3 in [18] for any πœ€, there exists πœ‚=πœ‚(πœ€)>0 such that ξ€œπ‘‘π‘‘βˆ’πœ‚||||𝑓(𝜎)2π‘‰β€²π‘‘πœŽ<πœˆπœ€;18(3.16) thus, we have 3πœˆξ€œπ‘‘π‘‘βˆ’πœ‚π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)||||𝑓(𝜎)2π‘‰β€²πœ€π‘‘πœŽβ‰€6,3(3.17)πœˆξ€œπ‘‘βˆ’πœ‚π‘‘βˆ’π‘ π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)||||𝑓(𝜎)2𝑉′≀3π‘‘πœŽπœˆξ€œπ‘‘βˆ’πœ‚π‘‘βˆ’πœ‚βˆ’1π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)||||𝑓(𝜎)2𝑉′+3π‘‘πœŽπœˆξ€œπ‘‘βˆ’πœ‚βˆ’1π‘‘βˆ’πœ‚βˆ’2π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)||||𝑓(𝜎)2𝑉′≀3π‘‘πœŽ+β‹…β‹…β‹…πœˆπ‘’βˆ’πœˆπœ†π‘š+1πœ‚ξ‚΅ξ€œπ‘‘βˆ’πœ‚π‘‘βˆ’πœ‚βˆ’1||||𝑓(𝜎)2π‘‰β€²π‘‘πœŽ+π‘’βˆ’πœˆπœ†π‘š+1ξ€œπ‘‘βˆ’πœ‚βˆ’1π‘‘βˆ’πœ‚βˆ’2||||𝑓(𝜎)2𝑉′≀3π‘‘πœŽ+β‹…β‹…β‹…πœˆπ‘’βˆ’πœˆπœ†π‘š+1πœ‚ξ€·1+π‘’βˆ’πœˆπœ†π‘š+1ξ€Έ+β‹…β‹…β‹…supπ‘ βˆˆπ‘…ξ€œπ‘ π‘ βˆ’1||||𝑓(𝜎)2π‘‰β€²β‰€π‘‘πœŽ(3/𝜈)π‘’βˆ’πœˆπœ†π‘š+1πœ‚1βˆ’π‘’βˆ’πœˆπœ†π‘š+1‖𝑓‖2𝐿2𝑏.(3.18) Using (1.11) and letting 𝑠1=(1/πœˆπœ†π‘š+1)ln(3𝜌20/πœ€), then 𝑠β‰₯𝑠1 implies that 3πœˆξ€œπ‘‘βˆ’πœ‚π‘‘βˆ’π‘ π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)||||𝑓(𝜎)2π‘‰β€²π‘‘πœŽβ‰€(3/𝜈)π‘’βˆ’πœˆπœ†π‘š+1πœ‚1βˆ’π‘’βˆ’πœˆπœ†π‘š+1‖𝑓‖2𝐿2𝑏(𝑅;𝑉′)β‰€πœ€6,(3.19)2𝑀1πœˆπœ†π‘š+1β‰€πœ€3,||𝑒2||(𝜏)2π‘”π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœ)β‰€πœŒ20π‘’βˆ’πœˆπœ†π‘š+1𝑠1β‰€πœ€3.(3.20) Therefore, we deduce from (3.14) that ||𝑒2||2π‘”β‰€πœ€,βˆ€π‘ β‰₯𝑠1,(3.21) which indicates {π‘ˆ(𝑑,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 {π‘ˆ(𝑑,𝜏)} are defined in 𝑉𝑔 and ℬ1 is a pullback absorbing set in 𝑉𝑔.

Theorem 3.4. If 𝑓(π‘₯,𝑑)∈𝐿2𝑏(𝑅;𝐻𝑔), then the processes {π‘ˆ(𝑑,𝜏)} corresponding to problem (1.16) possess compact pullback attractor π’œ1(𝑑) in 𝑉𝑔: π’œ1(𝑑)=ξšβ„¬1βˆˆβ„¬πœ”(ℬ1,𝑑),(3.22) where ℬ1 is the absorbing set in 𝑉𝑔.

Proof. Using Proposition 3.2, we have that the family of processes {π‘ˆ(𝑑,𝜏)} corresponding to (1.16) possess the pullback absorbing set in 𝑉𝑔.
Now we testify that the family of processes {π‘ˆ(𝑑,𝜏)} corresponding to (1.16) satisfies pullback Condition (C).
Multiplying (1.16) by 𝐴𝑔𝑒2(𝑑), we have 𝑑𝑣𝑑𝑑,𝐴𝑔𝑒2+ξ€·πœˆπ΄π‘”π‘’,𝐴𝑔𝑒2ξ€Έ+𝐡𝑔(𝑒,𝑒),𝐴𝑔𝑒2𝑔=𝑓,𝐴𝑔𝑒2ξ€Έβˆ’ξ€·πœˆπ‘…π‘’,𝐴𝑔𝑒2𝑔.(3.23) It follows that 12𝑑𝑑𝑑‖𝑒2β€–2𝑔||𝐴+πœˆπ‘”π‘’2||2𝑔≀|||𝐡𝑔(𝑒,𝑒),𝐴𝑔𝑒2𝑔|||+|||𝑓,𝐴𝑔𝑒2𝑔|||+|||ξ€·πœˆπ‘…π‘’,𝐴𝑔𝑒2𝑔|||.(3.24) To estimate (𝐡𝑔(𝑒,𝑒),𝐴𝑒2)𝑔, we recall some inequalities (see [22]), for every 𝑒,π‘£βˆˆπ·(𝐴𝑔), ||𝐡𝑔||ξƒ―(𝑒,𝑣)≀𝑐|𝑒|𝑔1/2‖𝑒‖𝑔1/2‖𝑣‖𝑔1/2||𝐴𝑔𝑣||𝑔1/2,|𝑒|𝑔1/2||𝐴𝑔𝑒||𝑔1/2‖𝑣‖𝑔,(3.25)|𝑀|𝐿∞(Ξ©)2≀𝑐‖𝑀‖𝑔||𝐴1+log𝑔𝑀||πœ†π‘”β€–π‘€β€–2𝑔ξƒͺ1/2,(3.26) from which we deduce that ||𝐡𝑔||(𝑒,𝑣)≀𝑐|𝑒|𝐿∞(Ξ©)||||βˆ‡π‘£π‘”|𝑒|𝑔||||βˆ‡π‘£πΏβˆž(Ξ©),(3.27) and using (3.26), ||𝐡𝑔||⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩(𝑒,𝑣)≀𝑐‖𝑒‖𝑔‖𝑣‖𝑔1+log|𝐴𝑔𝑒|2πœ†π‘”β€–π‘’β€–2𝑔ξƒͺ1/2,||𝑒|𝑔||𝐴𝑔𝑣|𝑔1+log|𝐴𝑔3/2𝑣|2πœ†π‘”β€–π΄π‘”π‘£β€–2𝑔ξƒͺ1/2.(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 ||𝐡𝑔(𝑒,𝑒),𝐴𝑔𝑒2ξ€Έ||≀||𝐡𝑔𝑒1,𝑒1+𝑒2ξ€Έ,𝐴𝑔𝑒2ξ€Έ||+||𝐡𝑔𝑒2,𝑒1+𝑒2ξ€Έ,𝐴𝑔𝑒2ξ€Έ||≀𝑐𝐿1/2‖𝑒1‖𝑔||𝐴𝑔𝑒2||𝑔‖𝑒1‖𝑔+‖𝑒2‖𝑔||𝑒+𝑐2||𝑔1/2||𝐴𝑔𝑒2||𝑔3/2β‰€πœˆ6||𝐴𝑔𝑒2||2𝑔+π‘πœˆπœŒ41𝑐𝐿+𝜈3𝜌20𝜌41,𝑑β‰₯𝑑0+1,(3.29) where we use ||𝐴𝑔𝑒1||2π‘”β‰€πœ†π‘šβ€–π‘’1β€–2𝑔(3.30) and set πœ†πΏ=1+logπ‘š+1πœ†π‘”.(3.31) Next, using the Cauchy inequality, |||𝑅𝑒,𝐴𝑔𝑒2𝑔|||=||||ξ‚΅πœˆπ‘”(βˆ‡π‘”β‹…βˆ‡)𝑒,𝐴𝑔𝑒2𝑔||||β‰€πœˆπ‘š0||||βˆ‡π‘”βˆžβ€–π‘’β€–π‘”||𝐴𝑔𝑒2||π‘”β‰€πœˆ6||𝐴𝑔𝑒2||2𝑔+3𝜈2||||βˆ‡π‘”2∞𝜌21.(3.32) Finally, we estimate |(𝑓,𝐴𝑔𝑒2)| by ||𝑓,𝐴𝑔𝑒2ξ€Έ||≀||𝑓||𝑔||𝐴𝑔𝑒2||2β‰€πœˆ6||𝐴𝑔𝑒2||2𝑔+3||𝑓||2𝜈2𝑔.(3.33) Putting (3.29)–(3.33) together, there exists a constant 𝑀2 such that 𝑑𝑑𝑑‖𝑒2β€–2𝑔+πœˆπœ†π‘š+1‖𝑒2β€–2𝑔≀3𝜈||𝑓||𝑔+𝑀2.(3.34) Here, 𝑀2=𝑀2(𝜌0,𝜌1,𝐿,𝜈,|βˆ‡π‘”|) depends on πœ†π‘š+1, is not increasing as πœ†π‘š+1 increasing. Therefore, by the Gronwall inequality, the above inequality implies that ‖𝑒2β€–2𝑔≀‖𝑒2(𝜏)β€–2π‘”π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœ)+2𝑀2πœˆπœ†π‘š+1+3πœˆξ€œπ‘‘πœπ‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’π‘ )||𝑓||2𝑔𝑑𝑠.(3.35) We consider the time π‘‘βˆ’π‘  instead of 𝜏. The following result is similar to (3.17)–(3.19), for any πœ€: 2π‘πœˆξ€œπ‘‘πœπ‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœŽ)/2||𝑓||2π‘”πœ€π‘‘πœŽβ‰€3.(3.36) Using (1.11) and letting 𝑠2=(2/πœˆπœ†π‘š+1)ln(3𝜌21/πœ€), then 𝑠β‰₯𝑠2 implies that 2𝑀2πœˆπœ†π‘š+1β‰€πœ€3,‖𝑒2(𝜏)β€–2π‘”π‘’βˆ’πœˆπœ†π‘š+1(π‘‘βˆ’πœ)β‰€πœŒ21π‘’βˆ’πœˆπœ†π‘š+1𝑠<πœ€3.(3.37) Therefore, we deduce from (3.35) that ‖𝑒2β€–2π‘”β‰€πœ€,βˆ€π‘ β‰₯𝑠1,(3.38) which indicates {π‘ˆ(𝑑,𝜏)} satisfying pullback Condition (C) in 𝑉𝑔.

4. The Dimension of the Pullback Attractor

To estimate the dimension of the pullback attractor π’œ0(𝑑), we will apply the abstract machinery in [18, 23]. Let πΉβˆΆπ‘‰π‘”Γ—β„β†’π‘‰ξ…žπ‘” be a given family of nonlinear operators such that, for all πœβˆˆβ„ and any π‘’πœβˆˆπ»π‘”, there exists a unique function 𝑒(𝑑)=𝑒(𝑑;𝜏,𝑒0) satisfyingπ‘’βˆˆπΏ2ξ€·πœ,𝑇;π‘‰π‘”ξ€Έξ€Ίβˆ©πΆπœ,𝑇;𝐻𝑔,𝐹(𝑒(𝑑),𝑑)∈𝐿1ξ€·πœ,𝑇;π‘‰ξ…žπ‘”ξ€Έ,βˆ€π‘‡>𝜏,𝑑𝑒𝑑𝑑=𝐹(𝑒(𝑑),𝑑),𝑑>𝜏,𝑒(𝜏)=π‘’πœ,(4.1) where 𝐹(𝑒)=βˆ’πœˆπ΄π‘”π‘’βˆ’π΅π‘”π‘’βˆ’πœˆπ‘…π‘’+𝑃𝑔𝑓.

Using the standard methods (see [17, 18]), we can show that {π‘ˆ(𝑑,𝜏)} is uniformly quasidifferentiable on {𝐡(𝑑)}π‘‘βˆˆβ„. Then, for all πœβ‰€π‘‡ and any π‘’πœ,π‘£πœβˆˆπ»π‘”, there exists a unique 𝑣(𝑑)=𝑣(𝑑;𝜏,π‘’πœ,π‘£πœ), which is a solution ofπ‘£βˆˆπΏ2ξ€·πœ,𝑇;π‘‰π‘”ξ€Έξ€Ίβˆ©πΆπœ,𝑇;𝐻𝑔,𝑑𝑣𝑑𝑑=πΉβ€²π‘ˆ(𝑑,𝜏)π‘’πœξ€Έ,𝑑𝑣,𝑣(𝜏)=π‘£πœ.(4.2) For all 𝜏<𝑇, we define the linear operator 𝐿(𝑑,π‘’πœ)βˆΆπ»π‘”β†’π»π‘” by𝑣𝑑;𝜏,π‘’πœ,π‘£πœξ€Έξ€·=𝐿𝑑,𝜏,π‘’πœξ€Έπ‘£πœ.(4.3)

Theorem 4.1. Suppose that 𝑓(𝑑) satisfies the assumptions of Theorem 3.3. Then, if 𝛾=1βˆ’(2|βˆ‡π‘”|∞/π‘š0πœ†π‘”1/2)>0, the Pullback attractor (uniformly in the past) π’œ0 defined by (3.4) satisfies π‘‘πΉξ€·π’œ0≀𝛽𝛼,(4.4) where 𝑐𝛼=2πœˆπ‘š0πœ†ξ…ž1𝛾2𝑀0,𝑐𝛽=1𝑑12𝜈3π‘š0𝛾supπœ‘π‘—βˆˆπ»π‘”,|πœ‘π‘—|≀1𝑗=1,2,…,π‘š1π‘‡ξ€œπœπœβˆ’π‘‡β€–π‘“(𝑠)β€–2𝑉′𝑔𝑑𝑠,(4.5) with the constant 𝑐1,𝑐2 of (3.29) and (3.32) of Chapter VI in [17], πœ†ξ…ž1 is the first eigenvalue of the Stokes operator and 𝑑1=|βˆ‡π‘”|2∞/4π‘š0+|βˆ‡π‘”|∞+𝑀0.

Proof. With Theorem 3.3 at our disposal we may apply the abstract framework in [17, 18, 23, 24].
For πœ‰1,πœ‰2,…,πœ‰π‘šβˆˆπ»π‘”, let 𝑣𝑗(𝑑)=𝐿(𝑑,π‘’πœ)β‹…πœ‰π‘—, where π‘’πœβˆˆπ»π‘”. Let {πœ‘π‘—(𝑠);𝑗=1,2,…,π‘š} be an orthonormal basis for span {𝑣𝑗;𝑗=1,2,…,π‘š}. Since 𝑣(𝑠;𝜏,π‘’πœ,π‘£π‘—πœ)βˆˆπ‘‰π‘” almost everywhere 𝑠β‰₯𝜏, we can also assume that πœ‘π‘—(𝑠)βˆˆπ‘‰π‘” almost everywhere 𝑠β‰₯𝜏. Then, similar to the proof process of Theorems 3.3 and 3.4, we may obtain π‘šξ“π‘–=1ξ«ξ€·πΉβ€²π‘ˆ(𝑠,𝜏)π‘’πœξ€Έπœ‘,𝑠𝑖,πœ‘π‘–ξ¬=βˆ’πœˆπ‘šξ“π‘–=1β€–πœ‘π‘—β€–2π‘”βˆ’π‘šξ“π‘–=1π‘π‘”ξ€·πœ‘π‘—,π‘ˆ(𝑠,𝜏)π‘’πœ,πœ‘π‘—ξ€Έβˆ’π‘šξ“π‘–=1ξ‚΅πœˆπ‘”(βˆ‡π‘”β‹…βˆ‡)πœ‘π‘—,πœ‘π‘—ξ‚Άπ‘”,(4.6) almost everywhere 𝑠β‰₯𝜏. From this equality, and in particular using the Schwarz and Lieb-Thirring inequality (see [17, 18, 23, 24]), one obtains π‘šξ“π‘–=1β€–πœ‘β€–2𝑔β‰₯πœ†1+β‹…β‹…β‹…+πœ†π‘šβ‰₯π‘š0𝑀0ξ€·πœ†ξ…ž1+β‹―+πœ†ξ…žπ‘šξ€Έβ‰₯π‘š0𝑀0𝑐2πœ†ξ…ž1π‘š2,π‘‡π‘Ÿπ‘—(πΉξ…žξ€·π‘ˆ(𝑠,𝜏)π‘’πœξ€Έξƒ©||||,π‘ β‰€βˆ’πœˆ1βˆ’βˆ‡π‘”βˆžπ‘š0πœ†11/2ξƒͺπ‘šξ“π‘–=1β€–πœ‘π‘—β€–2𝑔+β€–π‘ˆ(𝑠,𝜏)π‘’πœβ€–ξƒ©π‘1𝑑1π‘š0π‘šξ“π‘–=1β€–πœ‘π‘—β€–2𝑔ξƒͺ1/2πœˆβ‰€βˆ’22||||1βˆ’βˆ‡π‘”βˆžπ‘š0πœ†11/2ξƒͺπ‘šξ“π‘–=1β€–πœ‘π‘—β€–2𝑔+𝑐1𝑑12πœˆπ‘š0β€–π‘ˆ(𝑠,𝜏)π‘’πœβ€–2π‘”β‰€βˆ’πœˆπ‘š02𝑀02||||1βˆ’βˆ‡π‘”βˆžπ‘š0πœ†11/2ξƒͺ𝑐2πœ†ξ…ž1π‘š2+𝑐1𝑑12πœˆπ‘š0β€–π‘ˆ(𝑠,𝜏)π‘’πœβ€–2𝑔.(4.7) On the other hand, we can deduce that 𝑑||π‘‘π‘‘π‘ˆ(𝑠,𝜏)π‘’πœ||2𝑔+πœˆβ€–π‘ˆ(𝑠,𝜏)π‘’πœβ€–2𝑔≀‖𝑓‖2π‘‰β€²π‘”πœˆ+2πœˆπ‘š0πœ†π‘”1/2|βˆ‡π‘”|βˆžβ€–π‘ˆ(𝑠,𝜏)π‘’πœβ€–2𝑔(4.8) for πœ†π‘”=4πœ‹2π‘š0/𝑀0, and then ξ€œπ‘‘πœβ€–π‘ˆ(𝑠,𝜏)π‘’πœβ€–2𝑔1π‘‘π‘ β‰€πœˆ2ξ€œπ‘‘πœβ€–π‘“(𝑠)β€–2𝑉′𝑔||𝑒𝑑𝑠+𝜏||2𝜈2||||ξƒͺ1βˆ’βˆ‡π‘”βˆžπ‘š0πœ†π‘”1/2ξƒͺβˆ’1,𝑑β‰₯𝜏.(4.9)
Now we define π‘žπ‘š=supπœ‘π‘—βˆˆπ»π‘”,|πœ‘π‘—|≀1𝑗=1,2,…,π‘šξ‚΅1π‘‡ξ€œπœπœβˆ’π‘‡π‘‡π‘Ÿπ‘—ξ€·ξ€·πΉβ€²π‘ˆ(𝑠,𝜏)π‘’πœξ€Έξ€Έξ‚Ά,,π‘ π‘‘π‘ Μƒπ‘žπ‘šβ‰€βˆ’πœˆπ‘š02𝑀02||||1βˆ’βˆ‡π‘”βˆžπ‘š0πœ†11/2ξƒͺ𝑐2πœ†ξ…ž1π‘š2+𝑐1𝑑12πœˆπ‘š0βŽ›βŽœβŽœβŽœβŽsupπœ‘π‘—βˆˆπ»π‘”,|πœ‘π‘—|≀1𝑗=1,2,…,π‘šξ‚΅1π‘‡ξ€œπœπœβˆ’π‘‡β€–π‘ˆ(𝑠,𝜏)π‘’πœβ€–2π‘”ξ‚ΆβŽžβŽŸβŽŸβŽŸβŽ π‘‘π‘ β‰€βˆ’πœˆπ‘š02𝑀02||||1βˆ’βˆ‡π‘”βˆžπ‘š0πœ†11/2ξƒͺ𝑐2πœ†ξ…ž1π‘š2+𝑐1𝑑12πœˆπ‘š0βŽ›βŽœβŽœβŽœβŽ1𝜈2supπœ‘π‘—βˆˆπ»π‘”,|πœ‘π‘—|≀1𝑗=1,2,…,π‘š1π‘‡ξ€œπœπœβˆ’π‘‡β€–π‘“(𝑠)β€–2𝑉′𝑔||𝑒𝑑𝑠+𝜏||2βŽžβŽŸβŽŸβŽŸβŽ ξƒ©2||||πœˆπ‘‡1βˆ’βˆ‡π‘”βˆžπ‘š0πœ†π‘”1/2ξƒͺβˆ’1π‘ž),π‘š=limsupπ‘‡β†’βˆžΜƒπ‘žπ‘šβ‰€βˆ’π›Όπ‘š2+𝛽.(4.10) Hence, dimπΉπ’œ0ξ‚™(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).

References

  1. J. Roh, β€œDynamics of the g-Navier-Stokes equations,” Journal of Differential Equations, vol. 211, no. 2, pp. 452–484, 2005. View at Publisher Β· View at Google Scholar
  2. O. A. Ladyzhenskaya, β€œOn the dynamical system generated by the Navier-Stokes equations,” Zapiskii of Nauchnish Seminarovs LOMI, vol. 27, pp. 91–114, 1972, english translation in: Journal of Soviet Mathematics, vol. 3, 1975.
  3. C. Foiaş and R. Temam, β€œFinite parameter approximative structure of actual flows,” in Nonlinear Problems: Present and Future, A. R. Bishop, D. K. Campbell, and B. Nicolaenko, Eds., vol. 61, pp. 317–327, North-Holland, Amsterdam, The Netherlands, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. J. Mallet-Paret, β€œNegatively invariant sets of compact maps and an extension of a theorem of Cartwright,” Journal of Differential Equations, vol. 22, no. 2, pp. 331–348, 1976. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill, USA, 1988.
  6. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, vol. 2 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 3rd edition, 1984.
  7. T. Caraballo and J. Real, β€œNavier-Stokes equations with delays,” Proceedings of the Royal Society of London A, vol. 457, no. 2014, pp. 2441–2453, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. T. Caraballo and J. Real, β€œAsymptotic behaviour of two-dimensional Navier-Stokes equations with delays,” Proceedings of the Royal Society of London A, vol. 459, no. 2040, pp. 3181–3194, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. T. Caraballo and J. Real, β€œAttractors for 2D-Navier-Stokes models with delays,” Journal of Differential Equations, vol. 205, no. 2, pp. 271–297, 2004. View at Publisher Β· View at Google Scholar
  10. D. N. Cheban, P. E. Kloeden, and B. Schmalfuß, β€œThe relationship between pullback, forward and global attractors of nonautonomous dynamical systems,” Nonlinear Dynamics and Systems Theory, vol. 2, no. 2, pp. 125–144, 2002. View at Zentralblatt MATH
  11. H. Crauel, A. Debussche, and F. Flandoli, β€œRandom attractors,” Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp. 307–341, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. J. Mallet-Paret and G. R. Sell, β€œThe PoincarΓ©-Bendixson theorem for monotone cyclic feedback systems with delay,” Journal of Differential Equations, vol. 125, no. 2, pp. 441–489, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. B. Schmalfuss, β€œBackward cocycle and attractors of stochastic differential equations,” in Proceedings of the International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, V. Reitmann, T. Redrich, and N. J. Kosch, Eds., pp. 185–192, 1992.
  14. I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta Scientific, Kharkiv, Ukraine, 2002.
  15. Q. Ma, S. Wang, and C. Zhong, β€œNecessary and sufficient conditions for the existence of global attractors for semigroups and applications,” Indiana University Mathematics Journal, vol. 51, no. 6, pp. 1541–1559, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2002.
  17. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997.
  18. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2002.
  19. I. Moise, R. Rosa, and X. Wang, β€œAttractors for noncompact nonautonomous systems via energy equations,” Discrete and Continuous Dynamical Systems. Series A, vol. 10, no. 1-2, pp. 473–496, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. P. E. Kloeden and D. J. Stonier, β€œCocycle attractors in nonautonomously perturbed differential equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 4, no. 2, pp. 211–226, 1998. View at Zentralblatt MATH
  21. Y. Wang, C. Zhong, and S. Zhou, β€œPullback attractors of nonautonomous dynamical systems,” Discrete and Continuous Dynamical Systems A, vol. 16, no. 3, pp. 587–614, 2006. View at Zentralblatt MATH
  22. C. Foias, O. P. Manley, R. Temam, and Y. M. TrΓ¨ve, β€œAsymptotic analysis of the Navier-Stokes equations,” Physica D, vol. 9, no. 1-2, pp. 157–188, 1983. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  23. V. V. Chepyzhov and A. A. Ilyin, β€œOn the fractal dimension of invariant sets: applications to Navier-Stokes equations,” Discrete and Continuous Dynamical Systems A, vol. 10, no. 1-2, pp. 117–135, 2004. View at Zentralblatt MATH
  24. J. A. Langa, G. Łukaszewicz, and J. Real, β€œFinite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 3, pp. 735–749, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH