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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 503454, 12 pages
http://dx.doi.org/10.1155/2012/503454
Research Article

The Exponential Attractors for the g-Navier-Stokes Equations

1College of Modern Science and Technology, China Jiliang University, Hangzhou 310018, China
2College of Science, China Jiliang University, Hangzhou 310018, China

Received 6 February 2012; Accepted 2 May 2012

Academic Editor: Pankaj Jain

Copyright © 2012 Delin Wu and Jicheng Tao. 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 exponential attractors for the two-dimensional g-Navier-Stokes equations in bounded domain Ω. We establish the existence of the exponential attractor in L2(Ω).

1. Introduction

In this paper, we study the behavior of solutions of the 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 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 Roh [1] the author established the global regularity of solutions of the g-Navier-Stokes equations. One can refer to [2] for details. 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. Before we present the derivation of the g-Navier-Stokes equations, it is convenient to recall some relevant aspects of the classical theory of the Navier-Stokes equations. 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 Ladyzhenskaya [3] and independently by Foiaş and Temam [4]. The finite-dimensional property of the global attractor for general dissipative equations was first proved by Mallet-Paret [5]. For the analysis on the Navier-Stokes equations, one can refer to [6].

In the past decades, many papers in the literature show that the long-time behavior of dissipative systems can be understood through the concept of attractors, see [714]. In addition, in [15] the authors introduced the so-called exponential attractors, which is an interesting intermediate object between the usual (global) attractors and an inertial manifold and satisfies some nice properties like those of inertial manifolds (e.g., finite fractal dimension, exponential attracting, stable with respect to some perturbations). Indeed it now seems clear that the interesting object to investigate is the exponential attractor, rather than the usual (global) attractor (which is recovered as a byproduct). See [16, 17], and so forth. The exponential attractor is a compact and positively invariant set having finite fractal dimension which contains the global attractor and attracts every trajectory at an exponential rate. It is also known that the exponential attractor enjoys stronger robustness than the global attractor. When the semigroup of a dynamical system depends continuously on a parameter, the global attractor is in general only upper-semicontinuous. In turn, under some reasonable assumptions, if an exponential attractor exists, it can depend continuously on the parameter. Such a continuous dependence was recently studied by Efendiev and Yagi [18]. When the underlying space is a Hilbert space, it is known by the same reference [15] quoted above that the squeezing property of semigroup implies existence of exponential attractors and provides a sharp estimate of attractor dimensions. When the underlying space is a Banach space, it is known by Efendiev et al. [19] that the compact smoothing property of semigroup implies existence of exponential attractors (Theorem 2.3). Another construction of exponential attractors in Banach spaces was proposed by Dung and Nicolaenko in [20]. We also refer to [17, 2125] for more details.

In the paper, compared with the result obtained in [26], taking advantage of a recent result due to Efendiev et al. [19] (Theorem 2.3), we construct the exponential attractor. This paper is organized as follows. In Section 2, we first recall some basic results, and then, give an important technique tool [19], that is, Theorem 2.3. In Section 3, we study the existence of compact exponential attractor for the two-dimensional g-Navier-Stokes equations in the periodic boundary conditions Ω.

2. Preliminary Results

Let Ω=(0,1)×(0,1) and 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,(2.1) as well as 𝐻1(Ω,𝑔) with the norm𝑢𝐻1(Ω,𝑔)=(𝑢,𝑢)𝑔+2𝑖=1𝐷𝑖𝑢,𝐷𝑖𝑢𝑔1/2,(2.2) 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(2.3) 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.(2.4) Also, we define the orthogonal projection 𝑃𝑔 as𝑃𝑔𝐿2per(Ω,𝑔)𝐻𝑔(2.5) and we have that 𝑄𝐻𝑔, where𝑄=𝐶𝑙𝐿2per(Ω,𝑔)𝜙𝜙𝐶1Ω,.(2.6) Then, we define the 𝑔-Laplacian operatorΔ𝑔1𝑢𝑔1(𝑔)𝑢=Δ𝑢𝑔(𝑔)𝑢(2.7) to have the linear operator𝐴𝑔𝑢=𝑃𝑔1𝑔((𝑔𝑢)).(2.8) For the linear operator 𝐴𝑔, the following hold (see Roh [1]):(1)𝐴𝑔 is a positive, self-adjoint operator with compact inverse, where the domain of 𝐴𝑔,𝐷(𝐴𝑔)=𝑉𝑔𝐻2(Ω,𝑔).(2)There exist countable eigenvalues of 𝐴𝑔 satisfying0<𝜆𝑔𝜆1𝜆2𝜆3,(2.9) where 𝜆𝑔=4𝜋2𝑚/𝑀 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,(2.10) where 𝑢,𝑣,𝑤 lie in appropriate subspaces of 𝐿2(Ω,𝑔). Then, the form 𝑏𝑔 satisfies𝑏𝑔(𝑢,𝑣,𝑤)=𝑏𝑔(𝑢,𝑤,𝑣)for𝑢,𝑣,𝑤𝐻𝑔.(2.11)

We denote a linear operator 𝑅 on 𝑉𝑔 by𝑅𝑢=𝑃𝑔1𝑔(𝑔)𝑢for𝑢𝑉𝑔,(2.12) and have 𝑅 as a continuous linear operator from 𝑉𝑔 into 𝐻𝑔 such that||||||||(𝑅𝑢,𝑢)𝑔𝑚0𝑢𝑔|𝑢|𝑔||||𝑔𝑚0𝜆𝑔1/2𝑢𝑔for𝑢𝑉𝑔.(2.13)

We now rewrite (1.1) as abstract evolution equations,𝑑𝑢𝑑𝑡+𝜈𝐴𝑔𝑢+𝐵𝑔𝑢+𝜈𝑅𝑢=𝑃𝑔𝑓,𝑢(0)=𝑢0.(2.14)

Let us first recall some basic matters on the dynamical system. Let 𝐸 be a Banach space and let 𝐾 be a subset of 𝐸,𝐾 being a metric space equipped with the distance induced from the norm of 𝐸. Let 𝑆(𝑡),0𝑡< be a family of mappings from 𝐾 into itself having the following properties: (i) 𝑆(0)=𝐼 (the identity mapping); (ii) 𝑆(𝑡)𝑆(𝑠)=𝑆(𝑡+𝑠),0𝑡,𝑠< (the semigroup property); (iii) the mapping 𝐺[0,)×𝐾𝐾,(𝑡,𝑢0)𝑆(𝑡)𝑢0, is continuous. Such a family is called a continuous (nonlinear) semigroup acting on 𝐾. The image of 𝑆()𝑢0 drawn in 𝐾 is called the trajectory starting from 𝐾. The whole of such trajectories is the dynamical system (𝑆(𝑡),𝐾,𝐸), where 𝐾 and 𝐸 are called the phase-space and the universal space, respectively.

A subset 𝒜 of the phase-space 𝐾 is the global attractor of (𝑆(𝑡),𝐾,𝐸) if the following conditions are satisfied: (i) 𝒜 is a compact subset of 𝐸; (ii) 𝒜 is an invariant set, that is, 𝑆(𝑡)𝒜=𝒜 for every 0<𝑡<; (iii) 𝒜 attracts every bounded subset of 𝐾, namely, for any bounded subset 𝐵𝐾, it holds that lim𝑡dist(𝑆(𝑡)𝐵,𝒜)=0, where dist(𝐴,𝐵)=sup𝑥𝐴inf𝑦𝐵𝑥𝑦𝐸 denotes the Hausdorff pseudodistance between two sets 𝐴 and 𝐵.

We recall the definition of an exponential attractor (see, e.g., [15, 17]).

Definition 2.1. A compact set 𝔄𝐸 is an exponential attractor for 𝑆(𝑡) if
(1)it has finite fractal dimension, dim𝐹𝔄<+, (2)it is positively invariant, 𝑆(𝑡)𝔄𝔄,forall𝑡0, (3)it attracts exponentially the bounded subsets of 𝐸 in the following sense:𝐵𝐸bounded,dist(𝑆(𝑡)𝐵,𝔄)𝑄𝐵𝐸𝑒𝛼𝑡,𝑡0,(2.15) where the positive constant 𝛼, the monotonic function 𝑄 are independent of 𝐵.

Remark 2.2. We note that the existence of an exponential attractor 𝔄 for the semigroup 𝑆(𝑡) automatically implies the existence of the global attractor 𝒜 and the embedding 𝒜𝔄. We note however that, in contrast to the global attractor, an exponential attractor is not uniquely defined.

To construct an exponential attractor, we make use of the following result due to Efendiev et al. [19].

Theorem 2.3. Let 𝑋,𝑌 be two Banach spaces such that 𝑌 is compactly embedded in 𝑋. Let 𝑍 be a bounded closed subset of 𝑋. Assume that a semigroup (𝑆(𝑡))𝑡>0  on 𝑋 satisfies the following conditions: there exists a time 𝑡>0, constants 𝐿1,𝐿2>0, and exponents 𝛾1,𝛾2>0 such that 𝑆(𝑡) maps 𝑍 into itself and 𝑆(𝑡)𝑢0𝑆(𝑡)𝑣0𝑌𝐿1𝑢0𝑣0𝑋,𝑆(𝑠)𝑢0𝑆(𝑡)𝑣0𝑋𝐿2|𝑠𝑡|𝛾1+𝑢0𝑣0𝛾2𝑋(2.16) hold for any 𝑢0,𝑣0𝑍 and 𝑠,𝑡[0,𝑡]. Then the dynamical system ((𝑆(𝑡))𝑡>0,𝑍) admits an exponential attractor.

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.

3. Exponential Attractor of g-Navier-Stokes Equations

This section deals with the existence of the exponential attractor for the two-dimensional g-Navier-Stokes equations with periodic boundary condition.

In Roh [1], the authors have shown that the semigroup 𝑆(𝑡)𝐻𝑔𝐻𝑔(𝑡0) associated with the systems (2.14) possesses a global attractor in 𝐻𝑔 and 𝑉𝑔. The main objective of this section is to prove that the system (2.14) has exponential attractors in 𝐻𝑔.

To this end, we first state some of the following results of existence and uniqueness of solutions of (2.14).

Theorem 3.1. Let 𝑓𝑉𝑔 be given. Then for every 𝑢0𝐻𝑔 there exists a unique solution 𝑢=𝑢(𝑡) on [0,) of (2.14). Moreover, one has 𝑢(𝑡)𝐶0,𝑇;𝐻𝑔𝐿20,𝑇;𝑉𝑔,𝑇>0.(3.1) Finally, if 𝑢0𝑉𝑔, then 𝑢(𝑡)𝐶0,𝑇;𝑉𝑔𝐿2𝐴0,𝑇;𝐷𝑔,𝑇>0.(3.2)

Proof. The Proof of Theorem 3.1 is similar to Roh [1] and Kwaket al. [26] and Temam [12].

In a similar manner as in [13, 14], we can establish the following a priori estimate for for (2.14).

Lemma 3.2. Let be a bounded subset of 𝐻𝑔. The semigroup {𝑆(𝑡)}𝐻𝑔(𝑉𝑔)𝐻𝑔(𝑉𝑔) associated with (2.14) possesses absorbing sets 0=𝑢𝐻𝑔|𝑢|𝑔𝜌0𝑡𝑡0(),1=𝑢𝑉𝑔𝑢𝑔𝜌1𝑡𝑡1()=𝑡0()+1(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.

Let (𝑆(𝑡))𝑡0 be the semigroup associated with (2.14). Since Ω is bounded, 𝑉𝑔 is compactly embedded in 𝐻𝑔. Then we consider H𝑔,𝑉𝑔 as 𝑋,𝑌 in Theorem 2.3, respectively. The crucial point is the choice of the bounded subset 𝑍. Let𝔄=𝑡𝜏𝑆(𝑡)1,(3.4) where 𝐵 denote the closure of 𝐵 in 𝐻𝑔 and 𝜏 is the time when 1 absorbs itself. We claim that 𝔄 has all properties required for 𝑍. In fact, it is easy to see that 𝔄 is positively invariant under the semiflow 𝑆(𝑡). In order to see that 𝔄 has the other required properties, we begin with constructing uniform a priori estimates in time 𝑡 for the solution 𝑢 to (2.14).

Now we consider difference of two solutions of (2.14) starting from 0.

Proposition 3.3. Let the assumptions of Theorem 2.3 hold. Then, there exists a time 𝑡>0, constants 𝐿1>0, and exponents 𝛾1,𝛾2>0 such that 𝑆(𝑡) maps 𝔄 into itself and ||𝑆(𝑠)𝑢0𝑆(𝑡)𝑣0||𝑔𝐿1|𝑠𝑡|𝛾1+||𝑢0𝑣0||𝛾2𝑔(3.5) holds for any 𝑢0,𝑣0𝔄 and 𝑠,𝑡[0,𝑡].

Proof. Let 𝑢01,𝑢02𝔄 and let 𝑢1 and 𝑢2 be two solutions to (2.14) with 𝑢1(0)=𝑢01,𝑢2(0)=𝑢02, respectively.
Let ̃𝑢=𝑢1𝑢2 which satisfies𝑑̃𝑢𝑑𝑡+𝜈𝐴𝑔̃𝑢+𝐵𝑔𝑢1𝐵𝑔𝑢2+𝜈𝑅̃𝑢=0.(3.6) Multiplying (3.6) by ̃𝑢, we have 𝑑̃𝑢𝑑𝑡,̃𝑢𝑔+𝜈𝐴𝑔̃𝑢,̃𝑢𝑔+𝐵𝑢1,𝑢1𝑢𝐵2,𝑢2,̃𝑢𝑔+(𝑅̃𝑢,̃𝑢)𝑔1=0,(3.7)2𝑑||||𝑑𝑡̃𝑢2𝑔+𝜈̃𝑢𝑔2𝑔+𝑏𝑔̃𝑢,𝑢2,̃𝑢+(𝑅̃𝑢,̃𝑢)𝑔=0.(3.8) It follows that 12𝑑||||𝑑𝑡̃𝑢2𝑔+𝜈̃𝑢𝑔2𝑔||𝑏𝑔̃𝑢,𝑢2||,̃𝑢+(𝑅̃𝑢,̃𝑢)𝑔.(3.9) Since 𝑏𝑔 satisfies the following inequality (see Temam [12]): ||𝑏𝑔||(𝑢,𝑣,𝑤)𝑐|𝑢|𝑔1/2𝑢𝑔1/2𝑣𝑔|𝑤|𝑔1/2𝑤𝑔1/2,𝑢,𝑣,𝑤𝑉𝑔,(3.10) thus, ||𝑏𝑔̃𝑢,𝑢2||𝑢,̃𝑢𝑐2𝑔̃𝑢𝑔||||̃𝑢𝑔𝜈4̃𝑢2𝑔𝑢+𝑐22𝑔||||̃𝑢2𝑔.(3.11) Next, the Cauchy inequality, ||(𝑅̃𝑢,̃𝑢)𝑔||=||||𝜈𝑔(𝑔)̃𝑢,̃𝑢𝑔||||𝜈𝑚0||||𝑔̃𝑢𝑔||||̃𝑢𝑔𝜈4̃𝑢2𝑔||||+𝑐̃𝑢2𝑔.(3.12) Finally, putting (3.11)-(3.12) together, there exist constant 𝑀0=𝑀0(𝑚0,|𝑔|,𝜌0,𝜌1) such that 12𝑑||||𝑑𝑡̃𝑢2𝑔+12𝜈̃𝑢2𝑔𝑀0||||̃𝑢2𝑔.(3.13) Therefore, we deduce that 𝑑||||𝑑𝑡̃𝑢2𝑔||||𝑐̃𝑢2𝑔.(3.14) By the Gronwall inequality, the above inequality implies that ||||̃𝑢𝑔𝑒𝑐𝑡||𝑢01𝑢02||𝑔.(3.15)
Next we multiply (2.14) by 𝑢, and we have 𝑑𝑢𝑑𝑡,𝑢𝑔+𝜈𝐴𝑔𝑢,𝑢𝑔+(𝐵(𝑢,𝑢),𝑢)𝑔=(𝑓,𝑢)𝑔(𝑅𝑢,𝑢)𝑔.(3.16) Since Ω is bounded, the Poincaré inequality holds: 𝜆𝑔|𝑢|2𝑔𝑢2𝑔𝑢𝑉𝑔.(3.17) There exist constant 𝑀1=𝑀1(𝑚0,|𝑔|,𝜌0,𝜌1) such that we deduce that 𝑑𝑑𝑡|𝑢|2𝑔+𝜈𝜆|𝑢|2𝑔2𝑀1+3𝜈||𝑓||2𝑉𝑔.(3.18) Multiplying (2.14) by 𝐴𝑔𝑢, we have 12𝑑𝑑𝑡𝑢2𝑔||𝐴+𝜈𝑔𝑢||2𝑔|||𝐵𝑔(𝑢,𝑢),𝐴𝑔𝑢𝑔|||+|||𝑓,𝐴𝑔𝑢𝑔|||+|||𝑅𝑢,𝐴𝑔𝑢𝑔|||.(3.19) Expanding and using Young’s inequality, together with 𝑏𝑔 satisfying inequalities [12], there exists a constant 𝑀2=𝑀2(𝑚0,|𝑔|,𝜌0,𝜌1) such that 𝑑𝑑𝑡𝑢2𝑔+𝜈𝜆𝑔𝑢2𝑔3𝜈||𝑓||𝑔+𝑀2.(3.20) Since if 𝑢0𝔄𝑉𝑔 then the solution 𝑢 with 𝑢(0)=𝑢0 satisfies 𝑑𝑢/𝑑𝑡𝐿2(0,𝑇;𝐻𝑔), it holds that ||𝑢||(𝑠)𝑢(𝑡)𝑔||||𝑠𝑡||||𝑑𝑢||||𝑑𝑟(𝑟)𝑑𝑟|𝑠𝑡|1/2𝑑𝑢𝑑𝑡𝐿2(0,𝑇;𝐻𝑔).(3.21) From (3.18), (3.20) and Lemma 3.2 we can show that there exists a constant 𝑀>0 which satisfies 𝑑𝑢/𝑑𝑡𝐿2(0,𝑇;𝐻𝑔)𝑀 and depends on 𝑇 but not on 𝑢0. Putting (3.15) and (3.21) together, therefore (3.5) turns out to be valid with exponents 𝛾1=1/2 and 𝛾2=1.

Proposition 3.4. Let the assumptions of Theorem 2.3 hold. Then, there exists a time 𝑡>0 and constants 𝐿2>0 such that 𝑆(𝑡) maps 𝔄 into itself and 𝑆𝑡𝑢0𝑡𝑆𝑣0𝑔𝐿2||𝑢0𝑣0||𝑔(3.22) hold for any 𝑢0,𝑣0𝔄 and 𝑠,𝑡[0,𝑡].

Proof. Multiplying (3.6) by 𝑡𝐴𝑔̃𝑢, we have 𝑑̃𝑢𝑑𝑡,𝑡𝐴𝑔+̃𝑢𝜈𝐴𝑔̃𝑢,𝑡𝐴𝑔̃𝑢+𝑏𝑔̃𝑢,𝑢2,𝑡𝐴𝑔̃𝑢𝑔+𝑏𝑔𝑢1,̃𝑢,𝑡𝐴𝑔̃𝑢𝑔+𝑅̃𝑢,𝑡𝐴𝑔̃𝑢𝑔=0.(3.23) It follows that 12𝑑𝑡𝑑𝑡̃𝑢2𝑔||𝐴+𝜈𝑡𝑔||̃𝑢2𝑔12̃𝑢2𝑔|||𝑏𝑡𝑔̃𝑢,𝑢2,𝐴𝑔̃𝑢𝑔||||||𝑏+𝑡𝑔𝑢1,̃𝑢,𝐴𝑔̃𝑢𝑔||||||+𝑡𝑅̃𝑢,𝐴𝑔̃𝑢𝑔|||.(3.24) To estimate 𝑏𝑔, we recall some inequalities [12]: 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) Expanding and using Young’s inequality, together with (3.25), we have |||𝑏𝑔̃𝑢,𝑢2,𝐴𝑔̃𝑢𝑔|||||||̃𝑢𝑔1/2||𝐴𝑔||̃𝑢𝑔1/2𝑢2𝑔||𝐴𝑔||̃𝑢𝑔𝜈3||𝐴𝑔||̃𝑢2𝑔+𝑐𝜈||||̃𝑢2𝑔𝑢24𝑔,|||𝑏(3.26)𝑔𝑢1,̃𝑢,𝐴𝑔̃𝑢𝑔|||||𝑢1||𝑔1/2𝑢1𝑔1/2̃𝑢𝑔1/2||𝐴𝑔||̃𝑢𝑔1/2||𝐴𝑔||̃𝑢𝑔𝜈3||𝐴𝑔||̃𝑢2𝑔+𝑐𝜈||𝑢1||2𝑔𝑢12𝑔̃𝑢2𝑔.(3.27) Next, using the Cauchy inequality, |||𝑅̃𝑢,𝐴𝑔̃𝑢𝑔|||=||||1𝑔(𝑔)̃𝑢,𝐴𝑔̃𝑢𝑔||||||||𝑔𝑚0̃𝑢𝑔||𝐴𝑔||̃𝑢𝑔𝜈3||𝐴𝑔||̃𝑢2𝑔+3||||4𝜈𝑔2̃𝑢2𝑔.(3.28) Since (3.13), we have ̃𝑢2𝑔||||𝑐̃𝑢2𝑔,(3.29) Putting (3.26)–(3.29) together, therefore we have 𝑑𝑡𝑑𝑡̃𝑢2𝑔||𝑢𝑐1+1||2𝑔𝑢12𝑔𝑡̃𝑢2𝑔𝑡𝑢+𝑐24𝑔||||+1̃𝑢2𝑔.(3.30) By the Gronwall inequality and (3.15), the above inequality implies 𝑡̃𝑢2𝑔||𝑢𝐶(𝑡)01𝑢02||2𝑔,(3.31) where 𝐶(𝑡)=𝑡0𝑐exp𝑠+𝑡𝑠||𝑢1+1||2𝑔𝑢12𝑔𝑠𝑢𝑑𝑟24𝑔+1𝑑𝑠.(3.32) By taking 𝑡=𝑡1(1), we complete the proof.

Now, we give our main theorem which relies on the Propositions 3.3 and 3.4 to construct an exponential attractor.

Theorem 3.5. There exists a subset 𝔄 of 𝐻𝑔 such that 𝑆(𝑡) maps 𝔄 into itself and the dynamical system ((𝑆(𝑡))𝑡>0,𝔄) admits an exponential attractor.

Based on the abve results (Propositions 3.3 and 3.4) and applying Theorem 2.3, we can deduce Theorem 3.5.

Acknowledgment

The authors wish to thank the referees for their careful reading of this paper and useful comments. This work was partly supported by ZPNSFC grants Y6110078.

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