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, where is a suitable smooth real-valued function defined on and is a suitable bounded domain in . Notice that if , 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 [7–14]. 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, 21–25] 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 .

2. Preliminary Results

Let and we assume that the function satisfies the following properties:there exist constants and such that, for all . Note that the constant function satisfies these conditions.

We denote by the space with the scalar product and the norm given by as well as with the norm where .

Then for the functional setting of the problems (1.1), we use the following functional spaces: where is endowed with the scalar product and the norm in , and is the spaces with the scalar product and the norm given by Also, we define the orthogonal projection as and we have that , where Then, we define the -Laplacian operator to have the linear operator For the linear operator , the following hold (see Roh [1]): is a positive, self-adjoint operator with compact inverse, where the domain of .There exist countable eigenvalues of satisfying where and is the smallest eigenvalue of . In addition, there exists the corresponding collection of eigenfunctions which forms an orthonormal basis for .

Next, we denote the bilinear operator and the trilinear form where lie in appropriate subspaces of . Then, the form satisfies

We denote a linear operator on by and have as a continuous linear operator from into such that

We now rewrite (1.1) as abstract evolution equations,

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

A subset of the phase-space is the global attractor of if the following conditions are satisfied: (i) is a compact subset of ; (ii) is an invariant set, that is, for every ; (iii) attracts every bounded subset of , namely, for any bounded subset , it holds that , where denotes the Hausdorff pseudodistance between two sets and .

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

Definition 2.1. A compact set is an exponential attractor for if
(1)it has finite fractal dimension, , (2)it is positively invariant, , (3)it attracts exponentially the bounded subsets of in the following sense: where the positive constant , the monotonic function are independent of .

Remark 2.2. We note that the existence of an exponential attractor for the semigroup automatically implies the existence of the global attractor and the embedding . 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 be two Banach spaces such that is compactly embedded in . Let be a bounded closed subset of . Assume that a semigroup   on satisfies the following conditions: there exists a time , constants , and exponents such that maps into itself and hold for any and . Then the dynamical system admits an exponential attractor.

Hereafter 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 associated with the systems (2.14) possesses a global attractor in and . The main objective of this section is to prove that the system (2.14) has exponential attractors in .

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

Theorem 3.1. Let be given. Then for every there exists a unique solution on of (2.14). Moreover, one has Finally, if , then

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 . The semigroup associated with (2.14) possesses absorbing sets which absorb all bounded sets of . Moreover and absorb all bounded sets of and in the norms of and , respectively.

Let be the semigroup associated with (2.14). Since is bounded, is compactly embedded in . Then we consider as in Theorem 2.3, respectively. The crucial point is the choice of the bounded subset . Let where denote the closure of in and is the time when absorbs itself. We claim that has all properties required for . In fact, it is easy to see that is positively invariant under the semiflow . In order to see that has the other required properties, we begin with constructing uniform a priori estimates in time for the solution to (2.14).

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

Proposition 3.3. Let the assumptions of Theorem 2.3 hold. Then, there exists a time , constants , and exponents such that maps into itself and holds for any and .

Proof. Let and let and be two solutions to (2.14) with , respectively.
Let which satisfies Multiplying (3.6) by , we have It follows that Since satisfies the following inequality (see Temam [12]): thus, Next, the Cauchy inequality, Finally, putting (3.11)-(3.12) together, there exist constant such that Therefore, we deduce that By the Gronwall inequality, the above inequality implies that
Next we multiply (2.14) by , and we have Since is bounded, the Poincaré inequality holds: There exist constant such that we deduce that Multiplying (2.14) by , we have Expanding and using Young’s inequality, together with satisfying inequalities [12], there exists a constant such that Since if then the solution with satisfies , it holds that From (3.18), (3.20) and Lemma 3.2 we can show that there exists a constant which satisfies and depends on but not on . Putting (3.15) and (3.21) together, therefore (3.5) turns out to be valid with exponents and .

Proposition 3.4. Let the assumptions of Theorem 2.3 hold. Then, there exists a time and constants such that maps into itself and hold for any and .

Proof. Multiplying (3.6) by , we have It follows that To estimate , we recall some inequalities [12]: for every , Expanding and using Young’s inequality, together with (3.25), we have Next, using the Cauchy inequality, Since (3.13), we have Putting (3.26)–(3.29) together, therefore we have By the Gronwall inequality and (3.15), the above inequality implies where By taking , 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 of such that maps into itself and the dynamical system 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.