The Exponential Attractors for the g-Navier-Stokes Equations

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).


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 g g x 1 , x 2 is a suitable smooth real-valued function defined on Ω and Ω is a suitable bounded domain in R 2 .Notice that if g x 1 , x 2 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 Foias ¸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 Ω.

Preliminary Results
Let Ω 0, 1 × 0, 1 and we assume that the function g x g x 1 , x 2 satisfies the following properties: per Ω 2 there exist constants m 0 m 0 g and M 0 M 0 g such that, for all x ∈ Ω, 0 < m 0 ≤ g x ≤ M 0 .Note that the constant function g ≡ 1 satisfies these conditions.
We denote by L 2 Ω, g the space with the scalar product and the norm given by as well as H 1 Ω, g with the norm where ∂u/∂x i D i u.
Then for the functional setting of the problems 1.1 , we use the following functional spaces: where H g is endowed with the scalar product and the norm in L 2 Ω, g , and V g is the spaces with the scalar product and the norm given by Also, we define the orthogonal projection P g as and we have that Q ⊆ H ⊥ g , where Then, we define the g-Laplacian operator to have the linear operator For the linear operator A g , the following hold see Roh 1 : 1 A g is a positive, self-adjoint operator with compact inverse, where the domain of 2 There exist countable eigenvalues of A g satisfying where λ g 4π 2 m/M and λ 1 is the smallest eigenvalue of A g .In addition, there exists the corresponding collection of eigenfunctions {e 1 , e 2 , e 3 , . ..} which forms an orthonormal basis for H g .
Next, we denote the bilinear operator B g u, v P g u • ∇ v and the trilinear form where u, v, w lie in appropriate subspaces of L 2 Ω, g .Then, the form b g satisfies We denote a linear operator R on V g by 12 and have R as a continuous linear operator from V g into H g such that

2.13
We now rewrite 1.1 as abstract evolution equations, du dt νA g u B g u νRu P g f, u 0 u 0 .

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 , 0 ≤ t < ∞ 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 , 0 ≤ t, s < ∞ the semigroup property ; iii the mapping G : 0, ∞ × K → K, t, u 0 → S t u 0 , is continuous.Such a family is called a continuous nonlinear semigroup acting on K.The image of S • u 0 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 B ⊂ K, it holds that lim t → ∞ dist S t B, A 0, where dist A, B sup x∈A inf y∈B x − y E denotes the Hausdorff pseudodistance between two sets A and B.
We recall the definition of an exponential attractor see, e.g., 15, 17 .
2 it is positively invariant, S t A ⊆ A, for all t ≥ 0, 3 it attracts exponentially the bounded subsets of E in the following sense: 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 A ⊂ A. 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 L 1 , L 2 > 0, and exponents γ 1 , γ 2 > 0 such that S t * maps Z into itself and

2.16
hold for any u 0 , v 0 ∈ Z 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.

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 : H g → H g t ≥ 0 associated with the systems 2.14 possesses a global attractor in H g and V g .The main objective of this section is to prove that the system 2.14 has exponential attractors in H g .
To this end, we first state some of the following results of existence and uniqueness of solutions of 2.14 .Theorem 3.1.Let f ∈ V g be given.Then for every u 0 ∈ H g there exists a unique solution u u t on 0, ∞ of 2.14 .Moreover, one has 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 B be a bounded subset of H g .The semigroup {S t } : H g V g → H g V g associated with 2.14 possesses absorbing sets which absorb all bounded sets of H g .Moreover B 0 and B 1 absorb all bounded sets of H g and V g in the norms of H g and V g , respectively.
Let S t t≥0 be the semigroup associated with 2.14 .Since Ω is bounded, V g is compactly embedded in H g .Then we consider H g , V g as X, Y in Theorem 2.3, respectively.The crucial point is the choice of the bounded subset Z.Let where B denote the closure of B in H g and τ is the time when B 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 B 0 .
Proposition 3.3.Let the assumptions of Theorem 2.3 hold.Then, there exists a time t * > 0, constants L 1 > 0, and exponents γ 1 , γ 2 > 0 such that S t * maps A into itself and holds for any u 0 , v 0 ∈ A and s, t ∈ 0, t * .
Let u u 1 − u 2 which satisfies Multiplying 3.6 by u, we have Since b g satisfies the following inequality see Temam 12 :

3.15
Next we multiply 2.14 by u, and we have du dt , u g νA g u, u g B u, u , u g f, u g − Ru, u g .

3.18
Multiplying 2.14 by A g u, we have 1 2 ≤ B g u, u , A g u g f, A g u g Ru, A g u g .

3.19
Expanding and using Young's inequality, together with b g satisfying inequalities 12 , there exists a constant

3.24
To estimate b g , we recall some inequalities 12 : for every u, v ∈ D A g ,

3.25
Expanding and using Young's inequality, together with 3.25 , we have b g u, u 2 , A g u g ≤ | u| 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 H g 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.
the solution u with u 0 u 0 satisfies du/dt ∈ L 2 0, T; H g , it holds that |u s − u t | g ≤ From 3.18 , 3.20 and Lemma 3.2 we can show that there exists a constant M > 0 which satisfies du/dt L 2 0,T ;H g ≤ M and depends on T but not on u 0 .Putting 3.15 and 3.21 together, therefore 3.5 turns out to be valid with exponents γ 1 1/2 and γ 2 1.Let the assumptions of Theorem 2.3 hold.Then, there exists a time t * > 0 and constants L 2 > 0 such that S t * maps A into itself andS t * u 0 − S t * v 0 g ≤ L 2 |u 0 − v 0 | g 3.22hold for any u 0 , v 0 ∈ A and s, t ∈ 0, t * .