The Exponential Attractors for the g-Navier-Stokes Equations

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𝜕𝑡−𝜈Δ𝑢+(𝑢⋅∇)𝑢+∇𝑝=𝑓inΩ,𝑔(∇⋅𝑔𝑢)=∇𝑔𝑔⋅𝑢+∇⋅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 ℝ². 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 [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.

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)∈𝐶^∞_per(Ω)(2)there exist constants 𝑚₀=𝑚₀(𝑔) and 𝑀₀=𝑀₀(𝑔) such that, for all 𝑥∈Ω,0<𝑚₀≤𝑔(𝑥)≤𝑀₀. Note that the constant function 𝑔≡1 satisfies these conditions.

We denote by 𝐿²(Ω,𝑔) the space with the scalar product and the norm given by(𝑢,𝑣)_𝑔=_Ω(𝑢⋅𝑣)𝑔𝑑𝑥,|𝑢|^2𝑔=(𝑢,𝑢)_𝑔,(2.1) as well as 𝐻¹(Ω,𝑔) with the norm‖𝑢‖_𝐻¹_(Ω,𝑔)=(𝑢,𝑢)_𝑔+²_𝑖=1𝐷_𝑖𝑢,𝐷_𝑖𝑢_𝑔^1/2,(2.2) where 𝜕𝑢/𝜕𝑥_𝑖=𝐷_𝑖𝑢.

Then for the functional setting of the problems (1.1), we use the following functional spaces:𝐻_𝑔=𝐶𝑙_𝐿²_per(Ω,𝑔)𝑢∈𝐶^∞per𝑉(Ω)∶∇⋅𝑔𝑢=0,_Ω𝑢𝑑𝑥=0_𝑔,=𝑢∈𝐻¹_per(Ω,𝑔)∶∇⋅𝑔𝑢=0,_Ω𝑢𝑑𝑥=0,(2.3) where 𝐻_𝑔 is endowed with the scalar product and the norm in 𝐿²(Ω,𝑔), and 𝑉_𝑔 is the spaces with the scalar product and the norm given by((𝑢,𝑣))_𝑔=_Ω(∇𝑢⋅∇𝑣)𝑔𝑑𝑥,‖𝑢‖_𝑔=((𝑢,𝑢))_𝑔.(2.4) Also, we define the orthogonal projection 𝑃_𝑔 as𝑃_𝑔∶𝐿²_per(Ω,𝑔)⟶𝐻_𝑔(2.5) and we have that 𝑄⊆𝐻^⟂𝑔, where𝑄=𝐶𝑙_𝐿²_per(Ω,𝑔)∇𝜙∶𝜙∈𝐶¹Ω,ℝ.(2.6) Then, we define the 𝑔-Laplacian operator−Δ_𝑔𝑢≡1𝑔(∇⋅𝑔∇)𝑢=−Δ𝑢−𝑔1(∇𝑔⋅∇)𝑢(2.7) to have the linear operator𝐴_𝑔𝑢=𝑃_𝑔−𝑔1(∇⋅(𝑔∇𝑢)).(2.8) 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 𝐴_𝑔,𝐷(𝐴_𝑔)=𝑉_𝑔∩𝐻²(Ω,𝑔).(2)There exist countable eigenvalues of 𝐴_𝑔 satisfying0<𝜆_𝑔≤𝜆₁≤𝜆₂≤𝜆₃≤⋯,(2.9) where 𝜆_𝑔=4𝜋²𝑚/𝑀 and 𝜆₁ is the smallest eigenvalue of 𝐴_𝑔. In addition, there exists the corresponding collection of eigenfunctions {𝑒₁,𝑒₂,𝑒₃,…} 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 form𝑏_𝑔(𝑢,𝑣,𝑤)=²_𝑖_,𝑗=1_Ω𝑢_𝑖𝐷_𝑖𝑣_𝑗𝑤_𝑗𝑔𝑑𝑥=𝑃_𝑔(𝑢⋅∇)𝑣,𝑤_𝑔,(2.10) where 𝑢,𝑣,𝑤 lie in appropriate subspaces of 𝐿²(Ω,𝑔). Then, the form 𝑏_𝑔 satisfies𝑏_𝑔(𝑢,𝑣,𝑤)=−𝑏_𝑔(𝑢,𝑤,𝑣)for𝑢,𝑣,𝑤∈𝐻_𝑔.(2.11)

We denote a linear operator 𝑅 on 𝑉_𝑔 by𝑅𝑢=𝑃_𝑔𝑔1(∇𝑔⋅∇)𝑢for𝑢∈𝑉^𝑔,(2.12) and have 𝑅 as a continuous linear operator from 𝑉_𝑔 into 𝐻_𝑔 such that||(𝑅𝑢,𝑢)||≤||∇𝑔||^∞𝑚₀‖𝑢‖_𝑔|𝑢|_𝑔≤||∇𝑔||^∞𝑚₀𝜆^𝑔1/2‖𝑢‖_𝑔for𝑢∈𝑉_𝑔.(2.13)

We now rewrite (1.1) as abstract evolution equations,𝑑𝑢𝑑𝑡+𝜈𝐴^𝑔𝑢𝑢+(0𝐵)_𝑔=𝑢+𝑢₀𝜈𝑅𝑢=𝑃_𝑔𝑓,.(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,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 𝐵⊂𝐾, it holds that lim_𝑡→∞dist(𝑆(𝑡)𝐵,𝒜)=0, where dist(𝐴,𝐵)=sup_𝑥∈𝐴inf_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]).

Definition 2.1. A compact set 𝔄⊂𝐸 is an exponential attractor for 𝑆(𝑡) if
(1)it has finite fractal dimension, dim_𝐹𝔄<+∞, (2)it is positively invariant, 𝑆(𝑡)𝔄⊆𝔄,forall𝑡≥0, (3)it attracts exponentially the bounded subsets of 𝐸 in the following sense:∀𝐵⊂𝐸bounded,dist(𝑆(𝑡)𝐵,𝔄)≤𝑄‖𝐵‖_𝐸𝑒^−𝛼𝑡,𝑡≥0,(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 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 𝐿₁,𝐿₂>0, and exponents 𝛾₁,𝛾₂>0 such that 𝑆(𝑡_∗) maps 𝑍 into itself and ‖‖𝑆(𝑠)‖‖𝑢𝑆₀(𝑡_∗)𝑢₀−𝑆(𝑡_∗)𝑣₀‖‖_𝑌≤𝐿₁‖‖𝑢₀−𝑣₀‖‖_𝑋,−𝑆(𝑡)𝑣₀‖‖_𝑋≤𝐿₂|𝑠−𝑡|^𝛾1+‖‖𝑢₀−𝑣₀‖‖^𝛾2_𝑋(2.16) hold for any 𝑢₀,𝑣₀∈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.

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)∶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 𝑢₀∈𝐻_𝑔 there exists a unique solution 𝑢=𝑢(𝑡) on [0,∞) of (2.14). Moreover, one has 𝑢(𝑡)∈𝐶0,𝑇;𝐻_𝑔∩𝐿²0,𝑇;𝑉_𝑔,∀𝑇>0.(3.1) Finally, if 𝑢₀∈𝑉_𝑔, then 𝑢(𝑡)∈𝐶0,𝑇;𝑉_𝑔∩𝐿²0,𝑇;𝐷𝐴_𝑔,∀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 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 ℬℬ₀=𝑢∈𝐻_𝑔∣|𝑢|_𝑔≤𝜌₀∀𝑡≥𝑡₀(ℬ),₁=𝑢∈𝑉_𝑔∣‖𝑢‖_𝑔≤𝜌₁∀𝑡≥𝑡₁(ℬ)=𝑡₀(ℬ)+1(3.3) which absorb all bounded sets of 𝐻_𝑔. Moreover ℬ₀ and ℬ₁ 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. Let𝔄=_𝑡≥𝜏𝑆(𝑡)ℬ₁,(3.4) where 𝐵 denote the closure of 𝐵 in 𝐻_𝑔 and 𝜏 is the time when ℬ₁ 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 ℬ₀.

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)𝑢₀−𝑆(𝑡)𝑣₀||_𝑔≤𝐿₁|𝑠−𝑡|^𝛾1+||𝑢₀−𝑣₀||^𝛾2_𝑔(3.5) holds for any 𝑢₀,𝑣₀∈𝔄 and 𝑠,𝑡∈[0,𝑡_∗].

Proof. Let 𝑢⁰¹,𝑢⁰²∈𝔄 and let 𝑢₁ and 𝑢₂ be two solutions to (2.14) with 𝑢₁(0)=𝑢⁰¹,𝑢₂(0)=𝑢⁰², respectively.
Let 𝑢̃=𝑢₁−𝑢₂ which satisfies𝑑𝑢̃𝑑𝑡+𝜈𝐴^𝑔𝑢̃+𝐵_𝑔𝑢₁−𝐵_𝑔𝑢₂+𝜈𝑅𝑢̃=0.(3.6) Multiplying (3.6) by 𝑢̃, we have 𝑑𝑑𝑢𝑡̃,𝑢̃_𝑔+1𝜈𝐴_𝑔𝑢,̃𝑢̃_𝑔+𝐵𝑢₁,𝑢₁−𝐵𝑢₂,𝑢₂,𝑢̃_𝑔+(𝑅𝑢,̃𝑢)̃_𝑔=0,(3.7)2𝑑𝑑𝑡||𝑢||̃^2𝑔+𝜈‖‖𝑢_𝑔̃‖‖^2𝑔+𝑏_𝑔𝑢,̃𝑢₂,𝑢̃+(𝑅𝑢,̃𝑢)̃_𝑔=0.(3.8) It follows that 12𝑑𝑑𝑡||𝑢||̃^2𝑔+𝜈‖‖𝑢_𝑔̃‖‖^2𝑔≤||𝑏_𝑔𝑢,̃𝑢₂,𝑢̃||+(𝑅𝑢,̃𝑢)̃_𝑔.(3.9) Since 𝑏_𝑔 satisfies the following inequality (see Temam [12]): ||𝑏_𝑔(𝑢,𝑣,𝑤)||≤𝑐|𝑢|^𝑔1/2‖𝑢‖^𝑔1/2‖𝑣‖_𝑔|𝑤|^𝑔1/2‖𝑤‖^𝑔1/2,∀𝑢,𝑣,𝑤∈𝑉_𝑔,(3.10) thus, ||𝑏_𝑔𝑢,̃𝑢₂,𝑢̃||≤𝑐‖‖𝑢₂‖‖_𝑔‖𝑢‖̃_𝑔||𝑢||̃_𝑔≤𝜈4‖𝑢‖̃^2𝑔+𝑐‖‖𝑢₂‖‖^2𝑔||𝑢||̃^2𝑔.(3.11) Next, the Cauchy inequality, ||(𝑅𝑢,̃𝑢)̃_𝑔||=||||𝜈𝑔(∇𝑔⋅∇)𝑢,̃𝑢̃_𝑔≤𝜈||||𝑚₀||∇𝑔||_∞‖𝑢‖̃_𝑔||𝑢||̃_𝑔≤𝜈4‖𝑢‖̃^2𝑔+𝑐||𝑢||̃^2𝑔.(3.12) Finally, putting (3.11)-(3.12) together, there exist constant 𝑀₀=𝑀₀(𝑚₀,|∇𝑔|_∞,𝜌₀,𝜌₁) such that 12𝑑𝑑𝑡||𝑢||̃^2𝑔+12𝜈‖𝑢‖̃^2𝑔≤𝑀₀||𝑢||̃^2𝑔.(3.13) Therefore, we deduce that 𝑑𝑑𝑡||𝑢||̃^2𝑔≤𝑐||𝑢||̃^2𝑔.(3.14) By the Gronwall inequality, the above inequality implies that ||𝑢||̃_𝑔≤𝑒^𝑐𝑡||𝑢⁰¹−𝑢⁰²||_𝑔.(3.15)
Next we multiply (2.14) by 𝑢, and we have 𝑑𝑑𝑢𝑡,𝑢_𝑔+𝜈𝐴_𝑔𝑢,𝑢_𝑔+(𝐵(𝑢,𝑢),𝑢)_𝑔=(𝑓,𝑢)_𝑔−(𝑅𝑢,𝑢)_𝑔.(3.16) Since Ω is bounded, the Poincaré inequality holds: 𝜆_𝑔|𝑢|^2𝑔≤‖𝑢‖^2𝑔∀𝑢∈𝑉_𝑔.(3.17) There exist constant 𝑀₁=𝑀₁(𝑚₀,|∇𝑔|_∞,𝜌₀,𝜌₁) such that we deduce that 𝑑𝑑𝑡|𝑢|^2𝑔+𝜈𝜆|𝑢|^2𝑔≤2𝑀₁+3𝜈||𝑓||^2𝑉′𝑔.(3.18) Multiplying (2.14) by 𝐴_𝑔𝑢, we have 12𝑑𝑑𝑡‖𝑢‖^2𝑔+𝜈||𝐴_𝑔𝑢||^2𝑔≤|||𝐵_𝑔(𝑢,𝑢),𝐴_𝑔𝑢_𝑔|||+|||𝑓,𝐴_𝑔𝑢_𝑔|||+|||𝑅𝑢,𝐴_𝑔𝑢_𝑔|||.(3.19) Expanding and using Young’s inequality, together with 𝑏_𝑔 satisfying inequalities [12], there exists a constant 𝑀₂=𝑀₂(𝑚₀,|∇𝑔|_∞,𝜌₀,𝜌₁) such that 𝑑𝑑𝑡‖𝑢‖^2𝑔+𝜈𝜆_𝑔‖𝑢‖^2𝑔≤3𝜈||𝑓||^𝑔+𝑀₂.(3.20) Since if 𝑢₀∈𝔄⊂𝑉_𝑔 then the solution 𝑢 with 𝑢(0)=𝑢₀ satisfies 𝑑𝑢/𝑑𝑡∈𝐿²(0,𝑇;𝐻_𝑔), it holds that ||𝑢(𝑠)−𝑢(𝑡)||_𝑔≤||||^𝑠𝑡||||𝑑𝑢𝑑𝑟(𝑟)||𝑑𝑟||≤|𝑠−𝑡|^1/2‖‖‖𝑑𝑢𝑑𝑡‖‖‖_𝐿²_{(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/𝑑𝑡‖_𝐿²_{(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 ‖‖𝑆𝑡_∗𝑢₀−𝑆𝑡_∗𝑣₀‖‖_𝑔≤𝐿₂||𝑢₀−𝑣₀||_𝑔(3.22) hold for any 𝑢₀,𝑣₀∈𝔄 and 𝑠,𝑡∈[0,𝑡_∗].

Proof. Multiplying (3.6) by 𝑡𝐴_𝑔𝑢̃, we have 𝑑𝑑𝑢𝑡̃,𝑡𝐴^𝑔𝑢̃+𝜈𝐴_𝑔𝑢,̃𝑡𝐴_𝑔𝑢̃+𝑏_𝑔𝑢,̃𝑢₂,𝑡𝐴_𝑔𝑢̃_𝑔+𝑏_𝑔𝑢₁,𝑢,̃𝑡𝐴_𝑔𝑢̃_𝑔+𝑅𝑢,̃𝑡𝐴_𝑔𝑢̃_𝑔=0.(3.23) It follows that 12𝑑𝑑𝑡𝑡‖𝑢‖̃^2𝑔+𝜈𝑡||𝐴_𝑔𝑢||̃^2𝑔−12‖𝑢‖̃^2𝑔≤𝑡|||𝑏_𝑔𝑢,̃𝑢₂,𝐴_𝑔𝑢̃_𝑔|||+𝑡|||𝑏_𝑔𝑢₁,𝑢,̃𝐴_𝑔𝑢̃_𝑔|||+𝑡|||𝑅𝑢,̃𝐴_𝑔𝑢̃_𝑔|||.(3.24) To estimate 𝑏_𝑔, we recall some inequalities [12]: for every 𝑢,𝑣∈𝐷(𝐴_𝑔), ||𝐵_𝑔(𝑢,𝑣)||≤𝑐|𝑢|^𝑔1/2|𝑢|_𝑔1/2‖𝑢‖^𝑔1/2‖𝑣‖^𝑔1/2||𝐴_𝑔𝑣||^𝑔1/2,||𝐴_𝑔𝑢||^𝑔1/2‖𝑣‖_𝑔.(3.25) Expanding and using Young’s inequality, together with (3.25), we have |||𝑏_𝑔𝑢,̃𝑢₂,𝐴_𝑔𝑢̃_𝑔|||≤||𝑢||̃_𝑔1/2||𝐴_𝑔𝑢||̃^𝑔1/2‖‖𝑢₂‖‖_𝑔||𝐴_𝑔𝑢||̃_𝑔≤𝜈3||𝐴^𝑔𝑢||̃^2𝑔+𝑐𝜈||𝑢||̃^2𝑔|||𝑏‖‖𝑢₂‖‖^4𝑔,(3.26)_𝑔𝑢₁,𝑢,̃𝐴_𝑔𝑢̃_𝑔|||≤||𝑢₁||^𝑔1/2‖‖𝑢₁‖‖^𝑔1/2‖𝑢‖̃^𝑔1/2||𝐴_𝑔𝑢||̃^𝑔1/2||𝐴_𝑔𝑢||̃_𝑔≤𝜈3||𝐴^𝑔𝑢||̃^2𝑔+𝑐𝜈||𝑢¹||^2𝑔‖‖𝑢₁‖‖^2𝑔‖𝑢‖̃^2𝑔.(3.27) Next, using the Cauchy inequality, |||𝑅𝑢,̃𝐴_𝑔𝑢̃_𝑔|||=||||1𝑔(∇𝑔⋅∇)𝑢,̃𝐴^𝑔𝑢̃_𝑔≤||∇𝑔||_∞||||𝑚₀‖𝑢‖̃_𝑔||𝐴_𝑔𝑢||̃_𝑔≤𝜈3||𝐴^𝑔𝑢||̃^2𝑔+34𝜈||∇𝑔||^2∞‖𝑢‖̃^2𝑔.(3.28) Since (3.13), we have ‖𝑢‖̃^2𝑔≤𝑐||𝑢||̃^2𝑔,(3.29) Putting (3.26)–(3.29) together, therefore we have 𝑑𝑑𝑡𝑡‖𝑢‖̃^2𝑔≤𝑐1+||𝑢₁||^2𝑔‖‖𝑢₁‖‖^2𝑔𝑡‖𝑢‖̃^2𝑔+𝑐𝑡‖‖𝑢₂‖‖^4𝑔+1||𝑢||̃^2𝑔.(3.30) By the Gronwall inequality and (3.15), the above inequality implies 𝑡‖𝑢‖̃^2𝑔≤𝐶(𝑡)||𝑢⁰¹−𝑢⁰²||^2𝑔,(3.31) where 𝐶(𝑡)=^𝑡0𝑐exp𝑠+^𝑡𝑠1+||𝑢¹||^2𝑔‖‖𝑢₁‖‖^2𝑔𝑑𝑟𝑠‖‖𝑢²‖‖^4𝑔+1𝑑𝑠.(3.32) By taking 𝑡=𝑡₁(ℬ₁), 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 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.

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.