Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping

where D is the abounded domain with boundary ∂D, ( ) = u u u , ε ε ε T 1 2 is the velocity, which depends on the disturbance parameter ε. p is the pressure, > ν 0 is the kinematic viscosity, γuε is the linear damping which parallels the velocity uε, and the constant γ is positive. ( ) u x 0 is the initial velocity. The symbol ( ) W t is a real valued two-sided Wiener process. Equations (1.1) describe the movement of incompressible fluids in geophysical dynamics. The constant γ is Rayleigh’s friction coefficient or Ekman suction/dissipation constant. The linear damping γu is a simulation of the bottom friction in a two-dimensional ocean model or a line in a two-dimensional atmospheric model. Especially, = γ 0, equations (1.1) are the classical stochastic 2D Navier-Stokes equation. For = ε 0, equation (1.1) become non-stochastic system. In the past 20 years, many extensive and in-depth studies emerge. Ilyin et al. [1] studied the limit of small viscosity coefficient → + ν 0 and derived that the linear damping term γuε plays an important role in reducing the number of degrees of freedom in the twodimensional model. The estimates for the number of determining modes and nodes are comparable to the


Introduction
This paper considers the following stochastic Navier-Stokes equations with linear damping in a two-dimensional domain ⊂ D 2 , where D is the abounded domain with boundary ∂D, is the velocity, which depends on the disturbance parameter ε. p is the pressure, > ν 0 is the kinematic viscosity, γu ε is the linear damping which parallels the velocity u ε , and the constant γ is positive. ( ) u x 0 is the initial velocity. The symbol ( ) W t is a real valued two-sided Wiener process.
Equations (1.1) describe the movement of incompressible fluids in geophysical dynamics. The constant γ is Rayleigh's friction coefficient or Ekman suction/dissipation constant. The linear damping γu is a simulation of the bottom friction in a two-dimensional ocean model or a line in a two-dimensional atmospheric model. Especially, = γ 0, equations (1.1) are the classical stochastic 2D Navier-Stokes equation. For = ε 0, equation (1.1) become non-stochastic system. In the past 20 years, many extensive and in-depth studies emerge. Ilyin et al. [1] studied the limit of small viscosity coefficient → + ν 0 and derived that the linear damping term γu ε plays an important role in reducing the number of degrees of freedom in the twodimensional model. The estimates for the number of determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. For details, we can refer to the literature [2]. Constantin and Ramos [3] derived that in 2 space the rate of dissipation of enstrophy vanishes. The stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations, and the solutions obey the enstrophy balance [3]. On arbitrary open sets, Rosa [4] deduced the existence of the global attractor. Under the condition ( ) ( ( )) ∈ f x L 2 2 , Zhao and Zheng [5] proved the existence of global attractor and studied the deformations of the Navier-Stokes equation by limit behavior. Li [6] established the existence of uniform random attractor for stochastic Navier-Stokes equations in the space H .
The theory on the stochastic dynamical system is investigated in [7][8][9][10][11]. Our investigation of the Navier-Stokes equations with linear damping is inspired by [6,12,13]. We focus on the random attractor and its upper semicontinuity. By calculations, we derives the existence of random attractors and the upper semicontinuity for small random perturbations of Navier-Stokes equations with linear damping on the twodimensional space, which enriches the theoretical results of the model. This paper is arranged as follows. In Section 2, we recall some fundamental concepts and some lemmas which are used in the sequel. In Section 3, we conduce the existence of random attractors. In Section 4, we derive the upper semicontinuity for random attractors.

Preliminaries
This section introduces some basic related concepts for the random attractors, which were developed by Crauel and Flandoli [8,14].
is called a measurable dynamical system. For the integrity of knowledge, it introduces the following concepts. Ω satisfies the following conditions: , , , , for all ∈ ∈ + t s x X , , and P-almost every (a.e.) ∈ ω Ω, is continuous for all ∈ + t , the function Ω is called a continuous random dynamical system (RDS) on a metric dynamical system ( ( ) ) ∈ P θ Ω, , , t t .
Next is the concept of a random absorption set [14].
we called the random set { ( )} ∈ ∈ K ω ω Ω a random absorbing set for φ on RDS in .
The following three theorems were given and proved in [12].
K ω be a random compact set which absorbs every bounded non-random set ⊂ B X, the set is a random attractor for φ, where the union is taken over all ⊂ B X bounded, and ( ) B ω Λ , is the omega-limit set of B and is given by The upper semicontinuity of 2D Navier-Stokes equations  1529 When we add the random element which depends on a parameter perturbs the unperturbation system. According to the conventional theory, it derives an RDS which depends on the parameter uniformly on bounded sets of X.
be a random attractor of the system (1.1). Assume that there exists a compact set K such that, P-a.s. Then, Conditions ( ) C1 and ( ) C2 are necessary and sufficient for the upper semicontinuity property. Condition ( ) C2 is a similar property for the random absorbing sets, which are used to derive the random attractors (see [12] for more details). Here, we just list the result as follows.
be a family of random compact absorbing sets which are uniformly in disturbance parameter ε, that is, for P-a.e. ∈ ω Ω and all and there exists a compact set K such that P-a.s.
For convenience, it introduces the function spaces and inequalities. Denote by  For all ∈ u v w z V , , , , according to Sobolev's relevant knowledge, it has the following inequalities: (2.5) 3 Existence of random attractors We will derive computational inequalities and use Lemma 2.5 to study the random attractors of system (1.1) in this section. Assume ∈ f H, ( ) ∈ ϕ D A and set The constant in the above inequality > α 0 is large enough and fixed.
, , , , As we know, ( ) z t is a stationary process and its trajectories are P-a.s. continuous. Introducing the projection operator and the linear operator A, the first formula of equation (1.1) becomes the following form: The existence and uniqueness of solutions for (3.9) are derived by using the method which is similar to [11,10]. We omit it here. Define an RDS ( ( )) ≥ ∈ φ t τ ω , ; t τ ω , Ω associated with (3.9).
Taking the scalar product in (3.9), it derives Using the Young inequality with ε, it follows (3.14) Substitute (3.11)-(3.14) into (3.10) and deduce and λ 1 is the first eigenvalue of the operator A.
and substituting the above inequality into (3.15), it derives  Considering → −∞ s , the last part of the index in the first term which comes from (3.18) decays to 0. Now, we estimate the second term of (3.18).   4 Upper semicontinuity of attractors Section 3 derives that ( ) C2 which comes from Lemma 2.4 holds. In order to apply Lemma 2.4 to derive the upper semicontinuity of the random attractor, we just need to prove that C1 is established. By calculations, it has the following Theorem 4.1. Proof. The proof of the theorem is similar to Section 3.3 in [12]. In order to the completeness of the description, we describe as follows.
ε ε as the difference between the solutions of the perturbed and the unperturbed equation with the same initial condition u 0 at −t 0 . It is clear that v ε satisfies Considering the operator B is bilinear, it derives