Random Attractors for Stochastic Retarded 2 D-Navier-Stokes Equations with Additive Noise

In this paper, the existence and the upper semicontinuity of a pullback attractor for stochastic retarded 2D-Navier-Stokes equation on a boundeddomain are obtained.Wefirst transform the stochastic equation into a random equation and then obtain the existence of a randomattractor for randomequation.Then conjugation relation between two randomdynamical systems implies the existence of a random attractor for the stochastic equation. At last, we get the upper semicontinuity of random attractor.


Introduction
In this paper, we study the asymptotic behavior of the solutions for stochastic retarded Navier-Stokes equations with additive noise on a bounded domain in R 2 :  − Δ + ( ⋅ ∇)  = ( () +  (,  ( − ℎ)))  + where  is a bounded set in R 2 ,  is a positive constant, ℎ is the delay time of the system, , , ℎ  ( = 1, 2, . . ., ) are given functions,  > 0 is the strength of noise, and   ( = 1, 2, . . ., ) are two-sided Wiener processes on a probability space which will be specified later.
Navier-Stokes equations are basic equations of describing uncompressible viscous fluids, which are very important problems in fluid mechanics.Hence, Navier-Stokes equations have attracted many authors' attention.As all of us know, the asymptotic behavior of dynamical systems, which can be studied by attractor, is an important problem.Regarding the asymptotic profile of the solution, using similarity variables, some interesting results can be found in [1,2].The random attractor for stochastic system was introduced in [3,4] as an extension of the attractor for deterministic system in [5].The attractors for dynamical systems have been studied by many authors (see [3,4,[6][7][8][9][10][11][12][13][14][15][16][17][18][19]).The attractors are also used to study deterministic Navier-Stokes equations (see [9,[20][21][22]) and stochastic Navier-Stokes equations (see [4,23,24]).For the stochastic Navier-Stokes equations with additive noise and without time delay, the pullback attractors have been investigated in [23,24].However, as far as we know, there is no result on the random attractors for the stochastic retarded Navier-Stokes equation on bounded domains.So the main task of this paper is to study the asymptotic behavior of random attractor for stochastic Navier-Stokes equation (1).
To study the asymptotic behavior of random attractors, we shall establish a random dynamical system for (1) first.Then, we prove the existence and uniqueness of pullback random attractor.To achieve this, we derive the uniform estimates of solutions for Navier-Stokes equation with delays and additive noise.Finally, we study the convergence of random attractors as the noise intensity goes to zero.
The paper is organized as follows.In Section 2, we recall some notations and results on pullback attractors for random dynamical systems.And we show that the stochastic retarded Navier-Stokes equations generate a random dynamical system.In Section 3, we obtain some uniform estimates of solutions for system (14).These estimates are used to prove the existence of bounded absorbing sets and the asymptotic

RDS Generated by Stochastic 2D-Navier-Stokes Equation
In this section, we introduce some notations and recall some results about random attractors for random dynamical systems (RDS).The readers can refer to [3,4,6,8,10] for more details.
In the rest of this paper, we use the following notation.Let  ⊂ R 2 be an open bounded set with regular boundary Γ.Set and let  be the closure of V in ( 2 ()) 2 with norm ‖ ⋅ ‖ and inner product (⋅, ⋅).Let  be the closure of V in ( 1 0 ()) 2 with norm ‖ ⋅ ‖  , and inner product (⋅, ⋅)  , where for , Let   and   be dual space of  and .Then one has  ⊂  ≡   ⊂   , and the injections are dense and compact.
Let ⟨⋅, ⋅⟩ be the duality pairing between  and   .For , V ∈ ( 2 ()) 2 , define  :  →   , by Let () = { ∈ ( 2 ()) 2 :  ∈ }.Then  fl −Δ,  ∈ (), and  is a orthogonal projection from ( 2 ()) 2 to .Next, we define a trilinear form  on  ×  ×  by Then,  is a continuous bilinear form and satisfies the following condition (see [5,22]): By the Poincaré inequality and the homogeneous Dirichlet boundary, there is a constant  > 0 such that The norm of a general Banach space  is denoted by ‖⋅‖  .For ℎ > 0, we denote by In this section, we consider the following stochastic Navier-Stokes equation on  with additive white noise: Here  is a positive constant, ℎ > 0 is the delay time of the system, {  }  =1 are independent two-sided Wiener processes defined on a probability space on a probability space (Ω, F, P), with (15) the Borel -algebra F on Ω is generated by the compact open topology [6] and P is the corresponding Wiener measure on F; and  is a given function in [ 2 ()] 2 and ℎ  (1 ≤  ≤ ) are given functions in [ 2 0 ()] 2 with V ℎ  = 0, and  is a function that satisfies the following conditions. Let Then (Ω, F, P, (  ) ∈R ) is an ergodic metric dynamical system.Notice that the above probability space is canonical; one has To study the random attractor for system (14), we first transform that system into a deterministic system with random parameter.For  = 1, 2, . . ., , we introduce the Ornstein-Uhlenbeck process: Then, one has that   (    ) solves the following equation (see [11] for details): where () satisfies, for P-a.e.  ∈ Ω, Here  is a positive constant which will be fixed later in (43) and (44).Then, it follows that, for  > 0 and for P-a.e.

Uniform Estimates of Solutions
In this section, we prove the existence of the random attractor for the random dynamical system Ψ  associated with the stochastic Navier-Stokes equation (14).We first establish the existence of the random attractor for its conjugated random dynamical system Φ  ; then the existence of a D-random attractor for Ψ  follows from the conjugation relation between Φ  and Ψ  .
Proof.Taking the inner product of (31) with V, we obtain that Now, we estimate each term on the right-hand side of (36).
Proof.By (49) we have that Replacing  by  − , and by ( 54) and ( 55), one has that, for all  ≥   (), On the other hand, we have that Equations ( 60) and (61) imply that, for all  ≥   (), This ends the proof.
We use Ascoli theorem to prove the existence of a pullback random attractor.Before that, we need to prove the following result.Lemma 6.For all  ≥   ()+2ℎ+1, and for −ℎ ≤  1 ≤  2 ≤ 0, there exists a random variable  5 () such that Proof.By (31), Lemma 5 is used to estimate the first term on the right-hand side of (88).One has that, for  ≥   () + 2ℎ + 1, For the second term on the right-hand side of (88), we have that Next, we use Lemmas 3 and 5 to estimate the third and fourth terms on the right-hand side of (88).For  ≥   () + 2ℎ + 1, we can obtain that and In (92), we use condition (A1).For the last two terms on the right-hand side of (88), by (22) is equicontinuous.Ascoli theorem [27] implies that {V   (⋅,  −  , V 0 ( −  ))} ∞ = is relatively compact in ([−ℎ, 0];  2 ()).Thus, Proposition 1 implies the existence of a unique D-random attractor in S.
Notice that Φ  and Ψ  are conjugated random dynamical system.Hence, it follows from Theorem 7 that Ψ  has unique D− random attractors in S.

Upper Semicontinuity of Random Attractor
In this section, we consider the upper semicontinuity of random attractors of Navier-Stokes equation (31), when the intensity of noise tends to zero.We write the solution and the random dynamical system of (31) as V  and Φ  , respectively.Let with Set  2 0 () = 1 +  4  ℎ /.In Section 3, we prove that Φ  has a unique D-pullback random attractor   ().It follows from the invariance of the random attractor {  ()} ∈Ω that, for 0 <  ≤ 1,   () ⊆   ().Set with Then, for all 0 <  ≤ 1, This implies that We will show that, as  → 0, the solution of (31) converges to the limiting deterministic system: Lemma 8.For 0 <  ≤ 1, let V  and V be the solution of ( 31) and ( 101), respectively.Then, for  ∈ Ω,  > 0, there exists a positive constant  which is independent of , such that for all Proof.Let  = V  − V.By (31) and (101), we have that Taking the inner product of (103) with , we obtain that Now, we estimate each term on the right-hand side of (104).
Assume that {  ()} ∈Ω is defined by (95).From the proof of Lemma 3, we find that {  ()} ∈Ω is an absorbing set of Φ  in S. It is easy to see that lim sup By the conjugated relation between Φ  and Ψ  , it follows from Theorem 9 that the random attractor of Ψ  converges to the random attractor of Ψ, as  → 0.