Asymptotic Behavior for 2D Stochastic Navier-Stokes Equations with Memory in Unbounded Domains

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∞ ut ν∆u g(s)∆u(t s)ds − − −  Z0 x , t> 0,  dW ∈ O  + (u )u + p = f + εh ,  ·∇ ∇ dt  (1.1)  u = 0, x , t> 0,  ∇ · ∈ O u(x,t) = 0, x , t> 0,  | |→∞   u(x, 0) = u0(x), u(x, s)= ρ(x,s), x , s> 0,  − ∈ O  where u = u(x,t) is the unknown velocity, p = p(x,t) is the unknown pressure, while ν is the positive viscosity coefficient, f = f(x) [L2( )]2 is an assigned external forcing term, ∈ O h = h(x) [H1( )]2 [H2( )]2 is a given function and ε (0, 1] is a small parameter. W is ∈ 0 O ∩ O ∈ a two-side real-value Wiener process on a complete probability space which will be specified later. Concerning the kernel g : [0, ) R, we assume that it is convex, nonnegative, and ∞ → smooth on R+ = (0, ). Also it is supposed to satisfy ∞ lim g(s)=0 and ∞ g(s)ds = 1. s →∞ Z0 For deterministic case, Oskolkov [31] first studied the incompressible fluid with Kelvin- Voigt viscoelasticity which was illustrated by Navier-Stokes-Voigt system

ut ∆u α∆ut + (u )u + p = f, − − ·∇ ∇ (1.2) ( u = 0. ∇ · The existence of finite dimensional global attractors was investigated by Kalantarov and Titi in [23] and Anh and Trang [2] showed the existence of a weak solution to the problem by using the Faedo-Galerkin method in unbounded domains. After that, many authors considered system (1.2) in different aspects. Readers are referred to [34, 41] and references therein. Memory term arose in the description of several phenomena like, e.g., heat conduction in special materials (see e.g., [10, 22, 26, 27]), viscoelasticity of vibration in several materials (see e.g., [28, 30]). Actually, the presence of the memory destroys the parabolic character of the system and provides a more realistic description of the viscosity while ν∆u indicates − the instantaneous viscous effect. Astarita and Marucci [4, pp. 132] induced a first-order approximation to the constitutive equation of a simple fluid with fading memory

T = pI + ∞ f(s)Gt ds, (1.3) − Z0 where T is the stress and p is the pressure; Gt denotes the deformation history at some instant of observation t; f( ) is characteristic of the particular material. The constitutive equation · (1.3) was called “linear viscoelasticity” by the authors. In 2005, Gatti, Giorgi and Pata [20] proposed a Jeffreys type model depicting the motion of a two dimensional viscoelastic polymeric fluid with memory effect of the form

∞ ut ν∆u (1 ν) kε(s)∆η(s)ds + (u )u + p = f, − − − ·∇ ∇  Z0 ∂ η ∂ η u, (1.4)  t = s +  − u = 0, η = 0, ∇ · ∇ ·   2 where ν (0, 1) is a fixed parameter. In the system, the so-called memory kernel is defined ∈ as 1 s kε(s)= k , ε (0, 1]. ε2 ε ∈ They described the asymptotic dynamics and proved that when the scaling parameter ε in the memory kernel (physically, the Weissenberg number of the flow) tends to zero, the model converges to the Navier-Stokes equations in an appropriate sense. More recently, Gal and Medjo [18] put forward the following NSV system incorporating hereditary effects by adding the Voigt term α∆ut in (1.4) − ∞ ut ν∆u (1 ν) kε(s)∆η(s)ds α∆ut + (u )u + p = f, − − − − ·∇ ∇  Z0 ∂ η ∂ η u, (1.5)  t = s +  − u = 0, η = 0, ∇ · ∇ ·  for some ν  (0, 1). They took both Newtonian contributions and viscoelastic effects into ∈ account and considered a Cauchy stress tensor to derive the model. By the fact that the coupled effects of instantaneous viscous term ∆u, Voigt term ∆ut and hereditary kinematic viscous term ∞ g(s)∆u(t s)ds are strong enough to stabilize the system, Di Plinio, Giorgini, 0 − Pata and TemamR [15] investigated the long-time behavior of the following system without ∆u

∞ ut g(s)∆u(t s)ds α∆ut + (u )u + βu + p = f, − − − ·∇ ∇ (1.6)  Z0 u = 0,  ∇ · where βu is the Ekman term. They showed that the solution of (1.6) decays exponentially if f 0 and the system is dissipative from the viewpoint of dynamical systems. What’s more, ≡ they concluded that the system also possesses regular global and exponential finite fractal dimensional attractors. All the external forcing terms f above are deterministic, but actually, it is more meaningful that the system meets different random perturbations. So people considered different stochastic effects and combined the theory of random dynamics. Since 90s last century, there were many researches on stochastic Navier-Stokes equations. Flandoli and Schmalfuss [16] first combined random dynamical theory and Navier-Stokes equations and showed that there exist random attractors for 3D stochastic Navier-Stokes equations on bounded domains. However, they did not give the existence and uniqueness of solutions. Mar´ın-Rubio and Robinson [29] investigated the attractors for a 3D stochastic Navier-Stokes equations with additive white noise by a generalized semiflow. Bre´zniak and Li [8] proved the existence of the stochastic flow associated with 2D stochastic Navier-Stokes equations in possibly unbounded Poincaré domains by the classical Galerkin approximation. They then deduced the existence of an invariant measure for such system in two dimensional case. In [9], Bre´zniak, Carabollo, Lange and Li completed the result in [8]. They considered the random attractors of 2D Navier-Stokes equations in some unbounded domains and showed that the stochastic flow generated by the 2-dimensional Stochastic NavierStokes equations with rough noise on a Poincaré-like domain has a unique random attractor. It is known that the uniqueness of Navier-Stokes equations in three dimensional case is still open. So people turned to 3D Navier-Stokes-Voigt equations, 3 in which it has the term α∆ut, resulting in the uniqueness of the system. Gao and Sun − [19] examined the well-posedness of the 3D Navier-Stokes-Voigt system by classical Faedo- Galerkin method. They also investigated the random random attractors of three dimensional stochastic Navier-Stokes-Voigt equations. Further, Bao [6] continued corresponding works in unbounded domains. They use the so-called energy equation method, which was introduced by Ball [5] to establish the existence of random attractors. By means of the method in [39], they proved the upper semicontinuity of random attractors. For other works on stochastic Navier-Stokes-Voigt system please see [1, 25, 42] and the references therein. However, studies on stochastic Navier-Stokes equations with memory is still lack. Mo- tivated by the literature above, we investigate the well-posedness and asymptotic behaviors of two dimensional stochastic Navier-Stokes equations (1.1) in unbounded domains in this paper. More precisely, compared to (1.6), we have the viscosity dissipation, memory effects and random perturbation, while we do not have ∆ut (Voigt term) and βu (Ekman term). − The main features of our work are summarized as follows.

(i) Notice that Sobolev embeddings are no longer compact in unbounded domains. It leads to a major difficulty for us to prove the asymptotic compactness of solutions by standard method. To overcome this difficulty, we refer to [7, 38] which provide uniform estimates on the far-field values of solutions. Moreover, we establish a generalized Poincaréinequality to construct the weighted energy since we do not have βu in the system. (ii) The procedure in [7] indicates that we still need the compact embedding from higher regular space to common space in a bounded ball. Due to the memory term, the common regular space is = H and higher regular space is 1 = V 1. Though the H × M H × M embedding V ֒ H is compact, we can’t say that the embedding 1 ֒ is also → M → M compact. Nevertheless, we can recover the compactness with methods in [26, 32], for which we introduce a compact subspace and obtain a compact embedding N ⊂ M .in a bounded ball ֒ H →H (iii) In this manuscript, ψ , so we can not obtain the higher order estimate by using 0 ∈ H classicale energy method. To this end, we split the system into a “linear” system and a zero initial data nonlinear system [22, 23, 26, 35]. The energy of the “linear” system decays exponentially to 0 in while the energy of nonlinear system is bounded in 1. H H Then we can using this property to deduce the compactness.

Symbols above are all assigned in Section 2. The paper is arranged as follows. In Section 2, we recall the relevant mathematical framework for Navier-Stokes equations and memory kernel. In Section 3, we take some funda- mental results on the existence and semicontinuity of pullback random attractors for random dynamical systems, also we show that (1.1) generates a random dynamical system by several transformations. To derive the global well-posedness, we use classical Faedo-Galerkin method in Section 4. Some necessary uniform a prior and far-ﬁeld estimates are proposed in Section 5. By means of solution splitting method, far-ﬁeld estimates and a compact embedding we construct, we then prove the existence and uniqueness of random attractor for (1.1) in Sec- tion 5. In Section 6, we further show the upper semicontinuity of the attractors when the 4 stochastic perturbation parameters ε tends to zero. As usual, letter c in the paper represents generic positive constant which may change its value from line to line or even in the same line, unless we give a special declaration.

2 Mathematical Setting and Notation

In this section, we present some mathematical settings and notations as what in [15]. L2( ), H1( ), Hr( ) are standard Lebesgue-Sobolev spaces in . Let O 0 O O O 2 = u [C∞( )] : u = 0 . V ∈ 0 O ∇ · n o We denote by H the completion of in the norm of L2( ) 2 and by V the completion of V O V in the norm of H1( ) 2. Inner product and norm of H and V are 0 O u, v = (u, v) and u = (u, u), h i k k and u, v = u, v and u = u , h i1 h∇ ∇ i k k1 k∇ k respectively. We also denote by H′ the dual space of H and by V ′ the dual space of V . It follows that V H H V , where the injections are dense and continuous. We deﬁne ⊂ ≡ ′ ⊂ ′ the more regular space by 2 W = V H2( ) . ∩ O Recalling the Leray orthogonal projection : L2( ) 2 H from [24, 35, 36], we take P O → the Stokes operator A on H by A = ∆ with domain D(A) = W . Then A is a positive −P 1 1 self-adjoint operator with compact inverse and (Au, u) = (A 2 u, A 2 u) for all u V (see e.g., ∈ [37]). Hence, we deﬁne the compactly nested Hilbert spaces

r r V = D(A 2 ), r R, ∈ endowed with inner product and norm

r r r u, v = A 2 u, A 2 v and u r = A 2 u . h ir k k k k D E As usual, we deﬁne the continuous trilinear form b on V V V by × × 2 ∂vj b(u, v, w)= (u )v w dx = ui wj dx. ·∇ · ∂x i,j i ZO X=1 ZO By integration by parts, we easily prove that

b(u, v, w)= b(u, w, v), − and hence 2 b(u, v, v) = 0, u V , v H1( ) . ∀ ∈ ∈ 0 O The bilinear form B : V V V is deﬁned as × → ′ B(u, v), w = b(u, v, w). h i Then we introduce the common estimates for trilinear form b(u, v, w). 5 Lemma 2.1 (see e.g., [17, 35, 36, 37]). For all u, v, w V , we have ∈ 1 1 1 1 2 1 2 1 b(u, v, w) cˆ u 2 A 2 u A 2 v Av 2 w ; | |≤ k k k k k k 1 1 (2.1) 1 1 2 1 2 1 b(u, v, w) cˆ u 2 A 2 u Av A 2 w Aw 2 . | |≤ k k k k k k

In what follows, we describe the mathematical framework with respect to memory term. The function g is supposed to have the explicit form

µ(s)= g′(s). − We assume that µ here is nonnegative, absolutely continuous and decreasing. Hence µ 0 ′ ≤ for almost every s R+. Moreover, µ is summable on R+ with ∈ κ = ∞ µ(s)ds> 0. Z0 In our work, we consider the classical Dafermos condition (see e.g., [14])

µ′(s)+ δµ(s) 0, (2.2) ≤ for some δ > 0 and almost every s> 0. Then we deﬁne the weighted Hilbert space for memory on R+ = L2 (R+, V ), M µ endowed with inner product and norm

1 2 ∞ ∞ 2 η,ξ = µ(s) η(s),ξ(s) 1 ds and η = µ(s) η(s) 1 ds . h i h i k kM k k M Z0 Z0 The inﬁnitesimal generator of the right-translation semigroup on is the linear operator M

T η = ∂sη with domain D(T )= η : ∂sη , η(0) = 0 , − { ∈ M ∈ M } where ∂sη stands for the derivative of η(s) in regard to s. In the end, we introduce the phase space

= H , H × M endowed with norm (u, η) 2 = u 2 + η 2 . k kH k k k kM In this paper, we also utilize a more regular memory space denoted by

1 = L2 (R+, W ), M µ with norm analogous to that of . What’s more, the related higher order phase space is M denoted by 1 = V 1 H × M with norm 2 2 2 (u, η) 1 = u 1 + η 1 . k kH k k k kM 6 3 Random Dynamical System

3.1 Random attractors

In this subsection, we recall some basic concepts on the theory of random attractors for random dynamical systems. For a piece of detailed information and related applications, readers are referred to [3, 7, 39, 40].

Let (X, X ) be a separable Banach space with the Borel σ-algebra (X) and (Ω, , P) k · k B F be a probability space.

Definition 3.1. (Ω, , P, (θt)t R) is said to be a metric dynamical system if θ : R Ω Ω F ∈ × → is ( (R) , )-measurable and satisfies that θ is the identity on Ω, θt s = θt θs for all B × F F 0 + ◦ t,s R and θt(P)= P (measure preserved) for all t R. Here means composition. ∈ ∈ ◦ Definition 3.2. A mapping

Φ : R+ Ω X X, (t,ω,x) Φ(t,ω)x, × × → 7→ is known as a random dynamical system over a metric dynamical system (Ω, , P, (θt)t R) if F ∈ for P-a.e. ω Ω, ∈

(i) Φ(0,ω)=IdX on X;

+ (ii) Φ(t + s,ω)=Φ(t,θsω) Φ(s,ω), for all t,s R (cocycle property). ◦ ∈ A random dynamical system Φ is continuous if Φ(t,ω) : X X is continuous for all t R+, → ∈ ω Ω. ∈ Deﬁnition 3.3. A bounded random set is a random set B :Ω 2X which satisﬁes that there → is a random variable r(ω) [0, ), ω Ω, such that ∈ ∞ ∈

d(B(ω)) := sup x X : x B(ω) r(ω) for all ω Ω. {k k ∈ }≤ ∈

A bounded random set B(ω) ω Ω is said to be tempered in regard to the metric dynamical { } ∈ system (Ω, , P, (θt)t R) if for P-a.e. ω Ω, F ∈ ∈ µt lim e− d(B(θ tω)) = 0 for all µ> 0. t − →∞ In this manuscript, always denotes the collection of random sets of , i.e., D H

= B = B(ω) ω Ω : B(ω) ( ) and B is tempered . D { } ∈ ⊆H O

Deﬁnition 3.4. A random set K(ω) ω Ω is deﬁned as a random absorbing set for Φ { } ∈ ∈ D in if for every B and P-a.e. ω Ω, there is a T (B,ω) > 0 such that D ∈ D ∈

Φ(t,θ tω)B(θ tω) K(ω), for all t T (B,ω). − − ⊆ ≥ Definition 3.5. A random dynamical system Φ is ( -pullback) asymptotically compact in D P X if for -a.e. ω Ω, Φ(tn,θ tn ω)xn n∞=1 has a convergent subsequence in X whenever ∈ { − } tn , and xn B(θ tn ω) with B(ω) ω Ω . →∞ ∈ − { } ∈ ∈ D 7 Definition 3.6. A -pullback attractor for Φ is a random set (ω) ω Ω of X which satisfied D {A } ∈ that for P-a.e. ω Ω, ∈ (i) (ω) is compact, and ω d(x, (ω)) is measurable for every x X; A 7→ A ∈

(ii) (ω) ω Ω is invariant, i.e., {A } ∈

Φ(t,ω) (ω)= (θtω) for all t 0; A A ≥

(iii) (ω) ω Ω attracts every set in , i.e., for every B = B(ω) ω Ω , {A } ∈ D { } ∈ ∈ D

lim dist(Φ(t,θ tω)B(θ tω), (ω)) = 0, t − − →∞ A where dist( , ) is the Hausdorﬀ semi-distance denﬁned on X, i.e., for two nonempty sets · · Y,Z X, ⊆ dist(Y,Z) = sup inf y z X . y Y z Z k − k ∈ ∈

Recall that a collection of random sets in X is said to be inclusion-closed if E(ω)ω Ω D ∈ belong to when E(ω)ω Ω is a random set, and F (ω)ω Ω belongs to with E(ω) F (ω) for D ∈ ∈ D ⊂ all ω Ω. The following proposition with respect to the existence and uniqueness of random ∈ attractor can be found in [11, 13, 16, 39, 40].

Proposition 3.1. [39] Let be an inclusion-closed collection of random subsets of X. As- D sume that K(ω) ω Ω is a closed random absorbing set for Φ in and Φ is -pullback { } ∈ D D asymptotically compact in X. Then Φ has a unique -random attractor (ω) ω Ω given by D {A } ∈

(ω)= Φ(t,θ tω)K(θ tω), for every ω Ω. A − − ∈ τ 0 t τ \≥ [≥ 3.2 Upper semicontinuity of random attractors

In this subsection, we recall some results in [39] about the upper semicontinuity of random attractors when random disturbance vanishes. Given ε (0, 1] and let Φε be a random ∈ dynamical system with respect to (Ω, , P, (θt)t R) which has a random absorbing set Kε = F ∈ Kε(ω) ω Ω and a random attractor ε = ε(ω) ω Ω. Let (X, X ) be a Banach space { } ∈ A {A } ∈ k · k and Φ be a dynamical system deﬁned on X with the global attractor , which means that A0 is compact and invariant and attracts every bounded subset of X uniformly. A0

Deﬁnition 3.7. For 0 < ε 1, the family of random attractors ε(ω) ω Ω is said to be ≤ {A } ∈ upper semicontinuous when ε 0+ if →

lim dist( ε(ω), 0) = 0, for all ω Ω. ε 0+ A A ∈ → The following proposition is given and proved in [39].

Proposition 3.2. Suppose that the following conditions hold for P-a.e. ω Ω: ∈

8 (i) εn lim Φ (t,ω)xn = Φ(t)x n →∞ for all t 0, provided εn 0 and xn x in X; ≥ → → (ii)

lim sup Kε(ω) X c ε 0 k k ≤ →

for some deterministic positive constant c where Kε(ω) X = supx Kε(ω) x X ; k k ∈ k k (iii)

ε(ω) is precompact in X. A 0<ε 1 [≤ + Then the family of random attractors ε(ω) ω Ω is upper semicontinuous as ε 0 . {A } ∈ → 3.3 Stochastic Navier-Stokes equations with memory

In this subsection, we show that there is a continuous random dynamical system generated by the stochastic Navier-Stokes equations with memory in unbounded domains. νλ1 δ First, we set δ0 = min 2 , 2 and take a constant σ large enough such that n o c ε2 c ε4 σ> max 0 , 5 . (3.1) 2δ 2δ 0 0 2(ˆcc˜)2 3456(ˆcc˜)4 where c0 = ν and c5 = ν3 ,c ˜ is a constant which will be assigned later. Next, we consider the probability space (Ω, , P) where F Ω= ω C(R, R) : ω(0) = 0 , { ∈ } is the Borel σ-algebra induced by the compact-open topology of Ω and P is the corresponding F Wiener measure on (Ω, ). Then we identify W (t) with ω(t), i.e., ω(t) = W (t,ω), t R. F ∈ Deﬁne the time shift by

θtω( )= ω( + t) ω(t), ω Ω, t R. · · − ∈ ∈

Then (Ω, , P, (θt)t R) is an ergodic metric dynamical system (see e.g., [3]). F ∈ Moreover, by applying the Leray orthogonal projection to (1.1) , we have P 1 ∞ dW ut + νAu + g(s)Au(t s)ds + B(u, u)= f + εh (3.2) − dt Z0 subject to initial data u(0) = u H. Here we rewrite f and h as f and h respectively 0 ∈ P P and f(x) H, h(x) W . Then we introduce the past history variable ∈ ∈ s ηt(s)= u(t τ)dτ, − Z0 which satisﬁes the diﬀerential identity

t t ∂tη (s)= ∂sη (s)+ u(t). − 9 For readability, we will suppress t in the notation of η in the sequel. Combining the deﬁnition of T and integration by parts, one obtains that dW u + νAu + ∞ µ(s)Aη(s)ds + B(u, u)= f + εh , t dt  Z0  ∂tη = T η + u, (3.3)   s  0 u(0) = u0, η := η0 = ρ(σ)dσ. 0  Z  To derive a continuous random dynamical system related to Eq. (3.3), we consider the Ornstein-Uhlenbeck equation [6] and convert the stochastic equation to a deterministic equation with random parameters z which satisﬁes

dz + σz dt = dW. (3.4)

It is easy to check that a solution to (3.4) is given by t ν(t τ) z(t; ω)= e− − dW (τ) (ω). Z∞ One may obtain from [3, Proposition 4.3.3] that there exists a tempered function r(ω) > 0 such that 2 4 β (θtω)= z(θtω) + z(θtω) r(θtω), (3.5) 1 | | | | ≤ where r(ω) satisﬁes that for P-a.e. ω Ω, ∈ δ0 t r(θtω) e 2 | |r(ω), t R. (3.6) ≤ ∈ It follows from (3.5) and (3.6) that, for P-a.e. ω Ω, ∈ δ0 t β (θtω) e 2 | |r(ω), t R. (3.7) 1 ≤ ∈ Next, we set y(θtω)= h(x)z(θtω) and v(t)= u(t) εy(θtω) where u is a solution of (3.2). 1− Since h(x) W , we denote byc ˜ = max h(x) , A 2 h(x) , Ah(x) and it follows that ∈ k k k k n o y(θtω) h(x)z(θtω ) c˜ z (θtω) , 1k k ≤ k 1 k≤ | | A 2 y(θtω) A 2 h(x)z(θtω) c˜ z(θtω) , (3.8) k k ≤ k k≤ | | Ay(θtω) Ah(x)z(θtω) c˜ z(θtω) . k k ≤ k k≤ | | Then v satisﬁes ∞ vt + νAv + µ(s)Aη(s)ds + B(v + εy(θtω), v + εy(θtω))  Z0  = f + ε (σy(θtω) νAy(θtω)) ,  − (3.9)   ∂tη = T η + v + εy(θtω), 0  v(0) = v0 = u0 εy(ω), η = η0.  −  With (3.9)3, we denote τ ψ(t,ω,ψ (ω)) = v(t, ω, v ), ηt(ω, η ( )) , 0 0 0 · t τ φ(t,ω,φ (ω)) = v(t, ω, u εy(ω)) + εy(θtω), η (ω, η ) (3.10) 0 0 − 0 τ = ψ(t,ω,ψ0(ω)) + (εy(θtω), 0) . Thus we obtain that ψ 2 = v 2 + η 2 . k kH k k k kM 10 4 Global Well-posedness

We ﬁrst give the deﬁnition of weak solutions:

Deﬁnition 4.1 (Weak solution). Let f H and ψ = (v , η )τ , ψ = (v, η)τ is a weak ∈ 0 0 0 ∈ H solution of problem (3.9) provided that

2 2 (i) v L (0, T ; H) L (0, T ; V ), vt L (0, T ; V ), η L (0, T ; ); ∈ ∞ ∩ ∈ ′ ∈ ∞ M (ii) for all ϕ = (w,ξ)τ , we have ∈H ∞ (∂tv, w)+ ν(Av, w)+ µ(s)(Aη(s), w)ds + b(v + εy(θtω), v + εy(θtω), w) 0  Z (4.1)  = (f + ε (σy(θtω) νAy(θtω)) , w),  − ∂tη,ξ = Tη,ξ + v + εy(θtω),ξ . h iM h iM h iM   Then, the global well-posedness of (3.9) is stated in the following theorem.

Theorem 4.1. Assume that T R+ and ψ . For P-a.e. ω Ω, there exists a unique 0 ∈ 0 ∈ H ∈ solution ψ of (3.9) on the interval [T , ) satisfying 0 ∞ 2 v L∞(T , ; H) L (T , ; V ), η L∞(T , ; ) ∈ 0 ∞ ∩ 0 ∞ ∈ 0 ∞ M 2 and vt is uniformly bounded in L (T , ; V ). Moreover, the solution continuously depends 0 ∞ ′ on the initial data.

Proof. We divide the proof of Theorem 4.1 into three parts: existence, uniqueness and dependence. Part 1: Existence. First, we use the standard Faedo-Galerkin procedure to show the existence of weak solution to (3.9).

Step 1. The approximate system. Let wj j∞=1 be the normalized eigenfunction basis of the { } + Stokes operator A and ξj ∞ be the orthonormal basis of with all ξj C (R , V ) where { }j=1 M ∈ 0∞ R+ V C0∞( , ) is a compactly supported inﬁnitely diﬀerentiable function space respect to s. For any integer n, we denote by Pn and Qn the projections onto the subspaces

Hn := PnH = span w , w , . . . , wn H { 1 2 }⊂ and

n := Qn = span ξ ,ξ ,...,ξn , M M { 1 2 } ⊂ M t τ respectively. Then we deﬁne the approximate solutions ψn := vn, ηn as

n n n t n vn(t)= ak (t)wk, ηn(s)= bk (t)ξk(s). k k X=1 X=1 Hence we have n n n t n Avn(t)= λkak (t)wk, Aηn(s)= λkbk (t)ξk(s). Xk=1 Xk=1

11 Therefore, the nth-order Galerkin approximate system is

∞ t (∂tvn, w)+ ν(Avn, w)+ µ(s)(Aηn(s), w)ds 0 for all w H ,  Z n ˜ ∈  +b(vn + εy, vn + εy, w) = (f, w),   ∂ ηt ,ξ = T ηt ,ξ + v + εy,ξ , for all ξ , (4.2)  t n n n n  M M h iM ∈ M v n(0) = Pnv0, a.e. in , O  t 0 R+  ηn(0) = Qnη , a.e. in .  O×  ˜ ˜ where we denote f by f = Pn [f + ε (σy νAy)]. τ − τ Denoting ϕ := (w,ξ) and taking ϕ = (wk,ξk) , we can deduce that n d n n n ak + νλkak + bk ξk, wk dt h iM  k=1  X  n n  n n ˜  + b ak wk + εy, ak wk + εy, wk = (f, wk), (4.3)   k=1 k=1 n X Xn d n n n  bk = ak wk,ξk bk ξk′ ,ξk + εy,ξk ,  dt h iM − h iM  k=1 k=1 M  X X  subjected to the initial conditions n n n 0 n ak (0) = v0, ak and bk (0) = η , bk . h i M Thanks to the local existence theory of ordinary diﬀerential equations, we know that there t τ exists a solution to the problem (4.3) which means that solutions vn, ηn to the approximate problem (4.2) exist. t τ Step 2. Uniform a priori estimates. Let us take ϕ = vn, ηn in problem (4.2), then we have 2 2 1 d 2 t 2 1 1 ∞ 1 t t vn + ηn + ν A 2 vn = µ′(s) A 2 ηn ds + ε ηn,y(θtω) 2 dt k k M 2 0 M Z b(vn + εy( θtω), v n + εy(θtω ), vn) − (4.4) + (f,˜ vn)

=: I1 + I2 + I3 + I4. It is clear from the Poincar´einequality that 2 1 ν 2 νλ1 2 ν A 2 vn vn + vn . (4.5) ≥ 2 k k1 2 k k

Now we are ready to estimate Ii , i = 1, 2, 3, 4 respectively. For I1, from the Dafermos condition (2.2), we get 2 δ ∞ 1 t δ t 2 I1 µ(s) A 2 ηn ds = ηn . (4.6) ≤−2 0 −2 M Z

By utilizing the H¨older’s inequality, the Young’s inequality with ǫ and (3.8), one obtains that ∞ 1 t 1 I2 = ε µ(s)(A 2 ηn, A 2 y(θtω)) ds Z0 2 2 δ ∞ 1 t ε ∞ 1 2 µ(s) A 2 ηn ds + µ(s) A 2 y(θtω) ds (4.7) ≤ 4 0 δ 0 k k Z 2 Z δ t 2 cε κ ηn + β 1(θtω). ≤ 4 M δ

12 Since b(u, v, v) = 0, it follows from the H¨older’s inequality, the Young’s inequality with ǫ, Lemma 2.1 and (3.8) that

I b(vn, εy(θtω), vn) + b(εy(θtω), εy(θtω), vn) | 3| ≤ | | | | 1 1 2 1 1 1 cεˆ vn A 2 y(θtω) A 2 vn +cε ˆ y(θtω) 2 A 2 y(θtω) Ay(θtω) 2 vn ≤ k k k k k k k k (4.8) 2 2 2 2 2 4 4 cˆ c˜ ε 2 ν 1 cˆ c˜ ε νλ 1 2 β 1(θtω) vn + A 2 vn + β 1(θtω)+ vn . ≤ ν k k 4 νλ1 8 k k

As f H, it can be deduced from the Young’s inequality with ǫ and (3.8) that ∈ νλ1 2 2 2 2 2 2 I vn + f + 2cε y(θtω) + Ay(θtω) 4 ≤ 8 k k νλ k k k k k k 1 (4.9) νλ1 2 c 2 vn + 1+ ε β1(θtω) . ≤ 8 k k νλ1 Combining (4.4)–(4.9), we have

d 2 ν 2 ψn(t,ω,ψ0(ω)) + vn(t, ω, v0(ω)) 1 dtk kH 2 k k (4.10) 2 2 2 δ0 + c0ε β1(θtω) ψn(t,ω,ψ0(ω)) + c 1+ ε β1(θtω) . ≤ − k kH Applying the Gronwall’s inequality, one obtains

t 2 s 2 ν δ0(s t) c0ε R β1(θτ ω) dτ 2 ψn(t,ω,ψ0(ω)) + e − − t vn(s, ω, v0(ω)) 1 ds k kH 2 0 k k Z t (4.11) 2 t 2 s δ0t+c0ε R β1(θτ ω) dτ 2 δ0(s t) c0ε R β1(θτ ω) dτ 2 e− 0 ψ0(ω) + c e − − t 1+ ε β1(θsω) ds. ≤ k kH Z0 In (4.11), we replace ω by θ tω, then − t ν 2 s 2 δ0(s t) c0ε Rt β1(θτ tω) dτ 2 ψn(t,θ tω, ψ0(θ tω)) + e − − − vn(s,θ tω, v0(θ tω)) 1 ds k − − kH 2 0 k − − k Z t 2 t 2 s δ0t+c0ε R0 β1(θτ tω) dτ 2 δ0(s t) c0ε R β1(θτ tω) dτ 2 e− − ψ0(θ tω) + c e − − t − 1+ ε β1(θs tω) ds ≤ k − kH 0 − Z0 2 0 2 0 δ0t+c0ε R t β1(θτ ω) dτ 2 δ0s+c0ε Rs β1(θτ ω) dτ 2 = e− − ψ0(θ tω) + c e 1+ ε β1(θsω) ds. k − kH Z t − (4.12)

Since β1(θτ ω) is stationary and ergodic, it follows from the ergodic theorem in [12] that 1 0 1 lim β1(θτ ω)dτ = E (β1(ω)) . t + t t ≤ 4σ → ∞ Z− Hence, there exists a T > 0 such that for all t T , 1 ≥ 1 0 1 δ0 β1(θτ ω)dτ t 2 t. (4.13) t ≤ 4σ ≤ 2c0ε Z− Thus for all t T , ≥ 1 t 2 ν 2 ψn(t,θ tω, ψ0(θ tω)) + vn(s,θ tω, v0(θ tω)) 1 ds k − − kH 2 0 k − − k 0 Z (4.14) δ0 2 0 t 2 δ0s+c0ε R β1(θτ ω) dτ 2 e− 2 ψ0(θ tω) + c e s 1+ ε β1(θsω) ds. ≤ k − kH t Z− 13 Note that ψ0(θ tω) B(θ tω) , we see that there is a T2 = T2(B,ω) > 0, independent of − ∈ − ⊂ D ε, such that for all t T2, δ ≥ 0 t 2 e− 2 ψ0(θ tω) 1. k − kH ≤ Since t

0 2 0 ε δ0s+c0ε Rs β1(θτ tω) dτ 2 r1(θ tω)= e − 1+ ε β1(θs tω) ds − − Z−∞ 0 2 t δ0s+c0ε Rs− t β1(θτ ω) dτ 2 = e − 1+ ε β1(θs tω) ds − Z−∞ 0 δ0 2 t 2 s+c0ε Rs− t β1(θτ ω) dτ 2 e − 1+ ε β1(θs tω) ds ≤ − Z−∞t δ0 2 t − (s+t)+c0ε R − β1(θτ ω) dτ 2 = e 2 s 1+ ε β1(θsω) ds

Z−∞t δ0 2 0 − (s+t)+c0ε R β1(θτ ω) dτ 2 e 2 s 1+ ε β (θsω) ds ≤ 1 Z−∞ t δ0 δ0 δ0t − s 2 s e e 2 1+ ε e− 2 r(ω) ds ≤ Z−∞ 2 δ0 t δ t = e 2 + ε2r(ω)te 0 . δ0 Then we get 3δ0 2 δ0 2 t ε δ0t 2 2 t e− r1(θ tω)= e− + ε r(ω)te− 0, as t , − δ0 → →∞ which means that rε(ω) is a tempered function. Then for all t T (B,ω) = max T , T , 1 ≥ 3 { 1 2} t 2 ν 2 ε ψn(t,θ tω, ψ0(θ tω)) + vn(s,θ tω, v0(θ tω)) 1 ds c1(r1(ω) + 1). (4.16) k − − kH 2 k − − k ≤ Z0 where c1 is a constant independent of ε. Therefore, we deduce that for T > T3,

2 vn is uniformly bounded in L∞(T , ; H) L (T , ; V ), (4.17) 3 ∞ ∩ 3 ∞ t η is uniformly bounded in L∞(T , ; ). (4.18) n 3 ∞ M

Next, we estimate ∂tvn. Taking any w V in equation (4.2) , we have ∈ 1 ∞ t (∂tvn, w)= ν(Avn, w) µ(s)(Aη (s), w)ds b(vn + εy, vn + εy, w) + (f,˜ w), − − n − Z0 Since

(Avn, w) vn w V ; | | ≤k∇ k k k 14 ∞ t t µ(s)(Aηn(s), w)ds κ ηn w V ; 0 ≤ M k k Z 1 2 2 b(vn + εy, vn + εy, w) cβ ( θtω) ε + ε vn w V | |≤ 1 k∇ k k k + vn vn w V ; k 1 kk∇ k k k 2 (f,˜ w) c(1 + εβ (θtω)) w V , ≤ 1 k k we obtain 1 t 2 2 ∂tvn V vn + κ η + cβ (θtω) ε + ε vn ′ n 1 k k ≤ k∇ k M 1 k∇ k 2 + vn v n + c(1 + β (θtω)). k k k∇ k 1 It follows from (4.17) that for T > T3,

T T T 2 2 t 2 ∂tvn V ds c vn ds + c ηn ds k k ′ ≤ k∇ k ZT3 ZT3 ZT3 M T 4 2 2 + c β (θsω) ε + ε vn ds 1 k∇ k ZT3 T T 2 2 2 + c vn vn ds + c (1 + ε β (θsω)) ds k k k∇ k 1 ZT3 ZT3 ε cT (r (ω) + 1). ≤ 1 where cT is a ﬁnite constant depending on T . Then we deduce that

2 ∂tvn is uniformly bounded in L (T , ; V ′). (4.19) 3 ∞

Step 3. Passing to the limits. It follows from (4.17), (4.18) and (4.19) that

vn v strongly in V ; (4.20) → vn ⇀ v weakly* in L∞(T , ; H); (4.21) 3 ∞ 2 vn ⇀ v weakly in L (T , ; V ); (4.22) 3 ∞ t t η ⇀ η weakly* in L∞(T , ; ); (4.23) n 3 ∞ M 2 ∂tvn ⇀ ∂t weakly in L (T , ; V ′). (4.24) 3 ∞ By the Aubin-Lions Lemma [33], one can easily get that

2 vn v strongly in L (T , ; H). (4.25) → 3 ∞ Like the procedure in [21], we pass to the limits and obtain the global weak solutions of problem (3.9). τ τ Part 2: Uniqueness. Suppose that ψ1 = (v1, η1) and ψ2 = (v2, η2) are two solutions to (3.9) with the same initial data. Also we deﬁne ψˆ := (w,ξ)τ = ψ ψ . Then we have the 1 − 2 following system

∞ wt + νAw + µ(s)Aξ(s)ds + B(w, v1 + εy)+ B(v2 + εy, w) = 0,  Z0 ∂ ξ Tξ w, (4.26)  t = +  w(0) = 0, ξ0 = 0.   15 Taking inner product with (4.26) by ψˆ, we obtain H 2 2 1 d 2 2 1 1 ∞ 1 w + ξ + ν A 2 w = µ′(s) A 2 ξ(s) ds b(w, v1 + εy, w). (4.27) 2 dt k k k kM 2 0 − Z

As the similar procedure above, we get

2 2 1 ν 1 νλ1 2 ν A 2 w A 2 w + w ; ≥ 2 2 k k 1 ∞ 1 2 δ ∞ 1 2 µ′(s ) A 2 ξ (s) d s µ(s) A 2 ξ(s) ds; 2 ≤ 2 Z0 Z0 2 2 c0 ε 2 ν 1 b(w, εy, w) β (θtω) w + A 2 w ; | |≤ ν 1 k k 8 2 2 2ˆc 2 2 ν 1 b(w, v1, w) v1 w + A 2 w . | |≤ ν k∇ k k k 8

Hence 2 2 2 2 2 d ν 1 2c0ε 4ˆc 2 ψˆ + A 2 w 2δ + β (θtω)+ v ψˆ . (4.28) dt 2 ≤ − 0 ν 1 ν k∇ 1k H H

It can be deduced from (4.17) that

2 ε 2 δ0t+c(r (ω)+1) ψˆ(t,θ tω, ψˆ0) e− 1 ψˆ0 = 0, as ψˆ0 = 0, (4.29) − ≤ H H which means the solution is unique. τ τ Part 3: Dependence. We assume that ψ1 = (v1, η1) and ψ2 = (v2, η2) are two solutions to (3.9) subjected to diﬀerent initial datum ψ01 and ψ02, respectively. Reasoning as in uniqueness, we obtain the dependence of initial datum immediately.

Remark 4.1. Gao and Sun [19] showed the well-posedness of 3D stochastic Navier-Stokes- Voigt equations in bounded domains. If we consider 3D stochastic Navier-Stokes-Voigt equations in some unbounded domains, we can obtain the well-posedness thanks to the Voigt term

α∆ut by means of the arguments in [2, 19]. However, when we take (1.1) into account in − three dimensional case, in which the memory eﬀects is weaker than α∆ut, the uniqueness − is lost Since the uniqueness of classical 3D Navier-Stokes equations is still open. Fortunately, two dimensional incompressible Navier-Stokes equations are well-posedness, so in our work, we investigate system (1.1) in two dimension successfully.

5 Random attractors

In this section, we aim to establish uniform estimates for ψ and φ with respect to the small parameter ε, including long-time a priori estimates and far-ﬁeld estimates. We decompose the solution into two parts and get the higher order estimate. By means of a constructed compact subspace, we prove the compactness of solution and show the existence of random attractor.

16 5.1 Uniform estimates for solutions

First of all, we impose the uniform estimates for solutions in and get the absorbing set. H

Lemma 5.1. For every B(ω)ω Ω and for P-a.e. ω Ω, there exist T3 = T3(B,ω) > 0 ∈ ε ∈ D ∈ and a tempered function r1(ω), such that for all ψ0(θ tω) B(θ tω), − ∈ − 2 ε ψ(t,θ tω, ψ0(θ tω)) c1(r1(ω) + 1), t T3, k − − kH ≤ ∀ ≥ and 2 ε φ(t,θ tω, φ0(θ tω)) c2(r2(ω) + 1), t T3, k − − kH ≤ ∀ ≥ ε ε where c1, c2 are positive deterministic constants independent of ε, r2(ω)= r1(ω)+ r(ω) and r(ω) is the tempered function in (3.5).

Proof. Substituting ψn in (4.16) by ψ, we can easily get that

2 ε ψ(t,θ tω, ψ0(θ tω)) c1(r1(ω) + 1), t T3. k − − kH ≤ ∀ ≥ Relation (3.10) between ψ and φ implies that

2 2 ψ0(θ tω) = φ0(θ tω) (εy(ω), 0) k − kH k − − kH 2 2 2 2 φ0(θ tω) + 2ε y(ω) ≤ k − kH k k 2 2 2 2 φ0(θ tω) + 2cε z(ω) . ≤ k − kH | |

We know that φ0(θ tω) B(θ tω) is tempered and z(ω) is also tempered, then ψ0(θ tω) − ∈ − ⊂ D − is tempered. Therefore, by (3.5), (3.8), (3.10) and (4.16), we obtain that, for all t T , ≥ 3 2 2 φ(t,θ tω, φ0(θ tω)) = ψ(t,θ tω, ψ0(θ tω)) + (εy(ω), 0) k − − kH k − − kH 2 2 2 2 ψ(t,θ tω, ψ0(θ tω)) + 2ε y(ω) ≤ k − − kH k k 2c (rε(ω) + 1) + 2cε2 z(ω) 2 (5.1) ≤ 1 1 | | c (rε(ω)+ ε2r(ω) + 1) ≤ 2 1 ε =: c2(r2(ω) + 1), where c2 is a constant independent of ε. This completes the proof.

Corollary 5.1. Notice that Φ(t,ω)φ0(ω) = φ(t,ω,φ0(ω)). By (5.1), we know that for all t T3, ≥ 2 2 ε Φ(t,θ tω)φ0(θ tω) = φ(t,θ tω, φ0(θ tω)) c2(r2(ω) + 1). (5.2) k − − kH k − − kH ≤ Given ω Ω, we denote ∈ 2 ε K(ω)= φ ( ) : φ c2(r2(ω) + 1) . (5.3) ∈H O k kH ≤ It is clear that K(ω) ω Ω . Moreover, (5.2) indicates that K(ω) ω Ω is a random { } ∈ ∈ D { } ∈ absorbing set for Φ in . D

17 To prepare for the proof of the compactness of Φ, we decompose the system into two subproblem (see [22, 23, 35]): one decays exponentially and the other is bounded in a higher τ τ τ regular space. Since ψ = (v, η) , we split the solution as ψ = ψL +ψN = (vL, ηL) +(vN , ηN ) .

Also, φ = φL + φN . Then we have

∞ ∂tvL + νAvL + µ(s)AηL ds + B(v + εy, vL) = 0, 0  Z (5.4)  ∂tηL = T ηL + vL,   0 v0L = v0, η0L = η .  and  ∞ ∂tvN + νAvN + µ(s)AηN ds  Z0  + B(v + εy, vN + εy)= f + ε(σy νAy),  − (5.5)   ∂tηN = T ηN + vN + εy,  v0N = 0, η0N = 0.   First, we show that ψL and φL has an exponential decay.

Lemma 5.2. For every B(ω)ω Ω and for P-a.e. ω Ω, the solutions of problem (5.4) ∈ ∈ D ∈ satisfy the following exponential decay property, i.e., for t> 0 and ψ0L(θ tω) B(θ tω), − ∈ −

2 2δ0t 2 ψL(t,θ tω, ψ0L(θ tω)) e− ψ0L . (5.6) k − − kH ≤ k kH What’s more, 2 2δ0t 2 φL(t,θ tω, φ0L(θ tω)) e− φ0L , (5.7) k − − kH ≤ k kH τ where φ0L = ψ0L + (εy, 0) .

Proof. Taking inner product with system (5.4) by ψL and adding the equations, we have H 2 1 d 2 2 2 1 ∞ 1 vL + ηL + ν AvL = µ′(s) A 2 ηL ds. 2 dt k k k kM k k 2 0 Z

Same procedure as in Section 4 tells us that

d 2 2 ψL + 2δ0 ψL 0. dt k kH k kH ≤

Then by the Gronwall’s inequality, we get exponential decay (5.6). Note that φL = ψL, we obtain (5.7).

Remark 5.1. (5.7) shows that φL goes to 0 as t , i.e., given ζ > 0, there is a suﬃcient →∞ large T such that for t T , ∗ ≥ ∗ 2 φL(t,θ tω, φ0L(θ tω)) ζ. k − − kH ≤

Next, we give higher order estimates for ψN and φN .

18 Lemma 5.3. For every B(ω)ω Ω and for P-a.e. ω Ω, there exist T4 = T4(B,ω) > 0 ∈ ε ∈ D ∈ and a tempered function r3(ω) such that for all ψ0(θ tω) B(θ tω), − ∈ − 2 ε ψN (t,θ tω, ψ0N (θ tω)) 1 c3(r3(ω) + 1), t T4, k − − kH ≤ ∀ ≥ and 2 ε φN (t,θ tω, φ0N (θ tω)) 1 c4(r4(ω) + 1), t T4, k − − kH ≤ ∀ ≥ ε ε where c3, c4 are positive deterministic constants independent of ε, r4(ω)= r3(ω)+ r(ω) and r(ω) is the tempered function in (3.5).

τ Proof. Taking the inner product with system (5.5) by AψN = (AvN , AηN ) , we have

1 d 2 2 2 1 ∞ 2 vN 1 + ηN 1 + ν AvN = µ′(s) AηN ds + ε ηN ,y(θtω) 1 2 dt k k k kM k k 2 0 k k h iM Z b(v + εy(θtω), v + εy(θtω), AvN ) − (5.8) + (f + ε (σy(θtω) νAy(θtω)) , AvN ) − =: J1 + J2 + J3 + J4.

It is clear from the Poincar´einequality that

2 ν 2 νλ1 2 ν AvN AvN + vN . (5.9) k k ≥ 2 k k 2 k k1

Now we are ready to estimate Ji, i = 1, 2, 3, 4 respectively. For J1, from the Dafermos condition (2.2), we get

δ ∞ 2 δ 2 J µ(s) AηN ds = ηN 1 . (5.10) 1 ≤−2 k k −2 k k Z0 M By utilizing the H¨older’s inequality, the Young’s inequality with ǫ and (3.8), one obtains that

∞ J2 = ε µ(s)(AηN , Ay(θtω)) ds 0 Z 2 δ ∞ 2 ε ∞ 2 µ(s) AηN ds + µ(s) Ay(θtω) ds (5.11) ≤ 4 0 k k δ 0 k k Z 2 Z δ 2 cε κ ηN 1 + β1(θtω). ≤ 4 k kM δ A direct calculation deduces that

1 1 1 1 2 1 2 3 b(vL, vN , AvN ) cˆ vL 2 A 2 vL A 2 vN AvN 2 | |≤ k k k k 4 ν 2 11664ˆ c 2 2 2 AvN + vL vL vN , ≤ 48 k k ν3 k k k k1 k k1 19 1 1 1 1 2 1 2 1 b(vL, εy(θtω), AvN ) εcˆ vL 2 A 2 vL A 2 εy(θtω) Aεy(θtω) 2 AvN | |≤ k k k k k k ν 2 2 AvN + 6ε c vL vL β (θtω), ≤ 48 k k 0 k k k k1 1

1 1 3 b(vN , vN , AvN ) cˆ vN 2 A 2 vN AvN 2 | |≤ k k k k 4 ν 2 11664ˆc 2 2 2 AvN + vN vN vN , ≤ 48 k k ν3 k k k k1 k k1

1 1 1 1 2 1 2 1 b(vN , εy(θtω), AvN ) εcˆ vN 2 A 2 vN A 2 y(θtω) Ay(θtω) 2 AvN | |≤ k k k k k k 4 ν 2 νλ 1 2 144( εcˆc˜) 2 AvN ++ vN 1 + 3 β1(θtω) vN , ≤ 48 k k 4 k k ν λ1 k k

1 1 1 1 2 1 2 3 b(εy(θtω), vN , AvN ) εcˆ y(θtω) 2 A 2 y(θtω) A 2 vN AvN 2 | |≤ k k k k 4 ν 2 11664(εcˆc˜ ) 2 AvN + β (θtω) vN ≤ 48 k k ν3 1 k k1 and

2 1 1 1 b(εy(θtω), εy(θtω), AvN ) ε cˆ y(θtω) 2 A 2 y(θtω) Ay(θtω) 2 AvN | |≤ k k k k k k 4 2 ν 2 8(εcˆ) c˜ AvN + β (θtω). ≤ 48 k k ν 1 Hence ν 2 νλ1 2 2 2 J3 AvN + vN 1 + cε β1(θtω)( vN + vL vL 1) | |≤ 8 k k 4 k k k k k k k k (5.12) 2 2 2 2 2 + c β (θtω)+ c vN vN + c vL vL vN . 5 1 6 k k k k1 6 k k k k1 k k1 4 11664(ˆcc˜) 11664ˆc4 where c5 = ν3 and c6 = ν3 .

Since f H, it can be deduced from the Young’s inequality with ǫ that ∈ ν 2 2 2 2 2 2 J4 AvN + f + 2cε y(θtω) + Ay(θtω) ≤ 8 k k ν k k k k k k (5.13) ν 2 2 AvN + c 1+ ε β (θtω) . ≤ 8 k k 1 Then by adding (5.8), (5.10)–(5.13), we have

d 2 ν 2 ψN (t,ω,ψ0N (ω)) 1 + AvN dtk kH 2 k k 4 2 2 2 δ0 + c5ε β1(θtω)+ c6 vN vN 1 ψN (t,ω,ψ0N (ω)) 1 ≤ − k k k k k kH 2 2 2 + c 1+ ε β (θtω)+ ε β (θtω)( vN + vL vL ) . 1 1 k k k k k k1 2 τ 1 Since (vL + vN ) L (T , ; H) L (T , ; V ) from (4.17) and ψ N = (0, 0) , an ∈ ∞ 3 ∞ ∩ 3 ∞ 0 ∈ H argument similar to the one used in Lemma 5.1 shows that there exists a T = T (B,ω) T 4 4 ≥ 3 such that for all t T , ≥ 4 2 ε ψN (t,θ tω, ψ0N (θ tω)) 1 c3(r3(ω) + 1), (5.14) k − − kH ≤ 20 and 2 ε φN (t,θ tω, φ0N (θ tω)) 1 c4(r4(ω) + 1), (5.15) k − − kH ≤ where

0 4 0 ε δ0s+c5ε Rs β1(θτ ω) dτ 2 2 ε r3(ω)= e 1+ ε β1(θsω)+ c1ε β1(θsω)(r1(ω) + 1) ds. Z −∞ ε ε It is easy to show that r3(ω) is a tempered random variable (Analogously to r1(ω)) and ε ε 2 r4(ω)= r3(ω)+ ε r(ω). This completes the proof.

c 1 ֒ We denote QR = x : x < R and Q = QR. Note that the embedding { ∈ O | | } R O \ H →H is not compact any more in unbounded domains, it is hard to prove the exsitence of uniqueness of the random attractor. Inspired by [7, 38, 39, 40], we obtain the far-ﬁeld values of solutions which can be applied to get the asymptotic compactness in unbounded domains.

Lemma 5.4. Suppose that B = B(ω) ω Ω and ψ0(ω) B(ω). Then for every ζ > 0 { } ∈ ∈ D ∈ and P-a.e. ω Ω, there exist a T = T (B,ω,ζ) and a R = R (ω,ζ) such that for all t T , ∈ 7 7 3 3 ≥ 7 2 ψ(t,θ tω, ψ0(θ tω)) (Qc ) ζ. (5.16) k − − kH R3 ≤ Bates, Lu and Wang [7] introduced a cutoff function in for the first step. Let ρ be a O smooth function defined on R+ such that 0 ρ(s) 1 for all s R+, and ≤ ≤ ∈ 0 0

~~ρ′(s) λ ρ(s) λ | |≤ 1 ≤ 1 for all s R+. The assumption here is diﬀerentp from thatp in [7] but reasonable because ρ is ∈ a smooth function. For our convenience, we denote~~

~~2 2 2 x u 2 = u ρ | | dx, u H, k kLρ | | k2 ∀ ∈ ZO where k is a large positive constant. Thus we have the following generalized Poincar´einequality.~~

~~Lemma 5.5. For all u V , we have ∈~~

~~λ1 2 2 u 2 u 2 , 4 k kLρ ≤ k∇ kLρ where λ1 is the constant in the Poincar´einequality.~~

~~21 1 x 2 1 x 2 2 2 Proof. First, we show that ρ | 2| u V , that is ρ | 2| u < . We know that k ∈ k V ∞ ~~

~~1 2 1 1 2 λ1 λ1 ρ 2 (s) = ρ− 2 (s)ρ′(s) ρ(s) . ∇ 4 ≤ 4 ≤ 4~~

~~x 2 2x ν vρ v x ′ | 2| 2 d k x √2k k k ·∇ Z ≤| |≤ 2 2 √2ν x v ρ′ | | v dx ≤ k | | k2 |∇ | Zk x √2k ≤| |≤~~

~~22 2√2λ ν x 2 1 ρ | | v v dx ≤ k k2 | ||∇ | ZO √2λ1ν 2 2 v + v 2 . ≤ k k k k∇ kLρ Therefore, we ﬁnd that~~

ν 2 νλ1 2 √2λ1ν 2 K1 v 2 + v 2 v , (5.18) ≥ 4 k∇ kLρ 8 k kLρ − k k k where we have chosen k R := 4√2λ . ≥ 0 1 Next, for the third term of the left-hand side of (5.17), it follows from the Young’s inequality with ǫ that

2 ∞ x K := µ(s) Aη ρ | | v dx ds 2 · k2 Z0 Z O 2 ∞ x = µ(s) Aη ρ | | (∂tη + ∂sη εy(θtω)) dx ds · k2 − Z0 Z O 2 2 1 d ∞ x 2 ∞ x = µ(s) ρ | | η dx ds + µ(s) ηρ | | ∂s η dx ds 2 dt k2 |∇ | ∇ k2 · ∇ Z0 Z Z0 Z O 2 O 2 ∞ x ∞ x 2x ε µ(s) ηρ | | y(θtω)dx ds + µ(s) vρ′ | | η dx ds − ∇ k2 ·∇ k2 k2 ·∇ Z0 Z Z0 Z O 2 O 2 1 d ∞ x 2 1 ∞ x 2 µ(s) ρ | 2| η dx ds µ′(s) ρ | 2| η dx ds ≥ 2 dt 0 k |∇ | − 2 0 k |∇ | Z ZO Z ZO 2 ∞ x √2λ1 ∞ 2 2 ε µ(s) ρ | | η y(θtω) dx ds µ(s) η + v dx ds − k2 |∇ ||∇ | − k |∇ | | | Z0 Z Z0 Z O 2 2 O 1 d ∞ x δ ∞ x µ(s) ρ | | η 2 dx ds + µ(s) ρ | | η 2 dx ds ≥ 2 dt k2 |∇ | 2 k2 |∇ | Z0 Z Z0 Z O 2 2 O 2 δ ∞ x 2 ε ∞ x 2 µ(s) ρ | 2| η dx ds µ(s) ρ | 2| y(θtω) dx ds − 4 0 k |∇ | − δ 0 k |∇ | Z ZO Z ZO √2λ √2λ κ 1 η 2 1 v 2 − k k kM − k k k 2 2 1 d ∞ x 2 δ ∞ x 2 = µ(s) ρ | 2| η dx ds + µ(s) ρ | 2| η dx ds 2 dt 0 k |∇ | 4 0 k |∇ | Z ZO Z ZO 2 ε κ 2 √2λ1 2 √2λ1κ 2 y(θtω) L2 η v . (5.19) − δ k∇ k ρ − k k kM − k k k From the deﬁnition and properties of the trilinear form b( , , ), we obtain · · · x 2 x 2 b(v, v, ρ | | v) = (v )v ρ | | v dx k2 ·∇ · k2 Z O (v )v v dx ≤ ·∇ · Z O = b(v, v, v) = 0 | | and x 2 b(εy, v, ρ | | v) = 0, k2

~~23 For the last term of the left-hand side of (5.17), we know that~~

~~x 2 K := b(v + εy, v + εy, ρ | | v) | 3| k2 x 2 x 2 b(v, εy, ρ | | v) + b(εy, εy, ρ | | v) ≤ k2 k2 ~~

~~=: L1 + L2.~~

~~It follows from the the Weighted H¨older’s inequality and Lemma 2.1 that~~

2 x 1 1 L1 = (v )(εy) ρ | | v dx cε A 2 v 2 A 2 y(θtω) 2 v 2 ·∇ · k2 ≤ k kLρ k kLρ k kLρ Z O 1 1 cε A 2 v 2 A 2 y(θtω) 2 v ≤ k kLρ k kLρ k k and 2 x 2 1 1 L2 = (εy )(εy) ρ | | v dx cε A 2 y(θtω) 2 y(θtω) 2 A 2 v 2 . ·∇ · k2 ≤ k kLρ k kLρ k kLρ Z O

Then we see that 2 ε 1 2 2 ν 2 4 1 2 2 K3 A 2 y(θtω) 2 v + v 2 + cε A 2 y(θtω) 2 y(θtω) 2 . (5.20) | |≤ 2 k kLρ k k 8 k∇ kLρ k kLρ k kLρ Finally, we derive from the Young’s inequality with ǫ that

x 2 K := (f(x)+ εσy(θtω) εAy(θtω)) ρ | | v dx 4 − · k2 ZO x 2 ρ | | f(x)+ εσy(θtω) εAy(θtω) v dx (5.21) ≤ k2 | − | | | ZO νλ1 2 4 2 2 2 v L2 + f L2 + εσ y(θtω) L2 + ε Ay(θtω) L2 . ≤ 16 k k ρ νλ1 k k ρ k k ρ k k ρ Thus, a combination of (5.17)–(5.21) implies that

d ν 2 H(t,ω)+ δ2H(t,ω)+ v 2 dt 4 k∇ kLρ 2√2λ1 2 2 1 2 2 (ν + κ) v + ε A 2 y(θtω) 2 v ≤ k k k k kLρ k k 2√2λ1 1 2 4 2 2 2 + η + 2cε A y(θtω) L2 y(θtω) L2 k k kM k k ρ k k ρ 2 8 2 2 ε κνλ1 1 2 2 + f 2 + εσ y(θtω) 2 + A 2 y(θtω) 2 + ε Ay(θtω) 2 , (5.22) νλ k kLρ k kLρ 2δ k kLρ k kLρ 1

~~νλ1 δ where δ2 = min 8 , 2 and n o 2 2 x 2 ∞ x 2 H(t,ω)= ρ | |2 v dx + µ(s) ρ | 2| η dx ds. (5.23) k | | 0 k |∇ | ZO Z ZO Rearranging inequality (5.22), one obtains that~~

~~d ν 2 c 2 2 H(t,ω)+ δ2H(t,ω)+ v L2 ψ + F (t,θtω) ψ + cG(t,θtω), (5.24) dt 4 k∇ k ρ ≤ k k kH k kH 24 where 2 1 2 F (t,θtω)= ε A 2 y(θtω) 2 , k kLρ and~~

2 2 1 2 2 1 4 G(t,θtω)= f 2 + y(θtω) 2 + A 2 y(θtω) 2 + Ay(θtω) 2 + A 2 y(θtω) 2 . k kLρ k kLρ k kLρ k kLρ k kLρ Multiplying both sides of (5.24) by eδ2t and integrating it over (T ,t) where T = max T , T , 5 5 { 3 4} we get that, for all t T , ≥ 5 δ2(T5 t) H(t,ω) e − H(T ,ω) ≤ 5 t c δ2(s t) 2 + e − ψ(s,ω,ψ0(ω)) ds k k kH ZT5 t δ2(s t) 2 + e − F (s,θsω) ψ(s,ω,ψ0(ω)) ds k kH ZT5 t δ2(s t) + c e − G(s,θsω)ds. ZT5 Replacing ω by θ tω, we see that, for all t T5, − ≥ δ2(T5 t) H(t,θ tω) e − H(T5,θ tω) − ≤ − t c δ2(s t) 2 + e − ψ(s,θ tω, ψ0(θ tω)) ds k k − − kH ZT5 t δ2(s t) 2 + e − F (s,θs tω) ψ(s,θ tω, ψ0(θ tω)) ds − k − − kH ZT5 t δ2(s t) + c e − G(s,θs tω)ds − ZT5 =: Y1 + Y2 + Y3 + Y4. (5.25)

~~Next, we will estimate each terms in (5.25). First, we have from (4.16) and (5.23) that~~

δ2(T5 t) 2 Y1 e − ψ(T5,θ tω, ψ0(θ tω)) ≤ k − − kH δ2(T5 t) ε e − c (r (ω) + 1). ≤ 1 1 Thus for every given ζ > 0, there is a T = T (B,ω,ζ) > T such that for all t T , 6 6 5 ≥ 6 ζ Y . (5.26) 1 ≤ 4 Since (4.16) and (5.14) holds for all t T , we get that, for all t T , ≥ 4 ≥ 5 0 c δ2s ε Y2 e [c1(r1(ω) + 1)] ds ≤ k T5 t Z − c ε δ2(T5 t) [c1(r1(ω) + 1)] 1 e − ≤ kδ2 − c (rε(ω) + 1). ≤ k 1 Hence, there is a R = R (ω,ζ) > 0 such that for all t T and k R R , 1 1 ≥ 5 ≥ 1 ≥ 0 ζ Y . (5.27) 2 ≤ 4 25 Note that f H and h(x) W , there is a R = R (ω,ζ) such that for all k R , ∈ ∈ 2 2 ≥ 2 2 2 x 2 2 δ2ζ f(x) 2 = ρ | | f(x) dx f(x) dx (5.28) Lρ 2 k k k | | ≤ x 2 k2 | | ≤ 8c ZO Z| | ≥ and 2 1 2 2 1 4 h˜ := h(x) 2 + A 2 h(x) 2 + Ah(x) 2 + A 2 h(x) 2 k kLρ k kLρ k kLρ k kLρ 2 x 1 1 = ρ | | h(x) 2 + A 2 h(x) 2 + Ah(x) 2 + A 2 h(x) 4 dx k2 | | | | | | | | ZO 1 1 (5.29) h(x) 2 + A 2 h(x) 2 + Ah(x) 2 + A 2 h(x) 4 dx ≤ x 2 k2 | | | | | | | | Z| | ≥ δ ζ δ ζ min 2 , 2 , ≤ 8ε2c r(ω)(rε(ω) + 1) 16cr(ω) 1 1 where c is the constant in Y4. Therefore, by (3.7), (4.16) and (5.29), Y3 is bounded by t 1 2 δ2(s t) 2 2 4 ε Y3 ε e − A h(x) L2 z(θs tω) c1(r1(ω)+1)ds ≤ k k ρ | − | ZT5 t δ2ζ δ2(s t) 4 e − z(θs tω) ds ≤ 8r(ω) | − | ZT5 0 δ2ζ δ2s 4 = e z(θsω) ds 8r(ω) T5 t | | Z − 0 δ2 δ2 δ2ζ s ζ (T5 t) e 2 r(ω)ds = 1 e 2 − . ≤ 8r(ω) T5 t 4 − Z − Then for all t T5, ≥ ζ Y . (5.30) 3 ≤ 4 Similarly, by (3.7), (4.16), (5.28) and (5.29), Y4 is bounded by t δ2(s t) 2 ˜ Y4 c e − f(x) L2 + hβ1(θs tω) ds ≤ k k ρ − ZT5 t h t i δ2ζ δ2(s t) δ2ζ δ2(s t) e − ds + e − β1(θs tω)ds ≤ 8 16r(ω) − ZT5 ZT5 0 0 δ2ζ δ2s δ2ζ δ2s = e ds + e β1(θsω)ds 8 T5 t 16r(ω) T5 t Z − Z − 0 0 δ2ζ δ s δ2ζ δ2 s e 2 ds + e 2 r(ω)ds ≤ 8 T5 t 16r(ω) T5 t Z − Z − δ2 ζ δ2(T5 t) ζ (T5 t) = 1 e − + 1 e 2 − . 8 − 8 − Therefore, for all t T5, ≥ ζ Y . (5.31) 4 ≤ 4 Let T = max T , T . It follows from (5.25)–(5.31) that, for all t T and k R , 7 { 5 6} ≥ 7 ≥ 2 H(t,θ tω) ζ. − ≤ Consequently, for all t T and k R , ≥ 7 ≥ 2 2 ψ(t,θ tω, ψ0(θ tω))(x) (Qc ) ζ. k − − kH √2k ≤

~~Taking R3 = √2R2, we obtain (5.16) which completes the proof. 26 By using (3.10), we have next lemma.~~

Lemma 5.6. Suppose that B = B(ω) ω Ω and φ0(ω) B(ω). Then for every ζ > 0 and { } ∈ ∈ D ∈ P-a.e. ω Ω, there exist a T = T (B,ω,ζ) > 0 assigned in Lemma 5.4 and a R = R (ω,ζ) ∈ 7 7 5 5 such that for all t T , ≥ 7 2 φ(t,θ tω, φ0(θ tω))(x) (Qc ) ζ. (5.32) k − − kH R5 ≤ Proof. It can be deduced from Lemma 5.4 and (3.10) that for all t T , ≥ 7 2 φ(t,θ tω, φ0(θ tω)) (Qc ) k − − kH R3 2 2 2 2 ψ(t,θ tω, ψ0(θ tω)) (Qc ) + 2ε y(ω) dx. R3 ≤ k − − kH x R3 | | Z| |≥ Indeed, since y(ω)= h(x)z(ω) and h(x) W , there is a R = R (ω,ζ) > 0 such that ∈ 4 4 2ε2 y(ω) 2 dx ζ. x R4 | | ≤ Z| |≥ Hence, there exists a R = max R ,R such that for all t T , 5 { 3 4} ≥ 7 2 φ(t,θ tω, φ0(θ tω)) (Qc ) 3ζ. k − − kH R5 ≤ The proof is complete.

5.2 –pullback asymptotic compactness D In this section, we prove the existence and uniqueness of -random attractor for the D random dynamical system Φ corresponding to the stochastic Navier-Stokes equation with memory in unbounded domains. To overcome the diﬃculty caused by the lack of compactness .[of 1 ֒ (see e.g., [32]), we construct a new compact subspace as in [26 M → M First, we give a lemma on producing a compact subspace . N ⊂ M Lemma 5.7. Denote by

t = η (θ tω, η0), N − η0 K(θ tω) t T7 ω Ω ∈ [− ≥[ [∈ where K(ω) ω Ω is deﬁned by (5.3) and T7 is deﬁned in Lemma 5.4. Then is relatively { } ∈ N compact in . M Proof. See [22, 26].

Next, we follow the procedure in [7] with Lemma 5.3, Lemma 5.6 and Lemma 5.7 to get the -pullback asymptotic compactness of Φ. D Lemma 5.8. The random dynamical system Φ is -pullback asymptotically compact in ( ); D H O P that is, for -a.e. ω Ω, the sequence Φ(tn,θ tn ω)φ0,n(θ tn ω) has a convergent subsequence ∈ { − − } in ( ) provided tn , B = B(ω) ω Ω and φ0,n(θ tn ω) B(θ tn ω). H O →∞ { } ∈ ⊂ D − ∈ −

~~27 Proof. For tn , B = B(ω) ω Ω and φ0,n(θ tn ω) B(θ tn ω), from Lemma 5.1, we →∞ { } ∈ ∈ D − ∈ − have that~~

Φ(tn,θ tn ω)φ0,n(θ tn ω) ∞ is bounded in ( ). { − − }n=1 H O Then there exists φˆ ( ) such that, up to a subsequence, ∈H O ˆ Φ(tn,θ tn ω)φ0,n(θ tn ω) φ weakly in ( ). (5.33) − − → H O In what follows, we prove that the weak convergence in (5.33) is actually strong convergence, for which we need to prove

ˆ 2 φ(tn,θ tn ω, φ0,n(θ tn ω)) φ ( ) ζ, k − − − kH O ≤ for a given ζ > 0. Lemma 5.6 implies that there exist a T7 = T7(B,ω,ζ) and a R5 = R5(ω,ζ), such that for all t T , ≥ 7 2 ζ φ(t,θ tω, φ0(θ tω)) (Qc ) . (5.34) k − − kH R5 ≤ 8 Since tn , there is a N = N (B,ω,ζ) N such that for all n N , tn T , it can be → ∞ 1 1 ∈ ≥ 1 ≥ 7 deduced from (5.34) that

2 ζ φ(tn,θ tn ω, φ0,n(θ tn ω)) (Qc ) . (5.35) k − − kH R5 ≤ 8 Notice that φˆ ( ) and indeed there exists a R = R (ζ) > 0 such that ∈H O 6 6 ˆ 2 ζ φ(x) (Qc ) . (5.36) k kH R6 ≤ 8 1 It remains to be done with the problem that (QR) ֒ (QR) is not compact. Let M → M R = max R ,R , by Lemma 5.7, we know that is compact in where is the closure 7 { 5 6} N M N of . Deﬁne := V ( 1 ) 1 , then it follows from (5.15) that for all t T , N H × M ∩ N ⊂H ⊂H ≥ 7 φ (t,θ ω, φ (θ ω)) 2 c (rε(ω) + 1). e N t 0N t ( ) 4 4 k − − kHe O ≤

Hence, there is a N = N (B,ω) N large enough such that for all n N , tn T , 2 2 ∈ ≥ 2 ≥ 7 φ (t ,θ ω, φ (θ ω)) 2 c (rε(ω) + 1). N n tn 0N,n tn ( ) 4 4 k − − kHe O ≤ 1 ֒ ( With compact embedding V (QR ) ֒ H(QR ) and ( (QR ) (QR )) (QR 7 → 7 M 7 ∩ N 7 ⊂ N 7 → (QR ), we know that (QR ) is compact in (QR ). Consequently, up to a subsequence, M 7 H 7 H 7 we get that e ˆ φN (tn,θ tn ω)φ0N,n(θ tn ω) φ strongly in (QR7 ), − − → H which indicates that for a given ζ > 0, there exists a N = N (B,ω,ζ) N such that for all 3 3 ∈ n N3, ≥ ζ φ (t ,θ ω, φ (θ ω)) φˆ 2 . (5.37) N n tn 0N,n tn (QR ) k − − − kH 7 ≤ 8 It can be deduced from the Lemma 5.1 that for a given ζ > 0, there are a N (B,ω,ζ) N 4 ∈ suﬃciently large and a T > T such that for n N , we have tn > T and 8 7 ≥ 4 8 ζ φ (t ,θ ω, φ (θ ω)) 2 . (5.38) L n tn 0L,n tn (QR ) k − − kH 7 ≤ 8 28 Set N = max N ,N ,N , it follows from (5.35) – (5.38) that for all n N , tn T , 5 { 1 3 4} ≥ 5 ≥ 8 φ(t ,θ ω, φ (θ ω)) φˆ 2 φ(t ,θ ω, φ (θ ω)) φˆ 2 n tn 0,n tn ( ) n tn 0,n tn (QR ) k − − − kH O ≤ k − − − kH 7 ˆ 2 + φ(tn,θ tn ω, φ0,n(θ tn ω)) φ (Qc ) k − − − kH R7 2 φ (t ,θ ω, φ (θ ω)) φˆ 2 N n tn 0N,n tn (QR ) ≤ k − − − kH 7 + 2 φ (t ,θ ω, φ (θ ω)) 2 L n tn 0L,n tn (QR ) k − − kH 7 2 + 2 φ(tn,θ tn ω, φ0,n(θ tn ω)) (Qc ) k − − kH R7 ˆ 2 + 2 φ (Qc ) k kH R7 ζ. ≤ This completes the proof.

Since (5.3) implies a closed random absorbing set K(ω) ω Ω for Φ, and Φ is -pullback { } ∈ D asymptotically compact in ( ) from Lemma 5.8, we immediately get the following theorem H O by Proposition 3.1.

~~Theorem 5.1. The random dynamical system Φ has a unique -random attractor in ( ). D H O 6 Upper semicontinuity of random attractors~~

In this section, we prove the upper semicontinuity of random attractors for Navier-Stokes equations with memory in unbounded domains by Proposition 3.2 with the constructed compact embedding and the unifom estimates above. When ε 0, the limiting deterministic → system of (3.9) is ∞ ut + νAu + µ(s)Aη(s)ds + B(u, u)= f, 0  Z (6.1)  ∂tη = T η + u,   0 u(0) = u0, η = η0.  Lemma 6.1. For a given 0 < ε 1, let ψε = (vε, ηε)τ and φ = (u, η)τ be the solution of ≤ ε ε ε τ τ (3.9) and (6.1) with initial conditions ψ0 = (v0, η0) and φ0 = (u0, η0) , respectively. Then for P-a.e. ω Ω and t T , we have ∈ ≥ 4 ε ε 2 ct ε 2 2 ct ε ε ψ (t,ω,ψ0) φ(t,ω,φ0) ce ψ0 φ0 + cε e r(ω)(1 + r1(ω)+ r3(ω)), k − kH ≤ k − kH where c> 0 is independent of ε.

~~Proof. Let ϕ = (w,ξ)τ = ψε φ. Since ψε, φ satisfy (3.9) and (6.1), respectively, we deduce − that~~

∞ ε ε wt + νAw + µ(s)Aξ(s)ds + B(v + εy, v + εy) B(u, u)= ε(σy νAy), − −  Z0 ∂ ξ Tξ w εy, (6.2)  t = + +  w(0) = w = vε u , ξ = ηε η . 0 0 − 0 0 0 − 0   29 Taking the inner product with (6.2) by ϕ in , we have H

1 d 2 2 1 2 1 ∞ 1 2 w + ξ + ν A 2 w = µ′(s) A 2 ξ ds + ε(ξ,y(θtω)) 2 dt k k k kM k k 2 0 k k M Z ε ε (6.3) + b(u, u, w) b(v + εy(θtω), v + εy(θtω), w) − + (ε (σy(θtω) νAy(θtω)) , w). − The same procedure as in Lemma 5.1 yields

~~d 2 2 ϕ(t,ω,ϕ0(ω)) (t,ω) ϕ(t,ω,ϕ0(ω)) + (t,ω), (6.4) dtk kH ≤ Q k kH R where~~

~~(t,ω) := δ + cε2 vε 2 , Q − 0 k∇ k 2 ε 2 ε 2 (t,ω) := cε β1(θtω) 1+ v + v . R k k k∇ k Applying Gronwall’s inequality, we obtain~~

t t t 2 R (τ,ω) dτ 2 R (τ,ω) dτ ϕ(t,ω,ϕ0(ω)) e 0 Q ϕ0(ω) + c e s Q (s,ω)ds. (6.5) k kH ≤ k kH R Z0 By means of (4.16), we have t (τ,ω)dτ ct. Q ≤ Z0 It follows from (3.7) and Lemma 5.3 that for all t T , ≥ 4

~~t t R (τ,ω) dτ 2 ct ε ε e s Q (s,ω)ds cε e r(ω)(1 + r (ω)+ r (ω)). R ≤ 1 3 Z0 This completes the proof.~~

~~ε ε ε ε ε Let φ be the solution of (3.3) with initial conditions φ0 = (u0, η0) and Φ be the corresponding cocycle. By (3.10), one obtains that~~

ε ε 2 ε ε 2 Φ (t,ω)φ0 φ(t,ω,φ0) = φ (t,ω,φ0) φ(t,ω,φ0) k − kH k − kH ε ε 2 = ψ (t,ω,ψ0) + (εy(θtω), 0) φ(t,ω,φ0) k − kH δ0 ε ε 2 2 2 t 2 ψ (t,ω,ψ0) φ(t,ω,φ0) + 2ε e r(ω). ≤ k − kH Then, we immediately derive the following lemma which satisﬁes the ﬁrst condition of Propo- sition 3.2.

Lemma 6.2. For a given 0 < ε 1, assume that conditions on Lemma 6.1 hold, then for ≤ P-a.e. ω Ω and t T , we have ∈ ≥ 4 ε ε 2 ct ε 2 2 ct ε ε φ (t,ω,φ0) φ(t,ω,φ0) ce φ0 φ0 + cε e r(ω)(1 + r1(ω)+ r3(ω)), k − kH ≤ k − kH where c> 0 is independent of ε.

~~30 Next, we prove the third condition in Proposition 3.2. Given 0 < ε 1, from the proof of ≤ Lemma 5.3, we can rewrite (5.3) as~~

2 ε Kε(ω)= φ ( ) : φ c2(r2(ω) + 1) , (6.6) ∈H O k kH ≤ that is, for every B = B(ω) ω Ω and P-a.e. ω Ω, there is a T3 > 0, independent of ε, { } ∈ ∈ D ∈ such that for all t T , ≥ 3 ε 2 ε Φ (t,θ tω)B(θ tω) c2(r2(ω) + 1). k − − kH ≤ We also denote 2 1 K(ω)= φ ( ) : φ c2(r2(ω) + 1) . (6.7) ∈H O k kH ≤ To obtain the compactness in the next lemma, we deﬁne two sets Kε(ω) = Kε(ω) and ∩ H K(ω)= K(ω) where is assigned in Lemma 5.8. Then Kε(ω) ω Ω is a closed absorbing ∩H H { } ∈ set for Φε in and e e e D e e e Kε(ω) K(ω). ⊆ 0<ε 1 [≤ e e It follows from the invariance of the random attractor ε(ω) ω Ω that {A } ∈

ε(ω) Kε(ω) K(ω). (6.8) A ⊆ ⊆ 0<ε 1 0<ε 1 [≤ [≤ ε e e ε ε ε From (5.4) and (5.5) we know that Φ can be written as Φ = ΦN +ΦL. So we denote ε ε ε(ω) by ε(ω)= Nε(ω) Lε(ω) where Nε(ω) and Lε(ω) are generated by Φ and Φ A A A ∪A A A N L respectively. Consequently, by Lemma 5.3, there exists T4 > 0, independent of ε, such that for all t T , ≥ 4 Φε (t,θ ω) (θ ω) 2 c (rε(ω) + 1) c (r1(ω) + 1). N t Nε t ( ) 4 4 4 4 k − A − kHe O ≤ ≤ ε According to the invariance of attractor, i.e. ε(ω)=Φ (t,θ tω) ε(θ tω) for all t 0, P-a.e. A − A − ≥ P ω Ω, we have that for every φN 0<ε 1 Nε(ω), -a.e. ω Ω, ∈ ∈ ≤ A ∈ S φ 2 c (r1(ω) + 1). (6.9) N ( ) 4 4 k kHe O ≤

Lemma 6.3. The union 0<ε 1 ε(ω) is precompact in ( ) for every ω Ω. ≤ A H O ∈ We will prove LemmaS 6.3 by the method in [39] with some modiﬁcations in which we use .our constructed compact embedding ֒ in a bounded ball from the proof of Lemma 5.8 H →H

Proof of Lemma 6.3. To show that 0e<ε 1 ε(ω) is precompact in ( ), that is, given ζ > 0, ≤ A H O 0<ε 1 ε(ω) has a ﬁnite covering of balls of radii less than ζ, we establish the far-ﬁeld ≤ A S Sestimate and the bounded ball estimate by the compact embedding we construct. Let Kε(ω) ω Ω be the random set given before. It follows from Lemma 5.6 that given { } ∈ P ε ζ > 0, -a.e. ω Ω, there exist T7 > 0 and R5 > 0, such that for all t T7 and φ0N (θ tω) e ∈ ≥ − ∈ K(θ tω), − 2 ε ε 2 ζ ΦN (t,θ tω)φ0N (θ tω) (Qc ) . e k − − kH R5 ≤ 64

31 ε ε Since (6.8) holds, φ0N (θ tω) Nε(θ tω) implies φ0N (θ tω) K(θ tω). Hence for every − ∈ A − − ∈ − P ε 0 < ε 1, -a.e. ω Ω, t T7 and φ0N (θ tω) Nε(θ tω), we have ≤ ∈ ≥ − ∈A − e 2 ε ε 2 ζ ΦN (t,θ tω)φ0N (θ tω) (Qc ) . k − − kH R5 ≤ 64

Due to the invariance of ε(ω) ω Ω, it is clear that for P-a.e. ω Ω and for all φN {A } ∈ ∈ ∈ 0<ε 1 Nε(ω), ≤ A 2 2 ζ S φN (x) (Qc ) , k kH R5 ≤ 64 which is equivalent to ζ φN (x) (Qc ) . (6.10) k kH R5 ≤ 8 P As (6.9) indicates that 0<ε 1 Nε(ω) is bounded in (QR5 ) for -a.e. ω Ω, by the ≤ A H ∈ (compactness of embedding (Q ) ֒ (Q ), we know that, for a given ζ, (ω S R5 R5 0<ε 1 Nε H →H ζ e ≤ A has finite covering of balls of radii less than 8 in (QR5 ). e H S On the other hand, Remark 5.1, (6.8) and the invariance of ε imply that given ζ > 0, A we have ζ φL(x) (Qc ) , k kH R5 ≤ 8 ζ which means that 0<ε 1 Lε(ω) has finite covering of balls of radii less than 8 in (QR5 ). ≤ A ζ H Then 0<ε 1 ε(ω) has finite covering of balls of radii less than 4 in (QR5 ). ≤ A S H In the end, 0<ε 1 ε(ω) has a finite covering of balls of radii less than ζ in ( ). S ≤ A H O S Theorem 6.1. For 0 < ε 1, the family of random attractors ε(ω) ω Ω is upper semi- ≤ {A } ∈ continuous at ε = 0, i.e., for P-a.e. ω Ω, ∈

lim dist( ε(ω), 0) = 0. ε 0 A A → ε Proof. We know that Kε(ω) ω Ω is a closed absorbing set for Φ in . From the deﬁnition { } ∈ D of Kε(ω), we ﬁnd that e 0 e lim sup Kε(ω) c2(r1(ω) + 1) ε 0 k k≤ → 0 e c eδ0s ds + 1 ≤ 2 (6.11) Z−∞ 1 c + 1 . ≤ 2 δ 0 By Lemma 6.2, we have that for P-a.e. ω Ω, and t T , ∈ ≥ 4

εn Φ (t,ω)φ ,n Φ(t)φ , (6.12) 0 → 0 provided εn 0 and φ ,n φ in ( ). Since (6.12), (6.11) and Lemma 6.3 verify three con- → 0 → 0 H O ditions in Proposition 3.2, respectively, we obtain the upper continuity of random attractors for Navier-Stokes equations with memory in unbounded domains.

~~32 Acknowledgements~~

The authors express their sincere thanks to Prof. Guangying Lv for his constructive comments and suggestions that helped to improve this paper. This work was supported by the National Natural Science Foundation of China [grant number 11771216], the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725], the Six Talent Peaks Project in Jiangsu Province [grant number 2015-XCL-020] and the Qing Lan Project of Jiangsu Province.

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