Asymptotic behavior of stochastic delay Navier-Stokes equations on unbounded domains

COMMUN. MATH. SCI. ©2025 International Press Vol. 23, No. 4, pp. 1139–1166 ASYMPTOTIC BEHAVIOR OF STOCHASTIC DELAY NAVIER-STOKES EQUATIONS ON UNBOUNDED DOMAINS∗ QIANGHENG ZHANG†, TOM´ AS CARABALLO‡,AND SHUANG YANG§ Abstract. In this paper, the random dynamics of non-autonomous stochastic Navier-Stokes equations with variable delays on unbounded Poincar´e domains is analysed. First, we establish the existence, uniqueness and backward compactness of pullback random attractors. Second, we study the upper semicontinuity of pullback random attractors as the delay time tends to zero. Finally, we investigate the backward asymptotic autonomy of pullback random attractors. Due to the non-compactness of Sobolev embeddings on unbounded domains, we introduce a stream function to prove backward uniform tailends smallest of solutions, and then establish the backward asymptotic compactness of the solution operators. Keywords. Navier-Stokes equations; Delay; Unbounded domain; Pullback random attractor; Stream function. AMS subject classifications. 35B40; 35B41; 37L55; 60H15. 1. Introduction In this paper, we consider the stability of pullback random attractors for the following stochastic non-autonomous Navier-Stokes equations with delays and multiplicative noise defined on unbounded Poincar´e domains:              du−(ν∆u−(u·∇)u−∇p)dt =(f(u(t−ρ(t)))+g(t,x))dt+u◦dW, ∇·u=0, x∈O, t > τ, u=0, x ∈∂O, t > τ, u(τ+ξ):=ψ(ξ), ξ ∈[−ϱ,0], τ ∈R, (1.1) where the two positive constants νand ϱstand for the kinematic viscosity and delay time of the fluid, respectively. u=(u1,u2) and pare the velocity field and pressure of the fluid, respectively. ρ(·) denotes the delay function. O ⊂R2is an unbounded Poincar´e domain with boundary ∂O, that is, there exists a positive constant λsuch that λZO |u|2dx≤ZO |∇u|2dx, ∀u∈H1 0(O).(1.2) fdenotes the delay forcing, which is Lipschitz continuous. gstands for the nonautonomous forcing. Wis a two-sided real-valued Wiener process on a probability space (Ω,F,P ), where Ω ={ω∈C(R,R): ω(0) =0},Fis the Borel σ-algebra induced by the compact-open topology of Ω and Pis the corresponding Wiener measure on (Ω,F). The symbol ◦denotes (1.1) is understood in the Stratonovich integration. ∗Received: June 05, 2024; Accepted (in revised form): December 02, 2024. Communicated by Alexis F. Vasseur. †School of Mathematics and Statistics, Heze University, Heze 274015, P.R. China (zqh math@ 126.com). ‡Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, C/Tarfia s/n, 41012-Sevilla, Spain; Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, P.R. China ([email protected]). §School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, P.R. China (shuang-y[email protected]). 1139 RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1140 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS Navier-Stokes equations are important models in fluid mechanics, which summarize the general law of viscous fluid flow. In past two decades, Navier-Stokes equations with delays have received much attention. Caraballo and Real [10,11] first considered the 2D Navier-Stokes equations with delays and proved the existence and uniqueness of solutions, and studied the convergence of solutions to the stationary one. Since then, the long-term behavior of 2D Navier-Stokes equations with delays has been widely studied (see [7,12,17–19,24–29,44] and the references therein). In [19], the authors established the existence and uniqueness of solutions as well as the exponential stability of stationary solutions for 2D Navier-Stokes equations with delays on unbounded Poincar´e domain. In [27], the authors used the energy equation method to prove the existence of pullback attractors for 2D Navier-Stokes equations with delays on unbounded Poincar´e domain. However, there is no result reported in the literature on the 2D stochastic Navier-Stokes equations with delays on unbounded Poincar´e domain. Motivated by [3,4,9,33,34,37, 38,44], we analyze the random dynamics of 2D stochastic Navier-Stokes equations with delays and non-autonomous forcing on unbounded Poincar´e domain. The first goal of this paper is to prove the existence and uniqueness of pullback random attractors Aϱ={Aϱ(τ,ω):τ∈R,ω ∈Ω}and that the backward compactness of Aϱ:Ss≤τAϱ(s,ω) is pre-compact for all τ∈R. As we know, the asymptotic compactness of solution operators is a key step in proving the existence of attractors. However, the Sobolev embedding is non-compact on unbounded domains. To overcome this difficulty, we usually use the energy equation method established in [1] and tail-ends estimates method initiated in [30]. In this work, we will use the backward uniform tailends estimates method. Note that Navier-Stokes equations are different from reactiondiffusion equations, and so we need to introduce a stream function to change the vector equation to a scalar equation. More precisely, let ηk(x) =η(|x|2 k2) for all x∈Ω, where η(·):[0,+∞)→[0,1] be a smooth function satisfying η(r)=(0,0≤r≤1, 1, r ≥4. If we multiply (1.1) by η2 ku, then we will encounter a new problem: the pressure term ROηku∇pdx. This is because the incompressible condition will be invalid, see Lemma 3.4 for more details. The second target of this paper is to consider the upper semicontinuity of pullback random attractors. It is easy to see that we can obtain different pullback random attractor for different delay time in (1.1). Then, a natural idea is to study the upper semicontinuity of pullback random attractors Aϱ={Aϱ(τ,ω): τ∈R,ω ∈Ω}: lim ϱ→0distϱ(Aϱ(τ,ω),A0(τ,ω))= 0,(1.3) where distϱ(A,B)=supa∈Ainfb∈Bsupξ∈[−ϱ,0] ∥a(ξ)−b∥2and A0={A0(τ,ω):τ∈R,ω ∈ Ω}is a pullback random attractor of limit equation for (1.1). Recently, the theoretical result of (1.3) was established in [36], and developed by [14,22,42] and the references therein. It is worth pointing out that we introduce an inverse function to overcome the difficulty caused by the variable delay. The last goal of this paper is concerned with the backward asymptotic autonomy of pullback random attractors Aϱ={Aϱ(τ,ω) :τ∈R,ω ∈Ω}: lim τ→−∞dist(Aϱ(τ,ω),A∞(ω))=0, RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1141 where dist(·,·) denotes the Hausdorff semi-distance and A∞={A∞(ω):ω∈Ω}is a random attractor of autonomous version of (1.1). The concept of pullback random attractors was introduced in [8,16,31,32], which is a family of sets parameterized by time tand sample ω. Then, an interesting problem is devoted to investigating their timedependent properties (see [5,15,20,21,23,34,35,39,40,42–44]). However, there is no paper on the asymptotic autonomy of pullback random attractors for (1.1). 2. Preliminaries In this paper, we identify W(t) with ω(t) for all t∈Ron the probability space (Ω,F,P ). Define a time shift θtby θtω(·)=ω(t+·)−ω(t) for all t∈Rand ω∈Ω, which along with (Ω,F,P ) forms a metric dynamical system (Ω,F,P,{θt}t∈R). We now establish a continuous random dynamical system for (1.1) over (Ω,F,P,{θt}t∈R). To this end, we first transform stochastic Equation (1.1) into a random equation. Let v(t,τ,ω,ϕ)=e−z(θtω)u(t,τ,ω,ψ),with ϕ(ξ)=e−z(θτ+ξω)ψ(ξ),∀ξ∈[−ϱ,0],(2.1) where z(θtω)=−R0 −∞ er(θtω)(r)dr for all t∈R, which is a solution of OrnsteinUhlenbeck equation: dz +zdt=dω(t). In addition, by [2, Proposition 5.1] there exists a θt-invariant subset Ω0⊂Ω (for convenience, still denoted as Ω) such that t7→z(θtω) is continuous and lim t→±∞ |z(θtω)| |t|=0,lim t→±∞ Rt 0z(θrω)dr t=0,∀ω∈Ω.(2.2) Substituting (2.1) into (1.1) yields the following random equation:                  ∂v ∂t −ν∆v+ez(θtω)(v·∇)v+e−z(θtω)∇p =e−z(θtω)(f(ez(θt−ρ(t)ω)v(t−ρ(t)))+g(t,x))+z(θtω)v, x ∈ O, t>τ, ∇·v=0, x ∈O, t > τ, v=0, x ∈∂O, t > τ, v(τ+ξ):=ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R, (2.3) where the delay function ρ(·) and delay term f(·,·) satisfy the following assumptions: (D) ρ(·)≥0 satisfies ρ(·)∈C1(R,R) and sup t∈R ρ(t)=ϱ > 0,sup t∈R d dtρ(t) =ρ∗<1.(2.4) (F) f(0)=0 and there exists a positive constant Lfsuch that |f(s1)−f(s2)|R2≤Lf|s1−s2|R2,for all s1,s2∈R2,(2.5) and ¯η:= νλ 2−2E(|z|)−Lf (1−ρ∗)1 2 eνλ 4ϱ(E(e2z(ω))+E(e−2z(ω)))>0,(2.6) where E(|z|) denotes the expectation of |z|. Let V={u∈(C∞ 0(O,R2))2:∇·u=0}. Suppose that Pis a Leray projection from L2(O) onto H, where His the closure of Vin L2(O,R2) with the inner product (·,·) and norm ∥·∥, defined by (u,v)=ZO uvdx, ∥u∥2= (u,u),for all u,v ∈H. RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1142 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS Let Vbe the closure of Vin H1 0(O,R2) with the inner product ((·,·)) and norm ∥·∥V, defined by ((u,v))= 2 X i,j=1ZO ∂ui ∂xj ∂vi ∂xj dx, ∥u∥2 V=((u,u)),for all u,v ∈V. We use V′to denote the dual space Vand ⟨·,·⟩ the duality pairing between Vand V′. Applying Pto (2.3), we obtain      ∂v ∂t +νAv +ez(θtω)B(v,v) =e−z(θtω)P(f(ez(θt−ρ(t)ω)v(t−ρ(t)))+g(t,x))+z(θtω)v, x ∈ O, t>τ, v(τ+ξ):=ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R, (2.7) where Av =−P∆vwith A:V→V′defined by ⟨Au,v⟩=((u,v)), and B(v,v)= P((v·∇)v) with B:V×V→V′and ⟨B(u,v),w⟩=b(u,v,w), where b(u,v,w)= 2 X i,j=1ZO ui ∂vj ∂xi wjdx, for all u,v,w ∈V, (2.8) which satisfies b(u,v,w)=−b(u,w,v) and b(u,v,v)=0.(2.9) Let CH=C([−ϱ,0];H) and CV=C([−ϱ,0];V) with norm ∥u∥CH= sup ξ∈[−ϱ,0] ∥u(ξ)∥and ∥u∥CV= sup ξ∈[−ϱ,0] ∥u(ξ)∥V. In [19], the authors proved the existence and uniqueness of solutions for 2D Navier-Stokes equations with the abstract delay term. Note that (2.7) is a deterministic equation parameterized by ω. Hence we need to show that the variable delay term fsatisfies the assumptions (I)-(IV) of the abstract delay term in [19]. By (D) and (F), it is easy to see that (I) and (II) hold. By (2.4) and (2.5) we obtain for all u,v ∈CH, ∥f(u(t−ρ(t)))−f(v(t−ρ(t)))∥2=ZO |f(u(t−ρ(t)))−f(v(t−ρ(t)))|2 R2dx ≤L2 f∥u(t−ρ(t))−v(t−ρ(t))∥2≤L2 f∥ut−vt∥2 CH, which implies that (III) holds. By (2.4) and (2.5) again, we obtain for all u,v ∈C([τ− h,t];H) and t≥τ, Zt τ ∥f(u(r−ρ(r)))−f(v(r−ρ(r)))∥2dr ≤L2 fZt τ ∥u(r−ρ(r))−v(r−ρ(r))∥2dr ≤L2 f 1−ρ∗Zt τ−ϱ ∥u(r)−v(r)∥2dr, which implies that (IV) holds. We remark that (2.7) is an abstract form of (2.3). Then we derive (2.3) has a unique solution v∈C([τ−ϱ,T ];H)∩L2(τ,T;V) for all T > τ when RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1143 ϕ∈CHand g∈L2 loc(R;(L2(O))2). We define a delay-shift vtby vt(ξ)=v(t+ξ) for all t≥τand ξ∈[−ϱ,0], and consider a mapping Φ:R+×R×Ω×CH→CHby Φ(t,τ,ω)ψ=ut+τ(·,τ,θ−τω,ψ)=vt+τ(·,τ,θ−τω,ϕ)ez(θt+·ω),(2.10) where vis the solution of (2.3), which satisfies v(τ+ξ,τ,ω,ϕ)=ϕ. In addition, v(t,τ,ω,ϕ) is (F,B(H))-measurable with respect to ω, where B(H) stands for the Borel σ-algebra of H. Then by (2.1) we imply the mapping Φ is a continuous non-autonomous random dynamical system associated with (2.1) in the sense of [31]. 3. Pullback random attractors In this section, we are concerned with the existence, uniqueness and backward compactness of pullback random attractors. For these purposes, we need to impose an assumption about the non-autonomous forcing g∈L2 loc(R;(L2(O))2), i.e., gis backward tempered: sup s≤τZs −∞ eκ(r−s)∥g(r)∥2dr :=G(τ)<+∞,∀τ∈R, κ > 0.(3.1) In order to treat the delay term, we now introduce a random variable η(ω), which is defined by η(ω)= νλ 2−2|z(ω)|− Lf (1−ρ∗)1 2 eνλ 4ϱ(e2z(ω)+e−2z(ω)).(3.2) Using the ergodic theorem [6, Theorem 2.1] to (3.2), we infer from (2.6) lim t→±∞ 1 tZt 0 η(θlω)dl = ¯η. (3.3) We consider two attraction universes Dand Bin CH, given by D={D={D(τ,ω):τ∈R,ω ∈Ω}: lim t→+∞e−κt∥D(τ−t,θ−tω)∥2 CH=0},(3.4) and B={B={B(τ,ω):τ∈R,ω ∈Ω}: lim t→+∞e−κt sup s≤τ ∥B(s−t,θ−tω)∥2 CH=0},(3.5) respectively. It is simple to imply that Dand Bare inclusion-closed, and B⊂D. Throughout this paper, we assume that the delay time ϱ∈(0,ϱ0] for some ϱ0>0. From now on, we assume that cand C(ω) are two generic positive constants, where cis independent of ϱand ω, and C(ω) is uniform with respect to ϱ. 3.1. Backward uniform estimates of solutions. Lemma 3.1. Suppose that (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backward tempered. Then we obtain the following conclusions: (i) For each τ∈R,ω∈Ωand D={D(τ,ω):τ∈R,ω ∈Ω}∈D, there exists a Td:= Td(τ,ω,D)≥3ϱ+3 such that sup ξ∈[−3ϱ−3,0] ∥u(τ+ξ,τ −t,θ−τω,ψ)∥2≤cecϱ(1+Rd(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω),(3.6) RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1144 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS for all t≥Tdand ψ∈D(τ−t,θ−tω), where Rd(τ,ω)=Z0 −∞ eRr 0η(θlω)dle−2z(θrω)∥g(r+τ)∥2dr. (3.7) (ii) For each τ∈R,ω∈Ωand B={B(τ,ω): τ∈R,ω ∈Ω}∈B, there exists a Tb:= Tb(τ,ω,B)≥3ϱ+3 such that sup s≤τ sup ξ∈[−3ϱ−3,0] ∥u(s+ξ,s−t,θ−sω,ψ)∥2≤cecϱ(1+Rb(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω),(3.8) for all t≥Tband ψ∈B(s−t,θ−tω), where Rb(τ,ω)=sup s≤τ Rd(s,ω).(3.9) Proof. Taking the inner product of (2.3) with v(r):=v(r,s−t,θ−sω,ϕ) in H, by the second equality of (2.9) and the incompressible condition we obtain 1 2 d dr ∥v(r)∥2+ν∥∇v(r)∥2 =e−z(θr−sω)(f(ez(θr−ρ(r)−sω)v(r−ρ(r)))+g(r),v(r))+z(θr−sω)∥v(r)∥2. Using the Young inequality and (2.5), we obtain e−z(θr−sω)(f(ez(θr−ρ(r)−sω)v(r−ρ(r)))+g(r),v(r)) ≤Lf(1−ρ∗)1 2 2eνλ 4ϱe2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2+Lfeνλ 4ϱ 2(1−ρ∗)1 2 e−2z(θr−sω)∥v(r)∥2 +νλ 4∥v(r)∥2+1 νλe−2z(θr−sω)∥g(r)∥2. By (1.2) we obtain ν∥∇v(r)∥2≥νλ 2∥v(r)∥2+ν 2∥∇v(r)∥2. Then we have d dr ∥v(r)∥2+ν∥∇v(r)∥2 ≤Lf(1−ρ∗)1 2 eνλ 4ϱe2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2+Lfeνλ 4ϱ (1−ρ∗)1 2 e−2z(θr−sω)∥v(r)∥2 +2 νλe−2z(θr−sω)∥g(r)∥2+(2z(θr−sω)−νλ 2)∥v(r)∥2.(3.10) Multiplying (3.10) by eRr s−tη(θl−sω)dl, and then integrating this result over [s−t,s+ξ] with ξ∈[−3ϱ−3,0] and t≥3ϱ+3 yields eRs+ξ s−tη(θl−sω)dl∥v(s+ξ)∥2 ≤∥ϕ∥2 CH+Lf(1−ρ∗)1 2 eνλ 4ϱZs+ξ s−t eRr s−tη(θl−sω)dle2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2dr RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1145 +Lfeνλ 4ϱ (1−ρ∗)1 2Zs+ξ s−t eRr s−tη(θl−sω)dle−2z(θr−sω)∥v(r)∥2dr +2 νλ Zs+ξ s−t eRr s−tη(θl−sω)dle−2z(θr−sω)∥g(r)∥2dr +Zs+ξ s−t (2z(θr−sω)−νλ 2+η(θr−sω))eRr s−tη(θl−sω)dl∥v(r)∥2dr. (3.11) It follows from (2.4) and (3.2) that Lf(1−ρ∗)1 2 eνλ 4ϱZs+ξ s−t eRr s−tη(θl−sω)dle2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2dr ≤Lf(1−ρ∗)1 2eνλ 4ϱZs+ξ s−t eRr−ρ(r) s−tη(θl−sω)dle2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2dr ≤Lfeνλ 4ϱ (1−ρ∗)1 2Zs+ξ s−t−ϱ eRr s−tη(θl−sω)dle2z(θr−sω)∥v(r)∥2dr =Lfeνλ 4ϱ (1−ρ∗)1 2 (Zs−t s−t−ϱ +Zs+ξ s−t )eRr s−tη(θl−sω)dle2z(θr−sω)∥v(r)∥2dr ≤Lfeνλ 4ϱ (1−ρ∗)1 2 ∥ϕ∥2 CHZs−t s−t−ϱ eRr s−tη(θl−sω)dle2z(θr−sω)dr +Lfeνλ 4ϱ (1−ρ∗)1 2Zs+ξ s−t eRr s−tη(θl−sω)dle2z(θr−sω)∥v(r)∥2dr. (3.12) Then by (3.2) (the definition of η(ω)) we obtain (2z(θr−sω)−νλ 2+η(θr−sω))+ Lfeνλ 4ϱ (1−ρ∗)1 2 (e−2z(θr−sω)+e2z(θr−sω)) =2z(θr−sω)−2|z(θr−sω)|≤0. Then by (3.11) and (3.12) we have ∥v(s+ξ)∥2≤ceR0 ξη(θlω)dl∥ϕ∥2 CH(e−R0 −tη(θlω)dl +eνλ 4ϱZ−t −t−ϱ eRr 0η(θlω)dl+2z(θrω)dr) +ceR0 ξη(θlω)dl Z0 −t eRr 0η(θlω)dle−2z(θrω)∥g(r+s)∥2dr ≤ce3νλ 2ϱe3νλ 2∥ϕ∥2 CH(e−R0 −tη(θlω)dl +eνλ 4ϱZ−t −t−ϱ eRr 0η(θlω)dl+2z(θrω)dr) +ce3νλ 2ϱe3νλ 2Z0 −t eRr 0η(θlω)dle−2z(θrω)∥g(r+s)∥2dr ≤ce2νλϱ∥ϕ∥2 CH(e−R0 −tη(θlω)dl +Z−t −t−ϱ eRr 0η(θlω)dl+2z(θrω)dr) +ce2νλϱ Z0 −t eRr 0η(θlω)dle−2z(θrω)∥g(r+s)∥2dr, (3.13) RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1146 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS where we used η(ω)≤νλ 2in (3.2). From (2.2) and (3.3), there exists a ˜ T:= ˜ T(¯η,ω)≥ 3ϱ+3 such that for all |t|≥ ˜ T, |z(θtω)|≤ ¯η 16|t|and Zt 0 (η(θlω)−¯η)dl ≤¯η 2|t|.(3.14) Then we have for all t≥˜ Tand ξ∈[−3ϱ−3,0], ∥v(s+ξ)∥2≤ce2νλϱ(e−¯η 2t+4 ¯ηe−¯ 3η 8t)∥ϕ∥2 CH+ce2νλϱ Z0 −∞ eRr 0η(θlω)dle−2z(θrω)∥g(r+τ)∥2dr. Hence by (2.1) we obtain sup ξ∈[−3ϱ−3,0] ∥u(s+ξ)∥2 = sup ξ∈[−3ϱ−3,0] e2z(θξω)∥v(s+ξ)∥2 ≤ce2νλϱ sup ξ∈[−3ϱ−3,0] e2z(θξω)e−2z(θ−t+ξω)(e−¯η 2t+4 ¯ηe−3¯η 8t)∥ψ∥2 CH +ce2νλϱ sup ξ∈[−3ϱ−3,0] e2z(θξω)Z0 −∞ eRr 0η(θlω)dle−2z(θrω)∥g(r+τ)∥2dr. (3.15) (i) When s=τ. If ψ∈D(τ−t,θ−tω), by (3.4) there exists a Td:=Td(τ,ω,D)≥˜ T such that for all t≥Td, ce2νλϱe−2z(θ−t+ξω)(e−¯η 2t+4 ¯ηe−3¯η 8t)∥ψ∥2 CH ≤ce(2νλ+3¯η 8)ϱe¯η 8t(e−¯η 2t+4 ¯ηe−¯η 4t)∥D(τ−t,θ−tω)∥2 CH≤ce(2νλ+¯η)ϱ, which implies that (3.6) holds. (ii) If ψ∈B(s−t,θ−tω) with s≤τ, by (3.5) there exists a Tb:=Tb(τ,ω,B)≥˜ Tsuch that for all t≥Td, ce2νλϱe−2z(θ−t+ξω)(e−¯η 2t+4 ¯ηe−¯η 4t)sup s≤τ ∥ψ∥2 CH ≤ce(2νλ+3¯η 8)ϱe¯η 8t(e−¯η 2t+4 ¯ηe−¯η 4t)sup s≤τ ∥B(τ−t,θ−tω)∥2 CH≤ce(2νλ+¯η)ϱ, which, along with (3.15), shows that (3.8) holds. The proof is complete. Lemma 3.2. Suppose that (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backward tempered. For each τ∈R,ω∈Ωand B={B(τ,ω) :τ∈R,ω ∈Ω}∈ B, we have sup s≤τ sup ξ∈[−2ϱ−2,0] ∥∇u(s+ξ,s−t,θ−sω,ψ)∥2≤C(ω)¯ Rb(τ,ω)eC(ω)¯ R2 b(τ,ω),(3.16) for all t≥Tb(Tbis given by Lemma 3.1) and ψ∈B(s−t,θ−tω)with s≤τ, where ¯ Rb(τ,ω)=1+Rb(τ,ω)+G(τ). Proof. Taking the inner product of (2.7) with Av(r) in H, by the incompressible condition we obtain 1 2 d dr ∥∇v(r)∥2+ν∥Av(r)∥2+ez(θr−s)b(v(r),v(r),Av(r)) RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1147 =e−z(θr−sω)(f(ez(θr−ρ(r)−sω)v(r−ρ(r))),Av(r)) e−z(θr−sω)(g(r,·),Av(r))+z(θr−sω)∥∇v(r)∥2. By the Young inequality and (2.5) we obtain e−z(θr−sω)(f(ez(θr−ρ(r)−sω)v(r−ρ(r))),Av(r)) ≤ν 6∥Av(r)∥2+ce−2z(θr−sω)e2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2, and e−z(θr−sω)(g(r,·),Av(r)) ≤ν 6∥Av(r)∥2+ce−2z(θr−sω)∥g(r)∥2. It follows from (2.8) and the Gagliardo-Nirenberg inequality that −ez(θr−s)b(v(r),v(r),Av(r))≤ez(θr−s)∥v(r)∥L4(O;R2)∥∇v(r)∥L4(O;R4)∥Av(r)∥ ≤ez(θr−s)∥v(r)∥1 2∥∇v(r)∥∥Av(r)∥3 2 ≤fracν6∥Av(r)∥2+ce4z(θr−sω)∥v(r)∥2∥∇v(r)∥4. Hence we have d dr ∥∇v(r)∥2+ν∥Av(r)∥2≤ce−2z(θr−sω)e2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2 +ce−2z(θr−sω)∥g(r)∥2+ce4z(θr−sω)∥v(r)∥2∥∇v(r)∥4.(3.17) Using the uniform Gronwall inequality to (3.17) we obtain for all ξ∈[−2ϱ−2,0], ∥∇v(s+ξ)∥2 ≤ceRs+ξ s+ξ−1e4z(θr−sω)∥v(r)∥2∥∇v(r)∥2dr Zs+ξ s+ξ−1 e−2z(θr−sω)e2z(θr−ρ(r)−sω)∥v(r−ρ(r))∥2dr +ceRs+ξ s+ξ−1e4z(θr−sω)∥v(r)∥2∥∇v(r)∥2dr Zs+ξ s+ξ−1 (e−2z(θr−sω)∥g(r)∥2+∥∇v(r)∥2)dr ≤C(ω)eC(ω)Rs s−2ϱ−3∥v(r)∥2∥∇v(r)∥2dr Zs s−2ϱ−3 (∥v(r−ρ(r))∥2+∥g(r)∥2+∥∇v(r)∥2)dr, which, along with (2.1), implies ∥∇u(s+ξ)∥2 ≤C(ω)eC(ω)Rs s−2ϱ−3∥u(r)∥2∥∇u(r)∥2dr Zs s−2ϱ−3 (∥u(r−ρ(r))∥2+∥g(r)∥2+∥∇u(r)∥2)dr. To show that ∥∇u(s+ξ)∥2is bounded for all s≤τ, we need prove Rs s−2ϱ−3∥∇v(r)∥2dr is bounded. To this end, integrating (3.10) over [s−2ϱ−3,s] by (2.1) we obtain sup s≤τZs s−2ϱ−3 ∥∇u(r)∥2dr ≤C(ω)sup s≤τ ∥u(s−2ϱ−3)∥2 +C(ω)sup s≤τZs s−2ϱ−3 (∥u(r−ρ(r))∥2+∥u(r)∥2+∥g(r)∥2)dr. RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1154 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS 3.3. Existence, uniqueness and backward compactness of pullback random attractors. We now establish the existence, uniqueness and backward compactness of pullback random attractors for (1.1). For the purpose of this subsection, we show that the Φ in (2.10) has a pullback random absorbing set. Theorem 3.1. Suppose that (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backward tempered and backward tail-small. Then we obtain the following results: (i) Φ has a D-pullback random absorbing set Kd={Kd(τ,ω) : τ∈R,ω ∈Ω}∈ D, defined by Kd(τ,ω)={ϕ∈CH:∥ϕ∥2 CH≤cecϱ(1+Rd(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω)}.(3.50) (ii) Φ has a B-pullback absorbing set Kb={Kb(τ,ω):τ∈R,ω ∈Ω}∈B, defined by Kb(τ,ω)={ϕ∈CH:∥ϕ∥2 CH≤cecϱ(1+Rb(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω)},(3.51) where Rd(τ,ω)and Rb(τ,ω)are given in (3.7)and (3.9), respectively. Proof. The proof is similar to [43, Lemma 3.3], and so is omitted. We remark that the measurability of the B-pullback absorbing set Kbis unknown, but we can show that the B-pullback attractor is measurable (see Theorem 3.4). We first prove the B-pullback asymptotic compactness of Φ in (2.10). Theorem 3.2. Suppose that (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backward tempered and backward tail-small. Then Φin (2.10)is backward Bpullback asymptotically compact, that is, for each τ∈R,ω∈Ωand B∈B, the sequence {Φ(tn,sn−tn,θ−tnω)ψn}n∈Nis pre-compact in CH, whenever tn→+∞,sn≤τ and ψn∈B(sn−tn,θ−tnω). Proof. Based on the Ascoli-Arzel`a theorem, the proof is split into two steps. Step 1. For each ξ∈[−ϱ,0], we prove {(Φ(tn,sn−tn,θ−tnω)ψn)(ξ)}n∈Nhas a convergent subsequence in H. Since tn→+∞as n→+∞, we assume that tn≥Tbfor all n∈N. From (3.8), we obtain {(Φ(tn,sn−tn,θ−tnω)ψn)(ξ)}n∈Nis bounded in H, and so {(Φ(tn,sn− tn,θ−tn)ψn)(ξ)}n∈Nhas a weakly convergent subsequence (not relabelled), that is, there exists a ˜v∈Hsuch that (Φ(tn,sn−tn,θ−tnω)ψn)(ξ)= u(sn+ξ,sn−tn,θ−snω,ψn)→˜uweakly in H. (3.52) We now show that the weak convergence of (3.52) is strong. Since ˜u∈H, for any ε>0 there exists a ˜ K:= ˜ K(ε)>0 such that Z|x|≥ ˜ K |˜u|2dx < ε 5.(3.53) Let ¯ K=max{K, ˜ K}, where Kis given in Lemma 3.4. From (3.35), we have Z|x|≥ ¯ K |u(sn+ξ,sn−tn,θ−snω,ψn)|2dx< ε 5.(3.54) RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1155 By (3.16), {(Φ(tn,sn−tn,θ−tnω)ψn)(ξ)}n∈Nis bounded in V, and so it is bounded in H1(O¯ K), where O¯ K={x:|x|<¯ K}. Hence {(Φ(tn,sn−tn,θ−tnω)ψn)(ξ)}n∈Nis precompact in L2(O¯ K), that is, there exists a N > 0 such that for all n≥N ∥u(sn+ξ,sn−tn,θ−snω,ψn)−˜u∥2 L2(O¯ K)<ε 5.(3.55) It follows from (3.53)-(3.55) that ∥u(sn+ξ,sn−tn,θ−snω,ψn)−˜u∥2 =Z|k|≥ ¯ K |v(sn+ξ,sn−tn,θ−snω,ψn)−˜u|2dx +∥u(sn+ξ,sn−tn,θ−snω,ψn)−˜u∥2 L2(O¯ K) ≤2Z|k|≥ ¯ K |u(sn+ξ,sn−tn,θ−snω,ψn)|2dx+2Z|k|≥ ¯ K |˜u|2dx +∥u(sn+ξ,sn−tn,θ−snω,ψn)−˜u∥2 L2(O¯ K)<ε, which implies the weak convergence of (3.52) is strong convergence. The proof of Step 1is complete. Step 2. We prove {Φ(tn,sn−tn,θ−tnω)ψn}n∈Nin CHis equi-continuous from [−ϱ,0] to H. For each ξ1,ξ2∈[−ϱ,0] with ξ1< ξ2, by (3.21) we deduce |(Φ(tn,sn−tn,θ−tnω)ψn)(ξ1)−(Φ(tn,sn−tn,θ−tnω)ψn)(ξ2)∥ =∥u(sn+ξ1,sn−tn,θ−snω,ψn)−u(sn+ξ2,sn−tn,θ−snω,ψn)∥ ≤ez(θξ1ω)∥v(sn+ξ1,sn−tn,θ−snω,ϕn)−v(sn+ξ2,sn−tn,θ−snω,ϕn)∥ +|ez(θξ1ω)−ez(θξ2ω)|∥v(sn+ξ2,sn−tn,θ−snω,ϕn)∥ ≤C(ω)Zsn+ξ2 sn+ξ1 ∥∂ ∂r v(r,sn−tn,θ−snω,ϕn)∥dr +C(ω)ecϱ(1+Rb(τ,ω))|ez(θξ1ω)−ez(θξ2ω)| ≤C(ω)Zsn sn−ϱ ∥∂ ∂r v(r,sn−tn,θ−snω,ϕn)∥2dr1 2 |ξ1−ξ2|1 2 +C(ω)ecϱ(1+Rb(τ,ω))|ez(θξ1ω)−ez(θξ2ω)| ≤C(ω)¯ R3 b(τ,ω)eC(ω)¯ R2 b(τ,ω)|ξ1−ξ2|1 2 +C(ω)ecϱ(1+Rb(τ,ω))|ez(θξ1ω)−ez(θξ2ω)|, which implies that {Φ(tn,sn−tn,θ−tnω)ψn}n∈Nis equi-continuous. The proof of Step 1is complete. We obtain that all conditions of the Ascoli-Arzel`a theorem are fulfilled, and so the proof is complete. We then show the D-pullback asymptotic compactness of Φ in (2.10). Theorem 3.3. Suppose that (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is tempered and tail-small. Then Φin (2.10)is D-pullback asymptotically compact, that is, for each τ∈R,ω∈Ωand D∈D, the sequence {Φ(tn,τ −tn,θ−tnω)ψn}n∈Nis pre-compact in CH, whenever tn→+∞and ψn∈D(τ−tn,θ−tnω). RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1156 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS Proof. The asymptotic compactness has been proved for the family Bin Theorem 3.2, this result also holds for the family D. The proof is similar to that of Theorem 3.2, and we omit the details. Now, we obtain the main result of this section. Theorem 3.4. Suppose that (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backward tempered and backward tail-small. Then we obtain the following results: (i) Φ has a unique D-pullback random attractor Ad={Ad(τ,ω):τ∈R,ω ∈Ω}∈D, defined by Ad(τ,ω)= \ T≥0[ t≥T Φ(t,τ −t,θ−tω)Kd(τ−t,θ−tω). (ii) Φ has a unique B-pullback attractor Ab={Ab(τ,ω):τ∈R,ω ∈Ω}∈B, defined by Ab(τ,ω)= \ T≥0[ t≥T Φ(t,τ −t,θ−tω)Kb(τ−t,θ−tω). In addition, Abis backward compact, that is, Ss≤τAb(s,ω)is compact in H. (iii) Ad=Ab, and so Abis measurable and Adis backward compact. Proof. The proof is similar to [43, Theorem 3.10], and so is omitted. 4. Upper semicontinuity of pullback random attractor In this section, we consider the upper semicontinuity of the pullback random attractor as the delay time tends to zero. For this purpose, we introduce the limit equation of (1.1):              d¯u−(ν∆¯u−(¯u·∇)¯u−∇p)dt =(f(¯u(t))+g(t,x))dt+ ¯u◦dW, x∈O, t > τ, ∇· ¯u= 0, x∈O, t >τ, ¯u=0, x ∈∂O, t > τ, ¯u(τ):= ¯ ψ, τ ∈R. (4.1) Using (2.1) to (4.1), we obtain                  ∂¯v ∂t −ν∆¯v+ez(θtω)(¯v·∇)¯v+e−z(θtω)∇p =e−z(θtω)(f(ez(θtω)¯v(t))+g(t,x))+z(θtω)¯v, x ∈O, t > τ, ∇· ¯v= 0, x∈O, t >τ, ¯v= 0, x ∈∂O, t > τ, ¯v(τ) := ¯ ϕ, τ ∈R. (4.2) By the same method as in Section 3, we obtain that ¯ Φ generated by (4.2) has a unique D0-pullback random attractor A0={A0(τ,ω) :τ∈R,ω ∈Ω}∈ D0, and a D0-pullback random absorbing set K0={K0(τ,ω):τ∈R,ω ∈Ω}∈D0, where D0={D0={D0(τ,ω):τ∈R,ω ∈Ω}: lim t→+∞e−κt∥D0(τ−t,θ−tω)∥2=0}, RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1157 and K0(τ,ω)={ϕ∈CH:∥ϕ∥2 CH≤cecϱ0(1+Rd(τ,ω)) sup ξ∈[−3ϱ0−3,0] e2z(θξω)}.(4.3) We rewrite Kdand Ad(Kdis given by (3.50) and Adis given by Theorem 3.4) to Kϱ and Aϱ. Since the delay time ϱ∈(0,ϱ0], by (3.50) and (4.3) we have limsup ϱ→0 ∥Kϱ(τ,ω)∥2 CH=∥K0(τ,ω)∥2,∀τ∈R, ω ∈Ω.(4.4) We now consider the pointwise convergence of Aϱ: Lemma 4.1. Suppose that {ϕn}n∈N⊂ Aϱn(τ,ω)with ϱn→0, then there exist a subsequence {ϕnk}k∈Nof {ϕn}n∈Nand ϕ∈Hsuch that lim k→+∞sup ξ∈[−ϱnk,0] ∥ϕnk(ξ)−ϕ∥=0.(4.5) Proof. The proof is similar to [42, Lemma 5.2], and so is omitted. Next, we consider the convergence of solutions for the delay system to those of the non-delay one. Lemma 4.2. Suppose that ϕϱ∈CHand ϕ0∈Hsatisfy lim ϱ→0sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥=0,(4.6) then the solution vϱof (2.3)and the solution v0of (4.2)associated with the initial data ϕϱand ϕ0satisfy lim ϱ→0sup ξ∈[−ϱ,0] ∥vϱ(t+τ+ξ,τ,θ−τω,ϕϱ)−v0(t+τ,τ,θ−τω,ϕ0)∥=0,(4.7) for all t≥0,τ∈Rand ω∈Ω. Moreover, we have lim ϱ→0sup ξ∈[−ϱ,0] ∥uϱ(t+τ+ξ,τ,θ−τω,ψϱ)−u0(t+τ,τ,θ−τω,ψ0)∥=0.(4.8) Proof. For each τ∈R, we define ¯ Vϱ(r)=vϱ(r+ξ,τ,θ−τω,ϕϱ)−v0(r,τ,θ−τω,ϕ0),∀r≥τ. From (2.3) and (4.2), we obtain ∂¯ Vϱ(r+τ) ∂r −ν∆¯ Vϱ(r+τ)+ez(θr+ξω)(vϱ(r+τ+ξ)·∇)vϱ(r+τ+ξ) −ez(θrω)(v0(r+τ)·∇)v0(r+τ)+e−z(θr+ξω)∇pϱ−e−z(θrω)∇p0 =e−z(θr+ξω)(f(ez(θr+ξ−ρ(r+τ+ξ)ω)vϱ(r+τ+ξ−ρ(r+τ+ξ)))+g(r+τ+ξ,x)) −e−z(θrω)(f(ez(θr−ρ(r+τ)ω)v0(r+τ))+g(r+τ,x)) +z(θr+ξω)vϱ(r+τ+ξ)−z(θrω)v0(r+τ). Multiplying the above equality by ¯ Vϱ(r+τ), and then integrating this result over O, we deduce 1 2 d dr ∥¯ Vϱ(r+τ)∥2+ν∥∇ ¯ Vϱ(r+τ)∥2 RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1158 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS +(ez(θr+ξω)(vϱ(r+τ+ξ)·∇)vϱ(r+τ+ξ),¯ Vϱ(r+τ)) −(ez(θrω)(v0(r+τ)·∇)v0(r+τ),¯ Vϱ(r+τ)) =(e−z(θr+ξω)f(ez(θr+ξ−ρ(r+τ+ξ)ω)vϱ(r+τ+ξ−ρ(r+τ+ξ))),¯ Vϱ(r+τ)) −(e−z(θrω)f(ez(θr−ρ(r+τ)ω)v0(r+τ)),¯ Vϱ(r+τ)) +(e−z(θr+ξω)g(r+τ+ξ,·)−e−z(θrω)g(r+τ,·),¯ Vϱ(r+τ)) +(z(θr+ξω)vϱ(r+τ+ξ)−z(θrω)v0(r+τ),¯ Vϱ(r+τ)).(4.9) It follows from (2.8) and the Gagliardo-Nirenberg inequality that −(ez(θr+ξω)(vϱ(r+τ+ξ)·∇)vϱ(r+τ+ξ),¯ Vϱ(r+τ)) +(ez(θrω)(v0(r+τ)·∇)v0(r+τ),¯ Vϱ(r+τ)) =−ez(θr+ξω)((vϱ(r+τ+ξ)·∇)vϱ(r+τ+ξ),¯ Vϱ(r+τ)) −((v0(r+τ)·∇)v0(r+τ),¯ Vϱ(r+τ)) −(ez(θr+ξω)−ez(θrω))((v0(r+τ)·∇)v0(r+τ),¯ Vϱ(r+τ)) =−ez(θr+ξω)b(¯ Vϱ(r+τ),v0(r+τ),¯ Vϱ(r+τ)) +(ez(θr+ξω)−ez(θrω))b(v0(r+τ),¯ Vϱ(r+τ),v0(r+τ)) ≤ez(θr+ξω)∥¯ Vϱ(r+τ)∥2 L4∥∇v0(r+τ)∥ +|ez(θr+ξω)−ez(θrω)|∥v0(r+τ)∥2 L4∥∇ ¯ Vϱ(r+τ)∥ ≤ez(θr+ξω)∥¯ Vϱ(r+τ)∥∥∇ ¯ Vϱ(r+τ)∥∥∇v0(r+τ)∥ +|ez(θr+ξω)−ez(θrω)|∥v0(r+τ)∥∥∇v0(r+τ)∥∥∇ ¯ Vϱ(r+τ)∥ ≤ν 4∥∇ ¯ Vϱ(r+τ)∥2+cez(θr+ξω)∥¯ Vϱ(r+τ)∥2∥∇v0(r+τ)∥2 +c|ez(θr+ξω)−ez(θrω)|2∥v0(r+τ)∥2∥∇v0(r+τ)∥2.(4.10) By (2.5) we derive (e−z(θr+ξω)f(ez(θr+ξ−ρ(r+τ+ξ)ω)vϱ(r+τ+ξ−ρ(r+τ+ξ))),¯ Vϱ(r+τ)) −(e−z(θrω)f(ez(θr−ρ(r+τ)ω)v0(r+τ)),¯ Vϱ(r+τ)) ≤ν 4∥∇ ¯ Vϱ(r+τ)∥2 +ce−2z(θr+ξω)e2z(θr+ξ−ρ(r+τ+ξ)ω)∥vϱ(r+τ+ξ−ρ(r+τ+ξ))−v0(r+τ)∥2 +c(ez(θr+ξ−ρ(r+τ+ξ)ω)−ez(θr−ρ(r+τ)ω))2∥v0(r+τ)∥2 +c(e−z(θr+ξω)−e−z(θrω))2e2z(θr−ρ(r+τ)ω)∥v0(r+τ)∥2.(4.11) The Young inequality implies (e−z(θr+ξω)g(r+τ+ξ,·)−e−z(θrω)g(r+τ,·),¯ Vϱ(r+τ)) +(z(θr+ξω)vϱ(r+τ+ξ)−z(θrω)v0(r+τ),¯ Vϱ(r+τ)) ≤ν 4∥∇ ¯ Vϱ(r+τ)∥2+ce−2z(θr+ξω)∥g(r+τ+ξ)−g(r+τ)∥2 +c(e−z(θr+ξω)−e−z(θrω))2∥g(r+τ)∥2+|z(θr+ξω)|∥ ¯ Vϱ(r+τ)∥2 +c(z(θr+ξω)−z(θrω))2∥v0(r+τ)∥2.(4.12) RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1159 Substituting (4.10)-(4.12) into (4.9), and then integrating this result over [−ξ,t] with t∈[−ξ,T] and T > ϱ0we show ∥¯ Vϱ(t+τ)∥2≤ ∥ ¯ Vϱ(−ξ+τ)∥2+C(ω)Zt −ξ (1+∥∇v0(r+τ)∥2)∥¯ Vϱ(r+τ)∥2dr +C(ω)Zt −ξ ∥vϱ(r+τ+ξ−ρ(r+τ+ξ))−v0(r+τ)∥2dr +C(ω)ZT 0 (ez(θr+ξω)−ez(θrω))2∥v0(r+τ)∥2∥∇v0(r+τ)∥2dr +C(ω)ZT 0 (ez(θr+ξ−ρ(r+τ+ξ)ω)−ez(θr−ρ(r+τ)ω))2∥v0(r+τ)∥2dr +C(ω)ZT 0 ((e−z(θr+ξω)−e−z(θrω))2+(z(θr+ξω)−z(θrω))2)∥v0(r+τ)∥2dr +C(ω)ZT 0 ∥g(r+τ+ξ)−g(r+τ)∥2dr +C(ω)ZT 0 (e−z(θr+ξω)−e−z(θrω))2∥g(r+τ)∥2dr. (4.13) We now mainly treat the second line of (4.13). Let s=y(r)=r+ξ−ρ(r+ξ) for any r∈Rand fixed ξ∈[−ϱ,0]. Since y′(r)≥1−ρ∗>0, it has an inverse function such that r=y−1(s) for any s∈R. Then, Zt −ξ ∥vϱ(r+τ+ξ−ρ(r+τ+ξ))−v0(r+τ)∥2dr = Zy−1(τ) τ−ξ +Zt+τ y−1(τ)!∥vϱ(r+ξ−ρ(r+ξ))−v0(r)∥2dr =Zy−1(τ) τ−ξ ∥vϱ(r+ξ−ρ(r+ξ))−v0(r)∥2dr +Zt+τ−ρ(t+τ+ξ) τ−ξ ∥vϱ(r+ξ)−v0(y−1(r+ξ))∥2 1−d dr ρ(y−1(r+ξ)+ξ)dr ≤2Zy−1(τ) τ−ξ ∥vϱ(r+ξ−ρ(r+ξ))−ϕ0∥2dr +2Zy−1(τ) τ−ξ ∥v0(r)−ϕ0∥2dr +2 1−ρ∗Zτ+t τ−ξ ∥vϱ(r+ξ)−v0(r)∥2dr +2 1−ρ∗Zτ+t τ−ξ ∥v0(y−1(r+ξ))−v0(r)∥2dr ≤2 1−ρ∗Zτ τ−ρ(τ) ∥vϱ(r)−ϕ0∥2dr+2Zy−1(τ) τ−ξ ∥v0(r)−ϕ0∥2dr +2 1−ρ∗Zτ+t τ−ξ ∥vϱ(r+ξ)−v0(r)∥2dr +2 1−ρ∗Zτ+t τ−ξ ∥v0(y−1(r+ξ))−v0(r)∥2dr. RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1160 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS Note that Zτ τ−ρ(τ) ∥vϱ(r)−ϕ0∥2dr ≤cϱ sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2, and Zτ+t τ−ξ ∥vϱ(r+ξ)−v0(r)∥2dr =Zt −ξ ∥¯ Vϱ(r+τ)∥2dr. Then we imply Zt −ξ ∥vϱ(r+τ+ξ−ρ(r+τ+ξ))−v0(r+τ)∥2dr ≤cϱ sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2+cZy−1(τ) τ−ξ ∥v0(r)−ϕ0∥2dr +cZt −ξ ∥¯ Vϱ(r+τ)∥2dr+cZτ+t τ−ξ ∥v0(y−1(r+ξ))−v0(r)∥2dr. (4.14) By the same method as in Lemma 3.1 and Lemma 3.2 we obtain that ∥v0(r+τ)∥2and ∥∇v0(r+τ)∥2are bounded when r∈[0,T ]. Then, from (4.13) and (4.14) we derive ∥¯ Vϱ(t+τ)∥2 ≤∥ ¯ Vϱ(−ξ+τ)∥2+C(ω)Zt −ξ ∥¯ Vϱ(r+τ)∥2dr+C(ω) sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2 +C(ω)Zy−1(τ) τ−ξ ∥v0(r)−ϕ0∥2dr+C(ω)Zτ+t τ−ξ ∥v0(y−1(r+ξ))−v0(r)∥2dr +C(ω)ZT 0 ((ez(θr+ξω)−ez(θrω))2+(ez(θr+ξ−ρ(r+τ+ξ)ω)−ez(θr−ρ(r+τ)ω))2)dr +C(ω)ZT 0 ((e−z(θr+ξω)−e−z(θrω))2+(z(θr+ξω)−z(θrω))2)dr +C(ω)ZT 0 ∥g(r+τ+ξ)−g(r+τ)∥2dr +C(ω)ZT 0 (e−z(θr+ξω)−e−z(θrω))2∥g(r+τ)∥2dr. (4.15) By (4.6) and the continuity of v0at τwe obtain ∥¯ Vϱ ξ(−ξ+τ)∥2=∥vϱ(τ,τ,θ−τω,ϕϱ)−v0(−ξ+τ,τ,θ−τω,ϕ0)∥2 ≤2∥ϕϱ(0)−ϕ0∥2+2∥ϕ0−v0(−ξ+τ,τ,θ−τω,ϕ0)∥2 ≤2 sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2+2∥ϕ0−v0(−ξ+τ,τ,θ−τω,ϕ0)∥2→0 as ϱ→0.(4.16) By the continuity of v0at τwe show C(ω)Zy−1(τ) τ−ξ ∥v0(r)−ϕ0∥2dr+C(ω)Zτ+t τ−ξ ∥v0(y−1(r+ξ))−v0(r)∥2dr →0 (4.17) RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1161 as ϱ→0, where we use y−1(τ)≤τ+2ϱ. Using the continuity of r→z(θrω) yields C(ω)ZT 0 ((ez(θr+ξω)−ez(θrω))2+(ez(θr+ξ−ρ(r+τ+ξ)ω)−ez(θr−ρ(r+τ)ω))2)dr +C(ω)ZT 0 ((e−z(θr+ξω)−e−z(θrω))2+(z(θr+ξω)−z(θrω))2)dr →0 as ϱ→0.(4.18) From g∈L2 loc(R,(L2(O))2), we have C(ω)ZT 0 ∥g(r+τ+ξ)−g(r+τ)∥2dr +C(ω)ZT 0 (e−z(θr+ξω)−e−z(θrω))2∥g(r+τ)∥2dr →0 as ϱ→0.(4.19) Inserting (4.16)-(4.19) to (4.15), for any ε>0, there exists a ¯ϱ∈(0,ϱ0] such that for all ϱ< ¯ϱ, ∥¯ Vϱ(t+τ)∥2≤C(ω)ε+C(ω)Zt −ξ ∥¯ Vϱ(r+τ)∥2dr. (4.20) Using the Gronwall inequality (see [13, page 167]) to (4.20), we obtain for all ϱ< ¯ϱand t∈[−ξ,T] with ξ∈[−ϱ,0], ∥¯ Vϱ(t+τ)∥2<C(ω)eC(ω)Tε. (4.21) If t∈[0,−ξ], we have τ+ξ≤t+τ+ξ≤τ. Hence we deduce ∥vϱ(t+τ+ξ)−v0(t+τ)∥2≤2∥vϱ(t+τ+ξ)−ϕ0∥2+2∥v0(t+τ)−ϕ0∥2 ≤2 sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2 V+2∥v0(t+τ)−ϕ0∥2→0,as ϱ→0.(4.22) It follows from (4.21) and (4.22) that lim ϱ→0∥¯ Vϱ(t+τ)∥2=0,∀t∈[0,T ], which implies that (4.7) holds. By (2.1) we have ∥uϱ(t+τ+ξ,τ,θ−τω,ψϱ)−u0(t+τ,τ,θ−τω,ψ0)∥ ≤ez(θt+ξω)∥vϱ(t+τ+ξ,τ,θ−τω,ψϱ)−v0(t+τ,τ,θ−τω,ψ0)∥ +|ez(θt+ξω)−ez(θtω)|∥u0(t+τ,τ,θ−τω,ψ0)∥, which, along with (5.7) and the continuity of z(θ·ω), implies that (4.8) holds. The proof is complete. It follows from (4.4), Lemma 4.1 and Lemma 4.2 that all conditions of [36, Theorem 2.1] are satisfied. Then we obtain the main result of this section: Theorem 4.1. Suppose that (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backward tempered and backward tail-small. Then we have lim ϱ→0distϱ(Aϱ(τ,ω),A0(τ,ω))= 0,for all τ∈R, ω ∈Ω. RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1162 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS 5. Asymptotic autonomy of pullback random attractor Let ρ(t)=ϱin (1.1) for all t∈R, then we have              du−(ν∆u−(u·∇)u−∇p)dt =(f(u(t−ϱ))+g(t,x))dt+u◦dW, ∇·u=0, x ∈O, t > τ, u=0, x ∈∂O, t > τ, u(τ+ξ)=uτ(ξ):= ψ(ξ), ξ ∈[−ϱ,0], τ ∈R. (5.1) Using (2.1) to (5.1) we obtain the following random equation:                  ∂v ∂t −ν∆v+ez(θtω)(v·∇)v+e−z(θtω)∇p =e−z(θtω)(f(ez(θt−ϱω)v(t−ϱ))+g(t,x))+z(θtω)v, x ∈ O, t>τ, ∇·v=0, x ∈O, t > τ, v=0, x ∈∂O, t > τ, v(τ+ξ)=vτ(ξ):= ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R. (5.2) Using the same method as in Section 3, we obtain that the non-autonomous random dynamical systems Φ generated by (5.1) have a unique backward compact D-pullback random attractor Aϱ={Aϱ(τ,ω):τ∈R,ω ∈Ω}, and a D-pullback absorbing set K= {K(τ,ω): τ∈R,ω ∈Ω}, defined by K(τ,ω)={ϕ∈CH:∥ϕ∥2 CH≤cecϱ(1+Rb(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω)},(5.3) where Rb(τ,ω) is given by (3.9). We now introduce the autonomous version of (5.1):              d˜u−(ν∆˜u−(˜u·∇)˜u−∇p)dt =(f(˜u(t−ϱ))+g∞(x))dt+ ˜u◦dW, ∇· ˜u= 0, x∈O, t >0, ˜u=0, x ∈∂O, t > 0, ˜u(0+ξ) = ˜u0(ξ):= ˜ ψ(ξ), ξ ∈[−ϱ,0], (5.4) where the forcing g∞satisfies lim τ→−∞Zτ −∞ ∥g(r)−g∞∥2dr =0.(5.5) Using (2.1) to (5.4), we obtain                  ∂˜v ∂t −ν∆˜v+ez(θtω)(v·∇)v+e−z(θtω)∇˜p =e−z(θtω)(f(ez(θt−ϱω)˜v(t−ϱ))+g∞(x))+z(θtω)˜v, x∈O, t > τ, ∇· ˜v= 0, x∈O, t >τ, ˜v= 0, x ∈∂O, t > τ, ˜v(0+ξ)= ˜v0(ξ):= ˜ ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R. (5.6) Similar to Theorem 3.4, we obtain that the autonomous random dynamical system Φ∞ associated with (5.4) has a unique D∞-random attractor A∞={A∞(ω):ω∈Ω} ∈D∞ with D∞being the collection of all tempered families in CH, more precisely, D∞={D∞={D∞(ω):ω∈Ω}: lim t→+∞e−κt∥D∞(θ−tω)∥2 CH=0}.(5.7) RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1163 In this section, we consider the asymptotic autonomy of Aϱ: lim τ→−∞distCH(Aϱ(τ,ω),A∞(ω))=0,∀ω∈Ω.(5.8) Let Kbu ={Kbu(ω):ω∈Ω}with Kbu(ω)=Ss≤0K(s,ω) for all ω∈Ω. We now prove Kbu ∈ D∞. Let δ1=min{¯η 6,κ 8}, by (2.2) and (3.3) there exists a ˜ T:= ˜ T(δ1,ω)>0 such that e−2z(θtω)≤eδ1|t|,Zt 0 (η(θrω)−¯η)dr ≤δ1|t|,for all |t|≥ ˜ T. Hence, we obtain for all t≥˜ Tand r≤0 eRr 0η(θl−tω)dl+2|z(θr−tω)|=eRr−t −tη(θlω)dl+2|z(θr−tω)| ≤eRr−t 0(η(θlω)−¯η)dl+¯γ(r−t)−R−t 0(η(θlω)−¯η)dl+¯ηt+2|z(θr−tω)| ≤e|Rr−t 0(η(θlω)−¯η)dl|+|R−t 0(η(θlω)−¯η)dl|+¯ηr+δ1(t−r) ≤e3δ1te(¯η−2δ1)r≤e3δ1te2δ1r. By (3.9) and (5.3) we imply Kis closed and increasing. Then we have e−κt∥Kbu(θ−tω)∥2 CH =e−κt∥[ s≤0 K(s,θ−tω)∥2 CH=e−κt∥K(0,θ−tω)∥2 CH ≤ce−κteνλϱ(1+Rb(0,θ−tω)) sup ξ∈[−3ϱ−3,0] e2z(θ−t+ξω) =ce−(κ−2δ1)te(νλ+2δ1)ϱ(1+sup s≤0 Rd(s,θ−tω)) sup ξ∈[−3ϱ−3,0] e2z(θ−t+ξω) ≤ce−(κ−2δ1)te(νλ+2δ1)ϱ(1+sup s≤0Z0 −∞ eRr 0η(θl−tω)dle−2z(θr−tω)∥g(r+s)∥2dr) ≤ce−(κ−5δ1)te(νλ+2δ1)ϱ(1+sup s≤0Z0 −∞ e2δ1r∥g(r+s)∥2dr)→0,as t→+∞. Hence we have Kbu ∈D∞. Next, we prove the convergence of solutions for the nonautonomous system to those of the autonomous one. Lemma 5.1. Suppose that vτ,˜v0∈CHsatisfy lim τ→−∞∥vτ−˜v0∥2 CH=0,(5.9) then we have lim τ→−∞∥ut+τ(·,τ,θ−τω,uτ)−˜ut(·,ω, ˜u0)∥2 CH=0,(5.10) for all t≥0and ω∈Ω. Proof. For each τ∈R, let ˜ Vτ(r)=v(r+τ,τ,θ−τω,vτ)−˜v(r,ω,˜v0),∀r≥ −ϱ. RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466Nrwhro7OTAxeHNErkltXQJ6BsgJl3zskmMXfdqykiwc3rxnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL