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

COMMUN. MATH. SCI. ©2025 In e na ional P ess 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§ Abs ac . In his pape , he andom dynamics o non-au onomous s ochas ic Na ie -S okes equa- ions wi h a iable delays on unbounded Poinca ´e domains is analysed. Fi s , we es ablish he exis ence, uniqueness and backwa d compac ness o pullback andom a ac o s. Second, we s udy he uppe semi- con inui y o pullback andom a ac o s as he delay ime ends o ze o. Finally, we in es iga e he backwa d asymp o ic au onomy o pullback andom a ac o s. Due o he non-compac ness o Sobole embeddings on unbounded domains, we in oduce a s eam unc ion o p o e backwa d uni o m ail- ends smalles o solu ions, and hen es ablish he backwa d asymp o ic compac ness o he solu ion ope a o s. Keywo ds. Na ie -S okes equa ions; Delay; Unbounded domain; Pullback andom a ac o ; S eam unc ion. AMS subjec classi ica ions. 35B40; 35B41; 37L55; 60H15. 1. In oduc ion In his pape , we conside he s abili y o pullback andom a ac o s o he ollow- ing s ochas ic non-au onomous Na ie -S okes equa ions wi h delays and mul iplica i e noise de ined on unbounded Poinca ´e domains:              du−(ν∆u−(u·∇)u−∇p)d =( (u( −ρ( )))+g( ,x))d +u◦dW, ∇·u=0, x∈O, > τ, u=0, x ∈∂O, > τ, u(τ+ξ):=ψ(ξ), ξ ∈[−ϱ,0], τ ∈R, (1.1) whe e he wo posi i e cons an s νand ϱs and o he kinema ic iscosi y and delay ime o he luid, espec i ely. u=(u1,u2) and pa e he eloci y ield and p essu e o he luid, espec i ely. ρ(·) deno es he delay unc ion. O ⊂R2is an unbounded Poinca ´e domain wi h bounda y ∂O, ha is, he e exis s a posi i e cons an λsuch ha λZO |u|2dx≤ZO |∇u|2dx, ∀u∈H1 0(O).(1.2) deno es he delay o cing, which is Lipschi z con inuous. gs ands o he non- au onomous o cing. Wis a wo-sided eal- alued Wiene p ocess on a p obabili y space (Ω,F,P ), whe e Ω ={ω∈C(R,R): ω(0) =0},Fis he Bo el σ-algeb a induced by he compac -open opology o Ω and Pis he co esponding Wiene measu e on (Ω,F). The symbol ◦deno es (1.1) is unde s ood in he S a ono ich in eg a ion. ∗Recei ed: June 05, 2024; Accep ed (in e ised o m): Decembe 02, 2024. Communica ed by Alexis F. Vasseu . †School o Ma hema ics and S a is ics, Heze Uni e si y, Heze 274015, P.R. China (zqh ma h@ 126.com). ‡Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Facul ad de Ma em´a icas, Uni e sidad de Se illa, C/Ta ia s/n, 41012-Se illa, Spain; Depa men o Ma hema ics, Wenzhou Uni e si y, Wen- zhou, Zhejiang P o ince, 325035, P.R. China ([email p o ec ed]). §School o Ma hema ics and S a is ics, Huazhong Uni e si y o Science and Technology, Wuhan 430074, P.R. China (shuang-y[email p o ec ed]). 1139 RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1140 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS Na ie -S okes equa ions a e impo an models in luid mechanics, which summa ize he gene al law o iscous luid low. In pas wo decades, Na ie -S okes equa ions wi h delays ha e ecei ed much a en ion. Ca aballo and Real [10,11] i s conside ed he 2D Na ie -S okes equa ions wi h delays and p o ed he exis ence and uniqueness o solu ions, and s udied he con e gence o solu ions o he s a iona y one. Since hen, he long- e m beha io o 2D Na ie -S okes equa ions wi h delays has been widely s udied (see [7,12,17–19,24–29,44] and he e e ences he ein). In [19], he au ho s es ablished he exis ence and uniqueness o solu ions as well as he exponen ial s abili y o s a iona y solu ions o 2D Na ie -S okes equa ions wi h delays on unbounded Poinca ´e domain. In [27], he au ho s used he ene gy equa ion me hod o p o e he exis ence o pullback a ac o s o 2D Na ie -S okes equa ions wi h delays on unbounded Poinca ´e domain. Howe e , he e is no esul epo ed in he li e a u e on he 2D s ochas ic Na ie -S okes equa ions wi h delays on unbounded Poinca ´e domain. Mo i a ed by [3,4,9,33,34,37, 38,44], we analyze he andom dynamics o 2D s ochas ic Na ie -S okes equa ions wi h delays and non-au onomous o cing on unbounded Poinca ´e domain. The i s goal o his pape is o p o e he exis ence and uniqueness o pullback andom a ac o s Aϱ={Aϱ(τ,ω):τ∈R,ω ∈Ω}and ha he backwa d compac ness o Aϱ:Ss≤τAϱ(s,ω) is p e-compac o all τ∈R. As we know, he asymp o ic compac - ness o solu ion ope a o s is a key s ep in p o ing he exis ence o a ac o s. How- e e , he Sobole embedding is non-compac on unbounded domains. To o e come his di icul y, we usually use he ene gy equa ion me hod es ablished in [1] and ail-ends es ima es me hod ini ia ed in [30]. In his wo k, we will use he backwa d uni o m ail- ends es ima es me hod. No e ha Na ie -S okes equa ions a e di e en om eac ion- di usion equa ions, and so we need o in oduce a s eam unc ion o change he ec o equa ion o a scala equa ion. Mo e p ecisely, le ηk(x) =η(|x|2 k2) o all x∈Ω, whe e η(·):[0,+∞)→[0,1] be a smoo h unc ion sa is ying η( )=(0,0≤ ≤1, 1, ≥4. I we mul iply (1.1) by η2 ku, hen we will encoun e a new p oblem: he p essu e e m ROηku∇pdx. This is because he incomp essible condi ion will be in alid, see Lemma 3.4 o mo e de ails. The second a ge o his pape is o conside he uppe semicon inui y o pullback andom a ac o s. I is easy o see ha we can ob ain di e en pullback andom a ac o o di e en delay ime in (1.1). Then, a na u al idea is o s udy he uppe semicon inui y o pullback andom a ac o s Aϱ={Aϱ(τ,ω): τ∈R,ω ∈Ω}: lim ϱ→0dis ϱ(Aϱ(τ,ω),A0(τ,ω))= 0,(1.3) whe e dis ϱ(A,B)=supa∈Ain b∈Bsupξ∈[−ϱ,0] ∥a(ξ)−b∥2and A0={A0(τ,ω):τ∈R,ω ∈ Ω}is a pullback andom a ac o o limi equa ion o (1.1). Recen ly, he heo e ical esul o (1.3) was es ablished in [36], and de eloped by [14,22,42] and he e e ences he ein. I is wo h poin ing ou ha we in oduce an in e se unc ion o o e come he di icul y caused by he a iable delay. The las goal o his pape is conce ned wi h he backwa d asymp o ic au onomy o pullback andom a ac o s Aϱ={Aϱ(τ,ω) :τ∈R,ω ∈Ω}: lim τ→−∞dis (Aϱ(τ,ω),A∞(ω))=0, RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1141 whe e dis (·,·) deno es he Hausdo semi-dis ance and A∞={A∞(ω):ω∈Ω}is a an- dom a ac o o au onomous e sion o (1.1). The concep o pullback andom a ac- o s was in oduced in [8,16,31,32], which is a amily o se s pa ame e ized by ime and sample ω. Then, an in e es ing p oblem is de o ed o in es iga ing hei ime- dependen p ope ies (see [5,15,20,21,23,34,35,39,40,42–44]). Howe e , he e is no pape on he asymp o ic au onomy o pullback andom a ac o s o (1.1). 2. P elimina ies In his pape , we iden i y W( ) wi h ω( ) o all ∈Ron he p obabili y space (Ω,F,P ). De ine a ime shi θ by θ ω(·)=ω( +·)−ω( ) o all ∈Rand ω∈Ω, which along wi h (Ω,F,P ) o ms a me ic dynamical sys em (Ω,F,P,{θ } ∈R). We now es ab- lish a con inuous andom dynamical sys em o (1.1) o e (Ω,F,P,{θ } ∈R). To his end, we i s ans o m s ochas ic Equa ion (1.1) in o a andom equa ion. Le ( ,τ,ω,ϕ)=e−z(θ ω)u( ,τ,ω,ψ),wi h ϕ(ξ)=e−z(θτ+ξω)ψ(ξ),∀ξ∈[−ϱ,0],(2.1) whe e z(θ ω)=−R0 −∞ e (θ ω)( )d o all ∈R, which is a solu ion o O ns ein- Uhlenbeck equa ion: dz +zd =dω( ). In addi ion, by [2, P oposi ion 5.1] he e exis s a θ -in a ian subse Ω0⊂Ω ( o con enience, s ill deno ed as Ω) such ha 7→z(θ ω) is con inuous and lim →±∞ |z(θ ω)| | |=0,lim →±∞ R 0z(θ ω)d =0,∀ω∈Ω.(2.2) Subs i u ing (2.1) in o (1.1) yields he ollowing andom equa ion:                  ∂ ∂ −ν∆ +ez(θ ω)( ·∇) +e−z(θ ω)∇p =e−z(θ ω)( (ez(θ −ρ( )ω) ( −ρ( )))+g( ,x))+z(θ ω) , x ∈ O, >τ, ∇· =0, x ∈O, > τ, =0, x ∈∂O, > τ, (τ+ξ):=ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R, (2.3) whe e he delay unc ion ρ(·) and delay e m (·,·) sa is y he ollowing assump ions: (D) ρ(·)≥0 sa is ies ρ(·)∈C1(R,R) and sup ∈R ρ( )=ϱ > 0,sup ∈R d d ρ( ) =ρ∗<1.(2.4) (F) (0)=0 and he e exis s a posi i e cons an L such ha | (s1)− (s2)|R2≤L |s1−s2|R2, o all s1,s2∈R2,(2.5) and ¯η:= νλ 2−2E(|z|)−L (1−ρ∗)1 2 eνλ 4ϱ(E(e2z(ω))+E(e−2z(ω)))>0,(2.6) whe e E(|z|) deno es he expec a ion o |z|. Le V={u∈(C∞ 0(O,R2))2:∇·u=0}. Suppose ha Pis a Le ay p ojec ion om L2(O) on o H, whe e His he closu e o Vin L2(O,R2) wi h he inne p oduc (·,·) and no m ∥·∥, de ined by (u, )=ZO u dx, ∥u∥2= (u,u), o all u, ∈H. RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1142 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS Le Vbe he closu e o Vin H1 0(O,R2) wi h he inne p oduc ((·,·)) and no m ∥·∥V, de ined by ((u, ))= 2 X i,j=1ZO ∂ui ∂xj ∂ i ∂xj dx, ∥u∥2 V=((u,u)), o all u, ∈V. We use V′ o deno e he dual space Vand ⟨·,·⟩ he duali y pai ing be ween Vand V′. Applying P o (2.3), we ob ain      ∂ ∂ +νA +ez(θ ω)B( , ) =e−z(θ ω)P( (ez(θ −ρ( )ω) ( −ρ( )))+g( ,x))+z(θ ω) , x ∈ O, >τ, (τ+ξ):=ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R, (2.7) whe e A =−P∆ wi h A:V→V′de ined by ⟨Au, ⟩=((u, )), and B( , )= P(( ·∇) ) wi h B:V×V→V′and ⟨B(u, ),w⟩=b(u, ,w), whe e b(u, ,w)= 2 X i,j=1ZO ui ∂ j ∂xi wjdx, o all u, ,w ∈V, (2.8) which sa is ies b(u, ,w)=−b(u,w, ) and b(u, , )=0.(2.9) Le CH=C([−ϱ,0];H) and CV=C([−ϱ,0];V) wi h no m ∥u∥CH= sup ξ∈[−ϱ,0] ∥u(ξ)∥and ∥u∥CV= sup ξ∈[−ϱ,0] ∥u(ξ)∥V. In [19], he au ho s p o ed he exis ence and uniqueness o solu ions o 2D Na ie -S okes equa ions wi h he abs ac delay e m. No e ha (2.7) is a de e minis ic equa ion pa ame e ized by ω. Hence we need o show ha he a iable delay e m sa is ies he assump ions (I)-(IV) o he abs ac delay e m in [19]. By (D) and (F), i is easy o see ha (I) and (II) hold. By (2.4) and (2.5) we ob ain o all u, ∈CH, ∥ (u( −ρ( )))− ( ( −ρ( )))∥2=ZO | (u( −ρ( )))− ( ( −ρ( )))|2 R2dx ≤L2 ∥u( −ρ( ))− ( −ρ( ))∥2≤L2 ∥u − ∥2 CH, which implies ha (III) holds. By (2.4) and (2.5) again, we ob ain o all u, ∈C([τ− h, ];H) and ≥τ, Z τ ∥ (u( −ρ( )))− ( ( −ρ( )))∥2d ≤L2 Z τ ∥u( −ρ( ))− ( −ρ( ))∥2d ≤L2 1−ρ∗Z τ−ϱ ∥u( )− ( )∥2d , which implies ha (IV) holds. We ema k ha (2.7) is an abs ac o m o (2.3). Then we de i e (2.3) has a unique solu ion ∈C([τ−ϱ,T ];H)∩L2(τ,T;V) o all T > τ when RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1143 ϕ∈CHand g∈L2 loc(R;(L2(O))2). We de ine a delay-shi by (ξ)= ( +ξ) o all ≥τand ξ∈[−ϱ,0], and conside a mapping Φ:R+×R×Ω×CH→CHby Φ( ,τ,ω)ψ=u +τ(·,τ,θ−τω,ψ)= +τ(·,τ,θ−τω,ϕ)ez(θ +·ω),(2.10) whe e is he solu ion o (2.3), which sa is ies (τ+ξ,τ,ω,ϕ)=ϕ. In addi ion, ( ,τ,ω,ϕ) is (F,B(H))-measu able wi h espec o ω, whe e B(H) s ands o he Bo el σ-algeb a o H. Then by (2.1) we imply he mapping Φ is a con inuous non-au onomous andom dynamical sys em associa ed wi h (2.1) in he sense o [31]. 3. Pullback andom a ac o s In his sec ion, we a e conce ned wi h he exis ence, uniqueness and backwa d compac ness o pullback andom a ac o s. Fo hese pu poses, we need o impose an assump ion abou he non-au onomous o cing g∈L2 loc(R;(L2(O))2), i.e., gis backwa d empe ed: sup s≤τZs −∞ eκ( −s)∥g( )∥2d :=G(τ)<+∞,∀τ∈R, κ > 0.(3.1) In o de o ea he delay e m, we now in oduce a andom a iable η(ω), which is de ined by η(ω)= νλ 2−2|z(ω)|− L (1−ρ∗)1 2 eνλ 4ϱ(e2z(ω)+e−2z(ω)).(3.2) Using he e godic heo em [6, Theo em 2.1] o (3.2), we in e om (2.6) lim →±∞ 1 Z 0 η(θlω)dl = ¯η. (3.3) We conside wo a ac ion uni e ses Dand Bin CH, gi en by D={D={D(τ,ω):τ∈R,ω ∈Ω}: lim →+∞e−κ ∥D(τ− ,θ− ω)∥2 CH=0},(3.4) and B={B={B(τ,ω):τ∈R,ω ∈Ω}: lim →+∞e−κ sup s≤τ ∥B(s− ,θ− ω)∥2 CH=0},(3.5) espec i ely. I is simple o imply ha Dand Ba e inclusion-closed, and B⊂D. Th oughou his pape , we assume ha he delay ime ϱ∈(0,ϱ0] o some ϱ0>0. F om now on, we assume ha cand C(ω) a e wo gene ic posi i e cons an s, whe e cis independen o ϱand ω, and C(ω) is uni o m wi h espec o ϱ. 3.1. Backwa d uni o m es ima es o solu ions. Lemma 3.1. Suppose ha (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backwa d empe ed. Then we ob ain he ollowing conclusions: (i) Fo each τ∈R,ω∈Ωand D={D(τ,ω):τ∈R,ω ∈Ω}∈D, he e exis s a Td:= Td(τ,ω,D)≥3ϱ+3 such ha sup ξ∈[−3ϱ−3,0] ∥u(τ+ξ,τ − ,θ−τω,ψ)∥2≤cecϱ(1+Rd(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω),(3.6) RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1144 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS o all ≥Tdand ψ∈D(τ− ,θ− ω), whe e Rd(τ,ω)=Z0 −∞ eR 0η(θlω)dle−2z(θ ω)∥g( +τ)∥2d . (3.7) (ii) Fo each τ∈R,ω∈Ωand B={B(τ,ω): τ∈R,ω ∈Ω}∈B, he e exis s a Tb:= Tb(τ,ω,B)≥3ϱ+3 such ha sup s≤τ sup ξ∈[−3ϱ−3,0] ∥u(s+ξ,s− ,θ−sω,ψ)∥2≤cecϱ(1+Rb(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω),(3.8) o all ≥Tband ψ∈B(s− ,θ− ω), whe e Rb(τ,ω)=sup s≤τ Rd(s,ω).(3.9) P oo . Taking he inne p oduc o (2.3) wi h ( ):= ( ,s− ,θ−sω,ϕ) in H, by he second equali y o (2.9) and he incomp essible condi ion we ob ain 1 2 d d ∥ ( )∥2+ν∥∇ ( )∥2 =e−z(θ −sω)( (ez(θ −ρ( )−sω) ( −ρ( )))+g( ), ( ))+z(θ −sω)∥ ( )∥2. Using he Young inequali y and (2.5), we ob ain e−z(θ −sω)( (ez(θ −ρ( )−sω) ( −ρ( )))+g( ), ( )) ≤L (1−ρ∗)1 2 2eνλ 4ϱe2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2+L eνλ 4ϱ 2(1−ρ∗)1 2 e−2z(θ −sω)∥ ( )∥2 +νλ 4∥ ( )∥2+1 νλe−2z(θ −sω)∥g( )∥2. By (1.2) we ob ain ν∥∇ ( )∥2≥νλ 2∥ ( )∥2+ν 2∥∇ ( )∥2. Then we ha e d d ∥ ( )∥2+ν∥∇ ( )∥2 ≤L (1−ρ∗)1 2 eνλ 4ϱe2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2+L eνλ 4ϱ (1−ρ∗)1 2 e−2z(θ −sω)∥ ( )∥2 +2 νλe−2z(θ −sω)∥g( )∥2+(2z(θ −sω)−νλ 2)∥ ( )∥2.(3.10) Mul iplying (3.10) by eR s− η(θl−sω)dl, and hen in eg a ing his esul o e [s− ,s+ξ] wi h ξ∈[−3ϱ−3,0] and ≥3ϱ+3 yields eRs+ξ s− η(θl−sω)dl∥ (s+ξ)∥2 ≤∥ϕ∥2 CH+L (1−ρ∗)1 2 eνλ 4ϱZs+ξ s− eR s− η(θl−sω)dle2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2d RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1145 +L eνλ 4ϱ (1−ρ∗)1 2Zs+ξ s− eR s− η(θl−sω)dle−2z(θ −sω)∥ ( )∥2d +2 νλ Zs+ξ s− eR s− η(θl−sω)dle−2z(θ −sω)∥g( )∥2d +Zs+ξ s− (2z(θ −sω)−νλ 2+η(θ −sω))eR s− η(θl−sω)dl∥ ( )∥2d . (3.11) I ollows om (2.4) and (3.2) ha L (1−ρ∗)1 2 eνλ 4ϱZs+ξ s− eR s− η(θl−sω)dle2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2d ≤L (1−ρ∗)1 2eνλ 4ϱZs+ξ s− eR −ρ( ) s− η(θl−sω)dle2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2d ≤L eνλ 4ϱ (1−ρ∗)1 2Zs+ξ s− −ϱ eR s− η(θl−sω)dle2z(θ −sω)∥ ( )∥2d =L eνλ 4ϱ (1−ρ∗)1 2 (Zs− s− −ϱ +Zs+ξ s− )eR s− η(θl−sω)dle2z(θ −sω)∥ ( )∥2d ≤L eνλ 4ϱ (1−ρ∗)1 2 ∥ϕ∥2 CHZs− s− −ϱ eR s− η(θl−sω)dle2z(θ −sω)d +L eνλ 4ϱ (1−ρ∗)1 2Zs+ξ s− eR s− η(θl−sω)dle2z(θ −sω)∥ ( )∥2d . (3.12) Then by (3.2) ( he de ini ion o η(ω)) we ob ain (2z(θ −sω)−νλ 2+η(θ −sω))+ L eνλ 4ϱ (1−ρ∗)1 2 (e−2z(θ −sω)+e2z(θ −sω)) =2z(θ −sω)−2|z(θ −sω)|≤0. Then by (3.11) and (3.12) we ha e ∥ (s+ξ)∥2≤ceR0 ξη(θlω)dl∥ϕ∥2 CH(e−R0 − η(θlω)dl +eνλ 4ϱZ− − −ϱ eR 0η(θlω)dl+2z(θ ω)d ) +ceR0 ξη(θlω)dl Z0 − eR 0η(θlω)dle−2z(θ ω)∥g( +s)∥2d ≤ce3νλ 2ϱe3νλ 2∥ϕ∥2 CH(e−R0 − η(θlω)dl +eνλ 4ϱZ− − −ϱ eR 0η(θlω)dl+2z(θ ω)d ) +ce3νλ 2ϱe3νλ 2Z0 − eR 0η(θlω)dle−2z(θ ω)∥g( +s)∥2d ≤ce2νλϱ∥ϕ∥2 CH(e−R0 − η(θlω)dl +Z− − −ϱ eR 0η(θlω)dl+2z(θ ω)d ) +ce2νλϱ Z0 − eR 0η(θlω)dle−2z(θ ω)∥g( +s)∥2d , (3.13) RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1146 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS whe e we used η(ω)≤νλ 2in (3.2). F om (2.2) and (3.3), he e exis s a ˜ T:= ˜ T(¯η,ω)≥ 3ϱ+3 such ha o all | |≥ ˜ T, |z(θ ω)|≤ ¯η 16| |and Z 0 (η(θlω)−¯η)dl ≤¯η 2| |.(3.14) Then we ha e o all ≥˜ Tand ξ∈[−3ϱ−3,0], ∥ (s+ξ)∥2≤ce2νλϱ(e−¯η 2 +4 ¯ηe−¯ 3η 8 )∥ϕ∥2 CH+ce2νλϱ Z0 −∞ eR 0η(θlω)dle−2z(θ ω)∥g( +τ)∥2d . Hence by (2.1) we ob ain sup ξ∈[−3ϱ−3,0] ∥u(s+ξ)∥2 = sup ξ∈[−3ϱ−3,0] e2z(θξω)∥ (s+ξ)∥2 ≤ce2νλϱ sup ξ∈[−3ϱ−3,0] e2z(θξω)e−2z(θ− +ξω)(e−¯η 2 +4 ¯ηe−3¯η 8 )∥ψ∥2 CH +ce2νλϱ sup ξ∈[−3ϱ−3,0] e2z(θξω)Z0 −∞ eR 0η(θlω)dle−2z(θ ω)∥g( +τ)∥2d . (3.15) (i) When s=τ. I ψ∈D(τ− ,θ− ω), by (3.4) he e exis s a Td:=Td(τ,ω,D)≥˜ T such ha o all ≥Td, ce2νλϱe−2z(θ− +ξω)(e−¯η 2 +4 ¯ηe−3¯η 8 )∥ψ∥2 CH ≤ce(2νλ+3¯η 8)ϱe¯η 8 (e−¯η 2 +4 ¯ηe−¯η 4 )∥D(τ− ,θ− ω)∥2 CH≤ce(2νλ+¯η)ϱ, which implies ha (3.6) holds. (ii) I ψ∈B(s− ,θ− ω) wi h s≤τ, by (3.5) he e exis s a Tb:=Tb(τ,ω,B)≥˜ Tsuch ha o all ≥Td, ce2νλϱe−2z(θ− +ξω)(e−¯η 2 +4 ¯ηe−¯η 4 )sup s≤τ ∥ψ∥2 CH ≤ce(2νλ+3¯η 8)ϱe¯η 8 (e−¯η 2 +4 ¯ηe−¯η 4 )sup s≤τ ∥B(τ− ,θ− ω)∥2 CH≤ce(2νλ+¯η)ϱ, which, along wi h (3.15), shows ha (3.8) holds. The p oo is comple e. Lemma 3.2. Suppose ha (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backwa d empe ed. Fo each τ∈R,ω∈Ωand B={B(τ,ω) :τ∈R,ω ∈Ω}∈ B, we ha e sup s≤τ sup ξ∈[−2ϱ−2,0] ∥∇u(s+ξ,s− ,θ−sω,ψ)∥2≤C(ω)¯ Rb(τ,ω)eC(ω)¯ R2 b(τ,ω),(3.16) o all ≥Tb(Tbis gi en by Lemma 3.1) and ψ∈B(s− ,θ− ω)wi h s≤τ, whe e ¯ Rb(τ,ω)=1+Rb(τ,ω)+G(τ). P oo . Taking he inne p oduc o (2.7) wi h A ( ) in H, by he incomp essible condi ion we ob ain 1 2 d d ∥∇ ( )∥2+ν∥A ( )∥2+ez(θ −s)b( ( ), ( ),A ( )) RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1147 =e−z(θ −sω)( (ez(θ −ρ( )−sω) ( −ρ( ))),A ( )) e−z(θ −sω)(g( ,·),A ( ))+z(θ −sω)∥∇ ( )∥2. By he Young inequali y and (2.5) we ob ain e−z(θ −sω)( (ez(θ −ρ( )−sω) ( −ρ( ))),A ( )) ≤ν 6∥A ( )∥2+ce−2z(θ −sω)e2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2, and e−z(θ −sω)(g( ,·),A ( )) ≤ν 6∥A ( )∥2+ce−2z(θ −sω)∥g( )∥2. I ollows om (2.8) and he Gaglia do-Ni enbe g inequali y ha −ez(θ −s)b( ( ), ( ),A ( ))≤ez(θ −s)∥ ( )∥L4(O;R2)∥∇ ( )∥L4(O;R4)∥A ( )∥ ≤ez(θ −s)∥ ( )∥1 2∥∇ ( )∥∥A ( )∥3 2 ≤ acν6∥A ( )∥2+ce4z(θ −sω)∥ ( )∥2∥∇ ( )∥4. Hence we ha e d d ∥∇ ( )∥2+ν∥A ( )∥2≤ce−2z(θ −sω)e2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2 +ce−2z(θ −sω)∥g( )∥2+ce4z(θ −sω)∥ ( )∥2∥∇ ( )∥4.(3.17) Using he uni o m G onwall inequali y o (3.17) we ob ain o all ξ∈[−2ϱ−2,0], ∥∇ (s+ξ)∥2 ≤ceRs+ξ s+ξ−1e4z(θ −sω)∥ ( )∥2∥∇ ( )∥2d Zs+ξ s+ξ−1 e−2z(θ −sω)e2z(θ −ρ( )−sω)∥ ( −ρ( ))∥2d +ceRs+ξ s+ξ−1e4z(θ −sω)∥ ( )∥2∥∇ ( )∥2d Zs+ξ s+ξ−1 (e−2z(θ −sω)∥g( )∥2+∥∇ ( )∥2)d ≤C(ω)eC(ω)Rs s−2ϱ−3∥ ( )∥2∥∇ ( )∥2d Zs s−2ϱ−3 (∥ ( −ρ( ))∥2+∥g( )∥2+∥∇ ( )∥2)d , which, along wi h (2.1), implies ∥∇u(s+ξ)∥2 ≤C(ω)eC(ω)Rs s−2ϱ−3∥u( )∥2∥∇u( )∥2d Zs s−2ϱ−3 (∥u( −ρ( ))∥2+∥g( )∥2+∥∇u( )∥2)d . To show ha ∥∇u(s+ξ)∥2is bounded o all s≤τ, we need p o e Rs s−2ϱ−3∥∇ ( )∥2d is bounded. To his end, in eg a ing (3.10) o e [s−2ϱ−3,s] by (2.1) we ob ain sup s≤τZs s−2ϱ−3 ∥∇u( )∥2d ≤C(ω)sup s≤τ ∥u(s−2ϱ−3)∥2 +C(ω)sup s≤τZs s−2ϱ−3 (∥u( −ρ( ))∥2+∥u( )∥2+∥g( )∥2)d . RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1154 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS 3.3. Exis ence, uniqueness and backwa d compac ness o pullback an- dom a ac o s. We now es ablish he exis ence, uniqueness and backwa d com- pac ness o pullback andom a ac o s o (1.1). Fo he pu pose o his subsec ion, we show ha he Φ in (2.10) has a pullback andom abso bing se . Theo em 3.1. Suppose ha (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backwa d empe ed and backwa d ail-small. Then we ob ain he ollowing esul s: (i) Φ has a D-pullback andom abso bing se Kd={Kd(τ,ω) : τ∈R,ω ∈Ω}∈ D, de- ined 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 abso bing se Kb={Kb(τ,ω):τ∈R,ω ∈Ω}∈B, de ined by Kb(τ,ω)={ϕ∈CH:∥ϕ∥2 CH≤cecϱ(1+Rb(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω)},(3.51) whe e Rd(τ,ω)and Rb(τ,ω)a e gi en in (3.7)and (3.9), espec i ely. P oo . The p oo is simila o [43, Lemma 3.3], and so is omi ed. We ema k ha he measu abili y o he B-pullback abso bing se Kbis unknown, bu we can show ha he B-pullback a ac o is measu able (see Theo em 3.4). We i s p o e he B-pullback asymp o ic compac ness o Φ in (2.10). Theo em 3.2. Suppose ha (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backwa d empe ed and backwa d ail-small. Then Φin (2.10)is backwa d B- pullback asymp o ically compac , ha is, o each τ∈R,ω∈Ωand B∈B, he se- quence {Φ( n,sn− n,θ− nω)ψn}n∈Nis p e-compac in CH, whene e n→+∞,sn≤τ and ψn∈B(sn− n,θ− nω). P oo . Based on he Ascoli-A zel`a heo em, he p oo is spli in o wo s eps. S ep 1. Fo each ξ∈[−ϱ,0], we p o e {(Φ( n,sn− n,θ− nω)ψn)(ξ)}n∈Nhas a con e gen subsequence in H. Since n→+∞as n→+∞, we assume ha n≥Tb o all n∈N. F om (3.8), we ob ain {(Φ( n,sn− n,θ− nω)ψn)(ξ)}n∈Nis bounded in H, and so {(Φ( n,sn− n,θ− n)ψn)(ξ)}n∈Nhas a weakly con e gen subsequence (no elabelled), ha is, he e exis s a ˜ ∈Hsuch ha (Φ( n,sn− n,θ− nω)ψn)(ξ)= u(sn+ξ,sn− n,θ−snω,ψn)→˜uweakly in H. (3.52) We now show ha he weak con e gence o (3.52) is s ong. Since ˜u∈H, o any ε>0 he e exis s a ˜ K:= ˜ K(ε)>0 such ha Z|x|≥ ˜ K |˜u|2dx < ε 5.(3.53) Le ¯ K=max{K, ˜ K}, whe e Kis gi en in Lemma 3.4. F om (3.35), we ha e Z|x|≥ ¯ K |u(sn+ξ,sn− n,θ−snω,ψn)|2dx< ε 5.(3.54) RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1155 By (3.16), {(Φ( n,sn− n,θ− nω)ψn)(ξ)}n∈Nis bounded in V, and so i is bounded in H1(O¯ K), whe e O¯ K={x:|x|<¯ K}. Hence {(Φ( n,sn− n,θ− nω)ψn)(ξ)}n∈Nis p e- compac in L2(O¯ K), ha is, he e exis s a N > 0 such ha o all n≥N ∥u(sn+ξ,sn− n,θ−snω,ψn)−˜u∥2 L2(O¯ K)<ε 5.(3.55) I ollows om (3.53)-(3.55) ha ∥u(sn+ξ,sn− n,θ−snω,ψn)−˜u∥2 =Z|k|≥ ¯ K | (sn+ξ,sn− n,θ−snω,ψn)−˜u|2dx +∥u(sn+ξ,sn− n,θ−snω,ψn)−˜u∥2 L2(O¯ K) ≤2Z|k|≥ ¯ K |u(sn+ξ,sn− n,θ−snω,ψn)|2dx+2Z|k|≥ ¯ K |˜u|2dx +∥u(sn+ξ,sn− n,θ−snω,ψn)−˜u∥2 L2(O¯ K)<ε, which implies he weak con e gence o (3.52) is s ong con e gence. The p oo o S ep 1is comple e. S ep 2. We p o e {Φ( n,sn− n,θ− nω)ψn}n∈Nin CHis equi-con inuous om [−ϱ,0] o H. Fo each ξ1,ξ2∈[−ϱ,0] wi h ξ1< ξ2, by (3.21) we deduce |(Φ( n,sn− n,θ− nω)ψn)(ξ1)−(Φ( n,sn− n,θ− nω)ψn)(ξ2)∥ =∥u(sn+ξ1,sn− n,θ−snω,ψn)−u(sn+ξ2,sn− n,θ−snω,ψn)∥ ≤ez(θξ1ω)∥ (sn+ξ1,sn− n,θ−snω,ϕn)− (sn+ξ2,sn− n,θ−snω,ϕn)∥ +|ez(θξ1ω)−ez(θξ2ω)|∥ (sn+ξ2,sn− n,θ−snω,ϕn)∥ ≤C(ω)Zsn+ξ2 sn+ξ1 ∥∂ ∂ ( ,sn− n,θ−snω,ϕn)∥d +C(ω)ecϱ(1+Rb(τ,ω))|ez(θξ1ω)−ez(θξ2ω)| ≤C(ω)Zsn sn−ϱ ∥∂ ∂ ( ,sn− n,θ−snω,ϕn)∥2d 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 ha {Φ( n,sn− n,θ− nω)ψn}n∈Nis equi-con inuous. The p oo o S ep 1is comple e. We ob ain ha all condi ions o he Ascoli-A zel`a heo em a e ul illed, and so he p oo is comple e. We hen show he D-pullback asymp o ic compac ness o Φ in (2.10). Theo em 3.3. Suppose ha (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is empe ed and ail-small. Then Φin (2.10)is D-pullback asymp o ically compac , ha is, o each τ∈R,ω∈Ωand D∈D, he sequence {Φ( n,τ − n,θ− nω)ψn}n∈Nis p e-compac in CH, whene e n→+∞and ψn∈D(τ− n,θ− nω). RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1156 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS P oo . The asymp o ic compac ness has been p o ed o he amily Bin Theo em 3.2, his esul also holds o he amily D. The p oo is simila o ha o Theo em 3.2, and we omi he de ails. Now, we ob ain he main esul o his sec ion. Theo em 3.4. Suppose ha (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backwa d empe ed and backwa d ail-small. Then we ob ain he ollowing esul s: (i) Φ has a unique D-pullback andom a ac o Ad={Ad(τ,ω):τ∈R,ω ∈Ω}∈D, de ined by Ad(τ,ω)= T≥0[ ≥T Φ( ,τ − ,θ− ω)Kd(τ− ,θ− ω). (ii) Φ has a unique B-pullback a ac o Ab={Ab(τ,ω):τ∈R,ω ∈Ω}∈B, de ined by Ab(τ,ω)= T≥0[ ≥T Φ( ,τ − ,θ− ω)Kb(τ− ,θ− ω). In addi ion, Abis backwa d compac , ha is, Ss≤τAb(s,ω)is compac in H. (iii) Ad=Ab, and so Abis measu able and Adis backwa d compac . P oo . The p oo is simila o [43, Theo em 3.10], and so is omi ed. 4. Uppe semicon inui y o pullback andom a ac o In his sec ion, we conside he uppe semicon inui y o he pullback andom a ac- o as he delay ime ends o ze o. Fo his pu pose, we in oduce he limi equa ion o (1.1):              d¯u−(ν∆¯u−(¯u·∇)¯u−∇p)d =( (¯u( ))+g( ,x))d + ¯u◦dW, x∈O, > τ, ∇· ¯u= 0, x∈O, >τ, ¯u=0, x ∈∂O, > τ, ¯u(τ):= ¯ ψ, τ ∈R. (4.1) Using (2.1) o (4.1), we ob ain                  ∂¯ ∂ −ν∆¯ +ez(θ ω)(¯ ·∇)¯ +e−z(θ ω)∇p =e−z(θ ω)( (ez(θ ω)¯ ( ))+g( ,x))+z(θ ω)¯ , x ∈O, > τ, ∇· ¯ = 0, x∈O, >τ, ¯ = 0, x ∈∂O, > τ, ¯ (τ) := ¯ ϕ, τ ∈R. (4.2) By he same me hod as in Sec ion 3, we ob ain ha ¯ Φ gene a ed by (4.2) has a unique D0-pullback andom a ac o A0={A0(τ,ω) :τ∈R,ω ∈Ω}∈ D0, and a D0-pullback andom abso bing se K0={K0(τ,ω):τ∈R,ω ∈Ω}∈D0, whe e D0={D0={D0(τ,ω):τ∈R,ω ∈Ω}: lim →+∞e−κ ∥D0(τ− ,θ− ω)∥2=0}, RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/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 ew i e Kdand Ad(Kdis gi en by (3.50) and Adis gi en by Theo em 3.4) o Kϱ and Aϱ. Since he delay ime ϱ∈(0,ϱ0], by (3.50) and (4.3) we ha e limsup ϱ→0 ∥Kϱ(τ,ω)∥2 CH=∥K0(τ,ω)∥2,∀τ∈R, ω ∈Ω.(4.4) We now conside he poin wise con e gence o Aϱ: Lemma 4.1. Suppose ha {ϕn}n∈N⊂ Aϱn(τ,ω)wi h ϱn→0, hen he e exis a subse- quence {ϕnk}k∈No {ϕn}n∈Nand ϕ∈Hsuch ha lim k→+∞sup ξ∈[−ϱnk,0] ∥ϕnk(ξ)−ϕ∥=0.(4.5) P oo . The p oo is simila o [42, Lemma 5.2], and so is omi ed. Nex , we conside he con e gence o solu ions o he delay sys em o hose o he non-delay one. Lemma 4.2. Suppose ha ϕϱ∈CHand ϕ0∈Hsa is y lim ϱ→0sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥=0,(4.6) hen he solu ion ϱo (2.3)and he solu ion 0o (4.2)associa ed wi h he ini ial da a ϕϱand ϕ0sa is y lim ϱ→0sup ξ∈[−ϱ,0] ∥ ϱ( +τ+ξ,τ,θ−τω,ϕϱ)− 0( +τ,τ,θ−τω,ϕ0)∥=0,(4.7) o all ≥0,τ∈Rand ω∈Ω. Mo eo e , we ha e lim ϱ→0sup ξ∈[−ϱ,0] ∥uϱ( +τ+ξ,τ,θ−τω,ψϱ)−u0( +τ,τ,θ−τω,ψ0)∥=0.(4.8) P oo . Fo each τ∈R, we de ine ¯ Vϱ( )= ϱ( +ξ,τ,θ−τω,ϕϱ)− 0( ,τ,θ−τω,ϕ0),∀ ≥τ. F om (2.3) and (4.2), we ob ain ∂¯ Vϱ( +τ) ∂ −ν∆¯ Vϱ( +τ)+ez(θ +ξω)( ϱ( +τ+ξ)·∇) ϱ( +τ+ξ) −ez(θ ω)( 0( +τ)·∇) 0( +τ)+e−z(θ +ξω)∇pϱ−e−z(θ ω)∇p0 =e−z(θ +ξω)( (ez(θ +ξ−ρ( +τ+ξ)ω) ϱ( +τ+ξ−ρ( +τ+ξ)))+g( +τ+ξ,x)) −e−z(θ ω)( (ez(θ −ρ( +τ)ω) 0( +τ))+g( +τ,x)) +z(θ +ξω) ϱ( +τ+ξ)−z(θ ω) 0( +τ). Mul iplying he abo e equali y by ¯ Vϱ( +τ), and hen in eg a ing his esul o e O, we deduce 1 2 d d ∥¯ Vϱ( +τ)∥2+ν∥∇ ¯ Vϱ( +τ)∥2 RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1158 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS +(ez(θ +ξω)( ϱ( +τ+ξ)·∇) ϱ( +τ+ξ),¯ Vϱ( +τ)) −(ez(θ ω)( 0( +τ)·∇) 0( +τ),¯ Vϱ( +τ)) =(e−z(θ +ξω) (ez(θ +ξ−ρ( +τ+ξ)ω) ϱ( +τ+ξ−ρ( +τ+ξ))),¯ Vϱ( +τ)) −(e−z(θ ω) (ez(θ −ρ( +τ)ω) 0( +τ)),¯ Vϱ( +τ)) +(e−z(θ +ξω)g( +τ+ξ,·)−e−z(θ ω)g( +τ,·),¯ Vϱ( +τ)) +(z(θ +ξω) ϱ( +τ+ξ)−z(θ ω) 0( +τ),¯ Vϱ( +τ)).(4.9) I ollows om (2.8) and he Gaglia do-Ni enbe g inequali y ha −(ez(θ +ξω)( ϱ( +τ+ξ)·∇) ϱ( +τ+ξ),¯ Vϱ( +τ)) +(ez(θ ω)( 0( +τ)·∇) 0( +τ),¯ Vϱ( +τ)) =−ez(θ +ξω)(( ϱ( +τ+ξ)·∇) ϱ( +τ+ξ),¯ Vϱ( +τ)) −(( 0( +τ)·∇) 0( +τ),¯ Vϱ( +τ)) −(ez(θ +ξω)−ez(θ ω))(( 0( +τ)·∇) 0( +τ),¯ Vϱ( +τ)) =−ez(θ +ξω)b(¯ Vϱ( +τ), 0( +τ),¯ Vϱ( +τ)) +(ez(θ +ξω)−ez(θ ω))b( 0( +τ),¯ Vϱ( +τ), 0( +τ)) ≤ez(θ +ξω)∥¯ Vϱ( +τ)∥2 L4∥∇ 0( +τ)∥ +|ez(θ +ξω)−ez(θ ω)|∥ 0( +τ)∥2 L4∥∇ ¯ Vϱ( +τ)∥ ≤ez(θ +ξω)∥¯ Vϱ( +τ)∥∥∇ ¯ Vϱ( +τ)∥∥∇ 0( +τ)∥ +|ez(θ +ξω)−ez(θ ω)|∥ 0( +τ)∥∥∇ 0( +τ)∥∥∇ ¯ Vϱ( +τ)∥ ≤ν 4∥∇ ¯ Vϱ( +τ)∥2+cez(θ +ξω)∥¯ Vϱ( +τ)∥2∥∇ 0( +τ)∥2 +c|ez(θ +ξω)−ez(θ ω)|2∥ 0( +τ)∥2∥∇ 0( +τ)∥2.(4.10) By (2.5) we de i e (e−z(θ +ξω) (ez(θ +ξ−ρ( +τ+ξ)ω) ϱ( +τ+ξ−ρ( +τ+ξ))),¯ Vϱ( +τ)) −(e−z(θ ω) (ez(θ −ρ( +τ)ω) 0( +τ)),¯ Vϱ( +τ)) ≤ν 4∥∇ ¯ Vϱ( +τ)∥2 +ce−2z(θ +ξω)e2z(θ +ξ−ρ( +τ+ξ)ω)∥ ϱ( +τ+ξ−ρ( +τ+ξ))− 0( +τ)∥2 +c(ez(θ +ξ−ρ( +τ+ξ)ω)−ez(θ −ρ( +τ)ω))2∥ 0( +τ)∥2 +c(e−z(θ +ξω)−e−z(θ ω))2e2z(θ −ρ( +τ)ω)∥ 0( +τ)∥2.(4.11) The Young inequali y implies (e−z(θ +ξω)g( +τ+ξ,·)−e−z(θ ω)g( +τ,·),¯ Vϱ( +τ)) +(z(θ +ξω) ϱ( +τ+ξ)−z(θ ω) 0( +τ),¯ Vϱ( +τ)) ≤ν 4∥∇ ¯ Vϱ( +τ)∥2+ce−2z(θ +ξω)∥g( +τ+ξ)−g( +τ)∥2 +c(e−z(θ +ξω)−e−z(θ ω))2∥g( +τ)∥2+|z(θ +ξω)|∥ ¯ Vϱ( +τ)∥2 +c(z(θ +ξω)−z(θ ω))2∥ 0( +τ)∥2.(4.12) RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1159 Subs i u ing (4.10)-(4.12) in o (4.9), and hen in eg a ing his esul o e [−ξ, ] wi h ∈[−ξ,T] and T > ϱ0we show ∥¯ Vϱ( +τ)∥2≤ ∥ ¯ Vϱ(−ξ+τ)∥2+C(ω)Z −ξ (1+∥∇ 0( +τ)∥2)∥¯ Vϱ( +τ)∥2d +C(ω)Z −ξ ∥ ϱ( +τ+ξ−ρ( +τ+ξ))− 0( +τ)∥2d +C(ω)ZT 0 (ez(θ +ξω)−ez(θ ω))2∥ 0( +τ)∥2∥∇ 0( +τ)∥2d +C(ω)ZT 0 (ez(θ +ξ−ρ( +τ+ξ)ω)−ez(θ −ρ( +τ)ω))2∥ 0( +τ)∥2d +C(ω)ZT 0 ((e−z(θ +ξω)−e−z(θ ω))2+(z(θ +ξω)−z(θ ω))2)∥ 0( +τ)∥2d +C(ω)ZT 0 ∥g( +τ+ξ)−g( +τ)∥2d +C(ω)ZT 0 (e−z(θ +ξω)−e−z(θ ω))2∥g( +τ)∥2d . (4.13) We now mainly ea he second line o (4.13). Le s=y( )= +ξ−ρ( +ξ) o any ∈Rand ixed ξ∈[−ϱ,0]. Since y′( )≥1−ρ∗>0, i has an in e se unc ion such ha =y−1(s) o any s∈R. Then, Z −ξ ∥ ϱ( +τ+ξ−ρ( +τ+ξ))− 0( +τ)∥2d = Zy−1(τ) τ−ξ +Z +τ y−1(τ)!∥ ϱ( +ξ−ρ( +ξ))− 0( )∥2d =Zy−1(τ) τ−ξ ∥ ϱ( +ξ−ρ( +ξ))− 0( )∥2d +Z +τ−ρ( +τ+ξ) τ−ξ ∥ ϱ( +ξ)− 0(y−1( +ξ))∥2 1−d d ρ(y−1( +ξ)+ξ)d ≤2Zy−1(τ) τ−ξ ∥ ϱ( +ξ−ρ( +ξ))−ϕ0∥2d +2Zy−1(τ) τ−ξ ∥ 0( )−ϕ0∥2d +2 1−ρ∗Zτ+ τ−ξ ∥ ϱ( +ξ)− 0( )∥2d +2 1−ρ∗Zτ+ τ−ξ ∥ 0(y−1( +ξ))− 0( )∥2d ≤2 1−ρ∗Zτ τ−ρ(τ) ∥ ϱ( )−ϕ0∥2d +2Zy−1(τ) τ−ξ ∥ 0( )−ϕ0∥2d +2 1−ρ∗Zτ+ τ−ξ ∥ ϱ( +ξ)− 0( )∥2d +2 1−ρ∗Zτ+ τ−ξ ∥ 0(y−1( +ξ))− 0( )∥2d . RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1160 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS No e ha Zτ τ−ρ(τ) ∥ ϱ( )−ϕ0∥2d ≤cϱ sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2, and Zτ+ τ−ξ ∥ ϱ( +ξ)− 0( )∥2d =Z −ξ ∥¯ Vϱ( +τ)∥2d . Then we imply Z −ξ ∥ ϱ( +τ+ξ−ρ( +τ+ξ))− 0( +τ)∥2d ≤cϱ sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2+cZy−1(τ) τ−ξ ∥ 0( )−ϕ0∥2d +cZ −ξ ∥¯ Vϱ( +τ)∥2d +cZτ+ τ−ξ ∥ 0(y−1( +ξ))− 0( )∥2d . (4.14) By he same me hod as in Lemma 3.1 and Lemma 3.2 we ob ain ha ∥ 0( +τ)∥2and ∥∇ 0( +τ)∥2a e bounded when ∈[0,T ]. Then, om (4.13) and (4.14) we de i e ∥¯ Vϱ( +τ)∥2 ≤∥ ¯ Vϱ(−ξ+τ)∥2+C(ω)Z −ξ ∥¯ Vϱ( +τ)∥2d +C(ω) sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2 +C(ω)Zy−1(τ) τ−ξ ∥ 0( )−ϕ0∥2d +C(ω)Zτ+ τ−ξ ∥ 0(y−1( +ξ))− 0( )∥2d +C(ω)ZT 0 ((ez(θ +ξω)−ez(θ ω))2+(ez(θ +ξ−ρ( +τ+ξ)ω)−ez(θ −ρ( +τ)ω))2)d +C(ω)ZT 0 ((e−z(θ +ξω)−e−z(θ ω))2+(z(θ +ξω)−z(θ ω))2)d +C(ω)ZT 0 ∥g( +τ+ξ)−g( +τ)∥2d +C(ω)ZT 0 (e−z(θ +ξω)−e−z(θ ω))2∥g( +τ)∥2d . (4.15) By (4.6) and he con inui y o 0a τwe ob ain ∥¯ Vϱ ξ(−ξ+τ)∥2=∥ ϱ(τ,τ,θ−τω,ϕϱ)− 0(−ξ+τ,τ,θ−τω,ϕ0)∥2 ≤2∥ϕϱ(0)−ϕ0∥2+2∥ϕ0− 0(−ξ+τ,τ,θ−τω,ϕ0)∥2 ≤2 sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2+2∥ϕ0− 0(−ξ+τ,τ,θ−τω,ϕ0)∥2→0 as ϱ→0.(4.16) By he con inui y o 0a τwe show C(ω)Zy−1(τ) τ−ξ ∥ 0( )−ϕ0∥2d +C(ω)Zτ+ τ−ξ ∥ 0(y−1( +ξ))− 0( )∥2d →0 (4.17) RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1161 as ϱ→0, whe e we use y−1(τ)≤τ+2ϱ. Using he con inui y o →z(θ ω) yields C(ω)ZT 0 ((ez(θ +ξω)−ez(θ ω))2+(ez(θ +ξ−ρ( +τ+ξ)ω)−ez(θ −ρ( +τ)ω))2)d +C(ω)ZT 0 ((e−z(θ +ξω)−e−z(θ ω))2+(z(θ +ξω)−z(θ ω))2)d →0 as ϱ→0.(4.18) F om g∈L2 loc(R,(L2(O))2), we ha e C(ω)ZT 0 ∥g( +τ+ξ)−g( +τ)∥2d +C(ω)ZT 0 (e−z(θ +ξω)−e−z(θ ω))2∥g( +τ)∥2d →0 as ϱ→0.(4.19) Inse ing (4.16)-(4.19) o (4.15), o any ε>0, he e exis s a ¯ϱ∈(0,ϱ0] such ha o all ϱ< ¯ϱ, ∥¯ Vϱ( +τ)∥2≤C(ω)ε+C(ω)Z −ξ ∥¯ Vϱ( +τ)∥2d . (4.20) Using he G onwall inequali y (see [13, page 167]) o (4.20), we ob ain o all ϱ< ¯ϱand ∈[−ξ,T] wi h ξ∈[−ϱ,0], ∥¯ Vϱ( +τ)∥2<C(ω)eC(ω)Tε. (4.21) I ∈[0,−ξ], we ha e τ+ξ≤ +τ+ξ≤τ. Hence we deduce ∥ ϱ( +τ+ξ)− 0( +τ)∥2≤2∥ ϱ( +τ+ξ)−ϕ0∥2+2∥ 0( +τ)−ϕ0∥2 ≤2 sup ξ∈[−ϱ,0] ∥ϕϱ(ξ)−ϕ0∥2 V+2∥ 0( +τ)−ϕ0∥2→0,as ϱ→0.(4.22) I ollows om (4.21) and (4.22) ha lim ϱ→0∥¯ Vϱ( +τ)∥2=0,∀ ∈[0,T ], which implies ha (4.7) holds. By (2.1) we ha e ∥uϱ( +τ+ξ,τ,θ−τω,ψϱ)−u0( +τ,τ,θ−τω,ψ0)∥ ≤ez(θ +ξω)∥ ϱ( +τ+ξ,τ,θ−τω,ψϱ)− 0( +τ,τ,θ−τω,ψ0)∥ +|ez(θ +ξω)−ez(θ ω)|∥u0( +τ,τ,θ−τω,ψ0)∥, which, along wi h (5.7) and he con inui y o z(θ·ω), implies ha (4.8) holds. The p oo is comple e. I ollows om (4.4), Lemma 4.1 and Lemma 4.2 ha all condi ions o [36, Theo em 2.1] a e sa is ied. Then we ob ain he main esul o his sec ion: Theo em 4.1. Suppose ha (D) and (F) hold, and g∈L2 loc(R;(L2(O))2)is backwa d empe ed and backwa d ail-small. Then we ha e lim ϱ→0dis ϱ(Aϱ(τ,ω),A0(τ,ω))= 0, o all τ∈R, ω ∈Ω. RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL 1162 STOCHASTIC DELAY NAVIER-STOKES EQUATIONS 5. Asymp o ic au onomy o pullback andom a ac o Le ρ( )=ϱin (1.1) o all ∈R, hen we ha e              du−(ν∆u−(u·∇)u−∇p)d =( (u( −ϱ))+g( ,x))d +u◦dW, ∇·u=0, x ∈O, > τ, u=0, x ∈∂O, > τ, u(τ+ξ)=uτ(ξ):= ψ(ξ), ξ ∈[−ϱ,0], τ ∈R. (5.1) Using (2.1) o (5.1) we ob ain he ollowing andom equa ion:                  ∂ ∂ −ν∆ +ez(θ ω)( ·∇) +e−z(θ ω)∇p =e−z(θ ω)( (ez(θ −ϱω) ( −ϱ))+g( ,x))+z(θ ω) , x ∈ O, >τ, ∇· =0, x ∈O, > τ, =0, x ∈∂O, > τ, (τ+ξ)= τ(ξ):= ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R. (5.2) Using he same me hod as in Sec ion 3, we ob ain ha he non-au onomous andom dynamical sys ems Φ gene a ed by (5.1) ha e a unique backwa d compac D-pullback andom a ac o Aϱ={Aϱ(τ,ω):τ∈R,ω ∈Ω}, and a D-pullback abso bing se K= {K(τ,ω): τ∈R,ω ∈Ω}, de ined by K(τ,ω)={ϕ∈CH:∥ϕ∥2 CH≤cecϱ(1+Rb(τ,ω)) sup ξ∈[−3ϱ−3,0] e2z(θξω)},(5.3) whe e Rb(τ,ω) is gi en by (3.9). We now in oduce he au onomous e sion o (5.1):              d˜u−(ν∆˜u−(˜u·∇)˜u−∇p)d =( (˜u( −ϱ))+g∞(x))d + ˜u◦dW, ∇· ˜u= 0, x∈O, >0, ˜u=0, x ∈∂O, > 0, ˜u(0+ξ) = ˜u0(ξ):= ˜ ψ(ξ), ξ ∈[−ϱ,0], (5.4) whe e he o cing g∞sa is ies lim τ→−∞Zτ −∞ ∥g( )−g∞∥2d =0.(5.5) Using (2.1) o (5.4), we ob ain                  ∂˜ ∂ −ν∆˜ +ez(θ ω)( ·∇) +e−z(θ ω)∇˜p =e−z(θ ω)( (ez(θ −ϱω)˜ ( −ϱ))+g∞(x))+z(θ ω)˜ , x∈O, > τ, ∇· ˜ = 0, x∈O, >τ, ˜ = 0, x ∈∂O, > τ, ˜ (0+ξ)= ˜ 0(ξ):= ˜ ϕ(ξ), ξ ∈[−ϱ,0], τ ∈R. (5.6) Simila o Theo em 3.4, we ob ain ha he au onomous andom dynamical sys em Φ∞ associa ed wi h (5.4) has a unique D∞- andom a ac o A∞={A∞(ω):ω∈Ω} ∈D∞ wi h D∞being he collec ion o all empe ed amilies in CH, mo e p ecisely, D∞={D∞={D∞(ω):ω∈Ω}: lim →+∞e−κ ∥D∞(θ− ω)∥2 CH=0}.(5.7) RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL QIANGHENG ZHANG, TOM´ AS CARABALLO, AND SHUANG YANG 1163 In his sec ion, we conside he asymp o ic au onomy o Aϱ: lim τ→−∞dis CH(Aϱ(τ,ω),A∞(ω))=0,∀ω∈Ω.(5.8) Le Kbu ={Kbu(ω):ω∈Ω}wi h Kbu(ω)=Ss≤0K(s,ω) o all ω∈Ω. We now p o e Kbu ∈ D∞. Le δ1=min{¯η 6,κ 8}, by (2.2) and (3.3) he e exis s a ˜ T:= ˜ T(δ1,ω)>0 such ha e−2z(θ ω)≤eδ1| |,Z 0 (η(θ ω)−¯η)d  ≤δ1| |, o all | |≥ ˜ T. Hence, we ob ain o all ≥˜ Tand ≤0 eR 0η(θl− ω)dl+2|z(θ − ω)|=eR − − η(θlω)dl+2|z(θ − ω)| ≤eR − 0(η(θlω)−¯η)dl+¯γ( − )−R− 0(η(θlω)−¯η)dl+¯η +2|z(θ − ω)| ≤e|R − 0(η(θlω)−¯η)dl|+|R− 0(η(θlω)−¯η)dl|+¯η +δ1( − ) ≤e3δ1 e(¯η−2δ1) ≤e3δ1 e2δ1 . By (3.9) and (5.3) we imply Kis closed and inc easing. Then we ha e e−κ ∥Kbu(θ− ω)∥2 CH =e−κ ∥[ s≤0 K(s,θ− ω)∥2 CH=e−κ ∥K(0,θ− ω)∥2 CH ≤ce−κ eνλϱ(1+Rb(0,θ− ω)) sup ξ∈[−3ϱ−3,0] e2z(θ− +ξω) =ce−(κ−2δ1) e(νλ+2δ1)ϱ(1+sup s≤0 Rd(s,θ− ω)) sup ξ∈[−3ϱ−3,0] e2z(θ− +ξω) ≤ce−(κ−2δ1) e(νλ+2δ1)ϱ(1+sup s≤0Z0 −∞ eR 0η(θl− ω)dle−2z(θ − ω)∥g( +s)∥2d ) ≤ce−(κ−5δ1) e(νλ+2δ1)ϱ(1+sup s≤0Z0 −∞ e2δ1 ∥g( +s)∥2d )→0,as →+∞. Hence we ha e Kbu ∈D∞. Nex , we p o e he con e gence o solu ions o he non- au onomous sys em o hose o he au onomous one. Lemma 5.1. Suppose ha τ,˜ 0∈CHsa is y lim τ→−∞∥ τ−˜ 0∥2 CH=0,(5.9) hen we ha e lim τ→−∞∥u +τ(·,τ,θ−τω,uτ)−˜u (·,ω, ˜u0)∥2 CH=0,(5.10) o all ≥0and ω∈Ω. P oo . Fo each τ∈R, le ˜ Vτ( )= ( +τ,τ,θ−τω, τ)−˜ ( ,ω,˜ 0),∀ ≥ −ϱ. RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL RJ+p466N wh o7OTAxeHNE kl XQJ6BsgJl3zskmMX dqykiwc3 xnmdadVIa0udgdaQiPTwB3DDx0XF8NVCK/CjwH/AHjixQjh5EVJGqheRRAnGUX8I6NkHIQYjxzZL