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
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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,
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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.
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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
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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)
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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
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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},
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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
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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)
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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 .
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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)
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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.
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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)
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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.
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