Asymptotical behavior of the 2D stochastic partial dissipative Boussinesq system with memory

Asymp o ical beha io o he 2D s ochas ic pa ial
dissipa i e Boussinesq sys em wi h memo y
Hao an Dai⇤
,BoYou
†
School o Ma hema ics and S a is ics, Xi’an Jiao ong Uni e si y
Xi’an, 710049, P. R. China
Tom´as Ca aballo‡
Depa amen o de 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
Janua y 22, 2025
Abs ac
The objec i e o his pape is o conside he asymp o ical beha io o solu ions o
he wo-dimensional pa ial dissipa i e Boussinesq sys em wi h memo y and addi i e
noise. We i s es ablish he exis ence o a andom abso bing se in he phase space.
Howe e , due o he p esence o he memo y e m, we canno ob ain some kind o
compac ness o he co esponding cocycle h ough Sobole compac ness embedding
heo em o by e i ying he pullback la ening p ope y. To o e come his di icul y,
we i s p o e he asymp o ical compac ness o he eloci y componen o weak solu-
ions, and hen we p o e he asymp o ical compac ness o o he componen s based on
some ene gy es ima es and he Aubin-Lions compac ness lemma, which implies he
asymp o ical compac ness o he co esponding cocycle. Thus, he exis ence o a an-
dom a ac o is ob ained. Finally, we es ablish an abs ac esul abou some kind o
uppe semi-con inui y o he andom a ac o , which is applied o he wo-dimensional
pa ial dissipa i e Boussinesq sys em.
Keywo ds: Random a ac o ; Pa ial dissipa i e Boussinesq sys em; Memo y
e m; Addi i e noise; Uppe semi-con inui y.
Ma hema ics Subjec Classi ica ion (2020) : 35B40, 35B41, 35Q35, 37L55,
60H15.
⇤Email add ess: [email p o ec ed]
†Email add ess: y[email p o ec ed]
‡Email add ess: [email p o ec ed]
1
1 In oduc ion
I is well-known ha he Boussinesq sys em plays an impo an ole in modelling geo-
physical lows, such as a mosphe ic on s and oceanic ci cula ion (see, e.g., [20, 28, 31]).
Howe e , in ce ain physical egimes, he dynamical sys em go e ning geophysical lows is
desc ibed o be only pa ial dissipa i e o semi-dissipa i e. Fo example, he au ho s in
[1] conside ed a 2D Boussinesq sys em wi h dissipa ion only in he eloci y a iable, while
a semi-dissipa i e Boussinesq sys em wi hou dissipa ion in empe a u e a iable was con-
side ed in [6]. Ac ually, he Boussinesq equa ions wi h pa ial dissipa ion has been widely
used o model he dynamics o geophysical lows in which he dissipa ion in some pa icula
di ec ion domina es (see, e.g., [12, 13, 29]).
In his pape , we conside he ollowing 2D pa ial dissipa i e Boussinesq sys em wi h
memo y and addi i e noise:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
@ u1+u· u1+@xp⌫@2
yu1=0,(x, )2Q,
@ u2+u· u2+@yp⌫@2
xu2=',(x, )2Q,
·u=0,(x, )2Q,
d''d +u· 'd +R
1 (  )'( )d d = d +dW ,(x, )2Q,
u(x, 0)=u 0,'(x, 0+ )='0(x, ),(x, )2D⇥(1,0]
(1.1)
equipped wi h pe iodic bounda y condi ions, whe e D=T2=[0,L]2,Q:= D⇥[ 0,+1),
02R.The eloci yu=(u1,u
2), he p essu e pand he empe a u e 'a e he unknown
unc ions. ⌫and a e he iscosi y and he he mal di↵usi i y coe icien s, espec i ely.
In he ollowing, we will always assume ha ⌫== 1, which has no bea ing on he
ma hema ical analysis. (·)2C2(R+) is he so-called memo y ke nel, he memo y e m
R
1 (  )'( )d ep esen s he in eg a ed pas his o y o empe a u e a iable. =
(x) is an ex e nal o cing e m. W(x, ) is a Wiene p ocess in L2(D)de inedonacomple e
p obabili y space (⌦,F,P)wi hexpec a iondeno edbyE,whichcanbew i enas
W(x, )=
1
X
j=1
jBj( )ej,
whe e {Bj}j2Z+a e independen s anda d B ownian mo ions, {ej}j2Z+is an o hono mal
basis o L2(D) consis ing o he eigen unc ions o wi h pe iodic bounda y condi ions
imposed he ze o-a e age condi ion and he coe icien s {j}j2Z+sa is ies he ollowing con-
di ion 1
X
j=1
2
j

1
220
j
<1(1.2)
o some 0>1
2.Theexis enceo such0can be ob ained by assump ion (A2)below.
The wo-dimensional Boussinesq equa ions wi h pa ial dissipa ion has ecen ly a ac ed
conside able a en ion and made some p og ess, especially in e ms o long- ime beha io
2
o pa ial dissipa i e Boussinesq sys em (see, e.g., [2, 3, 8, 11, 23, 24, 26]). Fo example,
he au ho s in [6] s udied he long- ime beha io o a semi-dissipa i e Boussinesq sys em
which is dissipa i e only in he eloci y a iable, bu no in he empe a u e. They p o ed
such sys em has a global a ac o . In [22], he au ho s conside ed a 2D Boussinesq sys em
which is pa ially dissipa i e only in he eloci y a iable, hey p o ed ha such sys em is
global well-posed unde some weake assump ions on he ini ial da a. In addi ion, hey also
p o ed he exis ence o a weak sigma-a ac o and es ablished i s uppe semi-con inui y
unde small pe u ba ions.
I is widely accep ed ha i some e ms aking in o accoun he pas his o y o he sys em
a e inco po a ed in o he equa ions, many physical phenomena can be be e desc ibed, such
as hea conduc ion in special ma e ials (see, e.g., [21, 34, 35]), iscoelas ici y o ib a ion in
se e al ma e ials (see, e.g., [7, 19]). In he pas se e al decades, he e a e many wo ks abou
he well-posedness and long- ime beha io o solu ions o pa ial di↵e en ial equa ions wi h
memo y e m. Fo example, he au ho s in [30] es ablished he exis ence and uniqueness
o weak solu ions o eloci y- o ici y-Voig model o he 3D Na ie -S okes equa ions wi h
damping and memo y, hey also p o ed he exis ence o a uni o m a ac o o his sys em.
In [36], he au ho s s udied he dynamics o he h ee-dimensional globally modi ied Na ie -
S okes equa ions wi h double delay in he o cing and con ec i e e ms, hey es ablished he
exis ence o pullback a ac o s o he associa ed dynamical sys ems.
As we know, he models o ce ain phenomena om he eal wo ld a e mo e ealis ic i
some kind o unce ain y is also conside ed in he o mula ion, such as some andomness o
en i onmen al noise. Thus, i is meaning ul and necessa y o conside he well-posedness
and long- ime beha io o solu ions o s ochas ic pa ial di↵e en ial equa ions. Fo ex-
ample, he au ho s ha e p o ed he exis ence o andom a ac o o he 2D s ochas ic
Cahn-Hillia d-Na ie -S okes sys em wi h small addi i e noise and h ee dimensional damped
Na ie -S okes equa ions wi h addi i e noise in [27, 38], espec i ely. Also in [33] a s ochas ic
nonlocal eac ion-di↵usion equa ion pe u bed wi h addi i e and mul iplica i e noise is an-
alyzed. The au ho s in [37] in es iga ed mean dynamics and s abili y analysis o s ochas ic
3D Lag angian-a e aged Na ie -S okes equa ions wi h in ini e delay d i en by mul iplica i e
noise in unbounded domains, hey p o ed such a dynamical sys em possesses a unique weak
pullback mean andom a ac o , which is a minimal, weakly compac and weakly pullback
a ac ing se . In [10], he au ho s s udied he asymp o ic beha io o a non-au onomous
s ochas ic eac ion-di↵usion equa ion wi h memo y and p o ed he exis ence o a andom
pullback a ac o . Howe e , o he bes o ou knowledge, he e is no known esul s con-
ce ning he long- ime beha io o solu ions o he 2D pa ial dissipa i e Boussinesq sys em
wi h memo y and addi i e noise.
Le us now commen on he main ma hema ical di icul ies and no el ies o his wo k.
Due o he p esence o he memo y e m, we canno es ablish he exis ence o an abso bing
se in mo e egula phase space, such ha some kind o compac ness o he co esponding
cocycle canno be ob ained by e i ying he pullback la ening p ope y o using he Sobole
compac ness embedding heo em. Meanwhile, he assump ion on he memo y ke nel gi es
ise o ano he c ucial di icul y ha we canno p o e he asymp o ic compac ness o he
co esponding cocycle by he me hods o ene gy equa ion o semig oup decomposi ion. Fo -
3
una ely, he eloci y componen o he weak solu ions possesses smoo hing p ope y, such
ha we can easily ob ain he asymp o ic compac ness o he eloci y componen o weak
solu ions, and hen we p o e he co esponding one o o he componen s based on some
ene gy es ima es and he Aubin-Lions compac ness Lemma, which implies he asymp o ic
compac ness o he co esponding cocycle. I is wo h men ioning ha ano he ype o aux-
ilia y O ns ein-Uhlenbeck p ocess is in oduced such ha he s ochas ic e m dW( )canbe
loca ed in L2(D),which is di↵e en om he exis ing wo k abou addi i e noise.
This pape is o ganized as ollows. In Sec ion 2, we i s in oduce some no a ion and
unc ion spaces, hen we ecall some abs ac esul s abou andom dynamical sys ems and
some use ul lemmas ha will be used in his pape . In Sec ion 3, we i s p o ide a well-
posedness esul o he 2D pa ial dissipa i e Boussinesq sys em wi h memo y and addi i e
noise, hen we es ablish he exis ence o andom abso bing se s in an app op ia e phase
space. Wi h he help o he Aubin-Lions compac ness lemma and he ene gy es ima es
me hod, we p o e he asymp o ic compac ness o he co esponding cocycle, which implies
he exis ence o a andom a ac o . In Sec ion 4, we es ablish an abs ac esul abou he
uppe semi-con inui y o andom a ac o s, which is applied o he 2D pa ial dissipa i e
Boussinesq sys em.
Th oughou his pape , we use L (D)(1  <+1)andHm(D)(m2N) odeno e he
usual Lebesgue and Sobole spaces o e Dwi h ze o-a e age condi ion. Fo con enience,
we always use k k o deno e he L (D) no m o unc ion , C deno es a gene ic cons an
which may change om line o line. A.Bmeans ha he e exis s a gene ic cons an C,
which may be di↵e en on di↵e en lines, such as ACB.
2 P elimina ies
In his sec ion, we will i s s a e some assump ions on he memo y e m and he s ochas ic
e m, hen we in oduce some unc ion spaces on Dand ecall some abs ac esul s on
andom a ac o s.
(A1)Thememo yke nel(·)2C2(R+)sa is ies lim
s!1 (s) = 0 and he unc ion µ(s)=
0(s)possesses he ollowingp ope ies
µ(s)0,µ
0(s)+µ(s)0,8s2R+,
whe e is a posi i e cons an . A ypical example is (s)=0ed0swi h d0>0and0>0.
Ques ion: can ha e a singula i y a ze o, i.e. is he case (s)=0ed0s
s1↵,↵2(0,1) included
in his amewo k?
(A2){W( ); 2R}is a wo-sided L2(D)- alued Wiene p ocess wi h co a iance ope a o
K=K⇤0,such ha
KA2↵⇤1<1
o some ↵⇤1, whe e A=Pis he S okes ope a o and Pis he Le ay-Helmhol z
p ojec ion om (L2(D))2on o he ee di e gence subspace Ho (L2(D))2.Inou caseo
pe iodic bounda y condi ions, i is well known ha A=(see, e.g., [15]).
4
Rema k 2.1. The condi ion ↵⇤1in (A2) implies he exis ence o 0in (1.2). In ac ,
we ha e
KA2↵⇤1:=
1
X
j=1 ⌦KA2↵⇤1ej,e
j↵=
1
X
j=1 ⌦K2↵⇤1
jej,e
j↵
=
1
X
j=1
2↵⇤1
jhKej,e
ji=
1
X
j=1
2↵⇤1
j⌦2
jej,e
j↵
=
1
X
j=1
2
j2↵⇤1
j<1.
Thus, in o de o ensu e condi ion (1.2) holds, i.e.
1
P
j=1
2
j201
2
j<1, i is su icien o
equi e ha 0sa is ies he ollowing condi ion:
201
22↵⇤1,
i.e. 0↵⇤1
4.The e o e, he e always exis s 02(1
2,↵
⇤1
4],such ha condi ion (1.2)
holds.
Le Vbe he se o all ec o - alued L-pe iodic igonome ic polynomials om R2 o
R2 ha a e di e gence- ee and ha e ze o a e age. Deno e by H, V and Z he closu es o V
in L2(D),H
1(D)andH2(D), espec i ely. De ine he bilinea ope a o B:V⇥V!V0by
hB(u, ),wi=ZD
[(u· ) ]·wdx, 8u, , w 2V.
In o de o ca y ou he analysis o he memo y e m, le M:= L2
µ(R+;H1(D)) be he
Hilbe space wi h inne p oduc and no m gi en by
(⇠,⇣)M:= Z1
0
µ(s)(⇠(s),⇣(s))H1(D)ds
and
k⇠k2
M:= Z1
0
µ(s)k⇠(s)k2
H1(D)ds,
o any ⇠,⇣2H1(D). In ou case o pe iodic bounda y condi ions, he no m on spaces
H1(D)andH2(D) wi h ze o-a e age condi ion a e de ined by
k⇠kH1(D):= k ⇠k2,k⇣kH2(D):= k⇣k2
o any ⇠2H1(D), ⇣2H2(D), espec i ely.
5

Now, we de ine an auxilia y O ns ein-Uhlenbeck p ocess z( ). Fo any ↵0andany
!2⌦, le
z( ;!)=Z
1
e(A+↵)( s)dW(s)
be he solu ion o he s ochas ic equa ion
dz +(A+↵)zd =dW( ).
No e ha zis a s a iona y Gaussian p ocess, i s ajec o ies a e P-a.s. con inuous (see
[18]). We can also in ol e a pe ec ion p ocedu e o de ine z( ;!)=¯z(✓ !) o all!2⌦
(see P oposi ion 3.1 in [14]). Mo eo e , he mapping !¯z(✓ !) is con inuous om Rin o
D(A↵⇤) o each!2⌦andsa is ies he ollowingcondi ion:
sup
2R{kA↵⇤¯z(✓ !)k2e| |}<1 o any >0andany!2⌦.
By in oducing a new a iable #='z, p oblem (1.1) can be ew i en as
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
@ u1+u· u1+@xp@2
yu1=0,(x, )2Q,
@ u2+u· u2+@yp@2
xu2=#+z, (x, )2Q,
·u=0,(x, )2Q,
@ ##+u· #+R
1 (  )#( )d = u· z+z( )+↵z, (x, )2Q,
u(x, 0)=u 0,#(x, 0+ )='(x, 0+ )z( 0+ ),(x, )2D⇥(1,0],
(2.1)
whe e z( )isap ocessde inedby
z( , !):=Z
1
(  )z( , !)d =Z
1
(  )Az( , !)d .
He e, we p o ide an impo an esul o z( )whichplaysakey olein hep oo o he
exis ence o a andom a ac o .
Lemma 2.2. (see [10]) The p ocess !z( , !)is con inuous wi h alued in D(A↵⇤1).
Mo eo e , z( , !)=¯z(✓ !), whe e
¯z(!)=Z1
0
(s)A¯z(✓s!)ds
is a empe ed andom a iable wi h alued in D(A↵⇤1).
In wha ollows, we ecall a use ul lemma used in he sequel.
Lemma 2.3. (see [1]) Fo any u2V, we ha e
k uk2
2=k ⇥uk2
22k(@yu1,@
xu2)k2
2.
Mo eo e , o any u2V (H2(D))2, we ha e
kuk2
22ZD
@2
yu1u1dxdy +2ZD
@2
xu2u2dxdy.
6
Now, in o de o ca y ou ou analysis, we in oduce he new a iable which e lec s he
in eg a ed pas his o y o p oblem (2.1) gi en by
⌘ (x, s)=⌘(x, , s)=Zs
0
#(x,  )d =Z
s
#(x, )d , s 0,  0,
hen we ha e
@ ⌘ (x, s)=#(x, )@s⌘ (x, s)and⌘ (x, 0) := lim
s!0⌘ (x, s)=0.
Thus, p oblem (2.1) can be e o mula ed in o he ollowing o m:
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
@ u1+u· u1+@xp@2
yu1=0,(x, )2Q,
@ u2+u· u2+@yp@2
xu2=#+z, (x, )2Q,
·u=0,(x, )2Q,
@ ##+u· #R+1
0µ( )⌘ ( )d = u· z+z( )+↵z, (x, )2Q,
@ ⌘ (s)=@s⌘ (s)+#( ),(x, )2Q, s 0,
u(x, 0)=u 0,#(x, 0)='0( 0)z( 0),⌘
0(x, s)=⌘0(s),x2D, s 0.
(2.2)
Le (X, k·kX)beasepa ableBanachspacewi hBo el-algeb a B(X), we ecall some
abs ac esul s om andom dynamical sys em.
De ini ion 2.4. (see [4, 5]) (⌦,F,P,{✓ } 2R)is called a me ic dynamical sys em, i
✓:R⇥⌦!⌦is (B(R)⇥F,F)-measu able, ✓0is he iden i y on ⌦,✓s+ =✓s✓ o all
s, 2Rand ✓ P=P o all 2R.
De ini ion 2.5. (see [4, 16, 25]) A mapping :R+⇥⌦⇥X!Xis called a con inuous
cocycle on Xo e a me ic dynamical sys em (⌦,F,P,{✓ } 2R), i i is (B(R+)⇥F⇥
B(X),B(X))-measu able and sa is ies o P-a.e. !2⌦,
(i) (0,!)is he iden i y on X,
(ii) ( +s, !)=( , ✓s!)(s, !) o all , s 2R+,
(iii) ( , !): X!Xis con inuous o all 2R+.
De ini ion 2.6. (see [16, 25]) A andom subse {B(!):!2⌦}o Xis called empe ed
wi h espec o {✓ } 2R, i o P-a.e. !2⌦,
lim
!+1e d(B(✓ !)) = 0
o all >0, whe e d(B)=sup
x2BkxkX.
7
De ini ion 2.7. (see [9, 17]) Assume ha is a con inuous cocycle on a Banach space
Xo e a me ic dynamical sys em (⌦,F,P,{✓ } 2R),le Dbe a collec ion o andom subse s
o X. ˆ
K={K(!):!2⌦}is called a D- andom abso bing se o , i o e e y ˆ
B=
{B(!):!2⌦}2Dand P-a.e. !2⌦, he e exis s some T=T(!,ˆ
B)>0,such ha
( , ✓ !)B(✓ !)⇢K(!)
o any T.
De ini ion 2.8. (see [25]) Le Dbe a collec ion o andom subse s o X. The con inuous
cocycle is said o be D-pullback asymp o ically compac in X, i o P-a.s. !2⌦,any
ˆ
B={B(!):!2⌦}2D,any sequence n!+1and any sequence xn2B(✓ n!), he
sequence {( n,✓
 n!)xn}1
n=1 has a con e gen subsequence in X.
De ini ion 2.9. (see [9, 17]) Le Dbe a collec ion o andom subse s o X. A andom se
ˆ
A={A(!):!2⌦}o Xis called a D- andom a ac o o , i he ollowing condi ions
a e sa is ied, o P-a.e. !2⌦,
(i) A(!)is compac and !7! d(x, A(!)) is measu able o any x2X,
(ii) ˆ
Ais in a ian , i.e., ( , !)A(!)=A(✓ !) o any 0,
(iii) ˆ
Apullback a ac s e e y membe o D, i.e., o e e y ˆ
B={B(!):!2⌦}2D,
lim
!+1d(( , ✓ !)B(✓ !),A(!)) = 0,
whe e dis he Hausdo ↵semi-me ic gi en by d(Y,Z)=sup
y2Y
in
z2ZkyzkX o any
Y,Z ⇢X.
In wha ollows, we s a e he de ini ion o weak solu ions o p oblem (2.2).
De ini ion 2.10. Assume ha 2L2(D),(u 0,'
0( 0),⌘ 0)2H:= H⇥L2(D)⇥M
o any ixed 02R, le T>
0be any ixed ime. The unc ion (u, #,⌘ )is called a weak
solu ion o p oblem (2.2) on he ime in e al [ 0,T], i o P-a.s. !2⌦,
u2L1( 0,T;H) L2( 0,T;V) H1( 0,T;V⇤),
#2L1( 0,T;L2(D)) L2( 0,T;H1(D)) H1( 0,T;(H1(D))⇤),
⌘ 2L1( 0,T;M),
and i sa is ies P-a.s. !2⌦,
h@ u, ⇣i+hB(u, u),⇣i+⌦@yu1,@
y⇣1↵+⌦@xu2,@
x⇣2↵=⌦#+z,⇣2↵,
h@ #,⇠i+h #, ⇠i+hu· #,⇠i+Z+1
0⌦µ( ) ⌘ ( ), ⇠↵d
=h ,⇠ihu· z,⇠i+hz,⇠i+↵hz,⇠i,
@ ⌘ +@s⌘ , M=(#, )M
o any ⇣=(⇣1,⇣2)2V,⇠2H1(D)and 2M.
8
3 The exis ence o a andom a ac o
In his sec ion, we will p o e he exis ence o a andom a ac o o he wo-dimensional
s ochas ic pa ial dissipa i e Boussinesq sys em wi h memo y and addi i e noise (2.2).
3.1 The well-posedness o p oblem (2.2)
The well-posedness esul o p oblem (2.2) can be ob ained by he s anda d Faedo-Gale kin
me hods ([32]). He e, we only s a e i as ollows.
Theo em 3.1. Assume (A1)-(A2) hold and 2L2(D). Then, o any !2⌦and any
(u 0,'
0( 0),⌘ 0)2H, he e exis s a unique weak solu ion (u, #,⌘ ) o p oblem (2.2) in he
sense o De ini ion 2.10, de ined on [ 0,T].
By Theo em 3.1, we can de ine a mapping
:R+⇥⌦⇥H!H
by
(  0,✓
0!)(u 0(x),'
0(x),⌘ 0(x)) =(u( , 0;!),'( , 0;!),⌘ ( 0;!))
:=(u( , 0;!),#( , 0;!)+¯z(✓ !),⌘ ( 0;!))
o any  0,whe e(u( , 0;!),#( , 0;!),⌘ ( 0;!)) is he weak solu ion o p oblem (2.2)
wi h ini ial da a (u( 0,
0;!),#( 0,
0;!),⌘ 0( 0;!)) = (u 0(x),'
0( 0)¯z(✓ 0!),⌘ 0(x)) 2H.
Tha is, a amily o mappings :R+⇥⌦⇥H!Hsa is ies
(1) (0,!)is heiden i yonH,
(2) ( +s, !)=( , ✓s!)(s, !) o all , s 2R+,
(3) ( , !): H!His con inuous o all 2R+.
3.2 The exis ence o a andom a ac o
In his subsec ion, we will p o e he exis ence o a andom a ac o o p oblem (2.2). To
his end, le Dbe he class o all amilies ˆ
D={D(!):!2⌦}o nonemp y subse s o H
such ha
lim
!+1e3c1
2 [D(✓ !)] = 0
o any !2⌦, whe e c1:= min{1
2,},1is he i s eigen alue o he S okes ope a o A
and [D(!)] = sup{kukH+k#kL2(D)+k⌘ kM:(u, #,⌘ )2D(!)}. Ob iously, he uni e se o
ixed bounded se s is con ained in D, so he esul s ha hold o he empe ed uni e se also
hold o he uni e se o ixed bounded se s.
9
We de i e om Poinca ´e’s inequali y and Young’s inequali y ha
ZD
(u· ¯z(✓s!)) #dx
kuk4k #k2k¯z(✓s!)k4Ck¯z(✓s!)k2
4k uk2
2+1
2k #k2
2,
hen we a i e a
1
2
d
ds ✓k#(s,  ;!)k2
2+Z+1
0
µ( )k ⌘s( )k2
2d ◆+1
2k #k2
2+
2Z+1
0
µ( )k ⌘s( )k2
2d
k k2
2+Ck#k2
2+Ck¯z(✓s!)k2
4k uk2
2+k¯z(✓s!)k2
2+↵2k¯z(✓s!)k2
2.(3.12)
I ollows om inequali ies (3.11)-(3.12) ha
d
ds ✓k u(s,  ;!)k2
2+k#(s,  ;!)k2
2+Z+1
0
µ( )k ⌘s( )k2
2d ◆
+kuk2
2+k #k2
2+Z+1
0
µ( )k ⌘s( )k2
2d
.k¯z(✓s!)k2
4k uk2
2+k#k2
2+k k2
2+k¯z(✓s!)k2
2+k¯z(✓s!)k2
2,
hen we in e om he classical G onwall inequali y ha he e exis s a posi i e cons an
M1(!)such ha o any!2⌦,
sup
s2[0,T ]k u(s,  ;!)k2
2+sup
s2[0,T ]k#(s,  ;!)k2
2+sup
s2[0,T ]k⌘s( ;!)k2
M
+ZT
0ku(s)k2
2ds +ZT
0k #(s)k2
2ds +ZT
0k⌘sk2
Mds
.✓ku0k2
V+k#0k2
2+k⌘0k2
M+ZT
0
1+k¯z(✓s!)k2
2+k¯z(✓s!)k2
2ds◆eRT
0(1+k¯z(✓s!)k2
4)ds
M1(!),
whe e we used he ac ha !¯z(✓ !) is con inuous om Rin o D(A↵⇤)and !¯z(✓ !)
is con inuous om Rin o D(A↵⇤1).
In o de o p o e (3.8)-(3.10), i only emains o show ha @ u2L2(0,T;H).To do his,
o any 2Hwi h k kH1, we ha e
h@ u, i=hB(u, u), i+D@2
yu1,@2
xu2>, E+⌦#+¯z(✓s!), 2↵.
Since
|hB(u, u), i|Ckuk4k uk4k k2.kukVkukZ
and
D@2
yu1,@2
xu2>, E
⌦@2
yu1, 1↵+⌦@2
xu2, 2↵
.ku1k2k k2+ku2k2k k2
kuk2,
16

hen we ha e
k@ ukH.kukVkukZ+kuk2+k#k2+k¯z(✓s!)k2,
which en ails ha
k@ uk2
L2(0,T ;H).kuk2
L1(0,T ;V)+1
ZT
0kuk2
Zds +ZT
0k#k2
2+k¯z(✓s!)k2
2ds 1+M1(!)2.
In wha ollows, we will p o e he asymp o ic compac ness o he cocycle associa ed
wi h p oblem (2.2).
Theo em 3.4. Assume (A1)-(A2) hold and 2L2(D). The cocycle co esponding o
p oblem (2.2) is D-pullback asymp o ically compac in H.
P oo . Le ˆ
B0={ˆ
B0(!):!2⌦}be he D- andom abso bing se in Hes ablished in
Theo em 3.2, o any !2⌦,any sequence { n}1
n=1 wi h n!+1and any sequence
(u n,#
 n,⌘ n)2ˆ
B0(✓ n!),
we will p o e ha he sequence
{( n,✓
 n!)(u n,#
 n,⌘ n)}1
n=1 ={(un(0, n;!),#
n(0, n;!)+¯z(!),⌘0
n( n;!))}1
n=1
possesses a con e gen subsequence in H.
Fo any T0andanys2Rwi h sT+ n0,deno e by
(un(sT),#
n(sT)+¯z(✓sT!),⌘sT
n)=(sT+ n,✓
 n!)(u n,#
 n,⌘ n),
hen by he de ini ion o cocycle and andom abso bing se , we ha e
(un(sT),#
n(sT)+¯z(✓sT!),⌘sT
n)2ˆ
B0(✓sT!)
and
(un(s),#
n(s)+¯z(✓s!),⌘s
n):=(T,✓sT!)(un(sT),#
n(sT)+¯z(✓sT!),⌘sT
n).
Fo any n1andanys0,le
n(s)=un(sT),
hen i ollows om Theo em 3.2, Lemma 3.3 and he Aubin-Lions compac ness lemma ha
he e exis s a subsequence o { n}1
n=1 (s ill deno e by { n}1
n=1), such ha {un(0)}1
n=1 =
{ n(T)}1
n=1 is con e gen in Hand { n}1
n=1 is con e gen in L2(0,T;V).Fo any n, m 1,
de ine
unm(s):= n(s) m(s)=un(sT)um(sT),
#nm(s):=#n(sT)#m(sT),⌘
s
nm := ⌘sT
n⌘sT
m.
17
Then i is clea ha (unm(s),#
nm(s),⌘s
nm) sa is ies he ollowing equa ion:
@s#nm #nm +un· #nm +unm · #mZ+1
0
µ( )⌘s
nm( )d =unm · ¯z(✓sT!).
We in e om H¨olde ’s inequali y and Young’s inequali y ha
1
2
d
ds(k#nm(s)k2
2+k⌘s
nmk2
M)+k #nmk2
2+
2k⌘s
nmk2
M
ZD
((unm · )#m)·#nm dx ZD
(unm · ¯z(✓sT!)) #nm dx
Ckunmk2
4k#mk2
4+1
2k #nmk2
2+Ckunmk2
4k¯z(✓sT!)k2
4,
which implies ha
d
ds(k#nm(s)k2
2+k⌘s
nmk2
M)+k #nmk2
2+k⌘s
nmk2
M
Ckunmk2
4k#mk2
4+Ckunmk2
4k¯z(✓sT!)k2
4,
hen we de i e om Poinca ´e’s inequali y and he classical G onwall inequali y ha , o any
s2[0,T],
k#nm(s)k2
2+k⌘s
nmk2
M
.(k#nm(0)k2
2+k⌘0
nmk2
M)ec1T
+ZT
0kunmk2
4k#mk2
4+kunmk2
4k¯z(✓sT!)k2
4ec1(Ts)ds
(k#nm(0)k2
2+k⌘0
nmk2
M)ec1T
+kunmkL1(0,T ;L2(D))k#mkL1(0,T ;L2(D))k unmkL2(0,T ;L2(D))k #mkL2(0,T ;L2(D))
+sup
s2[0,T ]k¯z(✓sT!)k2
4kunmkL2(0,T ;L2(D))k unmkL2(0,T ;L2(D)),
whe e c1=min{1
2,}.Due o he ac ha
(un(sT),#
n(sT),⌘sT
n)2ˆ
B0(✓sT!)
o any s2Rand T0, i ollows ha o any ✏>0, he e exis s a T2>0 such ha
(k#nm(0)k2
2+k⌘0
nmk2
M)ec1T2<✏
2
o any n, m 1.
Since { n}1
n=1 ={un(·T2)}1
n=1 is con e gen in L2(0,T
2;V),we ob ain
k unmkL2(0,T2;L2(D)) !0
18
as n, m !1.The e o e, we in e om Lemma 3.3 and he con inui y o 7! ¯z(✓ !) omR
in o D(A↵⇤), ha o he gi en ✏as abo e, he e exis s a na u al numbe N1such ha
o any n, m N,
kunmkL1(0,T2;L2(D))k#mkL1(0,T2;L2(D))k unmkL2(0,T2;L2(D))k #mkL2(0,T2;L2(D))
+sup
s2[0,T2]k¯z(✓sT2!)k2
4kunmkL2(0,T2;L2(D))k unmkL2(0,T2;L2(D)) <✏
2.
The e o e, we conclude ha o any ✏>0, he e exis s a na u al numbe N1,such ha
o any n, m N, we ha e
k#nm(T2)k2
2+k⌘T2
nmk2
M<✏,
which implies ha
{(#(0, n;!),⌘0( n;!))}1
n=1
is a Cauchy sequence in L2(D)⇥M.The e o e, he sequence
{( n,✓
 n!)(u n,#
 n,⌘ n)}1
n=1 ={(un(0, n;!),#
n(0, n;!)+¯z(!),⌘0
n( n;!))}1
n=1
has a con e gen subsequence in H.
F om Theo em 3.2, Theo em 3.4 and he abs ac heo y abou andom a ac o p o-
posed in [17], we immedia ely conclude he ollowing esul .
Theo em 3.5. Assume ha (A1)-(A2) hold and 2L2(D). The cocycle co esponding
o he wo-dimensional s ochas ic pa ial dissipa i e Boussinesq sys em wi h memo y and
addi i e noise (2.2) has a andom a ac o ˆ
A={A(!):!2⌦}in H.
4 The uppe semi-con inui y o he andom a ac o
The main objec i e o his sec ion is o p o e he uppe semi-con inui y o andom a ac o s
o p oblem (2.2) unde small iscosi y pe u ba ions. Fi s o all, we es ablish an abs ac
esul abou he uppe semi-con inui y o andom a ac o s.
P oposi ion 4.1. Le >0be a posi i e cons an . Assume ha he con inuous cocycle
✏0on a Banach space (X, k·k)has a andom a ac o ˆ
A✏0={A✏0(!):!2⌦}and
he co esponding cocycle ✏o i s pe u bed dynamical sys em possesses a andom a ac o
ˆ
A✏={A✏(!):!2⌦} o any ✏2[✏0,✏0+].In addi ion, assume he ollowing
condi ions hold:
(i) The e exis s a empe ed se ˆ
D={D(!):!2⌦}2Dsuch ha o any !2⌦,
[
|✏✏0|<
ˆ
A✏(!)⇢D(!).
19
(ii) Fo any !2⌦,>0and any ixed T>0, he e exis s >0,such ha o any
|✏✏0|<,
sup
u02D(✓T!)k✏(T,✓T!)u0✏0(T,✓T!)u0k<.
Then ˆ
A✏0and ˆ
A✏ha e he ollowing p ope y o uppe semi-con inui y, i.e. o any !2⌦,
lim
✏!✏0
dis (ˆ
A✏(!),ˆ
A✏0(!)) = 0.
P oo . F om he de ini ion o andom a ac o , we conclude ha o any >0andany
!2⌦, he e exis s a ime T=T(ˆ
D, ),such ha
dis (✏0(T,✓T!)D(✓T!),ˆ
A✏0(!)) <
2.
F om assump ion (ii), we in e ha he e exis s >0,such ha o any |✏✏0|<,
dis (✏(T,✓T!)D(✓T!),
✏0(T,✓T!)D(✓T!))
sup
u02D(✓T!)k✏(T,✓T!)u0✏0(T,✓T!)u0k
<
2.
The e o e, o any >0, he e exis s >0,such ha o any |✏✏0|<,
dis (ˆ
A✏(!),ˆ
A✏0(!)) = d(✏(T,✓T!)ˆ
A✏(✓T!),ˆ
A✏0(!)) <.
In o de o p o e he uppe semi-con inui y o andom a ac o o p oblem (2.2), o
any ✏>0,le us conside he ollowing iscosi y pe u bed sys em o p oblem (2.2):
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
@ u1
✏+u✏· u1
✏+@xp✏@2
xu1
✏@2
yu1
✏=0,(x, )2Q,
@ u2
✏+u✏· u2
✏+@yp@2
xu2
✏✏@2
yu2
✏=#✏+z, (x, )2Q,
·u✏=0,(x, )2Q,
@ #✏#✏+u✏· #✏R+1
0µ( )⌘
✏( )d = u✏· z+z( )+↵z, (x, )2Q,
@ ⌘
✏(s)=@s⌘
✏(s)+#✏( ),(x, )2Q, s 0,
u✏(x, 0)=u 0,#
✏(x, 0)='0( 0)z( 0),⌘
0
✏(x, s)=⌘0(s),x2D, s 0.
(4.1)
We no ice ha he iscosi y e ms ✏@2
xu1
✏and ✏@2
yu2
✏ha e no e↵ec on he esul o Theo em
3.5. The e o e, i is easy o ob ain ha o each 0 <✏1, he cocycle ✏co esponding o
p oblem (4.1) has a andom a ac o ˆ
A✏={A✏(!):!2⌦}in H.
Now, we a e eady o s a e and p o e he main esul o his sec ion.
20
Theo em 4.2. Assume (A1)-(A2) hold and 2L2(D). The andom a ac o ˆ
A=
{A(!):!2⌦}o p oblem (2.2) and he andom a ac o ˆ
A✏={A✏(!):!2⌦}o p oblem
(4.1) ha e he uppe semi-con inui y p ope y, i.e. o any !2⌦,
lim
✏!✏0
dis (ˆ
A✏(!),ˆ
A✏0(!)) = 0.
P oo . Le ˆ
D={D(!):!2⌦}be he andom abso bing se es ablished in Theo em 3.2, i
is easy o deduce om he p oo o Theo em 3.2 and he p ope y o andom a ac o ha
o any !2⌦,
[
0<✏1
ˆ
A✏(!)⇢ˆ
D(!).
Le (u, #,⌘ ), (u✏,#
✏,⌘
✏) be he weak solu ions o p oblem (2.2) and (4.1), espec i ely.
Deno e by (˜u, ˜
#,˜⌘ )=(uu✏,##✏,⌘ ⌘
✏), hen (˜u, ˜
#,˜⌘ ) sa is ies he ollowing p oblem:
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
@ ˜u1+˜u· u1+u✏· ˜u1+✏@2
xu1
✏@2
y˜u1=0,(x, )2Q,
@ ˜u2+˜u· u2+u✏· ˜u2@2
x˜u2+✏@2
yu2
✏=˜
#,(x, )2Q,
·˜u=0,(x, )2Q,
@ ˜
#˜
#+˜u· #+u✏· ˜
#R+1
0µ( )˜⌘ ( )d =˜u· z, (x, )2Q,
@ ˜⌘ (s)=@s˜⌘ (s)+˜
#( ),(x, )2Q, s 0,
˜u 0(x)=0,˜
# 0(x)=0,˜⌘ 0(x, s)=0,x2D, s 0.
(4.2)
Mul iplying he i s and second equa ion o (4.2) by ˜u1and ˜u2, espec i ely, henin eg a ing
he esul ing equali ies o e D, we in e om Lemma 2.3 and Young’s inequali y ha
1
2
d
dsk˜u(s,  0;!)k2
2+1
2k ˜uk2
2
C✏2k u✏k2
2+1+Ck uk2
2k˜uk2
2+1
8k ˜uk2
2+k˜
#k2
2.(4.3)
Mul iplying now he ou h equa ion o (4.2) by ˜
#and in eg a ing o e D,weob ain
1
2
d
ds ✓k˜
#(s,  0;!)k2
2+Z+1
0
µ( )k ˜⌘s( )k2
2d ◆+k ˜
#k2
21
2Z+1
0
µ0( )k ˜⌘s( )k2
2d
=ZD
(˜u· #)˜
#dxZD
(˜u· ¯z(✓s!)) ˜
#dx.
Mo eo e , we in e om he in e pola ion and Young’s inequali ies ha
ZD
(˜u· #)˜
#dx
k˜uk4k #k2k˜
#k4
Ck˜uk
1
2
2k ˜uk
1
2
2k #k
1
2
2k #k
1
2
2k˜
#k
1
2
2k ˜
#k
1
2
2
Ck˜uk2k ˜uk2k #k2+Ck #k2k˜
#k2k ˜
#k2
Ck #k2
2k˜uk2
2+1
16k ˜uk2
2+Ck #k2
2k˜
#k2
2+1
4k ˜
#k2
2
21

and
ZD
(˜u· ¯z(✓s!)) ˜
#dx
k˜uk4k ¯z(✓s!)k2k˜
#k4
Ck˜uk
1
2
2k ˜uk
1
2
2k ¯z(✓s!)k2k ˜
#k2
Ck˜uk2k ˜uk2k ¯z(✓s!)k2
2+1
8k ˜
#k2
2
Ck ¯z(✓s!)k4
2k˜uk2
2+1
16k ˜uk2
2+1
8k ˜
#k2
2.
Then we a i e a
1
2
d
ds ✓k˜
#(s,  0;!)k2
2+Z+1
0
µ( )k ˜⌘s( )k2
2d ◆+k ˜
#k2
21
2Z+1
0
µ0( )k ˜⌘s( )k2
2d
Ck #k2
2k˜uk2
2+Ck #k2
2k˜
#k2
2+Ck ¯z(✓s!)k4
2k˜uk2
2+1
8k ˜uk2
2+3
8k ˜
#k2
2.(4.4)
We conclude om inequali ies (4.3)-(4.4) and he assump ion on he memo y ke nel ha
d
ds ✓k˜u(s,  0;!)k2
2+k˜
#(s,  0;!)k2
2+Z+1
0
µ( )k ˜⌘s( )k2
2d ◆
+k ˜uk2
2+k ˜
#k2
2+Z+1
0
µ( )k ˜⌘ ( )k2
2d
.1+k uk2
2+k #k2
2+k ¯z(✓s!)k4
2k˜uk2
2+1+k #k2
2k˜
#k2
2+✏2k u✏k2
2.
Thus, we de i e om he classical G onwall inequali y ha o any  0,
k˜u( ,  0;!)k2
2+k˜
#( ,  0;!)k2
2+k˜⌘ k2
M
.✏2Z
0k u✏(s)k2
2eR
s1+k u(⌧)k2
2+k #(⌧)k2
2+k ¯z(✓⌧!)k4
2d⌧ds !0
as ✏!0+.
Acknowledgemen
Pa ial inancial suppo was ecei ed om he Na ional Science Founda ion o China G an
(11871389), he Fundamen al Resea ch Funds o he Cen al Uni e si ies (xzy012022008),
Shaanxi Fundamen al Science Resea ch P ojec o Ma hema ics and Physics (22JSY032)
and he Spanish Minis e io de Ciencia e Inno aci´on, Agencia Es a al de In es igaci´on and
FEDER G an PID2021-122991NB-C21.
22
Re e ences
[1] D. Adhika i, C. S. Cao, H. Shang, J. Wu, X. Xu, and Z. Ye. Global egula i y esul s
o he 2D Boussinesq equa ions wi h pa ial dissipa ion. J. Di↵e en ial Equa ions,
260(2):1893–1917, 2016.
[2] D. Adhika i, C. S. Cao, and J. H. Wu. The 2D Boussinesq equa ions wi h e ical
iscosi y and e ical di↵usi i y. J. Di↵e en ial Equa ions, 249(5):1078–1088, 2010.
[3] D. Adhika i, C. S. Cao, and J. H. Wu. Global egula i y esul s o he 2D Boussinesq
equa ions wi h e ical dissipa ion. J. Di↵e en ial Equa ions, 251(6):1637–1655, 2011.
[4] L. A nold, C. Jones, K. Mischaikow, and G. Raugel. Random Dynamical Sys ems.
Sp inge , Be lin, Ge many, 1998.
[5] P. W. Ba es, K. N. Lu, and B. X. Wang. Random a ac o s o s ochas ic eac ion-
di↵usion equa ions on unbounded domains. J. Di↵e en ial Equa ions, 246(2):845–869,
2009.
[6] A. Biswas, C. Foias, and A. La ios. On he a ac o o he semi-dissipa i e Boussinesq
equa ions. Ann. Ins . Hen i Poinca ´e, Anal. Non Lin´eai e, 34(2):381–405, 2017.
[7] S. Bo ini and V. Pa a. Uni o m a ac o s o a s ongly damped wa e equa ion wi h
linea memo y. Asymp o . Anal., 20(3):263–277, 1999.
[8] C. S. Cao and J. H. Wu. Global egula i y o he wo-dimensional aniso opic Boussi-
nesq equa ions wi h e ical dissipa ion. A ch. Ra ion. Mech. Anal., 208(3):985–1004,
2013.
[9] T. Ca aballo and P. E. Kloeden. Non-au onomous a ac o o in eg o-di↵e en ial
e olu ion equa ions. Disc e e Con in. Dyn. Sys . Se . S, 2(1):17–36, 2009.
[10] T. Ca aballo, J. Real, and I. D. Chuesho . Pullback a ac o s o s ochas ic hea
equa ions in ma e ials wi h memo y. Disc e e Con in. Dyn. Sys . Se . B, 9(3-4):525539,
2008.
[11] D. Chae. Global egula i y o he 2D Boussinesq equa ions wi h pa ial iscosi y e ms.
Ad . Ma h., 203(2):497–513, 2006.
[12] J.-Y. Chemin, B. Desja dins, I. Gallaghe , and E. G enie . Fluids wi h aniso opic
iscosi y. Ma h. Model. Nume . Anal., 34(2):315–335, 2000.
[13] J.-Y. Chemin, B. Desja dins, I. Gallaghe , and E. G enie . Ma hema ical Geophysics:
An In oduc ion o Ro a ing Fluids and he Na ie -S okes Equa ions. Ox o d Uni .
P ess, 2006.
23
[14] I. D. Chuesho and M. Scheu zow. Ine ial mani olds and o ms o s ochas ically
pe u bed e a ded semilinea pa abolic equa ions. J. Dynam. Di↵e en ial Equa ions,
13(2):355–380, 2001.
[15] P. Cons an in and C. Foias. Na ie -S okes Equa ions, Chicago Lec u es in Ma hema ics.
Uni e si y o Chicago P ess, Chicago, 1988.
[16] H. C auel, A. Debussche, and F. Flandoli. Random a ac o s. J. Dynam. Di↵e en ial
Equa ions, 9:307–341, 1997.
[17] H. C auel and F. Flandoli. A ac o s o andom dynamical sys ems. P obab. Theo y
Rela ed Fields, 100:365–393, 1994.
[18] G. Da P a o and J. Zabczyk. S ochas ic Equa ions in In ini e Dimensions. Camb idge
Uni e si y P ess, 2014.
[19] S. Ga i, A. Mi an ille, V. Pa a, and S. Zelik. A ac o s o semi-linea equa ions
o iscoelas ici y wi h e y low dissipa ion. Rocky Moun ain J. Ma h., 38:1117–1138,
2008.
[20] A. E. Gill. A mosphe e-Ocean Dynamics. Academic P ess, London, 1982.
[21] C. Gio gi, V. Pa a, and A. Ma zocchi. Uni o m a ac o s o a non-au onomous semi-
linea hea equa ion wi h memo y. Qua . Appl. Ma h., 58(4):661683, 2000.
[22] J. F. He, S. Ma, and C. Y. Sun. Well-posedness and a ac o s o a 2D Boussinesq
sys em wi h pa ial dissipa ion. J. Di↵e en ial Equa ions, 319:1–40, 2022.
[23] T. Hmidi, S. Ke aani, and F. Rousse . Global well-posedness o a Boussinesq-Na ie -
S okes sys em wi h c i ical dissipa ion. J. Di↵e en ial Equa ions, 249(9):2147–2174,
2010.
[24] T. Hmidi, S. Ke aani, and F. Rousse . Global well-posedness o Eule -Boussinesq
sys em wi h c i ical dissipa ion. Comm. Pa ial Di↵e en ial Equa ions, 36(3):420–445,
2011.
[25] P. E. Kloeden and J. A. Langa. Fla ening, squeezing and he exis ence o andom
a ac o s. P oc. A, 463(2077):163–181, 2007.
[26] A. La ios, E. Lunasin, and E. S. Ti i. Global well-posedness o he 2D Boussinesq
sys em wi h aniso opic iscosi y and wi hou hea di↵usion. J. Di↵e en ial Equa ions,
255(9):2636–2654, 2013.
[27] F. Li and B. You. Random a ac o o he s ochas ic Cahn-Hillia d-Na ie -S okes
sys em wi h small addi i e noise. S och. Anal. Appl., 36(3):546–559, 2018.
[28] A. J. Majda. In oduc ion o PDEs and Wa es o he A mosphe e and Ocean. ol.9,
Ame ican Ma hema ical Soc., 2003.
24
[29] A. J. Majda and M. J. G o e. Model dynamics and e ical collapse in decaying s ongly
s a i ied lows. Phys. Fluids, 9(10):2932–2940, 1997.
[30] T. Nguyen Duong. Exis ence and long- ime beha io o solu ions o he eloci y-
o ici y-Voig model o he 3D Na ie -S okes equa ions wi h damping and memo y.
Ma h. Me hods Appl. Sci., 45(17):11635–11658, 2022.
[31] J. Pedlosky. Geophysical Fluid Dyanmics. Sp inge , New Yo k, 1987.
[32] R. Temam. In ini e-Dimensional Dynamical Sys ems in Mechanics and Physics. Ap-
plied Ma hema ical Sciences, ol. 68, Sp inge , New Yo k, 1997.
[33] J. H. Xu and T. Ca aballo. Long ime beha io o s ochas ic nonlocal pa ial di↵e en ial
equa ions and Wong-Zakai app oxima ions. SIAM J. Ma h. Anal., 54:2792–2844, 2002.
[34] J. H. Xu, T. Ca aballo, and J. Vale o. Asymp o ic beha io o a semilinea p oblem
in hea conduc ion wi h long ime memo y and non-local di↵usion. J. Di↵e en ial
Equa ions, 327(9):418–447, 2022.
[35] J. H. Xu, T. Ca aballo, and J. Vale o. Asymp o ic beha io o nonlocal pa ial di↵e en-
ial equa ions wi h long ime memo y. Disc e e Con in. Dyn. Sys . Se . S, 15(10):3059–
3078, 2022.
[36] D. D. Yang, Z. Chen, and T. Ca aballo. Dynamics o a globally modi ied Na ie -S okes
model wi h double delay. Z. Angew. Ma h. Phys., 73(5):216, 2022.
[37] S. Yang, T. Ca aballo, and Y. Li. Dynamics and s abili y analysis o s ochas ic 3D
Lag angian-a e aged Na ie -S okes equa ions wi h in ini e delay on unbounded do-
mains. Appl. Ma h. Op im., 89(1):11,43, 2024.
[38] B. You. The exis ence o a andom a ac o o he h ee dimensional damped Na ie -
S okes equa ions wi h addi i e noise. S och. Anal. Appl., 35(4):691–700, 2017.
25