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PureMathematicsnØêÆ,2022,12(8),1346-1359
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128148
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n‘‘Å•§|§Littlewood-PaleynاBony•È©)E|§Itˆoúª§N)
GlobalExistenceofThree-Dimensional
StochasticPrimitiveEquations
inFourier-BesovSpaces
NingLi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity, LanzhouGansu
Received:Jul.18
th
,2022;accepted:Aug.18
th
,2022;published:Aug.30
th
,2022
©ÙÚ^:ow.n‘‘Å•§|3Fourier-Besov˜m¥)N•35[J].nØêÆ, 2022,12(8):1346-1359.
DOI:10.12677/pm.2022.128148
ow
Abstract
Thispaperisdevotedtostudyingtheglobalexistenceofsolutionstoinitialvalue
problem ofthethree-dimensionalstochasticprimitiveequations, which are abasicsys-
temthatisusually usedtodescribe thedynamicbehavior oftheatmosphericandthe
oceanicflows.Firstly,byusingthe Littlewood-Paley theoryandBonypara-productde-
compositiontechnique,weestablishanewbilinearestimationfortheStokes-Coriolis-
Stratificationsemigroup.Then,byestablishingtheboundednessestimationsforsolu-
tionsofthecorrespondingstochasticlinearinitialvalueproblem,andcombiningthe
superpositionprincipleandBanach’sfixedpointtheorem,weprovetheglobalexis-
tenceanduniquenessofmildsolutionstothethree-dimensionalstochasticprimitive
equationswithsmallinitialvaluesandsmallrandomexternalforcesintheFourier-
Besov spaceframe. Ourmainresultisageneralizationoftheglobalexistenceofthe
solutionsfortheinitialvalueproblemoftheclassicalthree-dimensionalprimitivee-
quationsunderthestochasticcase.
Keywords
Three-DimensionalStochasticPrimitiveEquations,Littlewood-PaleyTheory,Bony
ParaproductDecompositionTechnique,ItˆoFormula,GlobalSolutions
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2022.1281481347nØêÆ
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DOI:10.12677/pm.2022.1281481348nØêÆ
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DOI:10.12677/pm.2022.1281481349nØêÆ
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DOI:10.12677/pm.2022.1281481350nØêÆ
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,ξ
3
) ∈R
3
,|ξ|=
p
ξ
2
1
+ξ
2
2
+ξ
2
3
,|ξ|
0
=
p
N
2
ξ
2
1
+N
2
ξ
2
2
+K
2
ξ
2
3
,
M
1
(ξ) =




K
2
ξ
2
3
|ξ|
02
0−
N
2
ξ
1
ξ
3
|ξ|
02
KNξ
2
ξ
3
|ξ|
02
0
K
2
ξ
2
3
|ξ|
02
−
N
2
ξ
2
ξ
3
|ξ|
02
−
KNξ
1
ξ
3
|ξ|
02
−
K
2
ξ
1
ξ
3
|ξ|
02
−
K
2
ξ
2
ξ
3
|ξ|
02
N
2
(ξ
2
1
+ξ
2
2
)
|ξ|
02
0
KNξ
2
ξ
3
|ξ|
02
−
KNξ
1
ξ
3
|ξ|
02
0
N
2
(ξ
2
1
+ξ
2
2
)
|ξ|
02




,(2.4)
M
2
(ξ) =




0−
Kξ
2
3
|ξ||ξ|
0
Kξ
2
ξ
3
|ξ||ξ|
0
Nξ
1
ξ
3
|ξ||ξ|
0
Kξ
2
3
|ξ||ξ|
0
0−
Kξ
1
ξ
3
|ξ||ξ|
0
Nξ
2
ξ
3
|ξ||ξ|
0
−
Kξ
2
ξ
3
|ξ||ξ|
0
Kξ
1
ξ
3
|ξ||ξ|
0
0−
N(ξ
2
1
+ξ
2
3
)
|ξ||ξ|
0
−
Nξ
1
ξ
3
|ξ||ξ|
0
−
Nξ
2
ξ
3
|ξ||ξ|
0
N(ξ
2
1
+ξ
2
3
)
|ξ||ξ|
0
0




,(2.5)
M
3
(ξ) =






N
2
ξ
2
2
|ξ|
02
−
N
2
ξ
1
ξ
2
|ξ|
02
0−
KNξ
2
ξ
3
|ξ|
02
−
N
2
ξ
1
ξ
2
|ξ|
02
N
2
ξ
2
1
|ξ|
02
0
KNξ
1
ξ
3
|ξ|
02
0000
−
KNξ
2
ξ
3
|ξ|
02
KNξ
1
ξ
3
|ξ|
02
0
K
2
ξ
2
3
|ξ|
02






.(2.6)
Š˜J´, N´y, éξ∈R
3
, M
l
(ξ) (l=1,2,3) z‡š"©þM
l
jk
(ξ) (j,k=1,2,3,4) ÷
v|M
l
jk
(ξ)|≤max{2,
|K|
N
,
N
|K|
}(j,k= 1,2,3,4).
½ÂHelmholtz ÝKŽfP:= (P
jk
)
4×4
,Ù¥
P
jk
:=

δ
jk
+R
j
R
k
,1 ≤j,k≤3,
δ
jk
,Ù¦,
DOI:10.12677/pm.2022.1281481351nØêÆ
ow
Ù¥δ
jk
L«KroneckerÎÒ,{R
j
}
1≤j≤3
L«Riesz C†.dDuhameln, •§|(1.1) duX
eÈ©•§:
v(t) = T
K,N
(t)v
0
−
R
t
0
T
K,N
(t−τ)P
e
∇·(v(τ)⊗v(τ))dτ+
R
t
0
T
K,N
(t−τ)PGdW.
(2.7)
3.V‚5OÚ‘ÅO
•?n•§|¥é6‘,·‚I‡3ƒA¼ê˜m¥ïá'uStokes-Coriolis-
StratificationŒ+V‚5O. •d,·‚k5ïáƒA¦È{K.
Ún3.1[¦È{K]2 ≤p≤∞,1 ≤r≤∞, K•3~êC
1
>0 ¦
kv
1
v
2
k
e
L
2
(0,∞;F
˙
B
−1+
3
p
0
p,r
)
≤C
1
kv
1
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
kv
2
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
.
y²d½Â2.2 ±9Bony•È©)E|Œ, ¤á
kv
1
v
2
k
e
L
2
(0,∞;F
˙
B
−1+
3
p
0
p,r
)
=



n
2
j(−1+
3
p
0
)
kϕ
j
F(v
1
v
2
)k
L
2
(0,∞;L
p
ξ
)
o
j∈Z



l
r
≤





2
j(−1+
3
p
0
)




X
|k−j|≤4
ϕ
j
(ψ
k
ˆv
1
∗ϕ
k
ˆv
2
)




L
2
(0,∞;L
p
ξ
)

j∈Z




l
r
+





2
j(−1+
3
p
0
)




X
|k−j|≤4
ϕ
j
(ψ
k
ˆv
2
∗ϕ
k
ˆv
1
)




L
2
(0,∞;L
p
ξ
)

j∈Z




l
r
+





2
j(−1+
3
p
0
)




X
k≥j−2
X
|k−k
0
|≤1
ϕ
j
(ϕ
k
ˆv
1
∗ϕ
k
0
ˆv
2
)




L
2
(0,∞;L
p
ξ
)

j∈Z




l
r
=: I
1
+I
2
+I
3
.
éuI
1
,½j, (ÜÚn2.1,Œ
2
j(−1+
3
p
0
)




X
|k−j|≤4
ϕ
j
(ψ
k
ˆv
1
∗ϕ
k
ˆv
2
)




L
2
(0,∞;L
p
ξ
)
≤C2
j(−1+
3
p
0
)
X
|k−j|≤4


kψ
k
ˆv
1
k
L
1
ξ
kϕ
k
ˆv
2
k
L
p
ξ


L
2
(0,∞)
≤C2
j(−1+
3
p
0
)
X
|k−j|≤4




X
k
0
≤k−2
2
3k
0
(1−
1
p
)
kϕ
k
0
ˆv
1
k
L
p
ξ
kϕ
k
ˆv
2
k
L
p
ξ




L
2
(0,∞)
≤C2
j(−1+
3
p
0
)
X
|k−j|≤4
X
k
0
≤k−2
2
3k
0
(1−
1
p
)
kϕ
k
0
ˆv
1
k
L
4
(0,∞;L
p
ξ
)
kϕ
k
ˆv
2
k
L
4
(0,∞;L
p
ξ
)
≤C2
j(−1+
3
p
0
)
X
|k−j|≤4




2
k
0
(−
1
2
+
3
p
0
)
kϕ
k
0
ˆv
1
k
L
4
(0,∞;L
p
ξ
)




l
r
(k
0
≤k−2)



2
1
2
k
0



l
r
0
(k
0
≤k−2)
kϕ
k
ˆv
2
k
L
4
(0,∞;L
p
ξ
)
≤Ckv
1
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
X
|k−j|≤4
2
k(−
1
2
+
3
p
0
)
kϕ
k
ˆv
2
k
L
4
(0,∞;L
p
ξ
)
2
(j−k)(−1+
3
p
0
)
.
DOI:10.12677/pm.2022.1281481352nØêÆ
ow
3dÄ:þ,2$^Young Øª,Œ
I
1
≤Ckv
1
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
kv
2
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
.
ÓnŒ,¤á
I
2
≤Ckv
1
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
kv
2
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
.
e¡OI
3
,éu½j, Œ
2
j(−1+
3
p
0
)




X
k≥j−2
X
|k−k
0
|≤1
ϕ
j
(ϕ
k
ˆv
1
∗ϕ
k
0
ˆv
2
)




L
2
(0,∞;L
p
ξ
)
≤C2
j(−1+
3
p
0
)
X
k≥j−2
X
|k−k
0
|≤1


ϕ
j
(ϕ
k
ˆv
1
∗ϕ
k
0
ˆv
2
)


L
2
(0,∞;L
p
ξ
)
≤C2
j(−1+
3
p
0
)
X
k≥j−2
X
|k−k
0
|≤1


kϕ
k
0
ˆv
2
k
L
1
ξ
kϕ
k
ˆv
1
k
L
p
ξ


L
2
(0,∞)
≤C2
j(−1+
3
p
0
)
X
k≥j−2
X
|k−k
0
|≤1


2
3k
0
(1−
1
p
)
kϕ
k
0
ˆv
2
k
L
p
ξ
kϕ
k
ˆv
1
k
L
p
ξ


L
2
(0,∞)
≤C2
j(−1+
3
p
0
)
X
k≥j−2
X
|k−k
0
|≤1
2
3k
0
(1−
1
p
)
kϕ
k
0
ˆv
2
k
L
4
(0,∞;L
p
ξ
)
kϕ
k
ˆv
1
k
L
4
(0,∞;L
p
ξ
)
≤C2
j(−1+
3
p
0
)
X
k≥j−2



2
k
0
(−
1
2
+
3
p
0
)
kϕ
k
0
ˆv
2
k
L
4
(0,∞;L
p
ξ
)



l
r
(|k−k
0
|≤1)



2
1
2
k
0



l
r
0
(|k−k
0
|≤1)
kϕ
k
ˆv
1
k
L
4
(0,∞;L
p
ξ
)
≤Ckv
2
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
X
k≥j−2
2
k(−
1
2
+
3
p
)
kϕ
k
ˆv
1
k
L
4
(0,∞;L
p
ξ
)
2
(j−k)(−1+
3
p
0
)
.
3dÄ:þ,$^Young Øª,Œ
I
3
≤Ckv
1
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
kv
2
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
.
nþŒ,•3C
1
>0 ¦
kv
1
v
2
k
e
L
2
(0,∞;F
˙
B
−1+
3
p
0
p,r
)
≤C
1
kv
1
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
kv
2
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
,
=(Øy.
Ún3.2[V‚5O]2 ≤p≤∞,1 ≤r≤∞, K•3~êC
2
>0, ¦




Z
t
0
T
K,N
(t−τ)P
e
∇(v
1
⊗v
2
)(τ)dτ




e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
≤C
2
kv
1
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
kv
2
k
e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
.
DOI:10.12677/pm.2022.1281481353nØêÆ
ow
y²dÚn2.1 ÚÚn3.1, Œ




Z
t
0
T
K,N
(t−τ)P
e
∇(v
1
⊗v
2
)(τ)dτ




e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
=





2
j(−
1
2
+
3
p
0
)




Z
t
0
ϕ
j
F[T
K,N
(t−τ)P
e
∇(v
1
⊗v
2
)(τ)]dτ




L
4
(0,∞;L
p
ξ
)

j∈Z




l
r
≤C





2
j(−
1
2
+
3
p
0
)




Z
t
0
ϕ
j
e
−(t−τ)2
2k
F[
e
∇(v
1
⊗v
2
)(τ)]dτ




L
4
(0,∞;L
p
ξ
)

j∈Z




l
r
≤C





2
j(−
1
2
+
3
p
0
)


e
−(t−τ)2
2k


L
4
3
(0,∞)


ϕ
j
F[
e
∇(v
1
⊗v
2
)(t)]


L
2
(0,∞;L
p
ξ
)

j∈Z




l
r
≤C





2
j(−2+
3
p
0
)
kϕ
j
F[
e
∇(v
1
⊗v
2
)(t)]k
L
2
(0,∞;L
p
ξ
)

j∈Z




l
r
)
≤C


e
∇(v
1
⊗v
2
)


L
2
(0,∞;F
˙
B
−2+
3
p
0
p,r
)
≤C


(v
1
⊗v
2
)


L
2
(0,∞;F
˙
B
−1+
3
p
0
p,r
)
≤C


v
1


e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)


v
2


e
L
4
(0,∞;F
˙
B
−
1
2
+
3
p
0
p,r
)
,
=(Øy.
,˜•¡,•¯K(1.1))N•35, dU\n,·‚•ÄXe‘Å‚5Њ¯K

dβ+Qβdt+Sβdt= GdW,(x,t,ω) ∈R
3
×(0,∞)×Ω,
β|
t=0
= v
0
,(x,ω) ∈R
3
×Ω.
(3.1)
w,, ‚5¯K(3.1) )N•3…•˜.¯¢þ,ÏLé¯K(3.1) ÒüàÓž'u˜mCþ
‰FourierC†, Œ†)äNLˆ/ª.
e¡,·‚ïá‘Å‚5¯K(3.1) 3Fourier-Besov˜mµeek.5O.
Ún3.3-r∈[2,+∞]. v
0
´F
0
Œÿ, …v
0
∈
˜
L
4
(Ω;F
˙
B
1
2
2,r
(R
3
)),©ÙL§G´ÌSŒ
ÿ,…G∈
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
(R
3
))).K¯K(3.1))β∈
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2,r
(R
3
))),…÷v
kβk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2,r
))
≤C
3
h
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
i
.(3.2)
y²é¯K(3.1) ü>Óž'u˜mCþ‰Fourier C†,,Ó¦ϕ
j
,Œ
d(ϕ
j
ˆ
β) = −[|ξ|
2
(ϕ
j
ˆ
β)+S(ϕ
j
ˆ
β)]dt+(ϕ
j
ˆ
G)dW.(3.3)
Äk,ékϕ
j
ˆ
βk
2
L
2
¦^Itˆo úª,Œ
dkϕ
j
ˆ
βk
2
L
2
= dhϕ
j
ˆ
β,ϕ
j
ˆ
βi
= 2hd(ϕ
j
ˆ
β),ϕ
j
ˆ
βi+hd(ϕ
j
ˆ
β),d(ϕ
j
ˆ
β)i
= 2hϕ
j
ˆ
β,−[|ξ|
2
(ϕ
j
ˆ
β)+S(ϕ
j
ˆ
β)]dt+(ϕ
j
ˆ
G)dWi+kϕ
j
ˆ
Gk
2
L
2
dt
= (−2k|·|ϕ
j
ˆ
βk
2
L
2
+kϕ
j
ˆ
Gk
2
L
2
)dt+2hϕ
j
ˆ
β,ϕ
j
ˆ
GidW,(3.4)
DOI:10.12677/pm.2022.1281481354nØêÆ
ow
Ù¥h·,·iL«L
2
(R
3
)SÈ.
,,é(kϕ
j
ˆ
βk
2
L
2
+)
2
¦^Itˆo úª,Ù¥>0, ¿(Ü(3.4) ª,Œ
d(kϕ
j
ˆ
βk
2
L
2
+)
2
= 2(kϕ
j
ˆ
βk
2
L
2
+)[(−2k|·|ϕ
j
ˆ
βk
2
L
2
+kϕ
j
ˆ
Gk
2
L
2
)dt+2hϕ
j
ˆ
β,ϕ
j
ˆ
GidW]
+4hϕ
j
ˆ
β,ϕ
j
ˆ
Gi
2
dt.(3.5)
•ÄÊžS
τ
n
=

inf{t≥0 : kϕ
j
ˆ
βk
L
2
>n},{t: kϕ
j
ˆ
βk
L
2
>n}6= Ø,
T,{t: kϕ
j
ˆ
βk
L
2
>n}= Ø,
Ù¥n= 1,2,3,···.3[0,t] (t≤min{T,τ
n
})þé(3.5)ªÈ©, ,é¤(JÏ",Œ
E(kϕ
j
ˆ
βk
2
L
2
+)
2
−E(kϕ
j
ˆv
0
k
2
L
2
+)
2
= 2E
Z
t
0
(kϕ
j
ˆ
βk
2
L
2
+)kϕ
j
ˆ
Gk
2
L
2
dτ−4E
Z
t
0
(kϕ
j
ˆ
βk
2
L
2
+)k|·|ϕ
j
ˆ
βk
2
L
2
dτ
+4E
Z
t
0
hϕ
j
ˆ
G,ϕ
j
ˆ
βi
2
dτ+4E
Z
t
0
(kϕ
j
ˆ
βk
2
L
2
+)hϕ
j
ˆ
β,ϕ
j
ˆ
GidW
=: J
1
+J
2
+J
3
+J
4
.
e5•gOJ
1
,J
2
,J
3
,J
4
.¯¢þ, •IOJ
1
,J
3
,J
4
,Ï•J
2
´·‚¤I‡/ª.
J
1
= 2E
Z
t
0
(kϕ
j
ˆ
βk
2
L
2
+)kϕ
j
ˆ
Gk
2
L
2
dτ
≤C2Esup
τ∈[0,t]
(kϕ
j
ˆ
βk
2
L
2
+)
Z
t
0
kϕ
j
ˆ
Gk
2
L
2
dτ
≤CEsup
τ∈[0,t]
(kϕ
j
ˆ
βk
2
L
2
+)
2
+C

tE
Z
t
0
kϕ
j
ˆ
Gk
4
L
2
dτ.
¿…,
J
3
≤CEsup
τ∈[0,t]
kϕ
j
ˆ
βk
4
L
2
+C

tE
Z
t
0
kϕ
j
ˆ
Gk
4
L
2
dτ.
d,dBurkholder-Davis-GundyØªÚYoung Øª,Œ
J
4
= 4E
Z
t
0
(kϕ
j
ˆ
βk
2
L
2
+)hϕ
j
ˆ
β,ϕ
j
ˆ
GidW
≤CEsup
τ
0
∈[0,t]




Z
τ
0
0
(kϕ
j
ˆ
βk
2
L
2
+)hϕ
j
ˆ
β,ϕ
j
ˆ
GidW




≤CE

Z
t
0
|(kϕ
j
ˆ
βk
2
L
2
+)hϕ
j
ˆ
β,ϕ
j
ˆ
Gi|
2
dτ

1
2
≤CE

Z
t
0
(kϕ
j
ˆ
βk
2
L
2
+)
2
kϕ
j
ˆ
βk
2
L
2
kϕ
j
ˆ
Gk
2
L
2
dτ

1
2
≤CEsup
τ∈[0,t]
(kϕ
j
ˆ
βk
2
L
2
+)kϕ
j
ˆ
βk
L
2

Z
t
0
kϕ
j
ˆ
Gk
2
L
2
dτ

1
2
≤CEsup
τ∈[0,t]
[(kϕ
j
ˆ
βk
2
L
2
+)kϕ
j
ˆ
βk
L
2
]
4
3
+C

tE
Z
t
0
kϕ
j
ˆ
Gk
4
L
2
dτ.
DOI:10.12677/pm.2022.1281481355nØêÆ
ow
(ÜJ
1
,J
2
,J
3
ÚJ
4
O,ÏL'u→0 4•,Œ
Esup
t∈[0,T∧τ
n
]
kϕ
j
ˆ
βk
4
L
2
+E
Z
T∧τ
n
0
k|·|ϕ
j
ˆ
βk
2
L
2
kϕ
j
ˆ
βk
2
L
2
dτ
≤CEkϕ
j
ˆv
0
k
4
L
2
+C(1+(T∧τ
n
))E
Z
T∧τ
n
0
kϕ
j
ˆ
Gk
4
L
2
dτ.(3.6)
?˜Ú, dv
0
ÚGb^‡Œ(3.6) ª¥Esup
t∈[0,T∧τ
n
]
kϕ
j
ˆ
βk
4
L
2
Œ˜‡†nÃ'~ê
››.Ïd, -n→∞,Œlim
n→∞
τ
n
= T,P−a.s.. 2ŠâÚn2.1, ¤á
Esup
t∈[0,T]
kϕ
j
ˆ
βk
4
L
2
+2
2j
E
Z
T
0
kϕ
j
ˆ
βk
4
L
2
dτ≤CEkϕ
j
ˆv
0
k
4
L
2
+C(1+T)E
Z
T
0
kϕ
j
ˆ
Gk
4
L
2
dτ.(3.7)
Ïd,
2
2j
E
Z
T
0
kϕ
j
ˆ
βk
4
L
2
dτ≤CEkϕ
j
ˆv
0
k
4
L
2
+C(1+T)E
Z
T
0
kϕ
j
ˆ
Gk
4
L
2
dτ.
éþªü>Ó¦2
1
2
j
,,l
r
‰ê,¤á
kβk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2,r
))
≤C
3
h
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
i
,
=y(ؤá.
5º3.4d(3.7)ª•Œ
Esup
t∈[0,T]
kϕ
j
ˆ
βk
4
L
2
≤CEkϕ
j
ˆv
0
k
4
L
2
+C(1+T)E
Z
T
0
kϕ
j
ˆ
Gk
4
L
2
dτ.
éþªü>Ó¦2
1
2
j
, ,l
r
‰ê, Œ¼¯K(3.1) )β∈
˜
L
4
(Ω;
˜
L
∞
(0,T;F
˙
B
1
2
2,r
(R
3
))), ¿…
3Ún3.3 ^‡e,¤á
kβk
˜
L
4
(Ω;
˜
L
∞
(0,T;F
˙
B
1
2
2,r
))
≤C
4
h
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
i
.
Ún3.53Ún3.3 b^‡e, K•3˜‡VÇ8Ü
e
Ω ¦¯K(3.1) )β(ω,·,·)∈
˜
L
4
(0,T;F
˙
B
1
2,r
(R
3
)),¿…é?¿ω∈
e
Ω,¤á
kβ(ω,·,·)k
˜
L
4
(0,T;F
˙
B
1
2,r
)
≤C
5
h
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
i
,
Ù¥~êC
5
>0.
y²•Ä8Ü
Ω
0
:=
n
ω: kβ(ω,·,·)k
˜
L
4
(0,T;F
˙
B
1
2,r
)
>C
5
h
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
io
,(3.8)
DOI:10.12677/pm.2022.1281481356nØêÆ
ow
Ù¥~êC
5
>0 –½.dÚn3.3 ÚTchebychevØª, Œ•
P(Ω
0
) ≤Ekβ(ω,·,·)k
4
˜
L
4
(0,T;F
˙
B
1
2,r
)
C
−4
5
h
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
i
−4
≤(C
3
C
−1
5
)
4
.

˜
Ω = Ω\Ω
0
…C
5
>C
3
,KŒ
P(
˜
Ω) = 1−P(Ω
0
) ≥1−(C
3
C
−1
5
)
4
>0.
K(Øy.
4.½n1.4y²
½n1.4 y²½Â)NΨ Ú)˜mZ©O•
Ψ(v)(t) := β−B(v,v)
Ú
Z:=
n
v∈
˜
L
4
(0,T;F
˙
B
1
2,r
(R
3
)) : kvk
˜
L
4
(0,T;F
˙
B
1
2,r
)
≤2
o
,
Ù¥>0 –½,
β= T
K,N
(t)v
0
+
Z
t
0
T
K,N
(t−τ)PGdW,
±9
B(v,v) =
Z
t
0
T
K,N
(t−τ)P
e
∇·(v(τ)⊗v(τ))dτ.
Kò¦)¯K(1.1)=†•ÏéNΨ(v)ØÄ:.•d,·‚òy•3>0,¦Ψ ´r
Banach˜mZNZØ N.
Äk,dÚn3.2Œ, ¤á
kB(v,v)k
Z
≤C
3
kvk
Z
kvk
Z
.(4.1)
qdÚn3.5 Œ•,•3VÇ‘Å8Ü
˜
Ω,¦é?¿ω∈
˜
Ω,¤á
kβk
Z
≤C
5
h
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
i
.
Ïd,é?¿v∈Z,¤á
kΨ(v)k
Z
≤



T
K,N
(t)v
0
+
Z
t
0
T
K,N
(t−τ)PGdW



Z
+kB(v,v)k
Z
≤C
5

kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))

+C
2
kvk
2
Z
.
DOI:10.12677/pm.2022.1281481357nØêÆ
ow
,,daqL§Œ, éu?¿v
1
,v
2
∈Z,¤á
kΨ(v
1
)−Ψ(v
2
)k
Z
=



Z
t
0
T
K,N
(t−τ)P
e
∇·

[v
1
(τ)⊗(v
1
−v
2
)(τ)]+[(v
1
−v
2
)⊗v
2
(τ)]

dτ



Z
≤C
2
(kv
1
k
Z
+kv
2
k
Z
)kv
1
−v
2
k
Z
.
yv, ¦
0 <<min

1
4C
2
,1

,
¿…bé?¿T>0, v
0
∈L
4
(Ω;F
˙
B
1
2
2,r
(R
3
))ÚG∈L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
(R
3
)))…÷v
kv
0
k
˜
L
4
(Ω;F
˙
B
1
2
2,r
)
+(1+T)kGk
˜
L
4
(Ω;
˜
L
4
(0,T;F
˙
B
1
2
2,r
))
≤

C
5
.
KŒ
kΨ(v)k
Z
≤+4C
2
·min

1
4C
2
,1

≤2,
¿…
kΨ(v
1
)−Ψ(v
2
)k
Z
≤
1
2
kv
1
−v
2
k
Z
.
KdBanachØ NnŒ,¯K(1.1)•3•˜N§Ú)(u,θ)(ω,·,·) ∈
˜
L
4
(0,T;F
˙
B
1
2,r
(R
3
)),
…÷v
k(u,θ)k
˜
L
4
(0,T;F
˙
B
1
2,r
)
≤2.

ë•©z
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AnalysisandRelatedTopics,Vol.1,AcademicPress,Cambridge,MA,1-43.
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