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PureMathematics
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,2022,12(8),1346-1359
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128148
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GlobalExistenceofThree-Dimensional
StochasticPrimitiveEquations
inFourier-BesovSpaces
NingLi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity, LanzhouGansu
Received:Jul.18
th
,2022;accepted:Aug.18
th
,2022;published:Aug.30
th
,2022
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DOI:10.12677/pm.2022.128148
o
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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,
Itˆo
Formula,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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p
“
_
C
†
.
2.
ý
•
£
Ä
k
,
0
Littlewood-Paley
n
Ø
Ú
Bony
•
È
©
)
n
Ø
±
9
†
à
g
Fourier-Besov
˜
m
ƒ
'
¼
ê
˜
m
½
Â
.
ä
N
Œ
ë
„
;
Í
[15,16].
S
(
R
3
)
•
Schwartz
˜
m
,
S
0
(
R
3
)
•
…
O
2
Â
¼
ê
˜
m
,
ψ,ϕ
∈
S
(
R
3
)
´
»
•
¼
ê
,
9
ˆ
ψ
Ú
ˆ
ϕ
÷
v
e
5
Ÿ
:
suppˆ
ϕ
⊂B
:=
{
ξ
∈
R
3
:
|
ξ
|≤
4
3
}
,
supp
ˆ
ψ
⊂C
:=
{
ξ
∈
R
3
:
3
4
≤|
ξ
|≤
8
3
}
,
…
X
j
∈
Z
ˆ
ϕ
(2
−
j
ξ
) = 1
,
∀
ξ
∈
R
3
\{
0
}
.
DOI:10.12677/pm.2022.1281481349
n
Ø
ê
Æ
o
w
-
ψ
j
(
x
) :=2
3
j
ψ
(2
j
x
),
ϕ
j
(
x
) :=2
3
j
ϕ
(2
j
x
),
K
à
g
ª
Ç
Û
Ü
z
Ž
f
∆
j
Ú
à
g
$
ª
ä
Ž
f
S
j
f
Œ
©
O
½
Â
•
∆
j
f
:=
ϕ
j
∗
f,S
j
f
=
ψ
j
∗
f
∀
j
∈
Z
,f
∈
S
0
(
R
3
)
.
-
S
0
h
(
R
3
) :=
S
0
(
R
3
)
/
P
[
R
3
],
Ù
¥
P
[
R
3
]
•
½
Â
3
R
3
þ
N
õ
‘
ª
¤
¤
‚
5
˜
m
.
¯
¤
±•
,
3
S
0
h
(
R
3
)
¥
k
X
e
©
)
¤
á
:
f
=
X
j
∈
Z
∆
j
f,S
j
f
=
X
j
0
≤
j
−
1
∆
j
0
f,
¿
…
ª
Ç
Û
Ü
z
Ž
f
∆
j
Ú
$
ª
ä
Ž
f
S
j
þ
•
L
p
L
p
‚
5
Ž
f
.
d
, Littlewood-Paley
©
)
÷
v
±
e
A
5
Ÿ
:
∆
j
∆
k
f
= 0
,
|
j
−
k
|≥
2
,
Ú
∆
j
(
S
k
−
1
f
∆
k
f
) = 0
,
|
j
−
k
|≥
5
.
Ú
n
2.1([16])
1
≤
p
≤
q
≤∞
,r
∈
(0
,R
)
,j
∈
Z
.
K
é
?
¿
õ
•
I
γ
∈
Z
3
+
S
{
0
}
,
±
e
O
¤
á
:
(1)
e
supp
ˆ
f
⊂{
ξ
∈
R
3
:
|
ξ
|≤
R
2
j
}
,
K
k
(
iξ
)
γ
ˆ
f
k
L
q
≤
C
2
j
|
γ
|
+3
j
(
1
q
−
1
p
)
k
ˆ
f
k
L
p
;
(2)
e
supp
ˆ
f
⊂{
ξ
∈
R
3
:
r
2
j
≤|
ξ
|≤
R
2
j
}
,
K
k
ˆ
f
k
L
p
≤
C
2
−
j
|
γ
|
sup
|
α
|
=
|
γ
|
k
(
iξ
)
α
ˆ
f
k
L
p
.
½
Â
2.2
(1)
s
∈
R
,
1
≤
p,r
≤∞
,
à
g
Fouier-Besov
˜
m
F
˙
B
s
p,r
(
R
3
)
½
Â
•
F
˙
B
s
p,r
(
R
3
) :=
n
f
∈
S
0
h
(
R
3
) :
k
f
k
F
˙
B
s
p,r
=
n
2
js
k
d
∆
j
f
k
L
p
ξ
o
j
∈
Z
l
r
<
∞
o
.
(2)
é
?
¿
0
<T
≤∞
,
s
∈
R
,
1
≤
δ,p,r
≤∞
, Chemin-Lerner
.
˜
m
˜
L
δ
(0
,T
;
F
˙
B
s
p,r
(
R
3
))
½
Â
•
˜
m
C
((0
,T
];
F
˙
B
s
p,r
(
R
3
))
3
‰
ê
k
f
k
˜
L
δ
(0
,T
;
F
˙
B
s
p,r
)
:=
n
2
js
k
d
∆
j
f
k
L
δ
(0
,T
;
L
p
ξ
(
R
3
))
o
j
∈
Z
l
r
e
z
˜
m
.
½
Â
2.3
é
?
¿
0
<T
≤∞
,
s
∈
R
,
1
≤
p,r,δ,ρ
≤∞
, Bochner
.
Chemin-Lerner
.
˜
m
˜
L
ρ
(Ω;
˜
L
δ
(0
,T
;
F
˙
B
s
p,r
(
R
3
)))
½
Â
•
˜
L
ρ
(Ω;
˜
L
δ
(0
,T
;
F
˙
B
s
p,r
(
R
3
))) :=
n
f
∈M
T
:
f
(
ω,t
)
∈
S
0
h
(
R
3
)
,
P
−
a.s,
…
k
f
k
˜
L
ρ
(Ω;
˜
L
δ
(0
,T
;
F
˙
B
s
p,r
))
=
n
2
js
[
E
(
k
d
∆
j
f
k
L
δ
(0
,T
;
L
p
ξ
(
R
3
))
)
ρ
]
1
ρ
o
j
∈
Z
l
r
<
∞
o
.
•
,
‰
Ñ
•
§
|
(1
.
1)
d
È
©•
§
Ú
Stokes-Coriolis-Stratification
Œ
+
{
T
K
,N
}
t
≥
0
½
DOI:10.12677/pm.2022.1281481350
n
Ø
ê
Æ
o
w
Â
.
µ
=
ν
=1,
v
:=(
u
1
,u
2
,u
3
,
√
gθ/
N
),
v
0
:=(
u
0
1
,u
0
2
,u
0
3
,
√
gθ
0
/
N
),
e
∇
:=(
∂
1
,∂
2
,∂
3
,
0),
N
:=
N
√
g
,
Ð
Š
¯
K
(1.1)
Œ
=
†
•
X
e
d
¯
K
:
dv
+(
Q
v
+
S
v
+
e
∇
p
)
dt
=
−
(
v
·
e
∇
)
vdt
+
Gd
W
,
(
x,t,ω
)
∈
R
3
×
(0
,
∞
)
×
Ω
,
e
∇·
v
=0
,
(
x,t,ω
)
∈
R
3
×
(0
,
∞
)
×
Ω
,
v
|
t
=0
=
v
0
,
(
x,ω
)
∈
R
3
×
Ω
,
(2
.
1)
Ù
¥
G
:= (
h,
√
gl/
N
),
Q
:=
−
∆000
0
−
∆00
00
−
∆0
000
−
∆
,
S
:=
0
−K
00
K
000
000
−
N
00
N
0
.
(2
.
2)
d
[17]
Œ
•
,
¯
K
(2.1)
‚
5
z
¯
K
¤
)
¤
Stokes-Coriolis-Stratification
Œ
+
{
T
K
,N
(
t
)
}
t
≥
0
Œ
ä
N
L
ˆ
•
:
T
K
,N
(
t
)
f
:=
F
−
1
cos(
|
ξ
|
0
|
ξ
|
t
)
e
−|
ξ
|
2
t
M
1
(
ξ
)
ˆ
f
+sin(
|
ξ
|
0
|
ξ
|
t
)
e
−|
ξ
|
2
t
M
2
(
ξ
)
ˆ
f
+
e
−|
ξ
|
2
t
M
3
(
ξ
)
ˆ
f
,
(2
.
3)
Ù
¥
ξ
= (
ξ
1
,ξ
2
,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
|
0
2
0
−
N
2
ξ
1
ξ
3
|
ξ
|
0
2
K
Nξ
2
ξ
3
|
ξ
|
0
2
0
K
2
ξ
2
3
|
ξ
|
0
2
−
N
2
ξ
2
ξ
3
|
ξ
|
0
2
−
K
Nξ
1
ξ
3
|
ξ
|
0
2
−
K
2
ξ
1
ξ
3
|
ξ
|
0
2
−
K
2
ξ
2
ξ
3
|
ξ
|
0
2
N
2
(
ξ
2
1
+
ξ
2
2
)
|
ξ
|
0
2
0
K
Nξ
2
ξ
3
|
ξ
|
0
2
−
K
Nξ
1
ξ
3
|
ξ
|
0
2
0
N
2
(
ξ
2
1
+
ξ
2
2
)
|
ξ
|
0
2
,
(2
.
4)
M
2
(
ξ
) =
0
−
K
ξ
2
3
|
ξ
||
ξ
|
0
K
ξ
2
ξ
3
|
ξ
||
ξ
|
0
Nξ
1
ξ
3
|
ξ
||
ξ
|
0
K
ξ
2
3
|
ξ
||
ξ
|
0
0
−
K
ξ
1
ξ
3
|
ξ
||
ξ
|
0
Nξ
2
ξ
3
|
ξ
||
ξ
|
0
−
K
ξ
2
ξ
3
|
ξ
||
ξ
|
0
K
ξ
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
|
0
2
−
N
2
ξ
1
ξ
2
|
ξ
|
0
2
0
−
K
Nξ
2
ξ
3
|
ξ
|
0
2
−
N
2
ξ
1
ξ
2
|
ξ
|
0
2
N
2
ξ
2
1
|
ξ
|
0
2
0
K
Nξ
1
ξ
3
|
ξ
|
0
2
0000
−
K
Nξ
2
ξ
3
|
ξ
|
0
2
K
Nξ
1
ξ
3
|
ξ
|
0
2
0
K
2
ξ
2
3
|
ξ
|
0
2
.
(2
.
6)
Š
˜
J
´
,
N
´
y
,
é
ξ
∈
R
3
,
M
l
(
ξ
) (
l
=1
,
2
,
3)
z
‡
š
"
©
þ
M
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).
½
Â
Helmholtz
Ý
K
Ž
f
P
:= (
P
jk
)
4
×
4
,
Ù
¥
P
jk
:=
δ
jk
+
R
j
R
k
,
1
≤
j,k
≤
3
,
δ
jk
,
Ù
¦
,
DOI:10.12677/pm.2022.1281481351
n
Ø
ê
Æ
o
w
Ù
¥
δ
jk
L
«
Kronecker
Î
Ò
,
{
R
j
}
1
≤
j
≤
3
L
«
Riesz
C
†
.
d
Duhamel
n
,
•
§
|
(1.1)
d
u
X
e
È
©•
§
:
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
)
P
Gd
W
.
(2
.
7)
3.
V
‚
5
O
Ú
‘
Å
O
•
?
n
•
§
|
¥
é
6
‘
,
·
‚
I
‡
3
ƒ
A
¼
ê
˜
m
¥
ï
á
'
u
Stokes-Coriolis-
Stratification
Œ
+
V
‚
5
O
.
•
d
,
·
‚
k
5
ï
á
ƒ
A
¦
È
{
K
.
Ú
n
3.1[
¦
È
{
K
]
2
≤
p
≤∞
,
1
≤
r
≤∞
,
K
•
3
~
ê
C
1
>
0
¦
k
v
1
v
2
k
e
L
2
(0
,
∞
;
F
˙
B
−
1+
3
p
0
p,r
)
≤
C
1
k
v
1
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
k
v
2
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
.
y
²
d
½
Â
2.2
±
9
Bony
•
È
©
)
E
|
Œ
,
¤
á
k
v
1
v
2
k
e
L
2
(0
,
∞
;
F
˙
B
−
1+
3
p
0
p,r
)
=
n
2
j
(
−
1+
3
p
0
)
k
ϕ
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
ˆ
v
1
∗
ϕ
k
ˆ
v
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
ˆ
v
2
∗
ϕ
k
ˆ
v
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
ˆ
v
1
∗
ϕ
k
0
ˆ
v
2
)
L
2
(0
,
∞
;
L
p
ξ
)
j
∈
Z
l
r
=:
I
1
+
I
2
+
I
3
.
é
u
I
1
,
½
j
,
(
Ü
Ú
n
2.1,
Œ
2
j
(
−
1+
3
p
0
)
X
|
k
−
j
|≤
4
ϕ
j
(
ψ
k
ˆ
v
1
∗
ϕ
k
ˆ
v
2
)
L
2
(0
,
∞
;
L
p
ξ
)
≤
C
2
j
(
−
1+
3
p
0
)
X
|
k
−
j
|≤
4
k
ψ
k
ˆ
v
1
k
L
1
ξ
k
ϕ
k
ˆ
v
2
k
L
p
ξ
L
2
(0
,
∞
)
≤
C
2
j
(
−
1+
3
p
0
)
X
|
k
−
j
|≤
4
X
k
0
≤
k
−
2
2
3
k
0
(1
−
1
p
)
k
ϕ
k
0
ˆ
v
1
k
L
p
ξ
k
ϕ
k
ˆ
v
2
k
L
p
ξ
L
2
(0
,
∞
)
≤
C
2
j
(
−
1+
3
p
0
)
X
|
k
−
j
|≤
4
X
k
0
≤
k
−
2
2
3
k
0
(1
−
1
p
)
k
ϕ
k
0
ˆ
v
1
k
L
4
(0
,
∞
;
L
p
ξ
)
k
ϕ
k
ˆ
v
2
k
L
4
(0
,
∞
;
L
p
ξ
)
≤
C
2
j
(
−
1+
3
p
0
)
X
|
k
−
j
|≤
4
2
k
0
(
−
1
2
+
3
p
0
)
k
ϕ
k
0
ˆ
v
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)
k
ϕ
k
ˆ
v
2
k
L
4
(0
,
∞
;
L
p
ξ
)
≤
C
k
v
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
)
k
ϕ
k
ˆ
v
2
k
L
4
(0
,
∞
;
L
p
ξ
)
2
(
j
−
k
)(
−
1+
3
p
0
)
.
DOI:10.12677/pm.2022.1281481352
n
Ø
ê
Æ
o
w
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d
Ä
:
þ
,
2$
^
Young
Ø
ª
,
Œ
I
1
≤
C
k
v
1
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
k
v
2
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
.
Ó
n
Œ
,
¤
á
I
2
≤
C
k
v
1
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
k
v
2
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
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.
e
¡
O
I
3
,
é
u
½
j
,
Œ
2
j
(
−
1+
3
p
0
)
X
k
≥
j
−
2
X
|
k
−
k
0
|≤
1
ϕ
j
(
ϕ
k
ˆ
v
1
∗
ϕ
k
0
ˆ
v
2
)
L
2
(0
,
∞
;
L
p
ξ
)
≤
C
2
j
(
−
1+
3
p
0
)
X
k
≥
j
−
2
X
|
k
−
k
0
|≤
1
ϕ
j
(
ϕ
k
ˆ
v
1
∗
ϕ
k
0
ˆ
v
2
)
L
2
(0
,
∞
;
L
p
ξ
)
≤
C
2
j
(
−
1+
3
p
0
)
X
k
≥
j
−
2
X
|
k
−
k
0
|≤
1
k
ϕ
k
0
ˆ
v
2
k
L
1
ξ
k
ϕ
k
ˆ
v
1
k
L
p
ξ
L
2
(0
,
∞
)
≤
C
2
j
(
−
1+
3
p
0
)
X
k
≥
j
−
2
X
|
k
−
k
0
|≤
1
2
3
k
0
(1
−
1
p
)
k
ϕ
k
0
ˆ
v
2
k
L
p
ξ
k
ϕ
k
ˆ
v
1
k
L
p
ξ
L
2
(0
,
∞
)
≤
C
2
j
(
−
1+
3
p
0
)
X
k
≥
j
−
2
X
|
k
−
k
0
|≤
1
2
3
k
0
(1
−
1
p
)
k
ϕ
k
0
ˆ
v
2
k
L
4
(0
,
∞
;
L
p
ξ
)
k
ϕ
k
ˆ
v
1
k
L
4
(0
,
∞
;
L
p
ξ
)
≤
C
2
j
(
−
1+
3
p
0
)
X
k
≥
j
−
2
2
k
0
(
−
1
2
+
3
p
0
)
k
ϕ
k
0
ˆ
v
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)
k
ϕ
k
ˆ
v
1
k
L
4
(0
,
∞
;
L
p
ξ
)
≤
C
k
v
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
)
k
ϕ
k
ˆ
v
1
k
L
4
(0
,
∞
;
L
p
ξ
)
2
(
j
−
k
)(
−
1+
3
p
0
)
.
3
d
Ä
:
þ
,
$
^
Young
Ø
ª
,
Œ
I
3
≤
C
k
v
1
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
k
v
2
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
.
n
þ
Œ
,
•
3
C
1
>
0
¦
k
v
1
v
2
k
e
L
2
(0
,
∞
;
F
˙
B
−
1+
3
p
0
p,r
)
≤
C
1
k
v
1
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
k
v
2
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
)
,
=
(
Ø
y
.
Ú
n
3.2[
V
‚
5
O
]
2
≤
p
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,
1
≤
r
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,
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•
3
~
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C
2
>
0,
¦
Z
t
0
T
K
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(
t
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τ
)
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e
∇
(
v
1
⊗
v
2
)(
τ
)
dτ
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
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C
2
k
v
1
k
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
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k
v
2
k
e
L
4
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,
∞
;
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1
2
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3
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0
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DOI:10.12677/pm.2022.1281481353
n
Ø
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y
²
d
Ú
n
2.1
Ú
Ú
n
3.1,
Œ
Z
t
0
T
K
,N
(
t
−
τ
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P
e
∇
(
v
1
⊗
v
2
)(
τ
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dτ
e
L
4
(0
,
∞
;
F
˙
B
−
1
2
+
3
p
0
p,r
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2
j
(
−
1
2
+
3
p
0
)
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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
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2
k
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
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2
k
L
4
3
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,
∞
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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
)
k
ϕ
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
•
3
5
,
d
U
\
n
,
·
‚
•
Ä
X
e
‘
Å
‚
5
Ð
Š
¯
K
dβ
+
Q
βdt
+
S
βdt
=
Gd
W
,
(
x,t,ω
)
∈
R
3
×
(0
,
∞
)
×
Ω
,
β
|
t
=0
=
v
0
,
(
x,ω
)
∈
R
3
×
Ω
.
(3
.
1)
w
,
,
‚
5
¯
K
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)
N
•
3
…
•
˜
.
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þ
,
Ï
L
é
¯
K
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Ò
ü
à
Ó
ž
'
u
˜
m
C
þ
‰
Fourier
C
†
,
Œ
†
)
ä
N
L
ˆ
/
ª
.
e
¡
,
·
‚
ï
á
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Å
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5
¯
K
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3
Fourier-Besov
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m
µ
e
e
k
.
5
O
.
Ú
n
3.3
-
r
∈
[2
,
+
∞
].
v
0
´
F
0
Œ
ÿ
,
…
v
0
∈
˜
L
4
(Ω;
F
˙
B
1
2
2
,r
(
R
3
)),
©
Ù
L
§
G
´
ÌS
Œ
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
k
β
k
˜
L
4
(Ω;
˜
L
4
(0
,T
;
F
˙
B
1
2
,r
))
≤
C
3
h
k
v
0
k
˜
L
4
(Ω;
F
˙
B
1
2
2
,r
)
+(1+
T
)
k
G
k
˜
L
4
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L
4
(0
,T
;
F
˙
B
1
2
2
,r
))
i
.
(3
.
2)
y
²
é
¯
K
(3.1)
ü
>
Ó
ž
'
u
˜
m
C
þ
‰
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
)
d
W
.
(3
.
3)
Ä
k
,
é
k
ϕ
j
ˆ
β
k
2
L
2
¦
^
Itˆo
ú
ª
,
Œ
d
k
ϕ
j
ˆ
β
k
2
L
2
=
d
h
ϕ
j
ˆ
β,ϕ
j
ˆ
β
i
= 2
h
d
(
ϕ
j
ˆ
β
)
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j
ˆ
β
i
+
h
d
(
ϕ
j
ˆ
β
)
,d
(
ϕ
j
ˆ
β
)
i
= 2
h
ϕ
j
ˆ
β,
−
[
|
ξ
|
2
(
ϕ
j
ˆ
β
)+
S
(
ϕ
j
ˆ
β
)]
dt
+(
ϕ
j
ˆ
G
)
d
Wi
+
k
ϕ
j
ˆ
G
k
2
L
2
dt
= (
−
2
k|·|
ϕ
j
ˆ
β
k
2
L
2
+
k
ϕ
j
ˆ
G
k
2
L
2
)
dt
+2
h
ϕ
j
ˆ
β,ϕ
j
ˆ
G
i
d
W
,
(3.4)
DOI:10.12677/pm.2022.1281481354
n
Ø
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Æ
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w
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¥
h·
,
·i
L
«
L
2
(
R
3
)
S
È
.
,
,
é
(
k
ϕ
j
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β
k
2
L
2
+
)
2
¦
^
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ú
ª
,
Ù
¥
>
0,
¿
(
Ü
(3.4)
ª
,
Œ
d
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
2
= 2(
k
ϕ
j
ˆ
β
k
2
L
2
+
)[(
−
2
k|·|
ϕ
j
ˆ
β
k
2
L
2
+
k
ϕ
j
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G
k
2
L
2
)
dt
+2
h
ϕ
j
ˆ
β,ϕ
j
ˆ
G
i
d
W
]
+4
h
ϕ
j
ˆ
β,ϕ
j
ˆ
G
i
2
dt.
(3.5)
•
Ä
Ê
ž
S
τ
n
=
inf
{
t
≥
0 :
k
ϕ
j
ˆ
β
k
L
2
>n
}
,
{
t
:
k
ϕ
j
ˆ
β
k
L
2
>n
}6
= Ø
,
T,
{
t
:
k
ϕ
j
ˆ
β
k
L
2
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}
= Ø
,
Ù
¥
n
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,
2
,
3
,
···
.
3
[0
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t
≤
min
{
T,τ
n
}
)
þ
é
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ª
È
©
,
,
é
¤
(
J
Ï
"
,
Œ
E
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
2
−
E
(
k
ϕ
j
ˆ
v
0
k
2
L
2
+
)
2
= 2
E
Z
t
0
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
k
ϕ
j
ˆ
G
k
2
L
2
dτ
−
4
E
Z
t
0
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
k|·|
ϕ
j
ˆ
β
k
2
L
2
dτ
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E
Z
t
0
h
ϕ
j
ˆ
G,ϕ
j
ˆ
β
i
2
dτ
+4
E
Z
t
0
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
h
ϕ
j
ˆ
β,ϕ
j
ˆ
G
i
d
W
=:
J
1
+
J
2
+
J
3
+
J
4
.
e
5
•
g
O
J
1
,J
2
,J
3
,J
4
.
¯¢
þ
,
•
I
O
J
1
,J
3
,J
4
,
Ï
•
J
2
´
·
‚
¤
I
‡
/
ª
.
J
1
= 2
E
Z
t
0
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
k
ϕ
j
ˆ
G
k
2
L
2
dτ
≤
C
2
E
sup
τ
∈
[0
,t
]
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
Z
t
0
k
ϕ
j
ˆ
G
k
2
L
2
dτ
≤
C
E
sup
τ
∈
[0
,t
]
(
k
ϕ
j
ˆ
β
k
2
L
2
+
)
2
+
C
t
E
Z
t
0
k
ϕ
j
ˆ
G
k
4
L
2
dτ.
¿
…
,
J
3
≤
C
E
sup
τ
∈
[0
,t
]
k
ϕ
j
ˆ
β
k
4
L
2
+
C
t
E
Z
t
0
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k
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k
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[0
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k
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DOI:10.12677/pm.2022.1281481355
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DOI:10.12677/pm.2022.1281481356
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1
2
2
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DOI:10.12677/pm.2022.1281481357
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k
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©
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