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PureMathematics
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,2021,11(12),1957-1966
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112218
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UniquenessofSolutionstoInitialValue
ProblemsofFractionalAnisotropic
Navier-StokesEquations
MixiuLiu,XiaochunSun
∗
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Nov.1
st
,2021;accepted:Dec.2
nd
,2021;published:Dec.9
th
,2021
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1957-1966.DOI:10.12677/pm.2021.1112218
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Abstract
Inthispaper,weprovedtheuniquenessofthesolutionoftheinitialvalueproblemof
incompressibleNavier-Stokesequationwithonlyhorizontalfractionaldissipationin
the anisotropic Sobolev function space
H
2
α
−
2
,s
(
R
3
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, where
1
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2
.The
keyoftheproofistogivetheproductformulaofthefunctionwhen
(
α,s,t
)
satisfies
theappropriaterange,andthentheconclusionisobtainedbyusingFourieranalysis
technique.
Keywords
FractionalAnisotropicNavier-StokesEquations,SobolevSpace,ProductFormula
Copyright
c
2021byauthor(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.2021.11122181960
n
Ø
ê
Æ
4
æ
D
§
š
S
e
|
η
h
|≤
|
ξ
h
|
2
,
K
h
ξ
h
+
η
h
i'h
ξ
h
i
.
e
ξ
h
2
≤|
η
h
|≤
2
|
ξ
h
|
,
K
h
η
h
i'h
ξ
h
i
.
u
´
,
Z
|
η
h
|≤
|
ξ
h
|
2
h
ξ
h
+
η
h
i
2(
s
+
t
−
1)
h
η
h
i
2
t
dη
h
'h
ξ
h
i
2(
s
+
t
−
1)
Z
|
η
h
|≤
|
ξ
h
|
2
1
h
η
h
i
2
t
dη
h
≤
C
1
−
t
h
ξ
h
i
2
s
…
Z
ξ
h
2
≤|
η
h
|≤
2
|
ξ
h
|
h
ξ
h
+
η
h
i
2(
s
+
t
−
1)
h
η
h
i
2
t
dη
h
'
1
h
ξ
h
i
2
t
Z
|
ξ
h
|
2
≤|
η
h
|≤
2
|
ξ
h
|
h
ξ
h
+
η
h
i
2(
s
+
t
−
1)
dη
h
≤
C
h
ξ
h
i
2
t
Z
|
ζ
|≤
3
|
ξ
h
|
h
ζ
i
2(
s
+
t
−
1)
dζ
≤
C
s
+
t
h
ξ
h
i
2
s
,
ù
p
·
‚
Š
C
þ
O
†
ζ
=
ξ
h
+
η
h
.
Š
â
I
3
½
Â
,
·
‚
I
3
≤
C
Z
h
ξ
h
i
2
s
|
b
f
(
ξ
)
|
2
dξ
h
,
'
u
C
þ
(
ξ
3
,η
3
)
$
^
H¨older
Ø
ª
,
d
d
I
1
O
:
I
1
≤
C
ZZ
Z
h
ξ
h
i
2
s
h
ξ
3
i
2
s
0
|
b
f
(
ξ
)
|
2
dξ
h
Z
|
h
(
ζ,ξ
3
+
η
3
)
|
2
dζ
1
2
×
h
ξ
3
+
η
3
i
−
α
h
ξ
3
i
−
2
s
0
Z
h
η
h
i
2
t
|
b
g
(
η
)
|
2
dη
h
1
2
dξ
3
dη
3
.
l
I
1
≤
C
k
f
k
s,s
0
k
h
k
L
2
(
Z
h
η
h
i
2
t
ϕ
(
η
3
)
|
b
g
(
η
)
|
2
dη
)
1
2
,
(2
.
2)
Ù
¥
ϕ
(
η
3
) =
Z
1
h
ξ
3
+
η
3
i
α
h
ξ
3
i
2
s
0
.
e
¡
·
‚
O
ϕ
,
Z
|
ξ
3
|≥
2
|
η
3
|
1
h
ξ
3
+
η
3
i
α
h
ξ
3
i
2
s
0
dξ
3
'
Z
∞
2
|
η
3
|
1
h
ξ
3
i
2
s
0
+1
dξ
3
'
Z
∞
2
|
η
3
|
1
h
(1+
ξ
3
)
i
2
s
0
+1
dξ
3
≤
C
h
η
3
i
2
s
0
≤
C
h
η
3
i
α
,
Z
|
ξ
3
|≤
|
η
3
|
2
1
h
ξ
3
+
η
3
i
α
h
ξ
3
i
2
s
0
dξ
3
'
1
h
η
3
i
α
Z
|
ξ
3
|≤
|
η
3
|
2
1
h
ξ
3
i
2
s
0
dξ
3
≤
C
(
s
0
−
1
2
)
h
η
3
i
α
,
Z
|
η
3
|
2
≤|
ξ
3
|≤
2
|
η
3
|
1
h
ξ
3
+
η
3
i
α
h
ξ
3
i
2
s
0
dξ
3
'
1
h
η
3
i
2
s
0
Z
|
η
3
|
2
≤|
ξ
3
|≤
2
|
η
3
|
1
h
ξ
3
+
η
3
i
α
dξ
3
≤
C
h
η
3
i
2
s
0
Z
|
ζ
|≤
3
|
η
3
|
1
h
ζ
i
α
dζ
≤
C
(
s
0
−
1
2
)
h
η
3
i
α
,
DOI:10.12677/pm.2021.11122181961
n
Ø
ê
Æ
4
æ
D
§
š
S
l
ϕ
(
η
3
)
≤
C
h
η
3
i
−
α
,
ò
þ
ª
“
\
(2.2),
±
e
O
I
1
≤
C
k
f
k
s,s
0
k
g
k
t,
−
α
2
k
h
k
L
2
.
(2
.
3)
•
O
I
2
,
d
H¨older
Ø
ª
,
I
2
≤
(
T
1
T
2
)
1
2
,
Ù
¥
T
1
=
ZZ
h
ξ
h
i
2
s
h
ξ
3
i
2
s
0
|
b
f
(
ξ
)
|
2
|
h
(
ξ
+
η
)
|
2
dξdη
…
T
2
=
ZZ
2
|
ξ
h
|≤|
η
h
|
h
ξ
h
+
η
h
i
2(
s
+
t
−
1)
h
ξ
h
i
2
s
h
ξ
3
+
η
3
i
−
α
h
ξ
3
i
2
s
0
|
b
g
(
η
)
|
2
dξdη.
w
,
,
T
1
=
k
h
k
2
L
2
k
f
k
2
s,s
0
.
•
O
T
2
,
Ä
k
'
u
ξ
h
Ú
ξ
3
È
©
.
Ó
I
3
O
,
k
Z
2
|
ξ
h
|≤|
η
h
|
h
ξ
h
+
η
h
i
2(
s
+
t
−
1)
h
ξ
h
i
2
s
dξ
h
'h
η
h
i
2(
s
+
t
−
1)
Z
2
|
ξ
h
|≤|
η
h
|
1
h
ξ
h
i
2
s
dξ
h
≤
C
1
2
−
s
h
η
h
i
2
t
.
Ï
d
,
T
2
≤
C
k
g
k
t,
−
α
2
.
u
´
I
2
≤
C
k
f
k
s,s
0
k
g
k
t,
−
α
2
k
h
k
L
2
,
(2
.
4)
(
Ü
(2.1)(2.3)(2.4),
Ú
n
2.3
y
.
3.
Ì
‡
½
n
9
y
²
½
n
3.1
1
2
<α
≤
1,
α
2
<s<
2
α
−
1
2
,
v
Ú
e
v
´
•
§
(1.1)
ƒ
Au
Ð
Š
v
0
∈
H
2
α
−
2
,s
(
R
3
)
ü
‡
)
,
…
υ,
e
υ
∈
L
∞
[0
,T
];
H
2
α
−
2
,s
∩
L
2
[0
,T
];
H
3
−
2
α,s
.
K
υ
=
e
υ
.
y
-
ω
=
υ
−
e
υ
,
3
•
§
ü
>
é
ω
¦
±
Λ
−
α
3
ω
,
,
3
(
ε,t
)
×
R
3
þ
?
1
È
©
,
-
ε
→
0
k
ω
(
t
)
2
0
,
−
α
2
+2
ν
h
Z
t
0
Λ
α
1
ω
(
τ
)
2
0
,
−
α
2
+
Λ
α
2
ω
(
τ
)
2
0
,
−
α
2
dτ
=
−
2
Z
t
0
Z
R
3
ω
(
τ,x
)
·∇
e
υ
(
τ,x
)
·
Λ
−
α
3
ω
(
τ,x
)
dτdx
−
2
Z
t
0
Z
R
3
υ
(
τ,x
)
·∇
ω
(
τ,x
)
·
Λ
−
α
3
ω
(
τ,x
)
dτdx.
(3
.
1)
DOI:10.12677/pm.2021.11122181962
n
Ø
ê
Æ
4
æ
D
§
š
S
•
{
z
Î
Ò
,
·
‚
P
υ
(
τ,x
)
•
υ
,
Ù
¦
Î
Ò
a
q
,
e
¡
O
Ž
Z
ω
·∇
e
υ
·
Λ
−
α
3
ωdx
=
Z
(
ω
1
∂
1
e
υ
+
ω
2
∂
2
e
υ
)
·
Λ
−
α
3
ωdx
|{z}
L
1
+
Z
ω
3
∂
3
e
υ
·
Λ
−
α
3
ωdx
|{z}
L
2
(3
.
2)
Ú
Z
υ
·∇
ω
·
Λ
−
α
3
ωdx
=
Z
(
υ
1
∂
1
ω
+
υ
2
∂
2
ω
)
·
Λ
−
α
3
ωdx
|{z}
L
3
+
Z
υ
3
∂
3
ω
·
Λ
−
α
3
ωdx
|{z}
L
4
.
(3
.
3)
L
1
O
.
Š
â
Ú
n
2.3,
|
L
1
|≤
ω
1
∂
1
e
υ
+
ω
2
∂
2
e
υ
0
,
−
α
2
Λ
−
α
3
ω
1
2
,
α
2
≤
C
e
v
3
−
2
α,s
ω
0
,
−
α
2
ω
α,
−
α
2
.
(3
.
4)
L
2
O
.
Š
â
½
n
2.2,
|
L
2
|≤
ω
3
∂
3
e
v
−
α,
2
s
−
1
−
α
2
Λ
−
α
3
ω
α,
−
2
s
+1+
α
2
≤
C
ω
3
α
−
1
,
s
2
−
1
4
∇
e
v
2
−
2
α,
s
2
−
α
2
+
1
4
ω
α,
−
2
s
+1
−
α
2
≤
C
e
v
3
−
2
α,s
ω
3
α
−
1
,
s
2
−
1
4
ω
α,
−
α
2
,
(3
.
5)
Ù
¥
,
ω
α
−
1
,
s
2
−
1
4
≤
ω
α
−
1
,
s
2
−
5
4
+
∂
3
ω
3
α
−
1
,
s
2
−
5
4
≤
ω
0
,
−
α
2
+
ω
α,
−
α
2
.
L
3
O
.
Š
â
Ú
n
2.3,
|
L
3
|≤
υ
1
∂
1
ω
+
υ
2
∂
2
ω
−
1
2
,
−
α
2
Λ
−
α
3
ω
1
2
,
α
2
≤
v
3
−
2
α,s
ω
0
,
−
α
2
ω
α,
−
α
2
.
(3
.
6)
L
4
O
.
Z
υ
3
∂
3
ω
·
Λ
−
α
h
ωdx
=(2
π
)
−
3
Z
\
v
3
∂
3
ω
(
ξ
)
·
\
Λ
−
α
3
ω
(
−
ξ
)
dξ
=(2
π
)
−
6
Z
1
h
ξ
3
i
α
b
v
3
∗
\
∂
3
ω
(
ξ
)
·
\
ω
(
−
ξ
)
dξ
=
i
(2
π
)
−
6
ZZ
η
3
h
ξ
3
i
α
b
v
3
(
ξ
−
η
)
b
ω
(
η
)
·
b
ω
(
−
ξ
)
dξdη.
(3
.
7)
DOI:10.12677/pm.2021.11122181963
n
Ø
ê
Æ
4
æ
D
§
š
S
Š
C
þ
O
†
(
ξ,η
)
↔
(
−
η,
−
ξ
),
Œ
L
4
=
i
2
(2
π
)
−
6
ZZ
η
3
h
ξ
3
i
α
−
ξ
3
h
η
3
i
α
b
v
3
(
ξ
−
η
)
b
ω
(
η
)
·
b
ω
(
−
ξ
)
dξdη.
(3
.
8)
Ï
•
,
é
∀
x,y
∈
R
,
1
2
<α
≤
1,
k
X
e
Ø
ª
¤
á
:
x
h
y
i
α
−
y
h
x
i
α
≤
x
−
y
1
h
x
i
α
−
1
h
y
i
α
.
d
(3.8)
Œ
|
L
4
|≤
1
2
(2
π
)
−
6
X
k
ZZ
|
ξ
3
−
η
3
|
1
h
ξ
3
i
α
+
1
h
η
3
i
α
|
b
v
3
(
ξ
−
η
)
||
c
ω
k
(
η
)
||
c
ω
k
(
−
ξ
)
|
dξdη.
2
Š
C
þ
O
†
(
ξ,η
)
↔
(
−
η,
−
ξ
),
Œ
|
L
4
|≤
(2
π
)
−
6
X
k
ZZ
|
ξ
3
−
η
3
|
h
ξ
3
i
α
|
b
v
3
(
ξ
−
η
)
||
c
ω
k
(
η
)
||
c
ω
k
(
−
ξ
)
|
dξdη.
Ï
•
div
v
= 0,
¤
±
ξ
3
b
v
3
=
−
ξ
1
b
v
1
(
ξ
)
−
ξ
2
b
v
2
(
ξ
),
K
|
ξ
3
||
b
v
3
|≤|
ξ
1
||
b
v
1
(
ξ
)
|
+
|
ξ
2
||
b
v
2
(
ξ
)
|
.
u
´
,
|
L
4
|≤
(2
π
)
−
6
X
k
ZZ
|
ξ
1
−
η
1
||
b
v
1
(
ξ
−
η
)
|
+
|
ξ
2
−
η
2
||
b
v
2
(
ξ
−
η
)
|
h
ξ
3
i
α
|
b
ω
k
(
η
)
||
b
ω
k
(
−
ξ
)
|
dξdη.
(3
.
9)
-
b
V
k
=
|
b
v
k
|
,
w
,
,
é
∀
r,r
0
,k
,
k
k
V
k
k
r,r
0
=
k
v
k
k
r,r
0
.
±
Ó
•
ª
,
½
Â
•
þ
W
.
$
^
(3.7)
ª
‡
•
(
Ø
,
†
(3.9)
d
ª
f
|
L
4
|≤
Z
(
|
D
1
|
V
1
+
|
D
2
|
V
2
)
W
·
Λ
−
α
3
Wdx,
Ù
¥
|
D
k
|
L
«
†
|
ξ
k
|
ƒ
'
Ž
f
.
du
é
∀
r,r
0
,k,
k|
D
k
|
V
k
k
H
r,r
0
=
k
∂
k
V
k
k
H
r,r
0
,
†
L
1
ƒ
Ó
O
|
L
4
|≤
Z
(
|
D
1
|
V
1
+
|
D
2
|
V
2
)
W
·
Λ
−
α
3
Wdx
≤
C
V
3
−
2
α,s
W
0
,
−
α
2
W
α,
−
α
2
=
C
v
3
−
2
α,s
ω
0
,
−
α
2
ω
α,
−
α
2
.
(3
.
10)
(
Ü'
X
ª
(3.1)-(3.6)
Ú
(3.10)
ω
(
t
)
2
0
,
−
α
2
+2
ν
h
Z
t
0
Λ
α
1
ω
2
0
,
−
α
2
+
Λ
α
2
ω
2
0
,
−
α
2
dτ
≤
C
Z
t
0
ω
0
,
−
α
2
ω
α,
−
α
2
(
e
v
3
−
2
α,s
+
v
3
−
2
α,s
)
dτ
+
C
Z
t
0
ω
0
,
−
α
2
ω
α,
−
α
2
e
v
3
−
2
α,s
dτ.
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DOI:10.12677/pm.2021.11122181966
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