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PureMathematicsnØêÆ,2021,11(12),1957-1966
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112218
©êˆ•É5Navier-Stokes•§ÐŠ¯K
)•˜5
444æææDDD§§§šššSSS
∗
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ÂvFϵ2021c111F¶¹^Fϵ2021c122F¶uÙFϵ2021c129F
Á‡
T©y²=kY²©êÑÑØŒØ Navier-Stokes•§ÐŠ¯K3ˆ•É5Sobolev¼
ê˜mH
2α−2,s
(R
3
)¥)•˜5§Ù¥
1
2
<α≤1,
α
2
<s<2α−
α
2
"y²'…´‰Ñ(α,s,t)
÷v·‰Œ¼ê¦Èúª§?|^Fourier©ÛE|Ñ(Ø"
'…c
©êˆ•É5Navier-Stokes•§§Sobolev ˜m§¦Èúª
UniquenessofSolutionstoInitialValue
ProblemsofFractionalAnisotropic
Navier-StokesEquations
MixiuLiu,XiaochunSun
∗
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Nov.1
st
,2021;accepted:Dec.2
nd
,2021;published:Dec.9
th
,2021
∗ÏÕŠö"
©ÙÚ^:4æD,šS.©êˆ•É5Navier-Stokes•§ÐŠ¯K)•˜5[J].nØêÆ,2021,11(12):
1957-1966.DOI:10.12677/pm.2021.1112218
4æD§šS
Abstract
Inthispaper,weprovedtheuniquenessofthesolutionoftheinitialvalueproblemof
incompressibleNavier-Stokesequationwithonlyhorizontalfractionaldissipationin
the anisotropic Sobolev function spaceH
2α−2,s
(R
3
), where
1
2
<α≤1,
α
2
<s<2α−
α
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.2021.11122181959nØêÆ
4æD§šS
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i
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b
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2
.
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α
2
=sup
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L
2
≤1
Z
hξ
h
i
s+t−1
hξ
3
i
−
α
2
b
f∗bg(ξ)h(ξ)dξ
=sup
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L
2
≤1
ZZ
hξ
h
+η
h
i
s+t−1
hξ
3
+η
3
i
−
α
2
b
f(ξ)bg(η)h(ξ+η)dξdη.
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(2π)
3
kfgk
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α
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≤ sup
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L
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≤1
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|
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hξ
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i
−
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hξ
h
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h
i
s+t−1
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h
i
t
b
f(ξ)hη
h
i
t
bg(η)h(ξ+η)dξdη
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3
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3
i
−
α
2

ZZ
2|ξ
h
|≥|η
h
|
hξ
h
+η
h
i
2(s+t−1)
hη
h
i
2t
|
b
f(ξ)|
2
dξ
h
dη
h
|{z}
I
3
×
ZZ
hη
h
i
2t
|bg(η)|
2
|h(ξ+η)|
2
dξ
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dη
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.
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hη
h
i
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h
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h
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h
|
2
hξ
h
+η
h
i
2(s+t−1)
hη
h
i
2t
dη
h
+
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ξ
h
2
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h
|≤2|ξ
h
|
hξ
h
+η
h
i
2(s+t−1)
hη
h
i
2t
dη
h
.
DOI:10.12677/pm.2021.11122181960nØêÆ
4æD§šS
e|η
h
|≤
|ξ
h
|
2
,Khξ
h
+η
h
i'hξ
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i.e
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h
2
≤|η
h
|≤2|ξ
h
|,Khη
h
i'hξ
h
i.u´,
Z
|η
h
|≤
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h
|
2
hξ
h
+η
h
i
2(s+t−1)
hη
h
i
2t
dη
h
'hξ
h
i
2(s+t−1)
Z
|η
h
|≤
|ξ
h
|
2
1
hη
h
i
2t
dη
h
≤
C
1−t
hξ
h
i
2s
…
Z
ξ
h
2
≤|η
h
|≤2|ξ
h
|
hξ
h
+η
h
i
2(s+t−1)
hη
h
i
2t
dη
h
'
1
hξ
h
i
2t
Z
|ξ
h
|
2
≤|η
h
|≤2|ξ
h
|
hξ
h
+η
h
i
2(s+t−1)
dη
h
≤
C
hξ
h
i
2t
Z
|ζ|≤3|ξ
h
|
hζi
2(s+t−1)
dζ≤
C
s+t
hξ
h
i
2s
,
ùp·‚ŠCþO†ζ= ξ
h
+η
h
.
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3
½Â,·‚
I
3
≤C
Z
hξ
h
i
2s
|
b
f(ξ)|
2
dξ
h
,
'uCþ(ξ
3
,η
3
)$^H¨olderØª,ddI
1
O:
I
1
≤C
ZZ

Z
hξ
h
i
2s
hξ
3
i
2s
0
|
b
f(ξ)|
2
dξ
h
Z
|h(ζ,ξ
3
+η
3
)|
2
dζ

1
2
×
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hξ
3
+η
3
i
−α
hξ
3
i
−2s
0
Z
hη
h
i
2t
|bg(η)|
2
dη
h

1
2
dξ
3
dη
3
.
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I
1
≤Ckfk
s,s
0
khk
L
2
(
Z
hη
h
i
2t
ϕ(η
3
)|bg(η)|
2
dη)
1
2
,(2.2)
Ù¥
ϕ(η
3
) =
Z
1
hξ
3
+η
3
i
α
hξ
3
i
2s
0
.
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Z
|ξ
3
|≥2|η
3
|
1
hξ
3
+η
3
i
α
hξ
3
i
2s
0
dξ
3
'
Z
∞
2|η
3
|
1
hξ
3
i
2s
0
+1
dξ
3
'
Z
∞
2|η
3
|
1
h(1+ξ
3
)i
2s
0
+1
dξ
3
≤
C
hη
3
i
2s
0
≤
C
hη
3
i
α
,
Z
|ξ
3
|≤
|η
3
|
2
1
hξ
3
+η
3
i
α
hξ
3
i
2s
0
dξ
3
'
1
hη
3
i
α
Z
|ξ
3
|≤
|η
3
|
2
1
hξ
3
i
2s
0
dξ
3
≤
C
(s
0
−
1
2
)hη
3
i
α
,
Z
|η
3
|
2
≤|ξ
3
|≤2|η
3
|
1
hξ
3
+η
3
i
α
hξ
3
i
2s
0
dξ
3
'
1
hη
3
i
2s
0
Z
|η
3
|
2
≤|ξ
3
|≤2|η
3
|
1
hξ
3
+η
3
i
α
dξ
3
≤
C
hη
3
i
2s
0
Z
|ζ|≤3|η
3
|
1
hζi
α
dζ
≤
C
(s
0
−
1
2
)hη
3
i
α
,
DOI:10.12677/pm.2021.11122181961nØêÆ
4æD§šS
l
ϕ(η
3
) ≤Chη
3
i
−α
,
òþª“\(2.2),±eO
I
1
≤Ckfk
s,s
0
kgk
t,−
α
2
khk
L
2
.(2.3)
•OI
2
,dH¨olderØª,
I
2
≤(T
1
T
2
)
1
2
,
Ù¥
T
1
=
ZZ
hξ
h
i
2s
hξ
3
i
2s
0
|
b
f(ξ)|
2
|h(ξ+η)|
2
dξdη
…
T
2
=
ZZ
2|ξ
h
|≤|η
h
|
hξ
h
+η
h
i
2(s+t−1)
hξ
h
i
2s
hξ
3
+η
3
i
−α
hξ
3
i
2s
0
|bg(η)|
2
dξdη.
w,,T
1
= khk
2
L
2
kfk
2
s,s
0
.•OT
2
,Äk'uξ
h
Úξ
3
È©.ÓI
3
O,k
Z
2|ξ
h
|≤|η
h
|
hξ
h
+η
h
i
2(s+t−1)
hξ
h
i
2s
dξ
h
'hη
h
i
2(s+t−1)
Z
2|ξ
h
|≤|η
h
|
1
hξ
h
i
2s
dξ
h
≤
C
1
2
−s
hη
h
i
2t
.
Ïd,
T
2
≤Ckgk
t,−
α
2
.
u´
I
2
≤Ckfk
s,s
0
kgk
t,−
α
2
khk
L
2
,(2.4)
(Ü(2.1)(2.3)(2.4),Ún2.3y.
3.̇½n9y²
½n3.1
1
2
<α≤1,
α
2
<s<2α−
1
2
,vÚev´•§(1.1)ƒAuЊv
0
∈H
2α−2,s
(R
3
)
ü‡),…
υ,eυ∈L
∞

[0,T];H
2α−2,s

∩L
2

[0,T];H
3−2α,s

.
Kυ= eυ.
y-ω= υ−eυ,3•§ü>éω¦±Λ
−α
3
ω,,3(ε,t)×R
3
þ?1È©,-ε→0k


ω(t)


2
0,−
α
2
+2ν
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)·∇eυ(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.2021.11122181962nØêÆ
4æD§šS
•{zÎÒ,·‚Pυ(τ,x) •υ,Ù¦ÎÒaq,e¡OŽ
Z
ω·∇eυ·Λ
−α
3
ωdx=
Z
(ω
1
∂
1
eυ+ω
2
∂
2
eυ)·Λ
−α
3
ωdx
|{z}
L
1
+
Z
ω
3
∂
3
eυ·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.ŠâÚn2.3,
|L
1
|≤


ω
1
∂
1
eυ+ω
2
∂
2
eυ


0,−
α
2


Λ
−α
3
ω


1
2
,
α
2
≤C


ev


3−2α,s


ω


0,−
α
2


ω


α,−
α
2
.
(3.4)
L
2
O.Šâ½n2.2,
|L
2
|≤


ω
3
∂
3
ev


−α,
2s−1−α
2


Λ
−α
3
ω


α,
−2s+1+α
2
≤C


ω
3


α−1,
s
2
−
1
4


∇ev


2−2α,
s
2
−
α
2
+
1
4


ω


α,
−2s+1−α
2
≤C


ev


3−2α,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.ŠâÚn2.3,
|L
3
|≤


υ
1
∂
1
ω+υ
2
∂
2
ω


−
1
2
,−
α
2


Λ
−α
3
ω


1
2
,
α
2
≤


v


3−2α,s


ω


0,−
α
2


ω


α,−
α
2
.
(3.6)
L
4
O.
Z
υ
3
∂
3
ω·Λ
−α
h
ωdx
=(2π)
−3
Z
\
v
3
∂
3
ω(ξ)·
\
Λ
−α
3
ω(−ξ)dξ
=(2π)
−6
Z
1
hξ
3
i
α
bv
3
∗
\
∂
3
ω(ξ)·
\
ω(−ξ)dξ
=i(2π)
−6
ZZ
η
3
hξ
3
i
α
bv
3
(ξ−η)bω(η)·bω(−ξ)dξdη.
(3.7)
DOI:10.12677/pm.2021.11122181963nØêÆ
4æD§šS
ŠCþO†(ξ,η) ↔(−η,−ξ),Œ
L
4
=
i
2
(2π)
−6
ZZ

η
3
hξ
3
i
α
−
ξ
3
hη
3
i
α

bv
3
(ξ−η)bω(η)·bω(−ξ)dξdη.(3.8)
Ï•,é∀x,y∈R,
1
2
<α≤1,kXeØª¤á:




x
hyi
α
−
y
hxi
α




≤


x−y



1
hxi
α
−
1
hyi
α

.
d(3.8)Œ
|L
4
|≤
1
2
(2π)
−6
X
k
ZZ
|ξ
3
−η
3
|

1
hξ
3
i
α
+
1
hη
3
i
α

|bv
3
(ξ−η)||cω
k
(η)||cω
k
(−ξ)|dξdη.
2ŠCþO†(ξ,η) ↔(−η,−ξ),Œ
|L
4
|≤(2π)
−6
X
k
ZZ
|ξ
3
−η
3
|
hξ
3
i
α
|bv
3
(ξ−η)||cω
k
(η)||cω
k
(−ξ)|dξdη.
Ï•divv= 0,¤±ξ
3
bv
3
= −ξ
1
bv
1
(ξ)−ξ
2
bv
2
(ξ),K|ξ
3
||bv
3
|≤|ξ
1
||bv
1
(ξ)|+|ξ
2
||bv
2
(ξ)|.u´,
|L
4
|≤(2π)
−6
X
k
ZZ
|ξ
1
−η
1
||bv
1
(ξ−η)|+|ξ
2
−η
2
||bv
2
(ξ−η)|
hξ
3
i
α
|bω
k
(η)||bω
k
(−ξ)|dξdη.(3.9)
-
b
V
k
=|bv
k
|,w,,é∀r,r
0
,k,kkV
k
k
r,r
0
=kv
k
k
r,r
0
.±Ó•ª,½Â•þW.$^(3.7)ª
‡•(Ø,†(3.9)dªf
|L
4
|≤
Z
(|D
1
|V
1
+|D
2
|V
2
)W·Λ
−α
3
Wdx,
Ù¥|D
k
|L«†|ξ
k
|ƒ'Žf.dué∀r,r
0
,k,k|D
k
|V
k
k
H
r,r
0
= k∂
k
V
k
k
H
r,r
0
,†L
1
ƒÓ
O
|L
4
|≤
Z
(|D
1
|V
1
+|D
2
|V
2
)W·Λ
−α
3
Wdx
≤C


V


3−2α,s


W


0,−
α
2


W


α,−
α
2
=C


v


3−2α,s


ω


0,−
α
2


ω


α,−
α
2
.
(3.10)
(Ü'Xª(3.1)-(3.6)Ú(3.10)


ω(t)


2
0,−
α
2
+2ν
h
Z
t
0



Λ
α
1
ω


2
0,−
α
2
+


Λ
α
2
ω


2
0,−
α
2

dτ
≤C
Z
t
0


ω


0,−
α
2


ω


α,−
α
2
(


ev


3−2α,s
+


v


3−2α,s
)dτ
+C
Z
t
0


ω


0,−
α
2


ω


α,−
α
2


ev


3−2α,s
dτ.
DOI:10.12677/pm.2021.11122181964nØêÆ
4æD§šS
|^YoungØªŒ
kω(t)k
2
0,−
α
2
+2ν
h
Z
t
0
(kΛ
1
ωk
2
0,−
α
2
+kΛ
2
ωk
2
0,−
α
2
)dτ
≤2ν
h
Z
t
0
kωk
2
α,−
α
2
dτ+C
Z
t
0
kωk
2
0,−
α
2
(kevk
2
3−2α,s
+kvk
2
3−2α,s
)dτ.
du
kωk
2
α,−
α
2
= kΛ
α
1
ωk
2
0,−
α
2
+kΛ
α
2
ωk
2
0,−
α
2
.
?˜ÚŒ
kω(t)k
2
0,−
α
2
≤
Z
t
0
kω(t)k
2
0,−
α
2
h(τ)dτ,
(3.11)
Ù¥
h(t) = C(2ν
h
+kevk
2
3−2α,s
+kvk
2
3−2α,s
).
éúª(3.11)$^GronwallØªω= 0,lk˜v= v.½n3.1y.
Ä7‘8
I[g,‰Æ“cÄ7]Ï(11601434).
ë•©z
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[7]Paicu, M.and Zhang,P. (2011)Global Solutionsto the3-D IncompressibleAnisotropicNavier-
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