关于3D不可压Navier-Stokes方程H1正则性的注记 A Note on the H1 Regularity of 3D Incompressible Navier-Stokes Equation

doi:10.12677/AAM.2021.107264

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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(7),2529-2552

PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/aam

https://doi.org/10.12677/aam.2021.107264

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K5§ÛÜ·½5Ún§3DØŒØNavier-Stokes•§

ANoteontheH

Regularityof3D

IncompressibleNavier-StokesEquation

ChengmingYang

∗

,ZhenqiongCui

ShanghaiNormalUniversity,Shanghai

∗ÏÕŠö"

©ÙÚ^:¤²,w .'u3D ØŒØNavier-Stokes•§H

K55P[J].A^êÆ?Ð,2021,10(7):2529-

2552.DOI:10.12677/aam.2021.107264

¤²§w

Received:Jun.21

,2021;accepted:Jul.11

,2021;published:Jul.23

,2021

Abstract

Inthispaper,wemainlystudytheH

regularityofthesolutionof3Dincompressible

Navier-Stokesequations.Firstly,thelocalwell-ﬁtlemmaforsolutionsof3Dincom-

pressible Navier-Stokesequationsisgiven andprovedindetail.Secondly,by applying

thelocalwell-ﬁtlemmaofthesolutionmentionedabove,ﬁrstly,theglobalregularity

ofthesolutioninthecaseofsmallinitialdatacanbeprovedstrictly.Second,theH

regularityofthesolutionof3DincompressibleNavier-Stokesequationsisprovedfor

allpossibleU

andallpossibleF.Inthispaper,itisemphasizedthatthesolution

isH

regularnotonlyforthemaximumpossibleU

andaﬁxedF,butalsoforthe

maximumpossibleU

andthemaximumpossibleF.

Keywords

Regularity, TheLocalWell-FitLemma,3DIncompressibleNavier-StokesEquations

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI:10.12677/aam.2021.1072642530A^êÆ?Ð

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DOI:10.12677/aam.2021.1072642531A^êÆ?Ð

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bη

−4

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−4

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DOI:10.12677/aam.2021.1072642532A^êÆ?Ð

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u)|

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u||+C

(Ω)||A

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u||

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4ν

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

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

u||

+ν||A

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u||

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∞

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.(4.1)

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≤(||A

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||P



f||

∞

t)[1−2

2ν

||A



u(t)||

||P



f||

∞

ds]

−

·‚-

∗∗∗

= sup



t∈[0,T]

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2ν

||A



u(t)||

||P

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ds<1

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DOI:10.12677/aam.2021.1072642533A^êÆ?Ð

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U||

(Ω)

+ν||AU||

(Ω)

≤|(PF,AU)|+|(B(U,U),AU)|

≤||PF||

(Ω)

||AU||

(Ω)

(Ω)||A

U||

(Ω)

||AU||

(Ω)

≤

||AU||

(Ω)

||PF||

(Ω)

4ν

||A

U||

(Ω)

||AU||

(Ω)

||A

U||

(Ω)

+λ

ν||A

U||

(Ω)

≤

||A

U||

(Ω)

+ν||AU||

(Ω)

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∞

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||A

U||

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ν||A

U||

(Ω)

≤

||PF||

(Ω)

2ν

||A

U||

(Ω)

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2ν

≤



||A

U||

(Ω)

||A

U||

(Ω)

≤

||PF||

(Ω)

.(4.5)

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||A

U||

(Ω)

≤e

−

νt

(||A

(Ω)

||PF||

(Ω)

ds)

≤max(1,

)(||A

(Ω)

+||PF||

(Ω)

)

<NR

(4.6)

•,A^‡y{5y²T

max

= ∞.

DOI:10.12677/aam.2021.1072642534A^êÆ?Ð

¤²§w

Äk,Ä½1˜«œ/µT

∞

= T

max

<∞,dž,[0,T

∞

) = [0,T

max

),Ï•T

max

<∞,¤±

lim

t→T

−

max

||A

U(t)||

(Ω)

= ∞.

=M= NR

ž,∃δ>0,∀t∈(0,T

max

),•‡T

max

−δ<t<T

max

,Òk

||A

U(t)||

>NR

=∃t

∈(T

max

−δ,T

max

) = (T

∞

−δ,T

∞

),¦

||A

U(t)||

(Ω)

>NR

ù†[0,T

∞

)½Âgñ.

Ùg,Ä½1«œ/µT

∞

max

<∞,du||A

U(t)||

(Ω)

,3[0,T

max

)þ'utëY,¤

±||A

U(t)||

(Ω)

•3[0,T

∞

)þëY¿…

||A

U(t)||

(Ω)

≤NR

,[0,T

∞

du||A

U(t)||

(Ω)

∞

ëY,¤±

||A

U(t)||

(Ω)

≤NR

,[0,T

∞

l(4.6)•

||A

U(t)||

(Ω)

<NR

,[0,T

∞

].(4.7)

é(4.7)ª|^3T

∞

ëY5,

||A

U(t)||

(Ω)

≤NR

,t∈(T

∞

+δ).

ù†[0,T

∞

)½Âgñ.

nþ¤ã,T

max

= ∞,y..

Ún4.2^‡H1Ú(3.1)¤á,@o•3k

,

>0,éu0<≤

,•3T

()>0,¦

u(t) ∈D(A



),0 ≤t≤T

…¤á











||A



v(T

)||

≤4η

−2

−4

||A



w(T

)||

≤k



−2

(4.8)

DOI:10.12677/aam.2021.1072642535A^êÆ?Ð

¤²§w

y².½Â

= η

−2

+

−2

+η

−4

+2η

−2

+2

−2

Ï•

||A



+||P



f||

∞

= ||A



+||A



+||(M+I−M)P



f||

∞

≤||A



+||A



+2||MP



f||

∞

+2||(I−M)P



f||

∞

≤η

−2

+

−2

+2η

−2

+2

−2

≤η

−2

+

−2

+2η

−2

+2

−2

+η

−4

= R

ÀN= max{4,

}+1 >1,A^Lemma4.1,•3T

>0,¦

||A



u(t)||

≤NR

,[0,T

].(4.9)

P[0,T

∞

)L«¦(4.23)¤á•Œžm«m.XJT

∞

<∞,@o

||A



u(t)||

= NR

.(4.10)

e5,rt•›3[0,T

∞

)þ?1?Ø

Ï•

(I−M)B



(v,v) = B



(v,v)−MB



(v,v) = B



(v,v)−B



(v,v) = 0,

@o'uω©þ•§•

dω

+νA



ω= (I−M)P



f−(I−M)(B



(ω,v)+B



(v,ω)+B



(ω,ω)).(4.11)

(4.11)ª†A



ωŠSÈ,

dω

||A



ω||

+ν||A



ω||

≤|((I−M)P



f,A



ω)|

+|(b



(ω,v,A



w))|

+|(b



(v,ω,A



w))|

+|(b



(ω,ω,A



w))|,

DOI:10.12677/aam.2021.1072642536A^êÆ?Ð

¤²§w

|^YoungØª

dω

||A



ω||

+ν||A



ω||

≤|((I−M)P



f,A



ω)|+|(b



(ω,v,A



w))|

+|(b



(v,ω,A



w))|+|(b



(ω,ω,A



w))|

≤

||A



ω||

2ν

||(I−M)P



f||

∞



||A



ω||

||A



v||||A



ω||



||A



v||||A



ω||

||A



ω||



||A



ω||

||A



ω||

duMω= 0,¦^(3.4)

||A



ω||

+ν||A



ω||

≤

||(I−M)P



f||∞

+2C



||A



v||||A



ω||

+2C



||A



v||||A



ω||

+2C

||A



ω||||A



ω||

5¿

||A



ω||≤||A



u||,||A



v||≤||A



u||,

¤±·‚

||A



ω||

+(ν−D



||A



u||)||A



ω||

≤

||(I−M)P



f||

∞

,(4.12)

Ù¥D

= 2C

++2C

+2C

éu0 ≤t<T

∞

,|^b^‡H1



||A



u||≤D



= D



(η

−1

+

−1

+η

−2

√

2η

−1

√

2

−1

) →0,→0.

Ïd,∃

>0,¦



≤

,0 <≤

.(4.13)

(Ü(4.12)Ú(4.13)

||A



ω||

||A



ω||

≤

||(I−M)P



f||

∞

,(4.14)

|^Øª(3.4)

||A



ω||

νC

−2



−2

||A



ω||

≤

||(I−M)P



f||

∞

,(4.15)

DOI:10.12677/aam.2021.1072642537A^êÆ?Ð

¤²§w

^GronwallØª,

||A



ω||

≤e

−

νC

−2



−2



||A



||(I−M)P



f||

∞



≤e

νC

−2



−2

t||A



||(I−M)P



f||

∞

(4.16)

•¦

νC

−2



−2

t||A



||(I−M)P



f||

∞

νC

−2



−2

t

−2



−2

,(4.17)

Œ±éT

>0,÷v(4.17),ÏdÀT

= T

() >0,

def

=2C



−1

Q(),

Ù¥

Q() = |ln(2C

−2



−p

−2

)|.(4.18)

•y(4,18)¤á,A÷ve¡‡¦



−p

−2

≤1,(4.19)

~X§•IÀη

= −ln,=Œ÷v‡¦.

Ïd,∃

>0,éu0 <≤

,•3T

= T

() >0÷v(4.19).

e5,äóT

∞

.éuT

≤t<T

∞

||A



ω||

≤e

νC

−2



−2

||A



||(I−M)P



f||

∞

−2



−2

= k



−2

Ù¥k

= 4C

−2

'uvO,·‚•›t3[0,T

]þ

+νA



v= MP



f−MB



(v,v)−MB



(ω,v)−B



(v,ω)−B



(ω,ω),(4.20)

DOI:10.12677/aam.2021.1072642538A^êÆ?Ð

¤²§w

(4.20)ª†A



vŠSÈ,

||A



v||

+ν||A



v||

≤|(MP



f,A



v)|+|(MB



(v,v),A



v)|+|(MB



(ω,ω),A



ω)|

≤|(MP



f,A



v)|+|b



(v,v,A



v)|+|



(w,w,A



v)|,

dub



(v,ω,A



v) = b



(ω,v,A



v) = 0äN(Ø•„[1]

éþª|^YoungØª,

||A



v||

+ν||A



v||

≤||MP



f||

∞

||A



v||+C

||v||

||A



v||||A



v||



||A



ω||

||A



ω||

||A



v||

≤

||A



v||

2ν

||MP



f||

∞

||v||

||A



v||||A



v||



||A



ω||

||A



ω||

||A



v||,

||A



v||

+ν||A



v||

≤

||MP



f||

∞

+2(

||A



v||

4ν

||v||

||A



v||

||A



v||

||A



ω||

||A



ω||)

≤

||MP



f||

∞

||A



v||

2ν

||v||

||A



v||

||A



v||

||A



ω||

||A



ω||,

X

||A



v||

≤

||MP



f||

∞

2ν

||v||

||A



v||

||A



ω||

||A



ω||

= (

2ν

||v||

||A



v||

)||A



v||

||MP



f||

∞

||A



ω||

||A



ω||,

éþªA^GronwallØª,

||A



v||

≤e

2ν

||v||

||A



v||

[||A



||MP



f||

∞

||A



ω||

||A



ω||ds]

≤e

G(t)

(||A



+H(t)),

(4.21)

Ù¥

H(t) =

||MP



f||

∞

||A



ω||

||A



ω||ds,

G(t) =

2ν

||v||

||A



v||

ds.

DOI:10.12677/aam.2021.1072642539A^êÆ?Ð

¤²§w

y3§OH(t),0 ≤t≤T

.é(4.14)ªÈ©

||A



ω||

ds≤

||(I−M)P



f||

∞

+||A



||A



ω||

ds≤

||(I−M)P



f||

∞

||A



(4.16)ü>ng•§Øª˜Œ•

||A



ω||

≤(e

−νC

−2



−2

t||A



||(I−M)P



f||

∞

)

≤4(e

−3νC

−2



−2

t||A



||(I−M)P



f||

∞

(4.22)

é(4.22)ªÈ©

||A



ω||

ds≤4

−3νC

−2



−2

t||A



||(I−M)P



f||

∞

)

≤4(



3ν

||A



t||(I−M)P



f||

∞

¦^H?lderØª

||A



ω||

||A



ω||ds≤(

||A



ω||ds)

(

||A



ω||

ds)

≤2(

||(I−M)P



f||

∞

||A



)

(



3ν

||A



t||(I−M)P



f||

∞

)

≤



(

||(I−M)P



f||

∞

+||A



||)

(

√

||A



||(I−M)P



f||

∞



||A



ω||

||A



ω||ds≤

−2



(

||(I−M)P



f||

∞

+||A



||)

(

√

||A



||(I−M)P



f||

∞

)

≤D

||(I−M)P



f||

∞

+||A



||)

(||A



+

||(I−M)P



f||

∞

DOI:10.12677/aam.2021.1072642540A^êÆ?Ð

¤²§w

Ù¥ D

−2

max(

√

)max(1,

•

H(t) ≤

−2



−1

+

−1

)(

−3



−3

)

≤



Q()η

−2

−4



Q()η

−4



Q()

−4

√



Q()

−1

−3

≤E

()+

−4

Ù¥

() = D

(

Qη

−2

+

Qη

−4



−1

−3

+

Qη

−4

…

= max(

−2

√

db^‡H1,ØJy→0

žk

() →0,

Ïd,

||A



v(t)||

≤e

G(t)

(η

−2

()+

−4

),0 ≤t≤T

e5,OG(t)§¿`²G(t)¿©.

(2.2)ª†uŠSÈ,|^b



(u,u,u) = 0,Ú©z[8]

||u||

+ν||A



u||

≤|(P



f,u)|

≤|(A

−



f,A



u)|

≤||A



u||||A

−



f||

∞

≤

||A



u||

2ν

||A

−



f||

∞

||u||

+ν||A



≤

||A

−



f||

∞

,(4.23)

DOI:10.12677/aam.2021.1072642541A^êÆ?Ð

¤²§w

È©

||u||

−||u

+ν

||A



ds≤

||A

−



f||

∞

≤

(||A

−



f||

∞

+||A

−



(I−M)P



f||

∞

)

≤

(||A

−



f||

∞

+||A

−



(I−M)P



f||

∞

)

≤



(||A

−



f||

∞

+||A

−



(I−M)P



f||

∞

¦^(3.3)†(3.4)

||u||

≤||u



||MP



f||

∞



||(I−M)P



f||

∞

= ||v

+||ω



||MP



f||

∞



||(I−M)P



f||

∞

≤

||A



||A



||MP



f||

∞



||(I−M)P



f||

∞

||v||

≤||u||

≤

||A



||A



||MP



f||

∞



||(I−M)P



f||

∞

≤D

(||A



+

||A



+

Q||MP



f||

∞

+

Q||(I−M)P



f||

∞

Ù¥

= max(

|^eã(Ø

||A



v(t)||

≤||A



u(t)||

≤NR

·‚Œ±

G(t) =

2ν

||v||

||A



v||

≤

2ν

(η

−2

+

2+p

−2

+

Qη

−4

+

Qη

−2

)NR

≤

27C



Q()(η

−2

+

2+p

−2

+

Qη

−4

+

Qη

−2

)(η

−2

+

−2

+η

−4

+2η

−2

+2

−2

)

≤E

(),

Ù¥

() = D



Q()(η

−2

+

2+p

−2

+

Qη

−4

+

Qη

−2

)(η

−2

+

−2

+η

−4

+2η

−2

+2

−2

DOI:10.12677/aam.2021.1072642542A^êÆ?Ð

¤²§w

…

27C

aq/§|^b^‡H1,ØJy

() →0as→0.

•,3N= 1+max(4,

)e,·‚À

>0,¦

()≤2

() ≤η

−2

,2C



≤ν

,0 <≤

•y²T

∞

,,·‚¦^‡y{bT

∞

=∞,@ow,kT

∞

=∞.Ïd·‚•Ib

T

∞

≤T

<∞,˜•¡,k

||A



ω(T

∞

)||

≤||A



||(I−M)P



f||

∞

≤

−2



−2

≤

−2



−2

≤

−2

+

−2

…

||A



v(T

∞

)||

≤e

()

(η

−2

()+

−4

)

≤4η

−2

−4

•,·‚

||A



u(T

∞

)||

= 4η

−2

−4

+

−2

+

−2

<(1+max(4,

))R

= NR

(4.24)

,˜•¡,XJT

∞

<∞,@o

||A



u(T

∞

)||

= NR

ù†(4.23)ªgñ.Ïd,T

∞

-k

= 4C

−2

,

def

=

,·‚k

||A



v(T

)||

≤4η

−2

−4

||A



ω(T

)||

≤k



−2

DOI:10.12677/aam.2021.1072642543A^êÆ?Ð

¤²§w

y..

Ún4.3^‡H1ÚH(a,b)¤á,Ù¥aÚb¿©Œ,@o•3

>0¦éz‡,0<≤

Ð©êâ÷vb^‡(3.1)§K•§(2.2))u(t)÷vµ0≤t≤2T

žku(t)∈D(A



),…

T

≤t≤2T

žk











||A



v(t)||

≤

(4η

−2

−4

||A



w(t)||

≤k



−2

(4.25)

y².·‚3Ún4.2Ä:þy²Ún4.3.½Â

= 1+(η

−2

+2η

−2

)[1+e

−4

]+2

−2

||A



+||P



f||

∞

≤||A



+||A



+2||MP



f||

∞

+2||(I−M)P



f||

∞

≤4η

−2

−4



−2

+2η

−2

+2

−2

≤R

½N= max(1,D

) >1,A^Lemma4.1,•3T

>0,¦

||A



u(t)||

≤NR

t∈[0,T

P[0,T

∞

)L«¦þ¡Øª¤á•Œžm«m.eT

∞

<∞,Kk

||A



u(t)||

= NR

.(4.26)

(4.11)ª†A



ω,(4.13)ªÚD

.éu0 ≤t<T

∞



||A



u||

≤D

N

→0,as→0

ba≥2D

,b≥4D

,lb^‡H1ÚH(a,b)µ→0ª,eªªu0



= 

(1+η

−2

+2η

−2

+η

−2

9η

−4

+2η

−2

−4

+2

−2

@o,•3

>0,¦



≤

,0 <≤

l(4.16)ª

||A



ω(t)||

≤e

−

νC

−2



−2

t||A



||(I−M)P



f||

∞

≤



−2

(4.27)

DOI:10.12677/aam.2021.1072642544A^êÆ?Ð

¤²§w

éþªl0tÈ©

||A



ω||

ds≤

||A



||(I−M)P



f||

∞

≤max(

)[

+t]

−2

≤D

[

+t]

−2

Ù¥0 ≤t<1.

éþªlt−1tÈ©

t−1

||A



ω||

ds≤

||A



ω(t−1)||

||(I−M)P



f||

∞

≤max(

)

−2

≤D



−2

Ù¥1 ≤t<T

∞

Ó§È©

||A



ω(s)||

ds≤4

−

3νC

−2



6+r

−6



−6

ds)

≤4

−

3νC

−2



)ds

6+r

−6

≤4k

max(

3ν

)[

+t]

6+r

−6

≤D

[

+t]

6+r

−6

Ù¥0 ≤t<1.

d0 <≤1k,aq

t−1

||A



ω(s)||

ds≤2D



6+r

−6

Ù¥1 ≤t<T

∞

¦^H?lderØª

||A



ω||

||A



ω||≤(

||A



ω||

)

(

||A



ω||

)

≤D

[

+t]



−1

[

+t]



−3

= D

[

+t]

−4

(4.28)

DOI:10.12677/aam.2021.1072642545A^êÆ?Ð

¤²§w

Ù¥0 ≤t<1.

Ó¦^H?lderØª

t−1

||A



ω||

||A



ω||≤(

t−1

||A



ω||

)

(

t−1

||A



ω||

)

≤D



−1

√



−3

≤2D



−4

(4.29)

Ù¥1 ≤t<T

∞

e5,(4.20)ª†vŠSÈ,(Üb



(v,v,v) = 0 !(3.3)ªÚ(3.4)ª

||v||

+λ

ν||v||

≤

||v||

+ν||A



v||

≤||MPf||

∞

||v||+C



||A



ω||

||A



ω||

||v||

≤

(||MPf||

∞



||A



ω||

||A



ω||

|^YoungØª

||v||

+ν||A



v||

≤

||A



v||

νλ

||MPf||

∞

||A



v||

νλ

||A



ω||

||A



ω||,

ØJyeã(Ø

||v||

+ν||A



v||

≤

νλ

(||MPf||

∞

||A



ω||

||A



ω||),

||v||

+λ

ν||A



v||

≤

νλ

(||MPf||

∞

||A



ω||

||A



ω||),

¦^GronwallØª

||v||

≤e

−λ

(||v

νλ

(||MPf||

∞

||A



ω||

||A



ω||))ds

≤e

−λ

||v

νλ

||MPf||

∞

−λ

νλ

e

−λ

||A



ω||

||A



ω||ds

≤e

−λ

||v

(νλ

)

||MPf||

∞

νλ



||A



ω||

||A



ω||ds

≤e

−λ

||A



(νλ

)

||MPf||

∞

νλ



||A



ω||

||A



ω||ds

≤e

−λ

(4η

−2

−4

(νλ

)

−2

(νλ

)

D

[

+t]

−4

≤max(

(νλ

)

(νλ

)

)(e

−νλ

(4η

−2

−4

)+η

−2

+[

+t]

−4

)

= D

γ(,t),

DOI:10.12677/aam.2021.1072642546A^êÆ?Ð

¤²§w

Ù¥

def

=max(

(νλ

)

(νλ

)

γ(,t)

def

−νλ

−2

+η

−2

+[

+t]

−4

aq/,

||v||

−||v

−ν

||A



v||

ds≤

νλ

(||MP



f||

∞

t+C



||A



ω||

||A



ω||ds),

db^‡(3.1)Ú(4.28)

||A



v||

ds≤

||MP



f||

∞

||v



||A



ω||

||A



ω||ds

≤

−2

νλ

(4η

−2

−4

D

[

+t]

−4

≤max(

νλ

)(e

−νλ

(4η

−2

−4

)+η

−2

+[

+t]

−4

)

≤D

γ(,t),

Ù¥

def

=max(

νλ

aq/

t−1

||A



v||

ds≤

||MP



f||

∞

||v(t−1)||



t−1

||A



ω||

||A



ω||ds

≤

γ(,t)+

−2

2D

[

+t]

−4

≤

γ(,t)+

−2



−4

)

≤

max(D

,2D

10D

)γ(,t)

≤D

γ(,t),

|^±þO§k

||v||

||A



v||

ds≤sup

0≤s≤t

||v||

||A



v||

≤D

(,t)

≤e

νλ

(,t).

DOI:10.12677/aam.2021.1072642547A^êÆ?Ð

¤²§w

aq/,

t−1

||v||

||A



v||

ds≤e

νλ

(,t).

l(4.21)ª•

||A



v||

≤e

G(t)

(||A



+H(t)),

Ù¥

G(t) =

2ν

||A



v||

||v||

2ν

νλ

(,t)

= D

(,t),

…

H(t) =

||MP



f||

∞



||A



ω||

||A



ω||

≤

−2

[

+t]

−4

(ÜþãO§

||A



v(t)||

≤e

(,t)(

−2

[

+t]

−4

+4η

−2

−4

)

≤max(e

νλ

)

= D

γ(,t)e

(,t)

5¿

||A



v||

≤(

2ν

||v||

||A



v||

)||A



v||

||MP



f||

∞



||A



ω||

||A



ω||

éþª3t∈[1,∞)þ¦^˜—GronwallØª§

||A



v||

≤[

t−1

||A



v||

ds+

t−1

(

||MP



f||

∞



||A



ω||

||A



ω||

)ds]e

2ν

νλ

(,t)

≤[D

γ(,t)+

−2



−4

2ν

νλ

(,t)

≤[D

γ(,t)+

−2



−4

2ν

νλ

(,t)

≤(D

+max(

))γ(,t)e

2ν

νλ

(,t),

DOI:10.12677/aam.2021.1072642548A^êÆ?Ð

¤²§w

@o·‚











||A



v(t)||

≤D

γ(,t)e

(,t)

||A



v(t)||

≤(D

+max(

))γ(,t)e

2ν

νλ

(,t),

…

||A



v(t)||

≤D

γ(,t)e

(,t)

Ù¥D

= max(D

+max(

)),D

27C

2ν

νλ

max(D

·‚Ž‡éT

= T

().Ï•η→∞§→0

,Œ±b

−4

>1,

‡¦

−2νλ

−4

≤1.

def

2νλ

ln(2D

−4

) >0,

…

()

def

=(

+2T

())

−4

ØJOE

()Ú



−2

() = (

+2T

())

−4

= (1+

νλ

ln(2D

)η

−4

)

−4

= (1+

νλ

(ln(2D

)+lnη

−4

))

−4

= (1+

νλ

max(ln(2D

,1)))(1+lnη

−4

)

−4

≤D

…



−2

≤

(

−4

)

≤

(1+lnη

−4

)

(

−4

)

≤

(1+lnη

−4

)(

−4

)

≤D

DOI:10.12677/aam.2021.1072642549A^êÆ?Ð

¤²§w

…k

||A



v(t)||

≤D

γ(,t)e

(,t)

≤D

−νλ

−2

+η

−2

+[+t]

−4

(2e

−2νλ

−4

+4η

−2

+4D

)

≤D

−νλ

−2

+η

−2

(

−4

+1)

≤D

−νλ

−2

+η

−2

(

−4

+1)

Ïd,

||A



u(t)||

= ||A



v(t)||

+||A



ω(t)||

≤D

−νλ

−2

+η

−2

−4

≤max(1,D

)((η

−2

+η

−2

−4

)

<NR

(4.30)

e5,·‚¦^‡y{y²2T

≤T

∞

.XJT

∞

= ∞,@o2T

≤T

∞

Ïd·‚bT

∞

<2T

<∞,

l(4.30)ª•

||A



u(T

∞

)||

<NR

ù†(4.26)ªgñ.Ïd2T

≤T

∞

•,·‚5y²(4.25)ª.•›t3[T

,2T

]þ

l(4.27)ª

||A



ω(t)||

≤[k

−

2νC

−2



−2

]

−2

•‡ÀT

≥

2ln2C



,@o∃

,0 <≤

,¦

−

νC

−2



−2

≤

,0 <≤

Ïd,

||A



ω(t)||

≤[k

−

νC

−2



−2

]

−2

≤[

]

−2

= k



−2

DOI:10.12677/aam.2021.1072642550A^êÆ?Ð

¤²§w

d®•OªŒ

||A



v(t)||

≤D

−νλ

−2

+η

−2

≤D

(η

−2

√

−2

≤D

(η

−2

= Γ(η

−2

@ok

Γ(η

−2

) = D

−2

−4

√

−4

.(4.31)

db^‡H(a,b),•∃

,fu0 <≤

−2

−4

= D

−2

−4

≤

(4η

−2

−4

aq/,∃

,éu0 <≤

√

−4

≤

(4η

−2

−4

aq/,∃

,éu0 <≤

−4

≤

(4η

−2

−4



= min(

,

),¦

Γ(η

−2

) ≤

(4η

−2

−4

),0 <≤

éu0 <≤

||A



v(t)||

≤

(4η

−2

−4

),T

≤t≤2T



= 

.y..

½n4.2(H

K5)

η

>0,k

>0,

Ú˜‡ëY¼êΓ∈C([0,∞),R),…é,0< ≤

,•3

()> 0¦é,0<

≤

,u

∈D(A



)…f∈L

∞

([0,∞),L

))÷vb^‡(3.1).@o(2.5)•3•˜)u÷

vC([0,∞),V



)∩L

∞

([0,∞),V



),=k

||A



u(t)||

≤K

,t≥0,

DOI:10.12677/aam.2021.1072642551A^êÆ?Ð

¤²§w

Ù¥K

•6ν,λ

,Úη

,i= 1,2,3,4,Ø•6t≥0.…vÚω÷v

||A



v(t)||

≤Γ(η

−2

),t≥

Ù¥Γd(4.31)ª‰Ñ§…

||A



ω(t)||

≤k



−2

,t≥

y².k|^Ún(4.2),2|^ Ún(4.3),•éÚn(4.3)|^êÆ8B{,ØJy½n¤á(

Ø¤á.

ë•©z

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JournalofSovietMathematics,3,458-479.https://doi.org/10.1007/BF01084684

[2]Temam,R. (2001)Navier-StokesEquations:Theoryand NumericalAnalysis.Vol.343, Amer-

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IndustrialandAppliedMathematics,Philadelphia.

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R.E.L.,Eds.,DirectionsinPartialDiﬀerentialEquations,AcademicPress,Cambridge,MA,

55-73.https://doi.org/10.1016/B978-0-12-195255-6.50011-8

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[6]Raugel,G.andSell,G.R.(1993)Navier-StokesEquationsonThin3DDomains.I.Global

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[7]Dragomir, S.S.(2003)Some GronwallType InequalitiesandApplications. Nova Science,Haup-

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DOI:10.12677/aam.2021.1072642552A^êÆ?Ð