Hele-Shaw流模型的Darcy-Cahn-Hilliard方程组解的研究 Study on Darcy-Cahn-Hilliard Equations of Hele-ShawFlow

doi:10.12677/AAM.2021.107255

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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(7),2428-2441

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

https://doi.org/10.12677/aam.2021.107255

Hele-Shaw6.Darcy-Cahn-Hilliard•

§|)ïÄ

‹‹‹‰‰‰§§§ÆÆÆ“““

oA“‰ŒÆêÆ‰ÆÆ§oA¤Ñ

ÂvFÏµ2021c619F¶¹^FÏµ2021c711F¶uÙFÏµ2021c722F

Á‡

©édDarcy•§ÚCahn-Hilliard•§ÍÜ¤üƒHele-Shaw6*Ñ.¡.?1ï

Ä"3d.¥§Darcy•§¥ƒ på‘ÚCahn-Hilliard•§¥6NpÑ$‘

ÍÜüƒ•§§©é•§|¥š‚5‘3÷v•˜„b^‡e§ïÄ•§|f)•

35Ú•˜59)UþO"

'…c

ÍÜ§•35§•˜5§UþO

StudyonDarcy-Cahn-Hilliard

EquationsofHele-Shaw

Flow

XiangyuXiao,ZhilinPu

SchoolofMathematicalScience,SichuanNormalUniversity,ChengduSichuan

Received:Jun.19

,2021;accepted:Jul.11

,2021;published:Jul.22

,2021

©ÙÚ^:‹‰,Æ“.Hele-Shaw6.Darcy-Cahn-Hilliard•§|)ïÄ[J].A^êÆ?Ð,2021,10(7):

2428-2441.DOI:10.12677/aam.2021.107255

‹‰§Æ“

Abstract

Inthispaper,westudythetwophaseHele-Shawﬂow,whichconsistsoftheCahn

HilliardequtionandtheDarcyequation.Inthismodel,anextraphaseinducedforce

termintheDarcyequationiscoupledwithaﬂuidinducedtransportterminthe

Cahn-Hilliardequation.Asthenon-lineartermsatisﬁesthemoregeneralcondition,

weshowtheexistenceoftheweaksolution,energyestimate,andtheuniqueness.

Keywords

Coupled,Existence,Uniqueness,EnergyEstimate

2021byauthor(s)andHansPublishersInc.

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

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

1.0

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u= −

12η

(∇p−ρg),(x,t) ∈Ω

\Γ

.(1)

divu= 0,(x,t) ∈Ω

\Γ

.(2)

[p] = γκ,(x,t) ∈Γ

.(3)

DOI:10.12677/aam.2021.1072552429A^êÆ?Ð

‹‰§Æ“

[u·n] = 0,(x,t) ∈Γ

.(4)

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3©¥§¤ïÄHele-Shaw*Ñ.¡.Xe¤«

u= −∇p−γϕ∇µ,(x,t) ∈Ω

.(5)

divu= 0,(x,t) ∈Ω

.(6)

+u·∇ϕ−ε∆µ= 0,(x,t) ∈Ω

.(7)

µ= −ε∆ϕ+

f(ϕ)(x,t) ∈Ω

.(8)

é•§(5)−(8)\þ>.^‡ÚÐŠ^‡µ

∂p

∂n

∂µ

∂n

∂ϕ

∂n

= 0,(x,t) ∈∂Ω

.(9)

ϕ(·,0) = ϕ

(·),x∈Ω.(10)

Ù¥Ω

= Ω×(0,T)¶∂Ω

= ∂Ω×(0,T),∂Ω“LΩ>.

éš‚5‘f(s),b÷ve^‡µ

f(0) = 0,

f(s) ≥c

≥0),

f(s)s≥c

F(s)−c

>0,c

≥0),

F(s) ≥−c

≥0).

Ù¥F(s)=

f(s)ds.~Xf(s)=

2p+1

i=1

2p+1

>0)÷vþã5Ÿ"w´„§f(s)=s

+sÒ

DOI:10.12677/aam.2021.1072552430A^êÆ?Ð

‹‰§Æ“

÷vþã^‡"

5¿µ3kHele-Shaw¯K¥§Ω´˜‡‘•;3©¥§·‚•ÄΩ⊂R

(d=

2,3),…´˜‡k.«•§Ï•n‘¯KäkêÆ¿Â§¿…3)Ôþ•kA^§•þu(x,t)∈

“L„Ý§Iþp(x,t)∈R“L3:(x,t)ž—N·ÜÔØå§Cþµ(x,t),ϕ(x,t)∈R“Lƒ

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•§(5)−(10)´dLee,Lowengrub,andGoodmanJÑBHSCH.A~"¦‚y²BHSCH

XÚ[13])ÂñuHele-Shaw.(1)−(4))"·‚5¿(5)¥Ørp†BHSCH.¥

ØrpkØÓ'~"•¼aq'~Øå§ÏL˜p=p+γϕµ§Œ ±{ü/3.¥Ú\-#

½ÂØå"·‚ò•§|(5)−(10)¡•DCH(Darcy-Cahn-Hilliard)XÚ"

½ÂXeCahn-HilliardUþ•§

(ϕ) =

Ω



|∇ϕ|

F(ϕ)



dx.(11)

†NõÑÑ.¡.aq[14]§DCHXÚ•´‡ÑÑXÚ§§÷vXeUþÑÑ5Æµ

(ϕ)

+εk∇µk

kuk

= 0.(12)

þãUþ•§3DCHXÚêÆ©Ûå–'-‡Š^"

©´3[15]Ä:þŠ•˜„b^ ‡"·‚ ò?ØÐ>Š¯K(5)-(10)f)•3•˜5§

f)˜5Ÿ"

IO˜mL

(Ω),H

(Ω)‰ê©O^k·k

,k·k

L«§Z

∗

“LBanach˜mZéó˜

m"ÎÒ(·,·)“L˜mL

(Ω)þSÈ§ÎÒh·,·i“L˜mH

(Ω)Ú˜mH

(Ω)

∗

éóÈ§˜

(Ω)“L˜mL

(Ω)f˜m§¿…3>.þŠ•""©¥§cÚC“L†p,µ,ϕ,uÚεÃ'

~ê¶XJ~êÚεk'{§·‚^C= C(ε)5L«§Ù¥0 <ε<1"

2.PDE©Û

·‚ò3p,µ,ϕ§ò•§|C/µ

div(∇p+γϕ∇µ) = 0,(x,t) ∈Ω

.(13)

−ε∆µ−div(ϕ[∇p+γϕ∇µ]) = 0,(x,t) ∈Ω

.(14)

¤±§©ïÄ ‡©•§/ªÒ´•§(13),(14)Ú•§(8):µ=−ε∆ϕ+

f(ϕ)±9Ð>Š

^‡(9),(10):

div(∇p+γϕ∇µ) = 0,(x,t) ∈Ω

−ε∆µ−div(ϕ[∇p+γϕ∇µ]) = 0,(x,t) ∈Ω

DOI:10.12677/aam.2021.1072552431A^êÆ?Ð

‹‰§Æ“

µ= −ε∆ϕ+

f(ϕ)(x,t) ∈Ω

∂p

∂n

∂µ

∂n

∂ϕ

∂n

= 0,(x,t) ∈∂Ω

ϕ(·,0) = ϕ

(·),x∈Ω.

·‚½ÂXe/ªf)[15]µ

½Â2.1bϕ

∈H

(Ω),XJn-ê|(p,µ,ϕ)÷ve^‡µ

p∈L

d+1

((0,T);H

(Ω)∩L

(Ω).(15)

µ∈L

((0,T);H

(Ω)).(16)

∇p+γϕ∇µ∈L

((0,T);L

(Ω)).(17)

ϕ∈L

∞

((0,T);H

(Ω)).(18)

∈L

d+1

((0,T);(H

(Ω))

∗

).(19)

¿…é¤kt∈(0,T),±eÑ¤áµ

(∇p+γϕ∇µ,∇q) = 0∀q∈H

(Ω).(20)

hϕ

,vi+ε(∇µ,∇v)+(ϕ[∇p+γϕ∇µ],∇v) = 0∀v∈H

(Ω).(21)

(µ,ψ)−ε(∇ϕ,∇ψ)−

(f(ϕ),ψ) = 0∀ψ∈H

(Ω).(22)

…÷vÐŠ^‡ϕ(0)=ϕ

Kn-ê|(p,µ,ϕ)¡••§(13),(14)±9•§(8)(9),(10)˜‡f

e¡?Ø)•35.

½n2.1E(t) = J

(ϕ),eÐŠ^‡÷vE(0) ≤C

ž§K•§•3½Â2.1¥f)"

y².|^³7Cq•{§=^k•‘%CÃ•‘§·‚éϕ,p,µEÑCq)§·‚ò¦

(Ω)˜|k•‘Ä•þ{ω

}

i=1...m

§ùÄ•þ¤Ü¤˜m·‚P•W

,Ù¥·‚

ép

,ϕ

,µ

: [0,T] →W

i=1

i,m

i=1

i,m

i=1

i,m

¤±·‚µ

(∇p

+γϕ

∇µ

,∇q) = 0∀q∈H

(Ω).(23)

DOI:10.12677/aam.2021.1072552432A^êÆ?Ð

‹‰§Æ“





+ε(∇µ

,∇v)+(ϕ

[∇p

+γϕ

∇µ

],∇v) = 0∀v∈H

(Ω).(24)

(µ

,ψ)−ε(∇ϕ

,∇ψ)−

(f(ϕ

),ψ) = 0∀ψ∈H

(Ω).(25)

3•§(23)¥-q=ω

,i=1,2...m¿Óž¦±

i,m

,,¦Ú¶3•§(24)¥-v=ω

,i=

1,2...m,¿Óž¦±µ

i,m

,2¦Ú¶ 3•§(25)¥-ψ=ω

i=1,2....m§Óž¦±

i,m

§2¦Ú§

•òn‡ªfÜ¿µ



εk∇ϕ

((F(ϕ

),1)



+εk∇µ

k∇p

+γϕ

∇µ

= 0.(26)

·‚3þ¡•§ü>ÓžÈ©§Kµ

(t)+

εk∇µ

k∇p

+γϕ

∇µ

= E

(0).(27)

Ù¥E

(t) =

Ω

|∇ϕ

F(ϕ

)

dx"

Ún2.1bE(t)=J

(ϕ),E

(t)=J

(ϕ

),,Ù¥m´k•ê§…W

´H

(Ω)¥˜|k•‘

Ä•þ¤Ü¤˜m§E(0) ≤C

ž,KE

(0) ≤C

(ÜÚn2.1§|^•§(27)Œk∇ϕ

≤C

k∇µ

≤C

,2ÏL

'X∇p

=−u

−γϕ

∇µ

§Äkéud=2,3ŠâSobolevi\½nµH

(Ω)→L

(Ω),2

dYoungØªÚþãOŒ

k∇p

≤C

,Ùg3•§(25),-ψ= ∆ϕ

,·‚Œ

εk∆ϕ

≤

k∇µ

k∇ϕ

−

k∇ϕ

?˜Úµ

εk∆ϕ

≤

k∇µ

k∇ϕ

≤C

.(28)

dGagliardo-NirenbergØª§·‚kkϕ

∞

≤Ck∆ϕ

kϕ

2d−1

(d= 2,3)d

k∇p

d+1

≤Cku

d+1

+Ckϕ

d+1

∞

k∇µ

d+1

≤Cku

+Ckϕ

∞

+Ck∇µ

kϕ

∞

≤Ck∆ϕ

kϕ

2d−1

≤Ck∆ϕ

kϕ

4d−1

2ÏLsup

t∈(0,T)

k∇ϕ

≤C

,µ

kϕ

≤C

,Ù¥C

´˜‡†Tk'~ê"

·‚2éþãªfÓžét¦È©§¿dþãOµ

k∇p

2d+1

≤C

†C

d,Tk

'"2ÏL•§(24),é?¿v∈H

(Ω)Œµ

,vi= −ε(∇µ

,∇v)+(ϕ

,∇v) ≤[εk∇µ

+kϕ

∞

]k∇vk

DOI:10.12677/aam.2021.1072552433A^êÆ?Ð

‹‰§Æ“

Šâéó˜m‰ê½Â§•kd

(Ω))

∗

≤C

,¦È©

(Ω))

∗

≤C

´˜‡~

ê"

Šâg‡˜m5Ÿ§Ïd·‚(J•3ϕ,p,µ,u§¦ϕ∈L

∞

((0,T);H

(Ω)),¦p∈

d+1

((0,T);H

(Ω))∩L

(Ω),¦µ∈L

((0,T);H

(Ω)),Šâ‚55ku=∇p+ γϕ∇µ,X

ku∈L

((0,T);L

(Ω))k:

→ϕ

→p

→µ

→u.

Šâ‚55§·‚kµ

(∇p+γϕ∇µ,∇q) = 0,

ŒdV‚››Âñ½n·‚kµ

hϕ

,vi+ε(∇µ,∇v)+(ϕ[∇p+γϕ∇µ],∇v) = 0

(µ,ψ)−ε(∇ϕ,∇ψ)−

(f(ϕ),ψ) = 0.

e¡?1UþO:

½n2.2bϕ

∈H

(Ω),f(ϕ)∈L

((0,T);H

(Ω)),Ω⊂R

(d=2,3)´˜‡Lipschitz«•§±

(ϕ

)≤C

,e(p,µ,ϕ)´•§(20) −(22)˜|f)§éuu=−(∇p+ γϕ∇µ)é¤kt∈

(0,T),•3˜‡Ø•6uε~êC= C(E(0)) >0k

Ω

ϕ(x,t)dx=

Ω

(x)dx.(29)

E(t)+



εk∇µ(s)k

ku(s)k



ds= E(0) <∞.(30)

max

0≤s≤t

kϕ(s)k

≤

|Ω|.(31)

kµ(s)k

ds≤

|Ω|.(32)



µ(s)−ε

−1

f(ϕ(s))



ds≤

2ε

|Ω|.(33)

kϕ

(s)+u(s)·∇ϕ(s)k

)

∗

ds≤

√

Cε.(34)

kϕ

(s)k

1,3

)

∗

ds≤C(

ε+c

|Ω|+

√

2(C

ε+c

|Ω|

).(35)

DOI:10.12677/aam.2021.1072552434A^êÆ?Ð

‹‰§Æ“

ùpE(t) = J

(ϕ(t)),Ù¥J

(·)´Uì•§(11)5½Â"

y².•§(29)´3•§(21)¥-v=1=Œ"••§(30),·‚©•ü«œ¹5‰µ

1˜«AÏœ¹§ϕ

∈L

((0,T);H

(Ω))ž§·‚3•§(20)¥½q=

,3•§(21)¥

½v= µ§3•§(22)¥½ψ= −ϕ

,¿rnöƒ\·‚µ



εk∇ϕk

(F(ϕ),1)



+εk∇µk

k∇p+γϕ∇µk

= 0.

3•§ü>Óž«m(0,t)þÈ©ÒŒ•§(30)"

éu˜„œ¹ϕ

∈L

d+1

((0,T);(H

(Ω))

∗

),XJdž·‚2-ψ= −ϕ

{§3•§(22)¥Òv

k¿Â§Ï·‚3ùpò^Steklov²þ{Eâ[16]"éut∈(0,T),δ>0´?¿ê§

½ÂϕSteklov²þϕ

ϕ(·,t) = S

(ϕ)(·,t) =

t+δ

ϕ(·,s)ds∀t∈(0,T).

éuvδ,

(·,t) := (ϕ

(·,t))

ϕ(·,t+δ)−ϕ(·,t)

Ïd§éuz˜‡t∈(0,T−δ),kϕ

(·,t) ∈H

(Ω)§Uì²þ{(ØÒkµ

(ϕ

) = (S

(ϕ))

= ϕ

.(36)

@o·‚y3òS

A^•§(20)−(22)¥§¿^s†t,·‚kµ

(∇p

+γ(ϕ∇µ)

,∇q) = 0∀q∈H

(Ω).(37)

(ϕ

,v)+ε(∇µ

,∇v)+((ϕ[∇p+γϕ∇µ])

,∇v) = 0∀v∈H

(Ω).(38)

(µ

,ψ)−ε(∇ϕ

,∇ψ)−

((f(ϕ))

,ψ) = 0∀ψ∈H

(Ω).(39)

3•§(39)¥-ψ= −ϕ

,3•§(38)¥-v= µ

,3•§(37)¥-q=

,¿òn‡•§ƒ\§

(ϕ

)+ε



∇µ



∇p

+γ(ϕ∇µ)



= R

(t).(40)

Ù¥

(t) =

(f(ϕ

)−(f(ϕ))

−(∇p

+γ(ϕ∇µ)

,ϕ∇µ

−(ϕ∇µ)

)

+(ϕ

∇p

+γ(ϕ∇µ)

−(ϕ[∇p+γϕ∇µ])

,∇µ

DOI:10.12677/aam.2021.1072552435A^êÆ?Ð

‹‰§Æ“

3•§(40)ü>ÓžétÈ©§·‚kµ

(ϕ

(s))+

(ε



∇µ

(t)



∇p

(t)+γ(ϕ∇µ)

(t)



)dt

= J

(ϕ

(0))+

(t)dt∀s∈(0,T).

5¿§éz˜‡½ε>0,duf(ϕ)∈L

((0,T);H

(Ω))¿…f(·)´‡ëY¼ê§¤±

kf(ϕ) ∈L

((0,T);H

(Ω)),-δ→0

,¿(ÜSteklov²þ5Ÿ§·‚kµ

lim

δ→0

(t)dt= 0,

(ϕ(s))+

(εk∇µ(t)k

k∇p(t)+γ(ϕ(t)∇µ(t)k

)dt= J

(ϕ(0).

Ïd§(30),l(30)¥•E(t)´'užmëY¼ê"

|^(30)§Ú·‚b:

F(s) ≥−c

≥0.(41)

·‚k

Ω

F(s)dx≥−c

|Ω|(c

≥0)§KŒ•§(31)"•§(32)Œd•§(30)†"•

§(33)Œd•§(22),•§(31),(32)±9HolderØªÚb^‡

f(s)≥c

"d˜m'Xµ

(Ω) ⊂L

(Ω)

∼

(Ω) ⊂(H

(Ω))

∗

Ú•§(21)Œ

hϕ

+u(s)·∇ϕ(s),vi= −ε(∇µ,∇v),

ŠâHolderØªµ

|hϕ

+u(s)·∇ϕ(s),vi|= ε|(∇µ,∇v)|≤εk∇µk

k∇vk

•2Š âéó‰ê±9d‰ê•§(34)"••§(35)§| ^•§(21)ÚSobolev

i\½nµd= 2,3,kH

(Ω) →L

(Ω),éuv∈W

1,3

(Ω),

hϕ

,vi= −ε(∇µ,∇v)+(ϕu,∇v)

≤εk∇µk

k∇vk

+kϕk

kuk

k∇vk

≤C[εk∇µk

+kϕk

kuk

]k∇vk

≤C



εk∇µk

√

ε+2c

|Ω|



•|^d‰êÚéó‰ê·‚(J[17]"

e50f)„kÙ¦K5"

DOI:10.12677/aam.2021.1072552436A^êÆ?Ð

‹‰§Æ“

Ún2.2bϕ

∈H

(Ω),f(ϕ)∈L

((0,T);H

(Ω))±9«•Ω⊂R

(d=2,3)´˜‡Lipschitz«

•,en-|(p,ϕ,µ)´½Â2.1˜|f),Kϕ∈L

((0,T);H

(Ω))"

y².·‚ò•§(22)-#¤Xe

ε(∇ϕ,∇ψ) = (µ−

f(ϕ),ψ)∀ψ∈H

(Ω).(42)

ùpϕ´‡aquPoisson•§,¿äkàgNeumann>.^‡˜‡f)§Ùmà‘¼ê•g= µ−

f(ϕ)§duf(ϕ) ∈L

((0,T);H

(Ω)),¤±mà¼êg∈L

((0,T);H

(Ω)),ϕ∈L

((0,T);H

(Ω)"

e¡·‚?Øf)•˜5µ

½n2.3bϕ

∈H

(Ω)ÚJ

(ϕ

) ≤C

, Ù¥C

Ø•6uε,¿…Ω ⊂R

(d= 2,3)´˜‡Lipschitz«

•§±9f(ϕ) ∈L

((0,T);H

(Ω)),˜|f)(p,µ,ϕ) ÷vK^‡∇p+γϕ∇µ∈L

6−d

((0,T);L

(Ω)),

µ∈L

6−d

((0,T);H

(Ω)),ϕ

∈L

((0,T);(H

(Ω)

∗

))ž§Ù¥p´˜‡k•ê§Kù|f)´•˜

"

y².b(p

,µ

,ϕ

),i=1,2´ü|f)§½Âu

=−∇p

−γϕ

∇µ

,i=1,2,-p=p

−p

,u=

−u

,µ= µ

−µ

,ϕ= ϕ

−ϕ

,(p

,µ

,ϕ

),(p

,µ

,ϕ

)÷véA•§(20)−(22)§òö(Üå

5§·‚•§µ

(u,∇q) = 0∀q∈H

(Ω).(43)

hϕ

,vi+ε(∇µ,∇v)−(ϕ

u+ϕu

,∇v) = 0∀v∈H

(Ω).(44)

(µ,ψ)−ε(∇ϕ,∇ψ)−

(f(ϕ

)−f(ϕ

),ψ) = 0∀ψ∈H

(Ω).(45)

du•§(29),·‚Œ|^

Ω

ϕ(x,t)dx=0,∀t∈(0,T),•ŒÏL˜m²£d(J"

3•§(44)¥§-v=ϕ,3•§(45)¥§-ψ=µ,òöÜ¿§2ÏL•§(u

,∇(ϕ

))=0=

(u,∇(ϕ

ϕ)),·‚µ

kϕk

+kµk

= −(u·∇ϕ

,ϕ)+

(g(ϕ

,ϕ

)ϕ,µ).(46)

Ù¥g(ϕ

,ϕ

)´ÏLf(ϕ

)−f(ϕ

)ü‡õ‘ªÏLng•úª5§=g(ϕ

,ϕ

) =

f(ϕ

)−f(ϕ

)

−ϕ

|^SchwarzØªÚGagliardo-NirenbergØª[18]§k

kϕk

∞

≤Ck∆ϕk

kϕk

4−d

(d= 2,3).

3•§(46)¥k

kϕk

+2kµk

≤2kuk

k∇ϕ

kϕk

∞

kg(ϕ

,ϕ

∞

kϕk

kµk

≤

4γ

kuk

k∇ϕ

k∆ϕk

kϕk

4−d

kg(ϕ

,ϕ

∞

kϕk

kµk

≤

4γ

kuk

k∆ϕk

+C(ε)k∇ϕ

4−d

kϕk

+kµk

kg(ϕ

,ϕ

∞

kϕk

DOI:10.12677/aam.2021.1072552437A^êÆ?Ð

‹‰§Æ“

Ù¥éumà,˜‘kg(ϕ

,ϕ

∞

,Äkéu

g(ϕ

,ϕ

) =

2p+1

k=1

(ϕ

k−1

+ϕ

k−2

+....+ϕ

k−1

+ϕ

·‚kµ

kg(ϕ

,ϕ

∞

≤

2p+1

k=1

kϕ

k−1

∞

+kϕ

k−2

∞

kϕ

∞

+......+kϕ

k−1

∞

duϕ∈L

∞

(0,T;H

(Ω)),Ïdϕ∈L

∞

(0,T;L

∞

(Ω)),Kkϕk

∞

≤sup

t∈(0,T)

kϕk

∞

≤C,Ù¥C´˜‡

~ê§Ïkg(ϕ

,ϕ

∞

≤C

,ùpC

´˜‡†pk'~ê"

ÏL(30)

kϕk

+kµk

≤

4γ

kuk

k∆ϕk

+C(ε,p)kϕk

.(47)

Ù¥C(ε,p) =

3•§(39)¥-ψ= ∆ϕ,

εk∆ϕk

= −(µ,∆ϕ)+

(g(ϕ

,ϕ

)ϕ,∆ϕ)

≤

k∆ϕk

kµk

kϕk

,

k∆ϕk

≤

kµk

+C(ε,p)kϕk

.(48)

Ù¥C(ε,p) >0

5¿

u= u

−u

= −∇p−γ(ϕ

∇µ

−ϕ

∇µ

) = −∇p−γϕ

∇µ−γϕ∇µ

3•§(37)¥-q= p

kuk

= −

(u,∇p)−(u,ϕ

∇µ)−(u,ϕ∇µ

)

= −(ϕ

u,∇µ)−(ϕu,∇µ

(49)

3•§(38)¥-v= µ,Œ

hϕ

,µi+εk∇µk

= (ϕ

u+ϕu

,∇µ).(50)

e5é•§(39)A^dA‰âÅ²þŽ{S

§Kkµ



,ψ



−ε(∇ϕ

,∇ψ)−

(f(ϕ

)−f(ϕ

)

,ψ) = 0∀ψ∈H

(Ω).

DOI:10.12677/aam.2021.1072552438A^êÆ?Ð

‹‰§Æ“

-ψ= −ϕ

,þ¡•§Œ

−(µ

,ϕ



∇ϕ



(f(ϕ

)−f(ϕ

))

,ϕ

) = 0.

éδ4•δ→0

,|^dA‰âÅ²þŽf5Ÿ§k

−hϕ

,µi+

k∇ϕk

hϕ

,f(ϕ

)−f(ϕ

)i= 0.(51)

•r•§(49),(50),(51)\å5§¿|^(u

,∇(ϕµ))=(u,∇(ϕµ

))=0ÚYoungØª±

9Gagliardo-NirenbergØª

kuk

+εk∇µk

k∇ϕk

hϕ

,f(ϕ

)−f(ϕ

= −(u

·∇ϕ,µ)+(u·∇ϕ,µ

)

≤C(ku

kµk

+kuk

kµ

)(k∆ϕk

k∇ϕk

6−d

≤

4γ

kuk

kµk

k∆ϕk

+C(ε)(ku

6−d

+kµ

6−d

)k∇ϕk

Ïd§

4γ

kuk

k∇µk

k∇ϕk

hf(ϕ

)−f(ϕ

),ϕ

≤

k∆ϕk

kµk

+C(ε)(ku

6−d

+kµ

6−d

)k∇ϕk

e5éf(ϕ

)−f(ϕ

)A^.‚KF¥Š½n§¿dš‚5‘b^‡kf(ϕ

)−f(ϕ

f(η)ϕ≥c

ϕ,þªŒµ

4γ

kuk

k∇µk

k∇ϕk

2ε

kϕk

≤

k∆ϕk

kµk

+C(ε)(ku

6−d

+kµ

6−d

)k∇ϕk

(52)

ò•§(47),(48)±9•§(53)εƒ\å5§kµ







kϕk

k∇ϕk



k∆ϕk

2γ

kuk

1−ε

kµk

k∇µk

≤C(ε)(ku

6−d

+kµ

6−d

)k∇ϕk

+C(ε,p)kϕk

(53)

(Ü½nb^‡§•§(53)Œ¤µ

((

+1)kϕk

k∇ϕk

k∆ϕk

2γ

kuk

1−ε

kµk

k∇µk

≤a(t)(kϕk

+k∇ϕk

(54)

DOI:10.12677/aam.2021.1072552439A^êÆ?Ð

‹‰§Æ“

Ù¥a(t) = C(ε)(ku

6−d

+kµ

6−d

)+C(ε,p)

(

+1)kϕk

k∇ϕk

≤

a(s)(kϕ(s)k

+k∇ϕ(s)k

)dt.(55)

ÏLGronwallØª§·‚µ

(

+1)kϕk

k∇ϕk

≤



(

+1)kϕ(0)k

k∇ϕ(0)k



exp

(

a(s)ds

)

.(56)

d½nb^‡Œ•§

a(s)ds<∞,Ïdkϕ(t) = 0,é?¿t∈(0,T),¤±½ny²¤"

ë•©z

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