一类序参数守恒的相场模型的弱解存在性 Existence of Weak Solutions for a Class of Phase-Field Models with Conservation of Order Parameter

doi:10.12677/PM.2021.116123

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PureMathematicsnØêÆ,2021,11(6),1084-1102

PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.116123

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ExistenceofWeakSolutionsforaClassof

Phase-FieldModelswithConservationof

OrderParameter

YananZhao

,XingzhiBian

1∗

,LeiYu

∗ÏÕŠö"

©ÙÚ^:ëä™,>1ƒ,u[.˜aSëêÅðƒ|.f)•35[J].nØêÆ,2021,11(6):1084-1102.

DOI:10.12677/pm.2021.116123

ëä™

MaterialsGenomeInstitute,ShanghaiUniversity,Shanghai

CollegeofScience,ShanghaiUniversity,Shanghai

Received:May7

,2021;accepted:Jun.8

,2021;published:Jun.16

,2021

Abstract

Inthispaper, ourresearch willbebasedontheAlber-Zhumodelwhichorderparam-

eterisconserved.Theevolutionequationinthismodelisafourthorder,nonlinear

degeneratepartialdiﬀerentialequationofparabolictype.Thismodelisaphase-

ﬁeldmodelwhichdescribestheinterfacemotionbyinterfacediﬀusioninelastically

deformedsolids.Forexample,thebasicphenomenaoccurringduringthisprocess,

calledsintering,aredensiﬁcationandgraingrowth.Sincenoatomexchangeoccurs

attheinterface,thevolumesofthediﬀerentregionsseparatedbytheinterfaceare

conserved.Weignoretheelasticeﬀectandreducetheoriginalinitialboundaryvalue

problemtoasinglenon-degenerateequationof1dimensionalsituation,andprovethe

existenceoftheweaksolutionofthereducedinitialboundaryvalueproblemofthis

model,wheretheboundaryconditionsoftheorderparametersareacombinationof

theNeumannboundaryconditionsandtheno-ﬂowconditions.TheGalerkinmethod

is usedtoprove theexistenceofweak solutionsforthereducedinitialboundary value

problemofthismodel.Althoughtheproblemconsideredisasingleequationoforder

parameters,itisinherentlydiﬃcultduetothepresencewhichisthegradientofun-

knownfunctionS.Hence,weneedtomollifythegradientterm,whichcausesalotof

diﬃcultiesonthetheoreticalanalysisandnumericalsimulation.

Keywords

MotionofGrainBoundaries,Phase-FieldModel,ParabolicEquation,

ExistenceofWeakSolutions

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

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

DOI:10.12677/pm.2021.1161231085nØêÆ

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:x∈Ω,?uƒ1½ƒ2.S

L«é¡3×3Ý8Ü,Ù¥˜‡™•þT(t,x) ∈S

“L…Ü

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L« £.

™•¼ê(u,T,S)L÷v±eO·•§:

−div

T(t,x)=b(t,x),(2.1)

T(t,x)=D(ε(∇

u)−¯εS)(t,x),(2.2)

(t,x)=cdiv

(∇

(ψ

(ε(∇

u),S)−ν∆

S)|∇

S|)(t,x),(2.3)

Ù¥(t,x) ∈(0,∞)×Ω.•§Ð>Š^‡´:

u(t,x)=0,(t,x) ∈[0,∞)×∂Ω,(2.4)

∂

∂n

S(t,x)=0,(t,x) ∈[0,∞)×∂Ω,(2.5)

∂

∂n

(ψ

(ε,S)−ν∆

S)|∇

S|(t,x)=0,(t,x) ∈[0,∞)×∂Ω,(2.6)

S(0,x)=S

(x),x∈

Ω.(2.7)

Ù¥,b: [0,∞)×Ω→R

‰NÈå,∇

uL«u˜ê´3×3Ý,(∇

L« £FÝ

=˜Ý,…

DOI:10.12677/pm.2021.1161231086nØêÆ

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ε(∇

u) =



∇

u+(∇



´ACÜþ.¯ε∈S

´˜‡‰½Ý,D:S

→S

´˜‡‚5,é¡,½N,“L

5Üþ.3gdU¥

ψ(ε,S) =



D(ε−¯εS)



·(ε−¯εS)+

ψ(S),(2.8)

ψ∈C

(R,[0,∞))L«V³²¼ê,Ù3S= 0ÚS= 1?•Š.©˜‡AÏV³²

¼ê:

ψ(S) =



S(1−S)



.(2.9)

ü‡ÝIþÈ•A·B=

.ψ

´'uS ê,c>0´˜‡~ê,ν˜‡v

~ê.ÐŠêâS

: Ω →R´‰½.

ùÒ¤Ð>Š¯Kïá.þã.´˜‡•¹5AÍÜXÚ.3©¥,ÑN

5A,=T= 0,·‚k

= −



D(¯ε)·(ε−¯εS)+D(ε−¯εS)·(¯ε)



= −D(ε−¯εS)·(¯ε)+

òù‡Ð>Š¯K{z•˜‘¯K(äNí„[27]).T.'uSëê•§ ´òz,Ù

ÌÜ¹k ™•¼êSFÝ,©·‚òFÝ‘ ?11?n,ïÄ1šòz•§.·‚

^šòz•§“OòzÔ•§(2.3)5LãCq¯K

= c((ψ

−νS

)

+cr|S

,(2.10)

Ù¥

|y|

+κ

,(2.11)

~êκ∈(0,1],·‚N´

|y|≤|y|

≤|y|+κ≤|y|+1.(2.12)

T{zAlber-Zhu.Œ±U•±e/ª:

= c((ψ

−νS

)

+cr|S

,(2.13)

DOI:10.12677/pm.2021.1161231087nØêÆ

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>.^‡ÚÐ©^‡´µ

=0,(t,x) ∈[0,T

]×∂Ω,(2.14)

(

−νS

)

=0,(t,x) ∈[0,T

]×∂Ω,(2.15)

S(0,x)=S

(x),x∈Ω.(2.16)

Ω = (a,d)´k.m«m…~êa<d.PQ

:= (0,T

)×Ω,…T

´˜‡~ê,½Â

(υ,ϕ)

υ(y)ϕ(y)dy,

Z= Ω½öZ= Q

½Â2.1r∈L

∞

(0,T

)…S

∈L

(Ω).¡¼êS(x,t)•Ð>Š¯K(2.13)–(2.16)˜‡f),

÷v

S∈L

∞

(0,T

(Ω))∩L

(0,T

(Ω)),(2.17)

…éu?¿ÿÁ¼êϕ∈C

∞

((−∞,T

)×R),÷v

(S,ϕ

)

+c(νS

xxx

,ϕ

)

= c((

)

,ϕ

)

−c(r|S

,ϕ)

−(S

,ϕ(0))

Ω

.(2.18)

½n2.1bS

∈H

(Ω),¯K(2.13)–(2.16)•3f)S÷vµ

S∈L

∞

(0,T

(Ω))∩L

(0,T

(Ω)),(2.19)

∈L

(0,T

−1,

(Ω)).(2.20)

3.ECq)

•y²¯K(2.13)–(2.16)f)•35,3!¥,|^Galerkin•{ET Ð>Š¯K

˜‡Cq),¿y²ù‡Cq)ÛÜf)•3.

Äk,ù˜‡Sω

,···,ω

,···,§÷v:ω

∈C

∞

(∀i),¿…ω

,···,ω

,´‚5Ã'.

é?¿m,Sáu˜mH

(Ω).3ùp,ω

´e•§):







−

=λ

dω



∂Ω

=0,

Ù¥ω

= 1.•Ä¯KCq):S

= S

(t),§k±e/ª:

(t) =

i=0

(t)ω

.(3.1)

DOI:10.12677/pm.2021.1161231088nØêÆ

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3ùp,g

´de~‡©•§|¤û½:

,ω

) = ((c(

)−νS

)

+cr|S

,ω

),1 ≤j≤m,(3.2)

((S

)

,ω

) = c(((

−νS

xxx

)|S

)

,ω

)+c(r|S

,ω

),1 ≤j≤m,(3.3)

Šâ©ÜÈ©úª,

(((

−νS

xxx

)|S

)

,ω

) = −((

−νS

xxx

)|S

,ω

),1 ≤j≤m.(3.4)

r(3.4)“\(3.3)¥,Œ±

((S

)

,ω

)+c((

−νS

xxx

)|S

,ω

)−c(r|S

,ω

) = 0,1 ≤j≤m.(3.5)

y3,I‡`²•§(3.5)´˜‡~‡©•§|.

d(S

)

i=1

(t)ω

¿…{ω

}3˜mL

(Ω)¥´IO,¤±



)

,ω



= (

i=1

(t)ω

,ω

)

Ω

i=1

(t)ω

(x)ω

(x)dx

i=1

(t)

Ω

(x)ω

(x)dx= g

(t),1 ≤j≤m,

(3.6)

•†Cþžmtk'.Ón,‡©•§|(3.5)•†Cþtk'.

|^V³²¼ê(2.9)Œ±:

((

−νS

xxx

)|S

,ω

j,x

)

=(((12(S

)

−12S

+2)S

−νS

xxx

)|S

,ω

j,x

)

=12((S

)

,ω

j,x

)−12(S

,ω

j,x

)

+2(S

,ω

j,x

)−ν(S

xxx

,ω

j,x

),1 ≤j≤m.

(3.7)

DOI:10.12677/pm.2021.1161231089nØêÆ

ëä™

du3Ω¥,−

= λ

ν(S

xxx

,ω

j,x

)

=ν

Ω

i=1

i,xxx



h=1

h,x



j,x

=−ν

Ω

i=1

i,x



h=1

h,x



j,x

=−ν

i=1

Ω



h=1

h,x



i,x

j,x

dx,1 ≤j≤m,

(3.8)

ò(3.8)“\(3.7)¥,Œ±

((

−νS

xxx

)|S

,ω

j,x

)

=12

i=1

k=1

l=1

Ω



h=1

h,x



l,x

j,x

−12

k=1

l=1

Ω



h=1

h,x



l,x

j,x

l=1

Ω



h=1

h,x



l,x

j,x

+ν

i=1

Ω



h=1

h,x



i,x

j,x

dx,1 ≤j≤m,

(3.9)

…

c(r|S

,ω

) = cr

Ω



h,x



dx,1 ≤j≤m.

(3.10)

‡©•§|(3.5)÷vÐ©^‡:

(0) = S

i=1

→S

∈H

per

(Ω),m→∞,(3.11)

ùp,α

= g

(0).

•ò(3.6)–(3.10)“\(3.5),K~‡©•§|ŒU•'u{g

}

j=1

~‡©•§|,Xe

¤«:







,...,g

,t),

(0)=(S

,ω

(3.12)

DOI:10.12677/pm.2021.1161231090nØêÆ

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Ù¥

,...,g

,t)

=−12

i=1

k=1

l=1

Ω



h=1

h,x



l,x

j,x

+12

k=1

l=1

Ω



h=1

h,x



l,x

j,x

−2

l=1

Ω



h=1

h,x



l,x

j,x

−ν

i=1

Ω



h=1

h,x



i,x

j,x

+cr

Ω



h=1

h,x



dx,

(3.13)

Ù¥j= 1,...,m.w,,(3.12)–(3.13)´˜‡š‚5‘F'u™•¼ê÷vÛÜLipschitzëY

š‚5~‡©•§|,Šâ~‡©•§|ÛÜ)•35½n,Œ•T)•3u[0,t

].Šâ±þ(Ø,

Œ±éu?¿‰½m,S

•Cq¯K(3.5)˜‡ÛÜ).3e ˜!¥,òí'uS

kO,y²t

Œ±í2?¿‰½~êT

4.˜—kO

3ù˜!¥,·‚—åuí'u)S

˜kO.

Ún4.1é?Ût∈[0,T

]

∈L

∞

(0,T

(Ω)),(4.1)

∈L

(0,T

(Ω)),(4.2)

)

dxdτ≤C,(4.3)

)

|dxdτ≤C,(4.4)

dxdτ≤C.(4.5)

y².é(3.5)1j‡•§¦±g

(t),,éj¦Ú,ùŒ±µ

((S

)

)+c((

−νS

xxx

)|S

),S

)−c(r|S

) = 0.(4.6)

Ù¥



)



(Ω)

.(4.7)

DOI:10.12677/pm.2021.1161231091nØêÆ

ëä™

ÏLéx?1©ÜÈ©,Œ±

xxx

) = −(S

)−(S

)

(|S

)

−1

).(4.8)

ò(4.7)Ú(4.8)“\(4.6)¥,

(Ω)

+c(

)+cν(S

)

+cν(S

)

(|S

)

−1

)−c(r|S

) = 0,

(4.9)

ùp

) = 12(S

)

−12S

+2.(4.10)

ò(4.10)“\(4.9),Œ

(Ω)

Ω

(12(S

)

−12S

+2)|S

+cν

Ω

)

dx+cν

Ω

)

(|S

)

−1

dx−cr

Ω

dx= 0,

(4.11)

Œò(4.11)U•

(Ω)

+12c

Ω

)

dx+2c

Ω

+cν

Ω

)

dx+cν

Ω

)

(|S

)

−1

=12c

Ω

dx+cr

Ω

= : I

(4.12)

(Ü(2.11),A^YoungØª,5?nI

Ü©,

= 12c

Ω

dx≤12c

Ω

dx≤12c

Ω

(ε(S

)

)|S

≤ε

Ω

)

dx+C

Ω

dx≤ε

Ω

)

(|S

+C)dx+C

Ω

(|S

+C)dx

≤ε

Ω

)

dx+ε

Ω

)

dx+C

Ω

dx+C,

(4.13)

2d©ÜÈ©,

Ω

dx=

Ω

|)S

dx= −2

Ω

dx.

(4.14)

DOI:10.12677/pm.2021.1161231092nØêÆ

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|^YoungØªŒ

Ω

dx=

Ω

|)S

dx= −2

Ω

≤2

Ω

||S

|dx= 2

Ω

||S

·|S

|dx

≤C

Ω

|dx+ξ

Ω

||S

≤C

Ω

(η|S

)dx+ξ

Ω

||S

≤η

Ω

dx+C

Ω

dx+ξ

Ω

||S

dx.

(4.15)

e5?nI

Ü©,

= cr

Ω

dx= cr

Ω

)

≤crγ

Ω

dx+crC

Ω

)

≤crγ

Ω

dx+crC

Ω

(ζ|S

)dx

≤γ

Ω

dx+ζ

Ω

dx+rC

≤γ

Ω

(|S

+C)|S

dx+ζ

Ω

dx+rC

≤γ

Ω

dx+(γ+ζ)

Ω

dx+rC,

(4.16)

ÏL(4.13)–(4.16),Œ±

(Ω)

+12c

Ω

)

dx+2c

Ω

+cν

Ω

)

dx+cν

Ω

)

(|S

)

−1

≤(ε+η+γ)

Ω

)

dx+(ε+C+γ+ζ)

Ω

dx+ξ

Ω

||S

dx+rC,

(4.17)

-ε,ξ,η,γ,ζv,¦

(ε+η+γ) ≤c,

(ε+C+γ+ζ) ≤2C,

ξ≤

cν

(4.18)

DOI:10.12677/pm.2021.1161231093nØêÆ

ëä™

,ÏL(4.17)Ú(4.18),

(Ω)

+11c

Ω

)

dx+2c

Ω

cν

Ω

)

dx+cν

Ω

)

(|S

)

−1

≤2C

Ω

dx+Cr.

(4.19)

é(4.19)'užmt‰È©,ÏLA^GronwallØª,Œ

(Ω)

+11c

Ω

)

dxdτ+2c

Ω

dxdτ

cν

Ω

)

dxdτ+cν

Ω

)

(|S

)

−1

dxdτ

≤C

(0)k

≤C

(4.20)

ÏL(4.20),(2.11)Ú(2.12),ŒÚn4.1(Ø.

Ún4.2é?¿t∈[0,T

]

∈L

∞

(0,T

(Ω)),(4.21)

∈L

∞

(0,T

(Ω)),(4.22)

xxx

)

dxdτ≤C,(4.23)

xxx

dxdτ≤C.(4.24)

y².ò(3.3)1j‡•§¦±λ

(t),,éjl0m¦Ú

((S

)

,−S

)−c(((

−νS

xxx

)|S

)

,−S

)−c(r|S

,−S

) = 0,(4.25)

((S

)

,−S

)−c((

−νS

xxx

)|S

xxx

)+c(r|S

) = 0.(4.26)

dut∈[0,T

],Œ±



)

,−S



(Ω)

,(4.27)

…

) = 12(S

)

−12S

+2.(4.28)

DOI:10.12677/pm.2021.1161231094nØêÆ

ëä™

r(4.27)“\(4.26),Œ±

(Ω)

+cν

Ω

xxx

)

Ω

xxx

dx−cr

Ω

=0,

(4.29)

,,ÏL(4.28)Œ±

(Ω)

+cν

Ω

xxx

)

=12c

Ω

)

xxx

dx−12c

Ω

xxx

+2c

Ω

xxx

dx−cr

Ω

≤12c

Ω

)

xxx

dx+12c

Ω

||S

xxx

|dx

+2c

Ω

xxx

dx+cr

Ω

|dx

=:I

(4.30)

(Ü(2.11),|^YoungØª,Œ

= 12c

Ω

)

xxx

= 12c

Ω

)

·|S

xxx

≤C

Ω

)

dx+ε

Ω

xxx

dx.

(4.31)

d®•Sobolevi\½nÚÚn4.1(Ø,

Ω

)

dx≤

Ω

)

dx·kS

∞

≤

Ω

)

dx(kS

+kS

) ≤

Ω

)

dx(kS

+C)

≤kS

Ω

)

dx+C

Ω

)

dx.

≤kS

Ω

)

(|S

+C)dx+C

Ω

)

(|S

+C)dx.

≤kS

Ω

)

dx+CkS

Ω

)

dx+C

Ω

)

dx+C

Ω

)

dx,

(4.32)

DOI:10.12677/pm.2021.1161231095nØêÆ

ëä™

|^YoungØª

= 12c

Ω

||S

xxx

|dx= 12c

Ω

||S

·|S

xxx

|dx

≤C

Ω

)

dx+ξ

Ω

xxx

dx,

≤C

Ω

)

dx+C

Ω

)

dx+ξ

Ω

xxx

dx,

(4.33)

= 2c

Ω

xxx

dx= 2c

Ω

||S

·|S

xxx

|dx

≤C

Ω

dx+η

Ω

xxx

≤C

Ω

dx+η

Ω

xxx

dx+C,

(4.34)

…

= cr

Ω

|dx= cr

Ω

|dx

≤crC

Ω

dx+crγ

Ω

≤crC

Ω

(µ|S

)dx+crγ

Ω

≤crµC

Ω

dx+crC

Ω

dx+crγ

Ω

≤µ

Ω

dx+γ

Ω

dx+rC

≤µ

Ω

(|S

+C)dx+γ

Ω

dx+rC.

≤µ

Ω

dx+γ

Ω

dx+rC.

(4.35)

(Ü(4.31)–(4.35),Œ

(Ω)

+cν

Ω

xxx

)

≤(ε+ξ+η)

Ω

xxx

dx+CkS

Ω

)

Ω

)

dx+C

Ω

dx+CkS

Ω

)

dx+C

Ω

)

+µ

Ω

dx+γ

Ω

dx+rC.

(4.36)

DOI:10.12677/pm.2021.1161231096nØêÆ

ëä™

-ε,ξ,η,µ,γv,¿…0 <ε+ξ+η<

cν

,(ÜÚn4.1,·‚k

(Ω)

cν

Ω

xxx

)

≤CkS

Ω

)

dx+C

Ω

)

Ω

dx+CkS

Ω

)

dx+C

Ω

)

dx+rC.

(4.37)

(t)k

(Ω)

≤C+CkS

(0)k

(Ω)

≤C

(4.38)

(Ω)

cν

Ω

xxx

)

dxdτ≤C

(4.39)

–dÚn4.2y²..

Ún4.3é?¿t∈[0,T

Ω

(|S

xxx

dxdτ≤C,(4.40)

k|S

xxx

)

≤C,(4.41)

k((

)

≤C.(4.42)

y².ÏLH¨olderØª,éu1 ≤p<2,q=

,…

= 1,

Ω

(|S

xxx

dxdτ

Ω

(|S

xxx

)dxdτ

≤



Ω

dxdτ





Ω

xxx

dxdτ



≤



Ω

2−p

dxdτ



2−p



Ω

xxx

dxdτ



(4.43)

Øª(4.43)¿›Xé

2−p

≤2,=p≤

,Øªm>´k.,p=

ž,Œ

Ω

(|S

xxx

dxdτ

≤



Ω

dxdτ





Ω

xxx

dxdτ



(4.44)

DOI:10.12677/pm.2021.1161231097nØêÆ

ëä™

dÚn4.1ÚÚn4.2(ØŒ(4.40),(4.41)y.

ÏLH¨olderØª,-p=

,±9dÚn4.1ÚÚn4.2(ØŒ±Ñ

Ω

((

)

dxdτ

Ω

((

)

dxdτ

≤C

Ω

(|S

)

dxdτ≤C.

(4.45)

nþÚn4.3y²¤.

Ún4.4•3˜‡~êC,¦

(0,T

−1,

(Ω))

≤C.(4.46)

y².(ÜÚn4.1ÚÚn4.3(Ø,·‚Œ±,é?¿ϕ∈C

∞

|(S

,ϕ)

|=|−c((

−νS

)

,ϕ

)

+c(r|S

|,ϕ)

≤ck(

−νS

)

kϕ

)

+c¯rk|S

)

kϕk

)

≤Ckϕk

(0,T

1,4

(Ω))

.(4.47)

Ún4.4y.

5.4•L§

S

m→∞žÂñ5.!òy²•3ÂñuÐ>Š¯Kf)fS.

Ún5.1(Aubin−Lions)[29]B

ÚB

´g‡,bB´˜‡Banach˜m,¦B

;i\

B¥,Bi\B

¥.éu1 ≤p

≤+∞,½Â

W= {f|f∈L

(0,T;B

),f

∈L

(0,T;B

)}.

(i)XJp

<+∞,KW;i\L

(0,T;B)¥.

(ii)XJp

= +∞Úp

>1,KW;i\C([0,T];B)¥.

ù‡Úny²,ë•[30]¥157•.±e´!Ì‡(J.

DOI:10.12677/pm.2021.1161231098nØêÆ

ëä™

(4.24),(4.46)9f;5½n,·‚Œ•3fSS

,ùp„^S

L«,÷vµ

*S,f∗ÂñuL

∞

(0,T

(Ω)),(5.1)

*S,fÂñuL

(0,T

(Ω)),(5.2)

,fÂñuL

(0,T

−1,

(Ω)).(5.3)

Ïd,d(5.1),(5.2)ŒíÑ(2.19),…(5.3)Œy(Ø(2.20)¤á.•,·‚ïÄ(2.18)Âñ5.

ŠâAubin−LionsÚn,·‚µ

= 2,p

= H

(Ω),B= C

1+α

(

Ω),B

= W

−1,

(Ω),

K•3SS

,¦

−Sk

(0,T

1+α

(

Ω))

→0,m→∞.(5.4)

d(5.4)·‚N´Ñµ

−S

(0,T

(

Ω))

→0,m→∞.(5.5)

k|S

−|S

(0,T

(Ω))

→0,m→∞.(5.6)

l,·‚ŒÏL(5.2),(5.6)

xxx

*|S

xxx

,fÂñuL

).(5.7)

d,ÏL(4.22),(5.4)Ú(5.5)ØJ

(

)

(S)S

,fÂñuL

).(5.8)

(

)

,fÂñuL

).(5.9)

ÏL(5.6)–(5.9)Œé?¿ϕ∈C

∞

((−∞,T

)×R)¤á

,ϕ

)

→(S,ϕ

)

,(5.10)

c(r|S

,ϕ)

→c(r|S

,ϕ)

,(5.11)

(|S

xxx

,ϕ

)

→(|S

xxx

,ϕ

)

,(5.12)

((

)

,ϕ

)

→((

)

,ϕ

)

.(5.13)

DOI:10.12677/pm.2021.1161231099nØêÆ

ëä™

=S÷vf/ª(2.18).ù¿›XS´¯K(2.13)–(2.16)˜‡f),½n2.1y.

6.o(

þ,Ñ5A,z•˜‘˜mšòzü‡•§.$^Galerkin•{ETÐ>Š¯K˜‡

Cq)¿y²ÙÛÜf)•3,ÏL˜—kO9IO4•L§,Tz¯KNf)

•35.

ë•©z

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