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PureMathematics
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,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
1
,XingzhiBian
1
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,LeiYu
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[J].
n
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,2021,11(6):1084-1102.
DOI:10.12677/pm.2021.116123
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1
MaterialsGenomeInstitute,ShanghaiUniversity,Shanghai
2
CollegeofScience,ShanghaiUniversity,Shanghai
Received:May7
th
,2021;accepted:Jun.8
th
,2021;published:Jun.16
th
,2021
Abstract
Inthispaper, ourresearch willbebasedontheAlber-Zhumodelwhichorderparam-
eterisconserved.Theevolutionequationinthismodelisafourthorder,nonlinear
degeneratepartialdifferentialequationofparabolictype.Thismodelisaphase-
fieldmodelwhichdescribestheinterfacemotionbyinterfacediffusioninelastically
deformedsolids.Forexample,thebasicphenomenaoccurringduringthisprocess,
calledsintering,aredensificationandgraingrowth.Sincenoatomexchangeoccurs
attheinterface,thevolumesofthedifferentregionsseparatedbytheinterfaceare
conserved.Weignoretheelasticeffectandreducetheoriginalinitialboundaryvalue
problemtoasinglenon-degenerateequationof
1
dimensionalsituation,andprovethe
existenceoftheweaksolutionofthereducedinitialboundaryvalueproblemofthis
model,wheretheboundaryconditionsoftheorderparametersareacombinationof
theNeumannboundaryconditionsandtheno-flowconditions.TheGalerkinmethod
is usedtoprove theexistenceofweak solutionsforthereducedinitialboundary value
problemofthismodel.Althoughtheproblemconsideredisasingleequationoforder
parameters,itisinherentlydifficultduetothepresencewhichisthegradientofun-
knownfunction
S
.Hence,weneedtomollifythegradientterm,whichcausesalotof
difficultiesonthetheoreticalanalysisandnumericalsimulation.
Keywords
MotionofGrainBoundaries,Phase-FieldModel,ParabolicEquation,
ExistenceofWeakSolutions
Copyright
c
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.2021.1161231085
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DOI:10.12677/pm.2021.1161231087
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d
x
2
=
λ
i
ω
i
,
d
ω
i
d
x
∂
Ω
=0
,
Ù
¥
ω
1
= 1.
•
Ä
¯
K
C
q
)
:
S
m
=
S
m
(
t
),
§
k
±
e
/
ª
:
S
m
(
t
) =
m
X
i
=0
g
im
(
t
)
ω
i
.
(3.1)
DOI:10.12677/pm.2021.1161231088
n
Ø
ê
Æ
ë
ä
™
3ù
p
,
g
im
´
d
e
~
‡
©•
§
|
¤
û
½
:
(
S
m
t
,ω
j
) = ((
c
(
ˆ
ψ
S
m
(
S
m
)
−
νS
m
xx
)
x
|
S
m
x
|
κ
)
x
+
cr
|
S
m
x
|
κ
,ω
j
)
,
1
≤
j
≤
m,
(3.2)
=
((
S
m
)
0
,ω
j
) =
c
(((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
)
x
,ω
j
)+
c
(
r
|
S
m
x
|
κ
,ω
j
)
,
1
≤
j
≤
m,
(3.3)
Š
â
©
Ü
È
©
ú
ª
,
(((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
)
x
,ω
j
) =
−
((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
,ω
jx
)
,
1
≤
j
≤
m.
(3.4)
r
(3
.
4)
“
\
(3
.
3)
¥
,
Œ
±
((
S
m
)
0
,ω
j
)+
c
((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
,ω
jx
)
−
c
(
r
|
S
m
x
|
κ
,ω
j
) = 0
,
1
≤
j
≤
m.
(3.5)
y
3
,
I
‡
`
²
•
§
(3
.
5)
´
˜
‡
~
‡
©•
§
|
.
d
(
S
m
)
0
=
m
P
i
=1
g
0
im
(
t
)
ω
i
¿
…
{
ω
i
}
3
˜
m
L
2
(Ω)
¥
´
I
O
,
¤
±
(
S
m
)
0
,ω
j
= (
m
X
i
=1
g
0
im
(
t
)
ω
i
,ω
j
)
=
Z
Ω
m
X
i
=1
g
0
im
(
t
)
ω
i
(
x
)
ω
j
(
x
)
dx
=
m
X
i
=1
g
0
im
(
t
)
Z
Ω
ω
i
(
x
)
ω
j
(
x
)
dx
=
g
0
jm
(
t
)
,
1
≤
j
≤
m,
(3.6)
•
†
C
þ
ž
m
t
k
'
.
Ó
n
,
‡
©•
§
|
(3
.
5)
•
†
C
þ
t
k
'
.
|
^
V
³
²
¼
ê
(2
.
9)
Œ
±
:
((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
,ω
j,x
)
=(((12(
S
m
)
2
−
12
S
m
+2)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
,ω
j,x
)
=12((
S
m
)
2
S
m
x
|
S
m
x
|
κ
,ω
j,x
)
−
12(
S
m
S
m
x
|
S
m
x
|
κ
,ω
j,x
)
+2(
S
m
x
|
S
m
x
|
κ
,ω
j,x
)
−
ν
(
S
m
xxx
|
S
m
x
|
κ
,ω
j,x
)
,
1
≤
j
≤
m.
(3.7)
DOI:10.12677/pm.2021.1161231089
n
Ø
ê
Æ
ë
ä
™
du
3
Ω
¥
,
−
d
2
ω
i
d
x
2
=
λ
i
ω
i
ν
(
S
m
xxx
|
S
m
x
|
κ
,ω
j,x
)
=
ν
Z
Ω
m
X
i
=1
g
im
ω
i,xxx
m
X
h
=1
g
hm
ω
h,x
κ
ω
j,x
dx
=
−
ν
Z
Ω
m
X
i
=1
g
im
λ
i
ω
i,x
m
X
h
=1
g
hm
ω
h,x
κ
ω
j,x
dx
=
−
ν
m
X
i
=1
g
im
λ
i
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
i,x
ω
j,x
dx,
1
≤
j
≤
m,
(3.8)
ò
(3.8)
“
\
(3.7)
¥
,
Œ
±
((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
,ω
j,x
)
=12
m
X
i
=1
g
im
m
X
k
=1
g
km
m
X
l
=1
g
lm
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
i
ω
k
ω
l,x
ω
j,x
dx
−
12
m
X
k
=1
g
km
m
X
l
=1
g
lm
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
k
ω
l,x
ω
j,x
dx
+2
m
X
l
=1
g
lm
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
l,x
ω
j,x
dx
+
ν
m
X
i
=1
g
im
λ
i
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
i,x
ω
j,x
dx,
1
≤
j
≤
m,
(3.9)
…
c
(
r
|
S
m
x
|
κ
,ω
j
) =
cr
Z
Ω
m
X
=1
g
hm
ω
h,x
κ
ω
j
dx,
1
≤
j
≤
m.
(3.10)
‡
©•
§
|
(3.5)
÷
v
Ð
©
^
‡
:
S
m
(0) =
S
m
0
=
m
X
i
=1
α
im
ω
j
→
S
0
∈
H
1
per
(Ω)
,m
→∞
,
(3.11)
ù
p
,
α
im
=
g
im
(0).
•
ò
(3.6)–(3.10)
“
\
(3.5),
K
~
‡
©•
§
|
Œ
U
•
'
u
{
g
jm
}
m
j
=1
~
‡
©•
§
|
,
X
e
¤
«
:
d
dt
g
jm
=
F
j
(
g
1
m
,...,g
mm
,t
)
,
g
jm
(0)=(
S
0
,ω
j
)
,
(3.12)
DOI:10.12677/pm.2021.1161231090
n
Ø
ê
Æ
ë
ä
™
Ù
¥
F
j
(
g
1
m
,...,g
mm
,t
)
=
−
12
m
X
i
=1
g
im
m
X
k
=1
g
km
m
X
l
=1
g
lm
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
i
ω
k
ω
l,x
ω
j,x
dx
+12
m
X
k
=1
g
km
m
X
l
=1
g
lm
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
k
ω
l,x
ω
j,x
dx
−
2
m
X
l
=1
g
lm
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
l,x
ω
j,x
dx
−
ν
m
X
i
=1
g
im
λ
i
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
i,x
ω
j,x
dx
+
cr
Z
Ω
m
X
h
=1
g
hm
ω
h,x
κ
ω
j
dx,
(3.13)
Ù
¥
j
= 1
,...,m.
w
,
,(3.12)–(3.13)
´
˜
‡
š
‚
5
‘
F
'
u
™
•
¼
ê
÷
v
Û
Ü
Lipschitz
ë
Y
š
‚
5
~
‡
©•
§
|
,
Š
â
~
‡
©•
§
|
Û
Ü
)
•
3
5
½
n
,
Œ
•
T
)
•
3
u
[0
,t
m
].
Š
â
±
þ
(
Ø
,
Œ
±
é
u
?
¿
‰
½
m
,
S
m
•
C
q
¯
K
(3.5)
˜
‡
Û
Ü
)
.
3
e
˜
!
¥
,
ò
í
'
u
S
m
k
'
k
O
,
y
²
t
m
Œ
±
í
2
?
¿
‰
½
~
ê
T
e
.
4.
˜
—
k
O
3ù
˜
!
¥
,
·
‚
—
å
u
í
'
u
)
S
m
˜
k
O
.
Ú
n
4.1
é
?
Û
t
∈
[0
,T
e
]
S
m
∈
L
∞
(0
,T
e
;
L
2
(Ω))
,
(4.1)
S
m
x
∈
L
3
(0
,T
e
;
L
3
(Ω))
,
(4.2)
R
Q
T
e
(
S
m
)
2
|
S
m
x
|
3
dxdτ
≤
C,
(4.3)
R
Q
T
e
(
S
m
xx
)
2
|
S
m
x
|
dxdτ
≤
C,
(4.4)
κ
R
Q
T
e
|
S
m
xx
|
2
dxdτ
≤
C.
(4.5)
y
²
.
é
(3
.
5)
1
j
‡
•
§¦
±
g
jm
(
t
),
,
é
j
¦
Ú
,
ù
Œ
±
µ
((
S
m
)
0
,S
m
)+
c
((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
)
,S
m
x
)
−
c
(
r
|
S
m
x
|
κ
,S
m
) = 0
.
(4.6)
Ù
¥
(
S
m
)
0
,S
m
=
1
2
d
dt
k
S
m
k
2
L
2
(Ω)
.
(4.7)
DOI:10.12677/pm.2021.1161231091
n
Ø
ê
Æ
ë
ä
™
Ï
L
é
x
?
1
©
Ü
È
©
,
Œ
±
(
S
m
xxx
|
S
m
x
|
κ
,S
m
x
) =
−
(
S
m
xx
,S
m
xx
|
S
m
x
|
κ
)
−
(
S
m
xx
,S
m
xx
(
S
m
x
)
2
(
|
S
m
x
|
κ
)
−
1
)
.
(4.8)
ò
(4
.
7)
Ú
(4
.
8)
“
\
(4
.
6)
¥
,
1
2
d
dt
k
S
m
k
2
L
2
(Ω)
+
c
(
ˆ
ψ
00
(
S
m
)
S
m
x
|
S
m
x
|
κ
,S
m
x
)+
cν
(
S
m
xx
,S
m
xx
|
S
m
x
|
κ
)
+
cν
(
S
m
xx
,S
m
xx
(
S
m
x
)
2
(
|
S
m
x
|
κ
)
−
1
)
−
c
(
r
|
S
m
x
|
κ
,S
m
) = 0
,
(4.9)
ù
p
ˆ
ψ
00
(
S
m
) = 12(
S
m
)
2
−
12
S
m
+2
.
(4.10)
ò
(4.10)
“
\
(4.9),
Œ
1
2
d
dt
k
S
m
k
2
L
2
(Ω)
+
c
Z
Ω
(12(
S
m
)
2
−
12
S
m
+2)
|
S
m
x
|
3
κ
dx
+
cν
Z
Ω
(
S
m
xx
)
2
|
S
m
x
|
κ
dx
+
cν
Z
Ω
(
S
m
xx
)
2
(
S
m
x
)
2
(
|
S
m
x
|
κ
)
−
1
dx
−
cr
Z
Ω
|
S
m
x
|
κ
S
m
dx
= 0
,
(4.11)
Œ
ò
(4.11)
U
•
1
2
d
dt
k
S
m
k
2
L
2
(Ω)
+12
c
Z
Ω
(
S
m
)
2
|
S
m
x
|
2
|
S
m
x
|
κ
dx
+2
c
Z
Ω
|
S
m
x
|
2
|
S
m
x
|
κ
dx
+
cν
Z
Ω
(
S
m
xx
)
2
|
S
m
x
|
κ
dx
+
cν
Z
Ω
(
S
m
xx
)
2
(
S
m
x
)
2
(
|
S
m
x
|
κ
)
−
1
dx
=12
c
Z
Ω
S
m
|
S
m
x
|
2
|
S
m
x
|
κ
dx
+
cr
Z
Ω
|
S
m
x
|
κ
S
m
dx
= :
I
1
+
I
2
.
(4.12)
(
Ü
(2.11),
A^
Young
Ø
ª
,
5
?
n
I
1
Ü
©
,
I
1
= 12
c
Z
Ω
S
m
|
S
m
x
|
2
|
S
m
x
|
κ
dx
≤
12
c
Z
Ω
S
m
|
S
m
x
|
3
κ
dx
≤
12
c
Z
Ω
(
ε
(
S
m
)
2
+
C
ε
)
|
S
m
x
|
3
κ
dx
≤
ε
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
κ
dx
+
C
Z
Ω
|
S
m
x
|
3
κ
dx
≤
ε
Z
Ω
(
S
m
)
2
(
|
S
m
x
|
3
+
C
)
dx
+
C
Z
Ω
(
|
S
m
x
|
3
+
C
)
dx
≤
ε
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
ε
Z
Ω
(
S
m
)
2
dx
+
C
Z
Ω
|
S
m
x
|
3
dx
+
C,
(4.13)
2
d
©
Ü
È
©
,
Z
Ω
|
S
m
x
|
3
dx
=
Z
Ω
(
S
m
x
|
S
m
x
|
)
S
m
x
dx
=
−
2
Z
Ω
S
m
|
S
m
x
|
S
m
xx
dx.
(4.14)
DOI:10.12677/pm.2021.1161231092
n
Ø
ê
Æ
ë
ä
™
|
^
Young
Ø
ª
Œ
Z
Ω
|
S
m
x
|
3
dx
=
Z
Ω
(
S
m
x
|
S
m
x
|
)
S
m
x
dx
=
−
2
Z
Ω
S
m
|
S
m
x
|
S
m
xx
dx
≤
2
Z
Ω
|
S
m
||
S
m
x
||
S
m
xx
|
dx
= 2
Z
Ω
|
S
m
||
S
m
x
|
1
2
·|
S
m
x
|
1
2
|
S
m
xx
|
dx
≤
C
ξ
Z
Ω
|
S
m
|
2
|
S
m
x
|
dx
+
ξ
Z
Ω
|
S
m
x
||
S
m
xx
|
2
dx
≤
C
ξ
Z
Ω
|
S
m
|
2
(
η
|
S
m
x
|
3
+
C
η
)
dx
+
ξ
Z
Ω
|
S
m
x
||
S
m
xx
|
2
dx
≤
η
Z
Ω
|
S
m
|
2
|
S
m
x
|
3
dx
+
C
Z
Ω
|
S
m
|
2
dx
+
ξ
Z
Ω
|
S
m
x
||
S
m
xx
|
2
dx.
(4.15)
e
5
?
n
I
2
Ü
©
,
I
2
=
cr
Z
Ω
|
S
m
x
|
κ
S
m
dx
=
cr
Z
Ω
|
S
m
x
|
κ
(
S
m
)
2
3
(
S
m
)
1
3
dx
≤
crγ
Z
Ω
|
S
m
x
|
3
κ
|
S
m
|
2
dx
+
crC
γ
Z
Ω
(
S
m
)
1
2
dx
≤
crγ
Z
Ω
|
S
m
x
|
3
κ
|
S
m
|
2
dx
+
crC
γ
Z
Ω
(
ζ
|
S
m
|
2
+
C
ζ
)
dx
≤
γ
Z
Ω
|
S
m
x
|
3
κ
|
S
m
|
2
dx
+
ζ
Z
Ω
|
S
m
|
2
dx
+
rC
≤
γ
Z
Ω
(
|
S
m
x
|
3
+
C
)
|
S
m
|
2
dx
+
ζ
Z
Ω
|
S
m
|
2
dx
+
rC
≤
γ
Z
Ω
|
S
m
x
|
3
|
S
m
|
2
dx
+(
γ
+
ζ
)
Z
Ω
|
S
m
|
2
dx
+
rC,
(4.16)
Ï
L
(4.13)–(4.16),
Œ
±
1
2
d
dt
k
S
m
k
2
L
2
(Ω)
+12
c
Z
Ω
(
S
m
)
2
|
S
m
x
|
2
|
S
m
x
|
κ
dx
+2
c
Z
Ω
|
S
m
x
|
2
|
S
m
x
|
κ
dx
+
cν
Z
Ω
(
S
m
xx
)
2
|
S
m
x
|
κ
dx
+
cν
Z
Ω
(
S
m
xx
)
2
(
S
m
x
)
2
(
|
S
m
x
|
κ
)
−
1
dx
≤
(
ε
+
η
+
γ
)
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+(
ε
+
C
+
γ
+
ζ
)
Z
Ω
|
S
m
|
2
dx
+
ξ
Z
Ω
|
S
m
x
||
S
m
xx
|
2
dx
+
rC,
(4.17)
-
ε,ξ,η,γ,ζ
v
,
¦
(
ε
+
η
+
γ
)
≤
c,
(
ε
+
C
+
γ
+
ζ
)
≤
2
C,
ξ
≤
cν
2
.
(4.18)
DOI:10.12677/pm.2021.1161231093
n
Ø
ê
Æ
ë
ä
™
,
Ï
L
(4.17)
Ú
(4.18),
1
2
d
dt
k
S
m
k
2
L
2
(Ω)
+11
c
Z
Ω
(
S
m
)
2
|
S
m
x
|
2
|
S
m
x
|
κ
dx
+2
c
Z
Ω
|
S
m
x
|
2
|
S
m
x
|
κ
dx
+
cν
2
Z
Ω
(
S
m
xx
)
2
|
S
m
x
|
κ
dx
+
cν
Z
Ω
(
S
m
xx
)
2
(
S
m
x
)
2
(
|
S
m
x
|
κ
)
−
1
dx
≤
2
C
Z
Ω
|
S
m
|
2
dx
+
Cr.
(4.19)
é
(4.19)
'
u
ž
m
t
‰
È
©
,
Ï
L
A^
Gronwall
Ø
ª
,
Œ
1
2
k
S
m
k
2
L
2
(Ω)
+11
c
Z
t
o
Z
Ω
(
S
m
)
2
|
S
m
x
|
2
|
S
m
x
|
κ
dxdτ
+2
c
Z
t
o
Z
Ω
|
S
m
x
|
2
|
S
m
x
|
κ
dxdτ
+
cν
2
Z
t
o
Z
Ω
(
S
m
xx
)
2
|
S
m
x
|
κ
dxdτ
+
cν
Z
t
o
Z
Ω
(
S
m
xx
)
2
(
S
m
x
)
2
(
|
S
m
x
|
κ
)
−
1
dxdτ
≤
C
t
+
C
t
k
S
m
(0)
k
2
L
2
≤
C
T
e
.
(4.20)
Ï
L
(4.20),(2.11)
Ú
(2.12),
Œ
Ú
n
4.1
(
Ø
.
Ú
n
4.2
é
?
¿
t
∈
[0
,T
e
]
S
m
x
∈
L
∞
(0
,T
e
;
L
2
(Ω))
,
(4.21)
S
m
∈
L
∞
(0
,T
e
;
H
1
(Ω))
,
(4.22)
R
Q
t
(
S
m
xxx
)
2
|
S
m
x
|
κ
dxdτ
≤
C,
(4.23)
κ
R
Q
T
e
|
S
m
xxx
|
2
dxdτ
≤
C.
(4.24)
y
²
.
ò
(3.3)
1
j
‡
•
§¦
±
λ
j
g
jm
(
t
),
,
é
j
l
0
m
¦
Ú
((
S
m
)
0
,
−
S
m
xx
)
−
c
(((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
)
x
,
−
S
m
xx
)
−
c
(
r
|
S
m
x
|
κ
,
−
S
m
xx
) = 0
,
(4.25)
=
((
S
m
)
0
,
−
S
m
xx
)
−
c
((
ˆ
ψ
00
(
S
m
)
S
m
x
−
νS
m
xxx
)
|
S
m
x
|
κ
,S
m
xxx
)+
c
(
r
|
S
m
x
|
κ
,S
m
xx
) = 0
.
(4.26)
du
t
∈
[0
,T
e
],
Œ
±
(
S
m
)
0
,
−
S
m
xx
=
1
2
d
dt
k
S
m
x
k
2
L
2
(Ω)
,
(4.27)
…
ˆ
ψ
00
(
S
m
) = 12(
S
m
)
2
−
12
S
m
+2
.
(4.28)
DOI:10.12677/pm.2021.1161231094
n
Ø
ê
Æ
ë
ä
™
r
(4.27)
“
\
(4.26),
Œ
±
1
2
d
dt
k
S
m
x
k
2
L
2
(Ω)
+
cν
Z
Ω
(
S
m
xxx
)
2
|
S
m
x
|
κ
dx
=
c
Z
Ω
ˆ
ψ
00
(
S
m
)
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
−
cr
Z
Ω
|
S
m
x
|
κ
S
m
xx
dx
=0
,
(4.29)
,
,
Ï
L
(4.28)
Œ
±
1
2
d
dt
k
S
m
x
k
2
L
2
(Ω)
+
cν
Z
Ω
(
S
m
xxx
)
2
|
S
m
x
|
κ
dx
=12
c
Z
Ω
(
S
m
)
2
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
−
12
c
Z
Ω
S
m
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
+2
c
Z
Ω
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
−
cr
Z
Ω
|
S
m
x
|
κ
S
m
xx
dx
≤
12
c
Z
Ω
(
S
m
)
2
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
+12
c
Z
Ω
|
S
m
||
S
m
x
|
2
κ
|
S
m
xxx
|
dx
+2
c
Z
Ω
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
+
cr
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
dx
=:
I
3
+
I
4
+
I
5
+
I
6
.
(4.30)
(
Ü
(2.11),
|
^
Young
Ø
ª
,
Œ
I
3
= 12
c
Z
Ω
(
S
m
)
2
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
= 12
c
Z
Ω
(
S
m
)
2
S
m
x
|
S
m
x
|
1
2
κ
·|
S
m
x
|
1
2
κ
S
m
xxx
dx
≤
C
Z
Ω
(
S
m
)
4
|
S
m
x
|
3
κ
dx
+
ε
Z
Ω
|
S
m
x
|
κ
|
S
m
xxx
|
2
dx.
(4.31)
d
®
•
Sobolev
i
\
½
n
Ú
Ú
n
4.1
(
Ø
,
Z
Ω
(
S
m
)
4
|
S
m
x
|
3
κ
dx
≤
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
κ
dx
·k
S
m
k
2
L
∞
≤
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
κ
dx
(
k
S
m
x
k
2
L
2
+
k
S
m
k
2
L
2
)
≤
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
κ
dx
(
k
S
m
x
k
2
L
2
+
C
)
≤k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
κ
dx
+
C
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
κ
dx.
≤k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
(
|
S
m
x
|
3
+
C
)
dx
+
C
Z
Ω
(
S
m
)
2
(
|
S
m
x
|
3
+
C
)
dx.
≤k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
C
k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
dx
+
C
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
C
Z
Ω
(
S
m
)
2
dx,
(4.32)
DOI:10.12677/pm.2021.1161231095
n
Ø
ê
Æ
ë
ä
™
|
^
Young
Ø
ª
I
4
= 12
c
Z
Ω
|
S
m
||
S
m
x
|
2
κ
|
S
m
xxx
|
dx
= 12
c
Z
Ω
|
S
m
||
S
m
x
|
3
2
κ
·|
S
m
x
|
1
2
κ
|
S
m
xxx
|
dx
≤
C
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
κ
dx
+
ξ
Z
Ω
|
S
m
x
|
κ
|
S
m
xxx
|
2
dx,
≤
C
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
C
Z
Ω
(
S
m
)
2
dx
+
ξ
Z
Ω
|
S
m
x
|
κ
|
S
m
xxx
|
2
dx,
(4.33)
I
5
= 2
c
Z
Ω
S
m
x
S
m
xxx
|
S
m
x
|
κ
dx
= 2
c
Z
Ω
|
S
m
x
||
S
m
x
|
1
2
κ
·|
S
m
x
|
1
2
κ
|
S
m
xxx
|
dx
≤
C
Z
Ω
|
S
m
x
|
3
κ
dx
+
η
Z
Ω
|
S
m
x
|
κ
|
S
m
xxx
|
2
dx
≤
C
Z
Ω
|
S
m
x
|
3
dx
+
η
Z
Ω
|
S
m
x
|
κ
|
S
m
xxx
|
2
dx
+
C,
(4.34)
…
I
6
=
cr
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
dx
=
cr
Z
Ω
|
S
m
x
|
1
2
κ
|
S
m
x
|
1
2
κ
|
S
m
xx
|
dx
≤
crC
γ
Z
Ω
|
S
m
x
|
κ
dx
+
crγ
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
2
dx
≤
crC
γ
Z
Ω
(
µ
|
S
m
x
|
2
κ
+
C
µ
)
dx
+
crγ
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
2
dx
≤
crµC
γ
Z
Ω
|
S
m
x
|
2
κ
dx
+
crC
γ
Z
Ω
C
µ
dx
+
crγ
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
2
dx
≤
µ
Z
Ω
|
S
m
x
|
2
κ
dx
+
γ
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
2
dx
+
rC
≤
µ
Z
Ω
(
|
S
m
x
|
2
+
C
)
dx
+
γ
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
2
dx
+
rC.
≤
µ
Z
Ω
|
S
m
x
|
2
dx
+
γ
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
2
dx
+
rC.
(4.35)
(
Ü
(4.31)–(4.35),
Œ
1
2
d
dt
k
S
m
x
k
2
L
2
(Ω)
+
cν
Z
Ω
(
S
m
xxx
)
2
|
S
m
x
|
κ
dx
≤
(
ε
+
ξ
+
η
)
Z
Ω
|
S
m
x
|
κ
|
S
m
xxx
|
2
dx
+
C
k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
C
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
C
Z
Ω
|
S
m
x
|
3
dx
+
C
k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
dx
+
C
Z
Ω
(
S
m
)
2
dx
+
µ
Z
Ω
|
S
m
x
|
2
dx
+
γ
Z
Ω
|
S
m
x
|
κ
|
S
m
xx
|
2
dx
+
rC.
(4.36)
DOI:10.12677/pm.2021.1161231096
n
Ø
ê
Æ
ë
ä
™
-
ε,ξ,η,µ,γ
v
,
¿
…
0
<ε
+
ξ
+
η<
cν
2
,
(
Ü
Ú
n
4.1,
·
‚
k
1
2
d
dt
k
S
m
x
k
2
L
2
(Ω)
+
cν
2
Z
Ω
(
S
m
xxx
)
2
|
S
m
x
|
κ
dx
≤
C
k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
C
Z
Ω
(
S
m
)
2
|
S
m
x
|
3
dx
+
C
Z
Ω
|
S
m
x
|
3
dx
+
C
k
S
m
x
k
2
L
2
Z
Ω
(
S
m
)
2
dx
+
C
Z
Ω
(
S
m
)
2
dx
+
rC.
(4.37)
,
,
Š
â
‡
©
/
ª
Gronwall
Ø
ª
,
±
9
(4.1)–(4.24),
Œ
±
1
2
k
S
m
x
(
t
)
k
2
L
2
(Ω)
≤
C
+
C
k
S
m
x
(0)
k
2
L
2
(Ω)
≤
C
T
e
.
(4.38)
é
(4.37)
'
u
ž
m
t
È
©
,
Š
â
(4.38),
±
9
Ú
n
4.1
Œ
±
1
2
k
S
m
x
k
2
L
2
(Ω)
+
cν
2
Z
t
0
Z
Ω
(
S
m
xxx
)
2
|
S
m
x
|
κ
dxdτ
≤
C
T
e
.
(4.39)
–
d
Ú
n
4.2
y
²
.
.
Ú
n
4.3
é
?
¿
t
∈
[0
,T
e
]
,
Z
t
0
Z
Ω
(
|
S
m
x
|
κ
|
S
m
xxx
|
)
4
3
dxdτ
≤
C,
(4.40)
k|
S
m
x
|
κ
S
m
xxx
k
L
4
3
(
Q
T
e
)
≤
C,
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DOI:10.12677/pm.2021.1161231097
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DOI:10.12677/pm.2021.1161231099
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