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AdvancesinAppliedMathematics
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,2021,10(7),2428-2441
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.107255
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StudyonDarcy-Cahn-Hilliard
EquationsofHele-Shaw
Flow
XiangyuXiao,ZhilinPu
SchoolofMathematicalScience,SichuanNormalUniversity,ChengduSichuan
Received:Jun.19
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,2021;accepted:Jul.11
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2428-2441.DOI:10.12677/aam.2021.107255
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Abstract
Inthispaper,westudythetwophaseHele-Shawflow,whichconsistsoftheCahn
HilliardequtionandtheDarcyequation.Inthismodel,anextraphaseinducedforce
termintheDarcyequationiscoupledwithafluidinducedtransportterminthe
Cahn-Hilliardequation.Asthenon-lineartermsatisfiesthemoregeneralcondition,
weshowtheexistenceoftheweaksolution,energyestimate,andtheuniqueness.
Keywords
Coupled,Existence,Uniqueness,EnergyEstimate
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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∈
L
2
d
d
+1
((0
,T
);(
H
1
(Ω))
∗
)
.
(19)
¿
…
é
¤
k
t
∈
(0
,T
)
,
±
e
Ñ
¤
á
µ
(
∇
p
+
γϕ
∇
µ,
∇
q
) = 0
∀
q
∈
H
1
(Ω)
.
(20)
h
ϕ
t
,v
i
+
ε
(
∇
µ,
∇
v
)+(
ϕ
[
∇
p
+
γϕ
∇
µ
]
,
∇
v
) = 0
∀
v
∈
H
1
(Ω)
.
(21)
(
µ,ψ
)
−
ε
(
∇
ϕ,
∇
ψ
)
−
1
ε
(
f
(
ϕ
)
,ψ
) = 0
∀
ψ
∈
H
1
(Ω)
.
(22)
…
÷
v
Ð
Š
^
‡
ϕ
(0)=
ϕ
0
K
n
-
ê
|
(
p,µ,ϕ
)
¡
•
•
§
(13)
,
(14)
±
9
•
§
(8)(9)
,
(10)
˜
‡
f
)
"
e
¡
?
Ø
)
•
3
5
.
½
n
2.1
E
(
t
) =
J
ε
(
ϕ
)
,
e
Ð
Š
^
‡
÷
v
E
(0)
≤
C
0
ž
§
K
•
§
•
3
½
Â
2.1
¥
f
)
"
y
²
.
|
^
³
7C
q
•{
§
=
^k
•
‘
%
C
Ã
•
‘
§
·
‚
é
ϕ,p,µ
E
Ñ
C
q
)
§
·
‚
ò
¦
^
H
1
(Ω)
˜
|
k
•
‘
Ä
•
þ
{
ω
i
}
i
=1
...m
§
ù
Ä
•
þ
¤
Ü
¤
˜
m
·
‚
P
•
W
m
,
Ù
¥
·
‚
é
p
m
,ϕ
m
,µ
m
: [0
,T
]
→
W
m
,
p
m
=
X
m
i
=1
p
i,m
ω
i
ϕ
m
=
X
m
i
=1
ϕ
i,m
ω
i
,
µ
m
=
X
m
i
=1
µ
i,m
ω
i
.
¤
±
·
‚
µ
(
∇
p
m
+
γϕ
m
∇
µ
m
,
∇
q
) = 0
∀
q
∈
H
1
(Ω)
.
(23)
DOI:10.12677/aam.2021.1072552432
A^
ê
Æ
?
Ð
‹
‰
§
Æ
“
d
dt
ϕ
m
,v
+
ε
(
∇
µ
m
,
∇
v
)+(
ϕ
m
[
∇
p
m
+
γϕ
m
∇
µ
m
]
,
∇
v
) = 0
∀
v
∈
H
1
(Ω)
.
(24)
(
µ
m
,ψ
)
−
ε
(
∇
ϕ
m
,
∇
ψ
)
−
1
ε
(
f
(
ϕ
m
)
,ψ
) = 0
∀
ψ
∈
H
1
(Ω)
.
(25)
3
•
§
(23)
¥
-
q
=
ω
i
,i
=1
,
2
...m
¿
Ó
ž
¦
±
p
i,m
γ
,
,
¦
Ú
¶
3
•
§
(24)
¥
-
v
=
ω
i
,i
=
1
,
2
...m
,
¿
Ó
ž
¦
±
µ
i,m
,
2
¦
Ú
¶
3
•
§
(25)
¥
-
ψ
=
ω
i,
i
=1
,
2
....m
§
Ó
ž
¦
±
d
dt
ϕ
i,m
§
2
¦
Ú
§
•
ò
n
‡
ª
f
Ü
¿
µ
d
dt
ε
k∇
ϕ
m
k
2
L
2
+
1
ε
((
F
(
ϕ
m
)
,
1)
+
ε
k∇
µ
m
k
2
L
2
+
1
γ
k∇
p
m
+
γϕ
m
∇
µ
m
k
2
L
2
= 0
.
(26)
·
‚
3
þ
¡
•
§
ü
>
Ó
ž
È
©
§
K
µ
E
1
(
t
)+
Z
t
0
ε
k∇
µ
m
k
2
L
2
+
Z
t
0
1
γ
k∇
p
m
+
γϕ
m
∇
µ
m
k
2
L
2
=
E
1
(0)
.
(27)
Ù
¥
E
1
(
t
) =
R
Ω
h
ε
2
|∇
ϕ
m
|
2
+
1
ε
F
(
ϕ
m
)
i
dx
"
Ú
n
2.1
b
E
(
t
)=
J
ε
(
ϕ
)
,E
1
(
t
)=
J
ε
(
ϕ
m
)
,
,
Ù
¥
m
´
k
•
ê
§
…
W
m
´
H
1
(Ω)
¥
˜
|
k
•
‘
Ä
•
þ
¤
Ü
¤
˜
m
§
E
(0)
≤
C
0
ž
,
K
E
1
(0)
≤
C
0
"
(
Ü
Ú
n
2.1
§
|
^
•
§
(27)
Œ
k∇
ϕ
m
k
2
L
2
≤
C
0
,
R
t
0
k∇
µ
m
k
2
L
2
≤
C
0
R
t
0
k
u
m
k
2
L
2
≤
C
0
,
2
Ï
L
'
X
∇
p
m
=
−
u
m
−
γϕ
m
∇
µ
m
§
Ä
k
é
u
d
=2
,
3
Š
â
Sobolev
i
\
½
n
µ
H
1
(Ω)
→
L
6
(Ω),
2
d
Young
Ø
ª
Ú
þ
ã
O
Œ
R
t
0
k∇
p
m
k
2
L
3
2
≤
C
1
,
Ù
g
3
•
§
(25),
-
ψ
= ∆
ϕ
m
,
·
‚
Œ
ε
k
∆
ϕ
m
k
2
L
2
≤
1
2
k∇
µ
m
k
2
L
2
+
1
2
k∇
ϕ
m
k
2
L
2
−
c
0
ε
k∇
ϕ
m
k
2
L
2
,
?
˜
Ú
µ
ε
k
∆
ϕ
m
k
2
L
2
≤
1
2
k∇
µ
m
k
2
L
2
+
1
2
k∇
ϕ
m
k
2
L
2
≤
C
0
.
(28)
d
Gagliardo-Nirenberg
Ø
ª
§
·
‚
k
k
ϕ
m
k
L
∞
≤
C
k
∆
ϕ
m
k
1
2
d
L
2
k
ϕ
m
k
2
d
−
1
2
d
L
6
(
d
= 2
,
3)
d
k∇
p
m
k
2
d
d
+1
L
2
≤
C
k
u
m
k
2
d
d
+1
L
2
+
C
k
ϕ
m
k
2
d
d
+1
L
∞
k∇
µ
m
k
2
d
d
+1
L
2
≤
C
k
u
m
k
2
L
2
+
C
k
ϕ
m
k
2
d
L
∞
+
C
k∇
µ
m
k
2
L
2
,
du
k
ϕ
m
k
2
d
L
∞
≤
C
k
∆
ϕ
m
k
L
2
k
ϕ
m
k
2
d
−
1
L
6
≤
C
k
∆
ϕ
m
k
2
L
2
+
C
4
k
ϕ
m
k
4
d
−
1
H
1
.
2
Ï
L
sup
t
∈
(0
,T
)
k∇
ϕ
m
k
2
L
2
≤
C
0
,
µ
R
t
0
k
ϕ
m
k
2
L
2
≤
C
T
,
Ù
¥
C
T
´
˜
‡
†
T
k
'
~
ê
"
·
‚
2
é
þ
ãª
f
Ó
ž
é
t
¦
È
©
§
¿
d
þ
ã
O
µ
R
t
0
k∇
p
m
k
4
d
2
d
+1
L
2
≤
C
2
,
C
2
†
C
0
,C
1
d,T
k
'
"
2
Ï
L
•
§
(24),
é
?
¿
v
∈
H
1
(Ω)
Œ
µ
h
d
t
ϕ
m
,v
i
=
−
ε
(
∇
µ
m
,
∇
v
)+(
ϕ
m
u
m
,
∇
v
)
≤
[
ε
k∇
µ
m
k
L
2
+
k
ϕ
m
k
L
∞
k
u
m
k
L
2
]
k∇
v
k
L
2
.
DOI:10.12677/aam.2021.1072552433
A^
ê
Æ
?
Ð
‹
‰
§
Æ
“
Š
â
é
ó
˜
m
‰
ê
½
Â
§
•
k
d
t
ϕ
m
k
(
H
1
(Ω))
∗
≤
C
3
,
¦
È
©
R
t
0
k
d
t
ϕ
m
k
2
(
H
1
(Ω))
∗
≤
C
3
,
C
3
´
˜
‡
~
ê
"
Š
â
g
‡
˜
m
5
Ÿ
§
Ï
d
·
‚
(
J
•
3
ϕ,p,µ,
u
§
¦
ϕ
∈
L
∞
((0
,T
);
H
1
(Ω)),
¦
p
∈
L
2
d
d
+1
((0
,T
);
H
1
(Ω))
∩
L
2
0
(Ω),
¦
µ
∈
L
2
((0
,T
);
H
1
(Ω)),
Š
â
‚
55
k
u
=
∇
p
+
γϕ
∇
µ
,
X
k
u
∈
L
2
((0
,T
);
L
2
(Ω))
k
:
ϕ
m
→
ϕ
p
m
→
p
µ
m
→
µ
u
m
→
u
.
Š
â
‚
55
§
·
‚
k
µ
(
∇
p
+
γϕ
∇
µ,
∇
q
) = 0
,
Œ
d
V
‚
›
›
Â
ñ
½
n
·
‚
k
µ
h
ϕ
t
,v
i
+
ε
(
∇
µ,
∇
v
)+(
ϕ
[
∇
p
+
γϕ
∇
µ
]
,
∇
v
) = 0
(
µ,ψ
)
−
ε
(
∇
ϕ,
∇
ψ
)
−
1
ε
(
f
(
ϕ
)
,ψ
) = 0
.
e
¡
?
1
U
þ
O
:
½
n
2.2
b
ϕ
0
∈
H
1
(Ω)
,
f
(
ϕ
)
∈
L
2
((0
,T
);
H
1
(Ω))
,
Ω
⊂
R
d
(
d
=2
,
3)
´
˜
‡
Lipschitz
«
•
§
±
9
J
ε
(
ϕ
0
)
≤
C
0
,
e
(
p,µ,ϕ
)
´
•
§
(20)
−
(22)
˜
|
f
)
§
é
u
u
=
−
(
∇
p
+
γϕ
∇
µ
)
é
¤
k
t
∈
(0
,T
)
,
•
3
˜
‡
Ø
•
6
u
ε
~
ê
C
=
C
(
E
(0))
>
0
k
Z
Ω
ϕ
(
x,t
)
dx
=
Z
Ω
ϕ
0
(
x
)
dx.
(29)
E
(
t
)+
Z
t
0
ε
k∇
µ
(
s
)
k
2
L
2
+
1
γ
k
u
(
s
)
k
2
L
2
ds
=
E
(0)
<
∞
.
(30)
max
0
≤
s
≤
t
k
ϕ
(
s
)
k
2
H
1
≤
2
C
0
ε
+
2
c
3
ε
2
|
Ω
|
.
(31)
Z
t
0
k
µ
(
s
)
k
2
H
1
ds
≤
C
0
ε
+
c
3
ε
2
|
Ω
|
.
(32)
Z
t
0
µ
(
s
)
−
ε
−
1
f
(
ϕ
(
s
))
2
L
2
ds
≤
3
2
C
0
+
3
c
3
2
ε
|
Ω
|
.
(33)
Z
t
0
k
ϕ
t
(
s
)+
u
(
s
)
·∇
ϕ
(
s
)
k
(
H
1
)
∗
ds
≤
√
Cε.
(34)
Z
t
0
k
ϕ
t
(
s
)
k
2
(
W
1
,
3
)
∗
ds
≤
C
(
q
C
0
ε
+
c
3
|
Ω
|
+
√
2(
C
0
ε
+
c
3
|
Ω
|
ε
3
2
)
.
(35)
DOI:10.12677/aam.2021.1072552434
A^
ê
Æ
?
Ð
‹
‰
§
Æ
“
ù
p
E
(
t
) =
J
ε
(
ϕ
(
t
))
,
Ù
¥
J
ε
(
·
)
´
U
ì
•
§
(11)
5
½
Â
"
y
²
.
•
§
(29)
´
3
•
§
(21)
¥
-
v
=1
=
Œ
"
•
•
§
(30),
·
‚
©
•
ü
«
œ
¹
5
‰
µ
1
˜
«
A
Ï
œ
¹
§
ϕ
t
∈
L
2
((0
,T
);
H
1
(Ω))
ž
§
·
‚
3
•
§
(20)
¥
½
q
=
p
γ
,
3
•
§
(21)
¥
½
v
=
µ
§
3
•
§
(22)
¥
½
ψ
=
−
ϕ
t
,
¿
r
n
ö
ƒ
\
·
‚
µ
d
dt
ε
k∇
ϕ
k
2
L
2
+
1
ε
(
F
(
ϕ
)
,
1)
+
ε
k∇
µ
k
2
L
2
+
1
γ
k∇
p
+
γϕ
∇
µ
k
2
L
2
= 0
.
3
•
§
ü
>
Ó
ž
«
m
(0
,t
)
þ
È
©
Ò
Œ
•
§
(30)
"
é
u
˜
„
œ
¹
ϕ
t
∈
L
2
d
d
+1
((0
,T
);(
H
1
(Ω))
∗
),
X
J
d
ž
·
‚
2
-
ψ
=
−
ϕ
t
{
§
3
•
§
(22)
¥
Ò
v
k
¿Â
§
Ï
·
‚
3ù
p
ò
^
Steklov
²
þ
{
E
â
[16]
"
é
u
t
∈
(0
,T
),
δ>
0
´
?
¿
ê
§
½
Â
ϕ
Steklov
²
þ
ϕ
δ
µ
ϕ
(
·
,t
) =
S
δ
+
(
ϕ
)(
·
,t
) =
1
δ
Z
t
+
δ
t
ϕ
(
·
,s
)
ds
∀
t
∈
(0
,T
)
.
é
u
v
δ
,
ϕ
δ
t
(
·
,t
) := (
ϕ
δ
(
·
,t
))
t
=
ϕ
(
·
,t
+
δ
)
−
ϕ
(
·
,t
)
δ
.
Ï
d
§
é
u
z
˜
‡
t
∈
(0
,T
−
δ
),
k
ϕ
δ
t
(
·
,t
)
∈
H
1
(Ω)
§
U
ì
²
þ
{
(
Ø
Ò
k
µ
S
δ
+
(
ϕ
t
) = (
S
δ
+
(
ϕ
))
t
=
ϕ
δ
t
.
(36)
@
o
·
‚
y
3
ò
S
δ
+
A^
•
§
(20)
−
(22)
¥
§
¿
^
s
†
t
,
·
‚
k
µ
(
∇
p
δ
+
γ
(
ϕ
∇
µ
)
δ
,
∇
q
) = 0
∀
q
∈
H
1
(Ω)
.
(37)
(
ϕ
δ
t
,v
)+
ε
(
∇
µ
δ
,
∇
v
)+((
ϕ
[
∇
p
+
γϕ
∇
µ
])
δ
,
∇
v
) = 0
∀
v
∈
H
1
(Ω)
.
(38)
(
µ
δ
,ψ
)
−
ε
(
∇
ϕ
δ
,
∇
ψ
)
−
1
ε
((
f
(
ϕ
))
δ
,ψ
) = 0
∀
ψ
∈
H
1
(Ω)
.
(39)
3
•
§
(39)
¥
-
ψ
=
−
ϕ
t
,
3
•
§
(38)
¥
-
v
=
µ
δ
,
3
•
§
(37)
¥
-
q
=
p
δ
γ
,
¿
ò
n
‡
•
§
ƒ
\
§
d
dt
J
ε
(
ϕ
δ
)+
ε
∇
µ
δ
2
L
2
+
1
γ
∇
p
δ
+
γ
(
ϕ
∇
µ
)
δ
2
L
2
=
R
δ
(
t
)
.
(40)
Ù
¥
R
δ
(
t
) =
1
ε
(
f
(
ϕ
δ
)
−
(
f
(
ϕ
))
δ
−
(
∇
p
δ
+
γ
(
ϕ
∇
µ
)
δ
,ϕ
∇
µ
δ
−
(
ϕ
∇
µ
)
δ
)
+(
ϕ
h
∇
p
δ
+
γ
(
ϕ
∇
µ
)
δ
i
−
(
ϕ
[
∇
p
+
γϕ
∇
µ
])
δ
,
∇
µ
δ
)
.
DOI:10.12677/aam.2021.1072552435
A^
ê
Æ
?
Ð
‹
‰
§
Æ
“
3
•
§
(40)
ü
>
Ó
ž
é
t
È
©
§
·
‚
k
µ
J
ε
(
ϕ
δ
(
s
))+
R
s
0
(
ε
∇
µ
δ
(
t
)
2
L
2
+
1
γ
∇
p
δ
(
t
)+
γ
(
ϕ
∇
µ
)
δ
(
t
)
2
L
2
)
dt
=
J
ε
(
ϕ
δ
(0))+
R
s
0
R
δ
(
t
)
dt
∀
s
∈
(0
,T
)
.
5
¿
§
é
z
˜
‡
½
ε>
0,
du
f
(
ϕ
)
∈
L
2
((0
,T
);
H
1
(Ω))
¿
…
f
(
·
)
´
‡
ë
Y
¼
ê
§
¤
±
k
f
(
ϕ
)
∈
L
2
((0
,T
);
H
1
(Ω)),
-
δ
→
0
+
,
¿
(
Ü
Steklov
²
þ
5
Ÿ
§
·
‚
k
µ
lim
δ
→
0
+
Z
s
0
R
δ
(
t
)
dt
= 0
,
J
ε
(
ϕ
(
s
))+
Z
s
0
(
ε
k∇
µ
(
t
)
k
2
L
2
+
1
γ
k∇
p
(
t
)+
γ
(
ϕ
(
t
)
∇
µ
(
t
)
k
2
L
2
)
dt
=
J
ε
(
ϕ
(0)
.
Ï
d
§
(30),
l
(30)
¥•
E
(
t
)
´
'
u
ž
m
ë
Y
¼
ê
"
|
^
(30)
§
Ú
·
‚
b
:
F
(
s
)
≥−
c
3
,c
3
≥
0
.
(41)
·
‚
k
R
Ω
F
(
s
)
dx
≥−
c
3
|
Ω
|
(
c
3
≥
0)
§
K
Œ
•
§
(31)
"
•
§
(32)
Œ
d
•
§
(30)
†
"
•
§
(33)
Œ
d
•
§
(22),
•
§
(31)
,
(32)
±
9
Holder
Ø
ª
Ú
b
^
‡
·
f
(
s
)
≥
c
0
"
d
˜
m
'
X
µ
H
1
(Ω)
⊂
L
2
(Ω)
∼
=
L
2
(Ω)
⊂
(
H
1
(Ω))
∗
Ú
•
§
(21)
Œ
h
ϕ
t
+
u
(
s
)
·∇
ϕ
(
s
)
,v
i
=
−
ε
(
∇
µ,
∇
v
)
,
Š
â
Holder
Ø
ª
µ
|h
ϕ
t
+
u
(
s
)
·∇
ϕ
(
s
)
,v
i|
=
ε
|
(
∇
µ,
∇
v
)
|≤
ε
k∇
µ
k
L
2
k∇
v
k
L
2
.
•
2
Š
â
é
ó
‰
ê
±
9
d
‰
ê
•
§
(34)
"
•
•
§
(35)
§
|
^
•
§
(21)
Ú
Sobolev
i
\
½
n
µ
d
= 2
,
3,
k
H
1
(Ω)
→
L
6
(Ω),
é
u
v
∈
W
1
,
3
(Ω),
h
ϕ
t
,v
i
=
−
ε
(
∇
µ,
∇
v
)+(
ϕ
u
,
∇
v
)
≤
ε
k∇
µ
k
L
2
k∇
v
k
L
2
+
k
ϕ
k
L
6
k
u
k
L
2
k∇
v
k
L
3
≤
C
[
ε
k∇
µ
k
L
2
+
k
ϕ
k
H
1
k
u
k
L
2
]
k∇
v
k
L
3
≤
C
ε
k∇
µ
k
L
2
+
√
2
C
0
ε
+2
c
3
|
Ω
|
ε
.
•
|
^
d
‰
ê
Ú
é
ó
‰
ê
·
‚
(
J
[17]
"
e
5
0
f
)
„
k
Ù
¦
K
5
"
DOI:10.12677/aam.2021.1072552436
A^
ê
Æ
?
Ð
‹
‰
§
Æ
“
Ú
n
2.2
b
ϕ
0
∈
H
1
(Ω)
,
f
(
ϕ
)
∈
L
2
((0
,T
);
H
1
(Ω))
±
9
«
•
Ω
⊂
R
d
(
d
=2
,
3)
´
˜
‡
Lipschitz
«
•
,
en
-
|
(
p,ϕ,µ
)
´
½
Â
2.1
˜
|
f
)
,
K
ϕ
∈
L
2
((0
,T
);
H
2
(Ω))
"
y
²
.
·
‚
ò
•
§
(22)
-
#
¤
X
e
ε
(
∇
ϕ,
∇
ψ
) = (
µ
−
1
ε
f
(
ϕ
)
,ψ
)
∀
ψ
∈
H
1
(Ω)
.
(42)
ù
p
ϕ
´
‡
a
q
u
Poisson
•
§
,
¿
ä
k
à
g
Neumann
>
.
^
‡
˜
‡
f
)
§
Ù
m
à
‘
¼
ê
•
g
=
µ
−
1
ε
f
(
ϕ
)
§
du
f
(
ϕ
)
∈
L
2
((0
,T
);
H
1
(Ω)),
¤
±
m
à
¼
ê
g
∈
L
2
((0
,T
);
H
1
(Ω)),
ϕ
∈
L
2
((0
,T
);
H
2
(Ω)
"
e
¡
·
‚
?
Ø
f
)
•
˜
5
µ
½
n
2.3
b
ϕ
0
∈
H
1
(Ω)
Ú
J
ε
(
ϕ
0
)
≤
C
0
,
Ù
¥
C
0
Ø
•
6
u
ε
,
¿
…
Ω
⊂
R
d
(
d
= 2
,
3)
´
˜
‡
Lipschitz
«
•
§
±
9
f
(
ϕ
)
∈
L
2
((0
,T
);
H
1
(Ω))
,
˜
|
f
)
(
p,µ,ϕ
)
÷
v
K
^
‡
∇
p
+
γϕ
∇
µ
∈
L
12
6
−
d
((0
,T
);
L
2
(Ω))
,
µ
∈
L
12
6
−
d
((0
,T
);
H
1
(Ω))
,ϕ
t
∈
L
2
((0
,T
);(
H
1
(Ω)
∗
))
ž
§
Ù
¥
p
´
˜
‡
k
•
ê
§
Kù
|
f
)
´
•
˜
"
y
²
.
b
(
p
i
,µ
i
,ϕ
i
)
,i
=1
,
2
´
ü
|
f
)
§
½
Â
u
i
=
−∇
p
i
−
γϕ
i
∇
µ
i
,i
=1
,
2,
-
p
=
p
1
−
p
2
,
u
=
u
1
−
u
2
,µ
=
µ
1
−
µ
2
,ϕ
=
ϕ
1
−
ϕ
2
,(
p
1
,µ
1
,ϕ
1
),(
p
2
,µ
2
,ϕ
2
)
÷
v
é
A
•
§
(20)
−
(22)
§
ò
ö
(
Ü
å
5
§
·
‚
•
§
µ
(
u
,
∇
q
) = 0
∀
q
∈
H
1
(Ω)
.
(43)
h
ϕ
t
,v
i
+
ε
(
∇
µ,
∇
v
)
−
(
ϕ
1
u
+
ϕ
u
2
,
∇
v
) = 0
∀
v
∈
H
1
(Ω)
.
(44)
(
µ,ψ
)
−
ε
(
∇
ϕ,
∇
ψ
)
−
1
ε
(
f
(
ϕ
1
)
−
f
(
ϕ
2
)
,ψ
) = 0
∀
ψ
∈
H
1
(Ω)
.
(45)
du
•
§
(29),
·
‚
Œ
|
^
R
Ω
ϕ
(
x,t
)
dx
=0
,
∀
t
∈
(0
,T
),
•
Œ
Ï
L
˜
m
²
£
d
(
J
"
3
•
§
(44)
¥
§
-
v
=
ϕ
,
3
•
§
(45)
¥
§
-
ψ
=
µ
,
ò
ö
Ü
¿
§
2
Ï
L
•
§
(
u
2
,
∇
(
ϕ
2
))=0=
(
u
,
∇
(
ϕ
1
ϕ
)),
·
‚
µ
1
2
d
dt
k
ϕ
k
2
L
2
+
k
µ
k
2
L
2
=
−
(
u
·∇
ϕ
1
,ϕ
)+
1
ε
(
g
(
ϕ
1
,ϕ
2
)
ϕ,µ
)
.
(46)
Ù
¥
g
(
ϕ
1
,ϕ
2
)
´
Ï
L
f
(
ϕ
1
)
−
f
(
ϕ
2
)
ü
‡
õ
‘
ª
Ï
L
n
g
•
ú
ª
5
§
=
g
(
ϕ
1
,ϕ
2
) =
f
(
ϕ
1
)
−
f
(
ϕ
2
)
ϕ
1
−
ϕ
2
|
^
Schwarz
Ø
ª
Ú
Gagliardo-Nirenberg
Ø
ª
[18]
§
k
k
ϕ
k
L
∞
≤
C
k
∆
ϕ
k
d
4
L
2
k
ϕ
k
4
−
d
4
L
2
(
d
= 2
,
3)
.
3
•
§
(46)
¥
k
d
dt
k
ϕ
k
2
L
2
+2
k
µ
k
2
L
2
≤
2
k
u
k
L
2
k∇
ϕ
1
k
L
2
k
ϕ
k
L
∞
+
2
ε
k
g
(
ϕ
1
,ϕ
2
)
k
L
∞
k
ϕ
k
L
2
k
µ
k
2
L
2
≤
ε
4
γ
k
u
k
2
L
2
+
C
ε
k∇
ϕ
1
k
2
L
2
k
∆
ϕ
k
d
2
L
2
k
ϕ
k
4
−
d
2
L
2
+
2
ε
k
g
(
ϕ
1
,ϕ
2
)
k
L
∞
k
ϕ
k
L
2
k
µ
k
L
2
≤
ε
4
γ
k
u
k
2
L
2
+
ε
2
16
k
∆
ϕ
k
2
L
2
+
C
(
ε
)
k∇
ϕ
1
k
8
4
−
d
L
2
k
ϕ
k
2
L
2
+
k
µ
k
2
L
2
+
1
ε
2
k
g
(
ϕ
1
,ϕ
2
)
k
2
L
∞
k
ϕ
k
2
L
2
.
DOI:10.12677/aam.2021.1072552437
A^
ê
Æ
?
Ð
‹
‰
§
Æ
“
Ù
¥
é
um
à
,
˜
‘
k
g
(
ϕ
1
,ϕ
2
)
k
L
∞
,
Ä
k
é
u
g
(
ϕ
1
,ϕ
2
) =
2
p
+1
X
k
=1
(
ϕ
k
−
1
1
+
ϕ
k
−
2
1
ϕ
2
+
....
+
ϕ
1
ϕ
k
−
1
2
+
ϕ
k
2
)
.
·
‚
k
µ
k
g
(
ϕ
1
,ϕ
2
)
k
L
∞
≤
2
p
+1
X
k
=1
k
ϕ
1
k
k
−
1
L
∞
+
k
ϕ
1
k
k
−
2
L
∞
k
ϕ
2
k
1
L
∞
+
......
+
k
ϕ
2
k
k
−
1
L
∞
.
du
ϕ
∈
L
∞
(0
,T
;
H
1
(Ω)),
Ï
d
ϕ
∈
L
∞
(0
,T
;
L
∞
(Ω)),
K
k
ϕ
k
L
∞
≤
sup
t
∈
(0
,T
)
k
ϕ
k
L
∞
≤
C
,
Ù
¥
C
´
˜
‡
~
ê
§
Ï
k
g
(
ϕ
1
,ϕ
2
)
k
L
∞
≤
C
p
,
ù
p
C
p
´
˜
‡
†
p
k
'
~
ê
"
Ï
L
(30)
d
dt
k
ϕ
k
2
L
2
+
k
µ
k
2
L
2
≤
ε
4
γ
k
u
k
2
L
2
+
ε
16
k
∆
ϕ
k
2
L
2
+
C
(
ε,p
)
k
ϕ
k
2
L
2
.
(47)
Ù
¥
C
(
ε,p
) =
C
p
2
ε
2
>
0
3
•
§
(39)
¥
-
ψ
= ∆
ϕ
,
ε
k
∆
ϕ
k
2
L
2
=
−
(
µ,
∆
ϕ
)+
1
ε
(
g
(
ϕ
1
,ϕ
2
)
ϕ,
∆
ϕ
)
≤
ε
2
k
∆
ϕ
k
2
L
2
+
1
ε
k
µ
k
2
L
2
+
C
p
2
ε
2
k
ϕ
k
2
L
2
.
,
ε
2
4
k
∆
ϕ
k
2
L
2
≤
1
2
k
µ
k
2
L
2
+
C
(
ε,p
)
k
ϕ
k
2
L
2
.
(48)
Ù
¥
C
(
ε,p
)
>
0
5
¿
u
=
u
1
−
u
2
=
−∇
p
−
γ
(
ϕ
1
∇
µ
1
−
ϕ
2
∇
µ
2
) =
−∇
p
−
γϕ
1
∇
µ
−
γϕ
∇
µ
2
.
3
•
§
(37)
¥
-
q
=
p
1
γ
k
u
k
2
L
2
=
−
1
γ
(
u
,
∇
p
)
−
(
u
,ϕ
1
∇
µ
)
−
(
u
,ϕ
∇
µ
2
)
=
−
(
ϕ
1
u
,
∇
µ
)
−
(
ϕ
u
,
∇
µ
2
)
.
(49)
3
•
§
(38)
¥
-
v
=
µ
,
Œ
h
ϕ
t
,µ
i
+
ε
k∇
µ
k
2
L
2
= (
ϕ
1
u
+
ϕ
u
2
,
∇
µ
)
.
(50)
e
5
é
•
§
(39)
A^
d
A
‰
â
Å
²
þ
Ž
{
S
δ
+
§
K
k
µ
µ
δ
,ψ
−
ε
(
∇
ϕ
δ
,
∇
ψ
)
−
1
ε
(
f
(
ϕ
1
)
−
f
(
ϕ
2
)
δ
,ψ
) = 0
∀
ψ
∈
H
1
(Ω)
.
DOI:10.12677/aam.2021.1072552438
A^
ê
Æ
?
Ð
‹
‰
§
Æ
“
-
ψ
=
−
ϕ
δ
t
,
þ
¡
•
§
Œ
−
(
µ
δ
,ϕ
δ
t
)+
ε
2
d
dt
∇
ϕ
δ
2
L
2
+
1
ε
(
f
(
ϕ
1
)
−
f
(
ϕ
2
))
δ
,ϕ
δ
t
) = 0
.
é
δ
4
•
δ
→
0
+
,
|
^
d
A
‰
â
Å
²
þ
Ž
f
5
Ÿ
§
k
−h
ϕ
t
,µ
i
+
ε
2
d
dt
k∇
ϕ
k
2
L
2
+
1
ε
h
ϕ
t
,f
(
ϕ
1
)
−
f
(
ϕ
2
)
i
= 0
.
(51)
•
r
•
§
(49)
,
(50)
,
(51)
\
å
5
§
¿
|
^
(
u
2
,
∇
(
ϕµ
))=(
u
,
∇
(
ϕµ
2
))=0
Ú
Young
Ø
ª
±
9
Gagliardo-Nirenberg
Ø
ª
1
γ
k
u
k
2
L
2
+
ε
k∇
µ
k
2
L
2
+
ε
2
d
dt
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