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AdvancesinAppliedMathematics
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,2022,11(1),526-536
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.111060
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Nirenb erg-Gagliado
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TheMaximumEstimateofSolutiontothe
Cahn-HilliardEquationandViscous
Cahn-HilliardEquation
ChunxiangXue,ZhilinPu
∗
SchoolofMathematicalSciences,SichuanNormalUniversity
§
ChengduSichuan
Received:Dec.26
th
,2021;accepted:Jan.21
st
,2022;published:Jan.28
th
,2022
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526-536.DOI:10.12677/aam.2022.111060
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Abstract
Inthispaper,ouraimistoprovethemaximumestimatesofsolutionstotheCahn-
HilliardequationandViscousCahn-Hilliardequationwithageneralnonlinearsource
term.Firstly,weobtaintheboundednessofthe
L
q
normbyusingenergyestimates.
Then,theboundednessofessentialsupremumisdemonstratedbyNirenb erg-Gagliado
inequality.
Keywords
Cahn-HilliardEquation,ViscousCahn-HilliardEquation,MaximumEstimates,
Nirenb erg-GagliadoInequality
Copyright
c
2022byauthor(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.2022.111060530
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(
¯
Q
T
)
§
K
−
Z
Ω
∆
|
u
|
∆
|
u
|
p
dx
≤
c
Z
Ω
∆
|
u
|
p
dx,
=
c
(
Z
Ω
p
(
p
−
1)
|
u
|
p
−
2
|∇
u
|
2
+
p
|
u
|
p
−
1
∆
|
u
|
dx
)
,
≤
cp
2
Z
Ω
h
(
u
)
dx.
(3.15)
Ù
¥
h
(
u
) = max
{|
u
|
p
−
2
,
|
u
|
p
−
1
}
.
d
Young
0
sinequality
§
·
‚
Z
Ω
p
2
|
u
|
p
−
2
dx
≤
c
+
c
Z
Ω
p
−
2
p
+1
(
p
2
|
u
|
p
−
2
)
p
+1
p
−
2
dx,
≤
c
+
c
Z
Ω
p
2(
P
+1)
P
−
2
|
u
|
p
+1
dx,
≤
c
+
c
Z
Ω
p
8
|
u
|
p
+1
dx.
(3.16)
DOI:10.12677/aam.2022.111060531
A^
ê
Æ
?
Ð
Å
S
†
§
Æ
“
Z
Ω
p
2
|
u
|
p
−
1
dx
≤
c
+
c
Z
Ω
p
−
1
p
+1
(
p
2
|
u
|
p
−
1
)
p
+1
p
−
1
dx,
≤
c
+
c
Z
Ω
p
2(
P
+1)
P
−
1
|
u
|
p
+1
dx,
≤
c
+
c
Z
Ω
p
8
|
u
|
p
+1
dx.
(3.17)
(
Ü
^
‡
f
0
(
u
)
≥
r
(2
r
−
1)
b
2
r
u
2
r
−
2
−
c
9
u
|
u
|
p
−
1
u
t
=
1
p
+1
d
d
t
|
u
|
p
+1
§
1
p
+1
d
d
t
Z
Ω
|
u
|
p
dx
+
cp
Z
Ω
|∇
u
|
2
|
u
|
2
r
+
p
−
3
dx
≤
c
(1+
p
8
)
Z
Ω
|
u
|
p
+1
dx.
(3.18)
Ï
•
|∇
u
|
2
|
u
|
2
r
+
p
−
3
=
4
(2
r
+
p
−
1)
2
|∇|
u
|
(2
r
+
p
−
1)
2
|
2
.
(3.19)
n
Ü
•
§
(3.18)
,
(3.19)
§
1
p
+1
d
d
t
Z
Ω
|
u
|
p
dx
+
4
cp
(2
r
+
p
−
1)
2
Z
Ω
|∇|
u
|
(2
r
+
p
−
1)
2
|
2
dx
≤
c
(1+
p
8
)
Z
Ω
|
u
|
p
+1
dx.
(3.20)
|
u
(
x,t
)
|
>
1
ž
§
Ï
•
2
r
+
p
−
3
>p
−
1
§
K
|
u
|
2
r
+
p
−
3
>
|
u
|
p
−
1
§
|∇
u
|
2
|
u
|
2
r
+
p
−
3
≥|∇
u
|
2
|
u
|
p
−
1
§
=
§
4
(
p
+1)
2
|∇|
u
|
(
p
+1)
2
|
2
≤
4
(2
r
+
p
−
1)
2
|∇|
u
|
(2
r
+
p
−
1)
2
|
2
.
(3.21)
d
•
§
(3.20)
Ú
(3.21)
§
d
d
t
Z
Ω
|
u
|
p
+1
dx
+
4
cp
(
p
+1)
Z
Ω
|∇|
u
|
(
p
+1)
2
|
2
dx
≤
C
(1+
p
)
9
Z
Ω
|
u
|
p
+1
dx.
(3.22)
3
(3.1)
¥
§
-
s
= 2
,j
= 0
,r
= 2
,m
= 1
§
||
ν
||
2
L
2
≤
C
1
||
Dν
||
2
a
L
2
||
ν
||
1
−
a
L
q
+
C
2
||
ν
||
2
L
q
.
(3.23)
ν
=
|
u
|
µ
k
+1
2
,µ
k
= 2
k
,q
=
2(
µ
k
−
1
+1)
µ
k
+1
§
d
(3.2)
a
=
n
(2
−
q
)
n
(2
−
q
)+2
q
=
n
n
+2+2
2
−
k
(3.24)
3
•
§
(3.23)
¥
|
^
Young
0
s
Ø
ª
Z
Ω
|
u
|
µ
k
+1
dx
≤
Z
Ω
|∇|
u
|
µ
k
+1
2
|
2
dx
+
c
−
a
1
−
a
(
Z
Ω
|
u
|
µ
k
−
1
+1
dx
)
µ
k
+1
µ
k
−
1
+1
.
(3.25)
DOI:10.12677/aam.2022.111060532
A^
ê
Æ
?
Ð
Å
S
†
§
Æ
“
3
•
§
(3.22)
¥
-
p
=
µ
k
9
|
^
•
§
(3.25),
·
‚
d
d
t
Z
Ω
|
u
|
µ
k
+1
dx
+
4
c
1
µ
k
(
µ
k
+1)
Z
Ω
|∇|
u
|
(
µ
k
+1)
2
|
2
dx,
≤
C
(1+
µ
k
)
9
(
Z
Ω
|∇|
u
|
µ
k
+1
2
|
2
dx
+
c
−
a
1
−
a
(
Z
Ω
|
u
|
µ
k
−
1
+1
dx
)
µ
k
+1
µ
k
−
1
+1
)
.
(3.26)
-
=
1
C
(
µ
k
+1)
9
.
c
1
µ
k
µ
k
+1
§
·
‚
d
d
t
Z
Ω
|
u
|
µ
k
+1
dx
+
C
1
(
k
)
Z
Ω
|∇|
u
|
(
µ
k
+1)
2
|
2
dx
≤
C
2
(
k
)(
Z
Ω
|
u
|
µ
k
−
1
+1
dx
)
µ
k
+1
µ
k
−
1
+1
.
(3.27)
Ù
¥
C
1
(
k
) =
c
1
µ
k
µ
k
+1
,C
2
(
k
) =
C
1
1
−
a
.c.
(
c
1
µ
k
µ
k
+1
)
−
1
1
−
a
.
(1+
µ
k
)
9
1
−
a
3
•
§
(3.25)
¥
-
= 1
9
d
•
§
(3.27)
§
·
‚
d
d
t
Z
Ω
|
u
|
µ
k
+1
dx
+
C
1
(
k
)
Z
Ω
|
u
|
µ
k
+1
dx
≤
C
4
(
k
)(
Z
Ω
|
u
|
µ
k
−
1
+1
dx
)
µ
k
+1
µ
k
−
1
+1
.
(3.28)
Ù
¥
C
4
(
k
) =
C
2
(
k
)+
c
.
|
^
Gronwall
Ø
ª
§
·
‚
Z
Ω
|
u
|
µ
k
+1
dx
Z
Ω
|
u
0
|
µ
k
+1
dx
+
C
4
(
k
)
C
1
(
k
)
(sup
t
≥
0
Z
Ω
|
u
|
µ
k
−
1
+1
dx
)
µ
k
+1
µ
k
−
1
+1
,
≤
δ
(
k
)max
{
M
µ
k
+1
0
|
Ω
|
,
(sup
t
≥
0
Z
Ω
|
u
|
µ
k
−
1
+1
dx
)
µ
k
+1
µ
k
−
1
+1
}
.
(3.29)
Ù
¥
§
δ
(
k
) =
c
(1+
µ
k
)
α
,α
=
2
1
−
a
,M
0
= sup
x
∈
Ω
|
u
0
|
.
d
H
¨
older
Ø
ª
§
·
‚
Z
Ω
|
u
|
µ
k
+1
dx
≤
δ
(
k
)max
{
M
µ
k
+1
0
|
Ω
|
,
(sup
t
≥
0
Z
Ω
|
u
|
µ
k
−
1
+1
dx
)
µ
k
+1
µ
k
−
1
+1
}
,
≤
δ
(
k
)
|
Ω
|
max
{
M
µ
k
+1
0
,
(sup
t
≥
0
Z
Ω
|
u
|
2
dx
)
µ
k
+1
2
}
,
≤
k
Y
i
=0
(
|
Ω
|
δ
(
k
−
i
))
µ
k
+1
µ
k
−
i
+1
max
{
M
µ
k
+1
0
,
(sup
t
≥
0
Z
Ω
|
u
|
2
dx
)
µ
k
+1
2
}
.
(3.30)
Ï
•
µ
k
+1
µ
k
−
i
+1
<
2
i
§
·
‚
δ
(
k
)
δ
(
k
−
1)
µ
k
+1
µ
k
−
1
+1
δ
(
k
−
2)
µ
k
+1
µ
k
−
2
+1
...δ
(0)
µ
k
+1
2
,
≤
c
2
k
+1
−
1
(2
α
)
−
k
+2
k
+2
−
2
.
(3.31)
Ú
|
Ω
|·|
Ω
|
µ
k
+1
µ
k
−
1
+1
...
|
Ω
|
µ
k
+1
2
≤|
Ω
|
2
k
+1
+1
.
(3.32)
DOI:10.12677/aam.2022.111060533
A^
ê
Æ
?
Ð
Å
S
†
§
Æ
“
Ï
d
(
Ü
•
§
(3.30)
§
(3.31),(3.32)
±
9
Ú
n
3
.
2
§
·
‚
(
Z
Ω
|
u
|
2
k
+1
dx
)
1
2
k
+1
≤
C
|
Ω
|
2
4
α
max
{
M
0
,
(sup
t
≥
0
(
Z
Ω
|
u
|
2
dx
)
1
2
}≤
¯
C
(
u
0
)
.
(3.33)
Ï
•
k
´
?
¿
§
3
•
§
(3.33)
¥
§
-
k
→∞
§
·
‚
k
u
k
∞
≤
¯
C
(
u
0
)
.
Ï
d
§
sup
0
≤
t
≤
T
k
u
k
∞
≤
¯
C
(
u
0
)
.
(3.34)
Ï
•
u
∈
C
2
,
1
(
¯
Q
T
)
§
@
o
max
Q
T
|
u
(
x,t
)
|≤
¯
C
(
u
0
)
.
u
(
x
)
≤
1
ž
§
du
2
r
+
p
−
2
>p
§
@
o
|
u
t
||
u
|
2
r
+
p
−
2
≤|
u
t
||
u
|
p
"
d
^
‡
u
∈
C
2
,
1
(
¯
Q
T
),
|
u
t
||
u
|
2
r
+
p
−
2
≤|
u
t
||
u
|
p
±
9
L
2
r
+
P
−
1
(Ω)
⊂
L
p
(Ω)
§
·
‚
1
2
r
+
p
−
1
d
d
t
Z
Ω
|
u
|
2
r
+
p
−
1
dx
≤
1
2
r
+
p
−
1
Z
Ω
|
d
d
t
|
u
|
2
r
+
p
−
1
|
dx,
≤
1
p
+1
Z
Ω
|
d
d
t
|
u
|
p
+1
|
dx,
=
Z
Ω
|
u
t
||
u
|
p
dx,
≤
c
Z
Ω
|
u
|
p
dx,
≤
c
Z
Ω
|
u
|
2
r
+
P
−
1
dx.
(3.35)
d
^
‡
u
∈
C
2
,
1
(
¯
Q
T
)
±
9
Young
0
s
Ø
ª
§
·
‚
Z
Ω
p
|∇|
2
|
u
|
2
r
+
p
−
3
dx
≤
c
Z
Ω
p
|
u
|
2
r
+
p
−
3
dx,
≤
c
Z
Ω
(
p
|
u
|
2
r
+
p
−
3)
2
r
+
p
−
1
2
r
+
p
−
3
dx
+
c,
≤
c
Z
Ω
p
2
|
u
|
2
r
+
p
−
1
dx.
(3.36)
(
Ü
•
§
(3.19),(3.35),(3.36)
§
·
‚
d
d
t
Z
Ω
|
u
|
2
r
+
p
−
1
dx
+
4
cp
(2
r
+
p
−
1)
Z
Ω
|∇|
u
|
(2
r
+
p
−
1)
2
|
2
≤
c
(2
r
+
p
)
3
Z
Ω
|
u
|
2
r
+
p
−
1
dx.
(3.37)
DOI:10.12677/aam.2022.111060534
A^
ê
Æ
?
Ð
Å
S
†
§
Æ
“
-
E
•
§
(3.23)
−
(3.33)
Ú
½
§
|
u
(
x,t
)
|≤
1
ž
§
•
k
max
Q
T
|
u
(
x,t
)
|≤
¯
C
(
u
0
)
.
(
ii
)
ν
∈
(0
,
1)
ž
§
Ú
ν
= 0
ž
?
n
•{
˜
§
•
k
max
Q
T
|
u
(
x,t
)
|≤
¯
C
(
u
0
)
.
n
þ
¤
ã
§
y
"
3.2.
½
n
2
.
4
y
²
y
²
µ
·
‚
3
•
§
(1.4)
¥
ü
>
Ó
ž
¦
±
u
|
u
|
p
−
1
,(
p
≥
3)
§
¿
…
3
Ω
þ
È
©
§
·
‚
Z
Ω
u
|
u
|
p
−
1
u
t
dx
=
Z
Ω
u
|
u
|
p
−
1
∆(∆
u
+
f
(
u
))+
ε
(∆
u
−
f
(
u
))
u
|
u
|
p
−
1
dx,
=
Z
Ω
∆
u
∆(
u
|
u
|
p
−
1
)
dx
−
Z
Ω
∇
f
(
u
)
∇
(
u
|
u
|
p
−
1
)
dx
−
ε
Z
Ω
∇
u.
∇
(
u
|
u
|
p
−
1
)
dx
−
ε
Z
Ω
f
(
u
)
u
|
u
|
p
−
1
dx,
=
−
Z
Ω
∆
|
u
|
∆
|
u
|
p
dx
−
p
Z
Ω
f
0
(
u
)
|∇
u
|
2
|
u
|
p
−
1
dx
−
εp
Z
Ω
|∇
u
|
2
|
u
|
p
−
1
dx
−
ε
Z
Ω
f
(
u
)
u
|
u
|
p
−
1
dx,
≤−
Z
Ω
∆
|
u
|
∆
|
u
|
p
dx
−
p
Z
Ω
f
0
(
u
)
|∇
u
|
2
|
u
|
p
−
1
dx
−
ε
Z
Ω
f
(
u
)
u
|
u
|
p
−
1
dx.
(3.38)
d
•
§
(2.4)
±
9
L
p
+1
(Ω)
⊂
L
p
(Ω)
§
·
‚
−
ε
Z
Ω
f
(
u
)
u
|
u
|
p
−
1
dx
≤−
Z
Ω
{
r
(2
r
−
1)
b
2
r
u
2
r
−
1
u
|
u
|
p
−
1
−
cu
|
u
|
p
−
1
}
dx,
=
Z
Ω
cu
|
u
|
p
−
1
dx
−
c
Z
Ω
u
2
r
|
u
|
p
−
1
dx,
≤
Z
Ω
c
|
u
|
p
dx,
≤
c
(
p
+1)
Z
Ω
|
u
|
p
+1
dx.
(3.39)
5
e
§
-
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