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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(1),526-536
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.111060
Cahn-HilliardÚÊ5Cahn-Hilliard•§)
•ŒŠO
ÅÅÅSSS†††§§§ÆÆÆ“““
∗
oA“‰ŒÆêƉÆÆ§oA¤Ñ
ÂvFϵ2021c1226F¶¹^Fϵ2022c121F¶uÙFϵ2022c128F
Á‡
©ïÄäk˜„š‚5^‡Cahn-Hilliard•§ÚÊ5Cahn-Hilliard•§)•ŒŠ
O"Äk§·‚^UþO•{)L
q
‰êk.",§|^Nirenberg-GagliadoØ
ª)5þ(.k."
'…c
Cahn-Hilliard•§§Ê5Cahn-Hilliard•§§•ŒŠO§Nirenb erg-GagliadoØª
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
∗ÏÕŠö"
©ÙÚ^:ÅS†,Æ“.Cahn-HilliardÚÊ5Cahn-Hilliard•§)•ŒŠO[J].A^êÆ?Ð,2022,11(1):
526-536.DOI:10.12677/aam.2022.111060
ÅS†§Æ“
Abstract
Inthispaper,ouraimistoprovethemaximumestimatesofsolutionstotheCahn-
HilliardequationandViscousCahn-Hilliardequationwithageneralnonlinearsource
term.Firstly,weobtaintheboundednessoftheL
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/
1.Úó
Cahn-Hillardu1958c[1]JÑ±e•§µ
u
t
+α∆
2
u−k∆f(u) = 0,α,k>0(1.1)
3á‰Æ¥ˆü-‡ڧϕ§£ãÜ7¥ƒ©lL§¥üƒX Ú-‡½5A"
Cahn-Hilliard•§ØA^uÜ7ƒ§5•A^uÙ¦‰Æ+•§X¬5)•[2,3]![
ÿ[4]"§•kNõ*ÐÚí2(„~[5,6])§Ù¥A.Novick-Cohen3[7]¥JÑ^u©Û˜
ƒCÄåÆÊ5Cahn-Hilliard•§
(1−ν)u
t
= ∆(−∆u+f(u)−νu
t
),x∈Ω(1.2)
Ù¥ν∈[0,1],Ω⊂R
n
´k.1w«•§•)Š •4•œ¹Cahn-Hilliard•§(ν=0)ÚŒ‚5
9•§(ν= 1)"
C5§ØÆöÏLéIOÊ5Cahn-Hilliard•§š‚5‘‰ØÓb§(J"~
X,4•S<3©z[8]¥•Äf(u) = −u+γ
1
u
2
+γ
2
u
3
œ¹§²;)Û•35Ú»
DOI:10.12677/aam.2022.111060527A^êÆ?Ð
ÅS†§Æ“
5¶¿…•Ñγ
2
ÎÒéu)Û•35–'-‡"AO/§γ
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<0ž§ KeÚYin3[9]
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2
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Њvž)Û•35ÚЊvŒž)»(J"Ødƒ§4•S<3[11]¥
Њvž)Û•35Úé?¿š²…Њ§)»˜(J"
©§·‚•ÄXe
Ê5Cahn-Hilliard•§Ð>Š¯K













(1−ν)u
t
=∆(−∆u+f(u)−νu
t
),x∈Ω ⊂R
n
,t>0,
u(0,x) =u
0
(x),x∈Ω,
∂u
∂N
=
∂(∆u)
∂N
= 0,x∈∂Ω,t>0.
(1.3)
±9í2Cahn-Hilliard•§Ð>Š¯K













u
t
= (−∆
2
+ε∆)u−(−∆+εI)f(u),
u(0,x) = u
0
(x),x∈Ω,
∂u
∂N
=
∂(∆u)
∂N
= 0,x∈∂Ω,t>0.
(1.4)
)•ŒŠO"©Ì‡|^©z[12]•{ÚE|?1y²"Äk§·‚‰Ñ̇(J½
n2.3Ú½n2.4¶,§·‚‰Ñ½n2.3Ú½n2.4y²"
3©¥§¤kcÑ´Œu0~꧿…ˆ?ŠØ¦ƒÓ"©Ù¥¤k‰êÑ´IO
Sobolev˜m‰ê§Ù¥NL«÷∂Ω{•þ§ν∈[0,1),ε>0,Q
T
= Ω×(0,T),T>0"
2.̇(J
Äk§·‚éš‚5‘f: Ω×R→R‰Xebµ
b2.1.f´2r−1gõ‘ª§F´f¼ê§fÄ‘Xêa
2r−1
>0§÷v
f(s) =
P
2r−1
j=1
a
j
s
j
,r≥2,
F(s) =
P
2r
j=2
b
j
s
j
.
(2.1)
Ù¥§a
j−1
= jb
j
≥0,r≥2
52.2.é∀u∈R§db2.1[13],
f
0
(u) ≥r(2r−1)b
2r
u
2r−2
−c,(2.2)
1
2
b
2r
u
2r
−c≤F(u),(2.3)
DOI:10.12677/aam.2022.111060528A^êÆ?Ð
ÅS†§Æ“
Ú
f(u) ≥rb
2r
u
2r−1
−c.(2.4)
©Ì‡(J´µ
½n2.3.XJu(x,t) ∈C
2,1
(
¯
Q
T
)´•§(1.3))§K•3~ê
¯
C(u
0
)÷v
max
Q
T
|u(x,t)|≤
¯
C(u
0
).(2.5)
½n2.4.XJu(x,t) ∈C
2,1
(
¯
Q
T
)´•§(1.4))§K•3~ê
¯
C(u
0
)÷v
max
Q
T
|u(x,t)|≤
¯
C(u
0
).(2.6)
3.y²
3y²½n2.3Ú½n2.4ƒc§·‚k‰ÑA‡-‡Ún.
Ún3.1.[12](Nirenberg-GagliadoØª)Ωk.½Ω = R
n
,1 ≤s,q≤∞,m,j•꧅k
j
m
≤a≤1,
1
s
=
j
n
+a(
1
r
−
m
n
)+(1−a)
1
q
.(3.1)
@o•3~êC
1
= C
1
(n,j,s,m,a,r,q),C
2
= C
2
(n,j,s,m,a,r,q)§¦
||D
j
ν||
L
s
≤C
1
||D
m
ν||
a
L
r
||ν||
1−a
L
r
+C
2
||ν||
L
q
.(3.2)
Ún3.2.XJu(x,t) ∈C
2,1
(
¯
Q
T
)´•§(1.3))§K•3~ê
¯
C(u
0
)÷v
sup
0≤t≤T
||u(x,t)||
q
≤
¯
C(u
0
),q≤2.(3.3)
y²µò•§(1.3)ü>Óž¦±u,¿3ΩþÈ©§·‚
Z
Ω
(1−ν)u
t
udx=
Z
Ω
∆(−∆u+f(u)−νu
t
)udx,
=
Z
Ω
−|∆u|
2
dx−
Z
Ω
f
0
(u)|∇u|
2
dx−
Z
Ω
u
t
∆udx.
(3.4)
|^^‡u(x,t) ∈C
2,1
(
¯
Q
T
)§·‚
Z
Ω
−|∆u|
2
dx≤c.(3.5)
−
Z
Ω
u
t
∆udx≤c.(3.6)
DOI:10.12677/aam.2022.111060529A^êÆ?Ð
ÅS†§Æ“
d•§(2.2)±9^‡u(x,t) ∈C
2,1
(
¯
Q
T
)§·‚
−
Z
Ω
f
0
(u)|∇u|
2
dx≤
Z
Ω
(c−r(2r−1)b
2r
u
2r−2
)|∇u|
2
dx,
≤
Z
Ω
c|∇u|
2
dx,
≤c.
(3.7)
Z
Ω
(1−ν)u
t
udx=
1−ν
2
d
dt
Z
Ω
|u|
2
dx,
≤c.
(3.8)
KdGronwall
0
sÚn§
||u||
L
2
≤c.
y"
Ún3.3.XJu(x,t) ∈C
2,1
(
¯
Q
T
)´•§(1.4))§K•3~ê
¯
C(u
0
)÷v
sup
0≤t≤T
||u(x,t)||
q
≤
¯
C(u
0
),q≤2r.(3.9)
y²µò•§(1.4)ü>Óž¦±−∆u+f(u)§¿3ΩþÈ©§·‚
Z
Ω
(−∆u+f(u))u
t
dx= −
Z
Ω
[(∇(∆u−f(u)))
2
+ε(−∆u+f(u))
2
]dx(3.10)
Ï•
d
dt
(
Z
Ω
F(u)+
1
2
|∇u|
2
dx) =
Z
Ω
[f(u)u
t
+∇u∇u
t
]dx,
=
Z
Ω
(−∆u+f(u))u
t
dx,
≤0.
(3.11)
-
E(t) = E(u(x,t)) =
Z
Ω
F(u)+
1
2
|∇u|
2
dx.(3.12)
d(3.11)§·‚§
E(u(t)) ≤E(u(0)),
=
Z
Ω
F(u)+
1
2
|∇u|
2
dx≤E(u(0)).
DOI:10.12677/aam.2022.111060530A^êÆ?Ð
ÅS†§Æ“
d•§(2.3)§·‚
Z
Ω
1
2
b
2r
u
2r
dx≤E(u
0
)+c.
@o
sup
0≤t≤T
||u||
q
≤c,q≤2r.(3.13)
3.1.½n2.3y²
y².∀p≥3,ò•§(1.3)¥ü>Óž¦±u|u|
p−1
,¿…3ΩþÈ©§·‚
Z
Ω
(1−ν)u
t
u|u|
p−1
dx=
Z
Ω
u|u|
p−1
∆(−∆u+f(u)−νu
t
)udx,
=
Z
Ω
∆u∆(u|u|
p−1
)dx−
Z
Ω
∇f(u)∇(u|u|
p−1
)dx−
Z
Ω
ν∆u
t
u|u|
p−1
dx,
= −
Z
Ω
∆|u|∆|u|
p
dx−p
Z
Ω
f
0
(u)|∇u|
2
|u|
p−1
dx−
Z
Ω
νu
t
∆u|u|
p−1
dx.
(3.14)
|^^‡u(x,t) ∈C
2,1
(
¯
Q
T
)§·‚
−
Z
Ω
νu
t
∆u|u|
p−1
dx≤c
(i)ν= 0ž§•§(1.3)C•IOCahn-Hillard•§
d^‡u(x,t) ∈C
2,1
(
¯
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
}.
dYoung
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.2022.111060531A^êÆ?Ð
ÅS†§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(2r−1)b
2r
u
2r−2
−c9u|u|
p−1
u
t
=
1
p+1
d
dt
|u|
p+1
§
1
p+1
d
dt
Z
Ω
|u|
p
dx+cp
Z
Ω
|∇u|
2
|u|
2r+p−3
dx≤c(1+p
8
)
Z
Ω
|u|
p+1
dx.(3.18)
Ï•
|∇u|
2
|u|
2r+p−3
=
4
(2r+p−1)
2
|∇|u|
(2r+p−1)
2
|
2
.(3.19)
nÜ•§(3.18),(3.19)§
1
p+1
d
dt
Z
Ω
|u|
p
dx+
4cp
(2r+p−1)
2
Z
Ω
|∇|u|
(2r+p−1)
2
|
2
dx≤c(1+p
8
)
Z
Ω
|u|
p+1
dx.(3.20)
|u(x,t)|>1ž§Ï•2r+p−3 >p−1§K|u|
2r+p−3
>|u|
p−1
§|∇u|
2
|u|
2r+p−3
≥|∇u|
2
|u|
p−1
§
=§
4
(p+1)
2
|∇|u|
(p+1)
2
|
2
≤
4
(2r+p−1)
2
|∇|u|
(2r+p−1)
2
|
2
.(3.21)
d•§(3.20)Ú(3.21)§
d
dt
Z
Ω
|u|
p+1
dx+
4cp
(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||
2a
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)+2q
=
n
n+2+2
2−k
(3.24)
3•§(3.23)¥|^Young
0
sØ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.2022.111060532A^êÆ?Ð
ÅS†§Æ“
3•§(3.22)¥-p= µ
k
9|^•§(3.25),·‚
d
dt
Z
Ω
|u|
µ
k
+1
dx+
4c
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
dt
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)¥-= 19d•§(3.27)§·‚
d
dt
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.
|^GronwallØ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
|.
dH¨olderØ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.2022.111060533A^êÆ?Ð
ÅS†§Æ“
Ïd(Ü•§(3.30)§(3.31),(3.32)±9Ún3.2§·‚
(
Z
Ω
|u|
2
k
+1
dx)
1
2
k
+1
≤C|Ω|2
4α
max{M
0
,(sup
t≥0
(
Z
Ω
|u|
2
dx)
1
2
}≤
¯
C(u
0
).(3.33)
Ï•k´?¿§3•§(3.33)¥§-k→∞§·‚
kuk
∞
≤
¯
C(u
0
).
Ïd§
sup
0≤t≤T
kuk
∞
≤
¯
C(u
0
).(3.34)
Ï•u∈C
2,1
(
¯
Q
T
)§@o
max
Q
T
|u(x,t)|≤
¯
C(u
0
)
.
u(x) ≤1ž§du2r+p−2 >p§@o|u
t
||u|
2r+p−2
≤|u
t
||u|
p
"
d^‡u∈C
2,1
(
¯
Q
T
),|u
t
||u|
2r+p−2
≤|u
t
||u|
p
±9L
2r+P−1
(Ω) ⊂L
p
(Ω)§·‚
1
2r+p−1
d
dt
Z
Ω
|u|
2r+p−1
dx≤
1
2r+p−1
Z
Ω
|
d
dt
|u|
2r+p−1
|dx,
≤
1
p+1
Z
Ω
|
d
dt
|u|
p+1
|dx,
=
Z
Ω
|u
t
||u|
p
dx,
≤c
Z
Ω
|u|
p
dx,
≤c
Z
Ω
|u|
2r+P−1
dx.
(3.35)
d^‡u∈C
2,1
(
¯
Q
T
)±9Young
0
sØª§·‚
Z
Ω
p|∇|
2
|u|
2r+p−3
dx≤c
Z
Ω
p|u|
2r+p−3
dx,
≤c
Z
Ω
(p|u|
2r+p−3)
2r+p−1
2r+p−3
dx+c,
≤c
Z
Ω
p
2
|u|
2r+p−1
dx.
(3.36)
(Ü•§(3.19),(3.35),(3.36)§·‚
d
dt
Z
Ω
|u|
2r+p−1
dx+
4cp
(2r+p−1)
Z
Ω
|∇|u|
(2r+p−1)
2
|
2
≤c(2r+p)
3
Z
Ω
|u|
2r+p−1
dx.(3.37)
DOI:10.12677/aam.2022.111060534A^êÆ?Ð
ÅS†§Æ“
-E•§(3.23)−(3.33)Ú½§|u(x,t)|≤1ž§•k
max
Q
T
|u(x,t)|≤
¯
C(u
0
).
(ii)ν∈(0,1)ž§Úν= 0ž?n•{˜§•k
max
Q
T
|u(x,t)|≤
¯
C(u
0
).
nþ¤ã§y"
3.2.½n2.4y²
y²µ·‚3•§(1.4)¥ü>Óž¦±u|u|
p−1
,(p≥3)§¿…3ΩþÈ©§·‚
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)±9L
p+1
(Ω) ⊂L
p
(Ω)§·‚
−ε
Z
Ω
f(u)u|u|
p−1
dx≤−
Z
Ω
{r(2r−1)b
2r
u
2r−1
u|u|
p−1
−cu|u|
p−1
}dx,
=
Z
Ω
cu|u|
p−1
dx−c
Z
Ω
u
2r
|u|
p−1
dx,
≤
Z
Ω
c|u|
p
dx,
≤c(p+1)
Z
Ω
|u|
p+1
dx.
(3.39)
5e§-E•§(3.15) −(3.33)Ú½§=Œ±½n(اØ2‰Ñ•[y²"
y"
ë•©z
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