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PureMathematics
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,2021,11(5),739-751
PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.115089
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EnergyConservationfortheWeak
SolutionstotheThree-Dimensional
IncompressibleMagnetohydrodynamic
EquationsofViscousNon-Resistive
FluidsinaBoundedDomain
XiongWang
SchoolofMathematics,SouthChinaUniversityofTechnology,GuangzhouGuangdong
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[J].
n
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,2021,11(5):739-751.
DOI:10.12677/pm.2021.115089
<
Received:Apr.7
th
,2021;accepted:May11
th
,2021;published:May18
th
,2021
Abstract
Inthispaper,wemainlystudytheenergyconservationfortheweaksolutionsto
thethree-dimensionalincompressiblemagnetohydrodynamic equationsofviscousnon-
resistivefluidsinaboundeddomain.Togetenergyconservation,wefirstusetheglobal
mollificationmethodtotheequation,nexttakecut-offfunction,thengetthelimitof
δ,ε,τ
.Weproposeaconditionfor
(
u
,
b
,P
)
:u
∈
L
p
t
L
q
x
,b
∈
L
4
t
L
4
x
,
∇
b
∈
L
2
t
L
2
x
andP
∈
L
2
t
L
2
x
.
Keywords
IncompressibleMagnetohydrodynamicEquations,ViscousNon-ResistiveFluids,
BoundedDomain,WeakSolutions,EnergyConservation
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2021.115089743
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<
∞
ž
,
r
=
r
¶
r
2
=
∞
ž
,
r<r
.
Ú
n
2.3.
(
©
Ù
[1]
¥
í
Ø
2.1 )
X
J
f
∈
W
1
,r
1
(Ω
×
[0
,T
]),
g
∈
L
r
2
(Ω
×
[0
,T
])
Ù
¥
1
≤
r,r
1
,r
2
≤
∞
,
1
r
1
+
1
r
2
=
1
r
.
K
k
∂
t
((
˜
f
˜
g
)
ε
i
−
f
˜
g
ε
i
)
k
L
r
loc
(0
,T
;
L
r
(
V
i
))
≤
C
k
g
k
L
r
2
(Ω
×
[0
,T
])
k
∂
t
f
k
L
r
1
(Ω
×
[0
,T
])
+
k∇
f
k
L
r
1
(Ω
×
[0
,T
])
…
k
∂
x
((
˜
f
˜
g
)
ε
i
−
f
˜
g
ε
i
)
k
L
r
loc
(0
,T
;
L
r
(
V
i
))
≤
C
k
g
k
L
r
2
(Ω
×
[0
,T
])
k
∂
t
f
k
L
r
1
(Ω
×
[0
,T
])
+
k∇
f
k
L
r
1
(Ω
×
[0
,T
])
,
d?
˜
g
ε
i
½
Â
´
d
(2.3)
‰
Ñ
.
•
?
˜
Ú
,
ε
→
0
ž
,
k
3
˜
m
L
r
loc
(0
,T
;
L
r
(
V
i
))
¥
,∂
((
˜
f
˜
g
)
ε
i
−
f
˜
g
ε
i
)
→
0
,
d?
,
r
2
<
∞
ž
,
r
=
r
¶
r
2
=
∞
ž
,
r<r
.
Ú
n
2.4.
(Aubin-Lions
Ú
n
,
©
Ù
[4])
X
J
X
´
g
‡
Banach
˜
m
,Y
´
Banach
˜
m
,
X
→
Y
,
Y
0
Œ
©
…
3
X
0
¥
È
—
.
b
¼
ê
S
{
f
n
}
÷
v
f
n
∈
L
∞
(0
,T
;
X
)
,∂
t
f
n
∈
L
p
(0
,T
;
Y
)
,
1
<p
≤∞
,
k
f
n
k
L
∞
(0
,T
;
X
)
,
k
∂
t
f
n
k
L
p
(0
,T
;
Y
)
≤
C,
∀
n
≥
1
.
K
f
n
3
C
0
([0
,T
]
,X
ω
)
¥
ƒ
é
;
.
3.
½
n
1.1
y
²
•
y
²
½
n
1.1.
U
þ
Å
ð
(
Ø
,
e
¡
·
‚
æ
^
©
Ù
[1]
¥
•{
ò
•
§
(1.1)
2
1
,
∂
t
ψ
0
u
ε
+
k
X
i
=1
ψ
i
˜
u
ε
i
+
ψ
0
div(
u
⊗
u
)
ε
+
k
X
i
=1
ψ
i
div(
˜
u
⊗
˜
u
)
ε
i
+
ψ
0
∇
P
ε
+
k
X
i
=1
ψ
i
∇
˜
P
ε
i
−
ψ
0
div(
b
⊗
b
)
ε
+
k
X
i
=1
ψ
i
div(
˜
b
⊗
˜
b
)
ε
i
+
ψ
0
∇
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
−
µ
ψ
0
∆
u
ε
+
k
X
i
=1
ψ
i
∆
˜
u
ε
i
= 0
.
(3.1)
DOI:10.12677/pm.2021.115089744
n
Ø
ê
Æ
<
ä
¼
ê
ξ
τ
η
δ
[
u
]
ε
,
Ù
¥
ξ
τ
(
t
)
∈C
1
0
(
τ,T
−
τ
)
,
η
δ
(
x
)
∈C
1
0
(Ω)
…
0
≤
η
δ
(
x
)
≤
1
,η
δ
(
x
) = 1
x
∈
Ω
…
dist
(
x,∂
Ω)
≥
δ
ž
;
η
δ
→
1
δ
→
0
ž
,
…
|∇
η
δ
|≤
2
dist
(
x,∂
Ω)
.
ò
•
§
(3.1)
ü
>
¦
þ
ξ
τ
η
δ
[
u
]
ε
¿
3
Ω
×
(0
,T
)
þ
È
©
,
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∂
t
ψ
0
u
ε
+
k
X
i
=1
ψ
i
˜
u
ε
i
d
x
d
t
+
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
div(
u
⊗
u
)
ε
+
k
X
i
=1
ψ
i
div(
˜
u
⊗
˜
u
)
ε
i
d
x
d
t
+
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
∇
P
ε
+
k
X
i
=1
ψ
i
∇
˜
P
ε
i
dxdt
(3.2)
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
div(
b
⊗
b
)
ε
+
k
X
i
=1
ψ
i
div(
˜
b
⊗
˜
b
)
ε
i
d
x
d
t
+
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
∇
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
−
µ
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
∆
u
ε
+
k
X
i
=1
ψ
i
∆
˜
u
ε
i
d
x
d
t
= 0
.
e
5
é
(3.2)
¥
ˆ
‘
'
u
δ
Ú
ε
4
•
.
Ù
¥
1
‘
,
1
n
‘
,
1
8
‘
©
Oë
•
©
Ù
[1]
¥
Ú
n
A.1,
Ú
n
A.2,
Ú
n
3.3
Ú
Ù
!
3.2,
lim
δ
→
0
lim
ε
→
0
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
div(
u
⊗
u
)
ε
+
k
X
i
=1
ψ
i
div(
˜
u
⊗
˜
u
)
ε
i
d
x
d
t
= 0;
(3.3)
lim
δ
→
0
lim
ε
→
0
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
∇
P
ε
+
k
X
i
=1
ψ
i
∇
˜
P
ε
i
d
x
d
t
= 0;
(3.4)
lim
δ
→
0
lim
ε
→
0
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
∆
u
ε
+
k
X
i
=1
ψ
i
∆
˜
u
ε
i
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
|∇
u
|
2
d
x
d
t.
(3.5)
(3.2)
¥
1
˜
‘
,
1
o
‘
,
1
Ê
‘
'
u
δ
Ú
ε
4
•
(
J
d
e
ˆ
Ú
n
‰
Ñ
.
Ú
n
3.1.
(3.2)
¥
1
˜
‘
÷
v
lim
δ
→
0
lim
ε
→
0
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∂
t
ψ
0
u
ε
+
k
X
i
=1
ψ
i
˜
u
ε
i
d
x
d
t
=
−
1
2
Z
T
0
Z
Ω
ξ
0
τ
|
u
|
2
d
x
d
t.
DOI:10.12677/pm.2021.115089745
n
Ø
ê
Æ
<
y
²
d
½
Â
ª
(2.5)
•
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∂
t
ψ
0
u
ε
+
k
X
i
=1
ψ
i
˜
u
ε
i
d
x
d
t
=
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∂
t
[
u
]
ε
d
x
d
t
=
1
2
Z
T
0
Z
Ω
ξ
τ
η
δ
∂
t
|
[
u
]
ε
|
2
d
x
d
t
=
−
1
2
Z
T
0
Z
Ω
ξ
0
τ
η
δ
|
[
u
]
ε
|
2
d
x
d
t.
é
þ
ª
‘
,
¿
-
ε
→
0
,δ
→
0,
Š
â
ξ
0
τ
,η
δ
k
.
5
9
·
K
2.3
lim
δ
→
0
lim
ε
→
0
1
2
Z
T
0
Z
Ω
ξ
0
τ
η
δ
|
[
u
]
ε
|
2
−|
u
|
2
d
x
d
t
= 0
.
l
,
lim
δ
→
0
lim
ε
→
0
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∂
t
ψ
0
u
ε
+
k
X
i
=1
ψ
i
˜
u
ε
i
d
x
d
t
=
−
1
2
Z
T
0
Z
Ω
ξ
0
τ
|
u
|
2
d
x
d
t.
Ú
n
3.1.
y
.
.
Ú
n
3.2.
(3.2)
¥
1
o
‘
÷
v
lim
δ
→
0
lim
ε
→
0
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
div(
b
⊗
b
)
ε
+
k
X
i
=1
ψ
i
div(
˜
b
⊗
˜
b
)
ε
i
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
b
·
div(
b
⊗
u
)d
x
d
t.
y
²
k
©
Ü
È
©
,
k
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
div(
b
⊗
b
)
ε
+
k
X
i
=1
ψ
i
div(
˜
b
⊗
˜
b
)
ε
i
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
⊗
[
u
]
ε
:
ψ
0
(
b
⊗
b
)
ε
+
k
X
i
=1
ψ
i
(
˜
b
⊗
˜
b
)
ε
i
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
∇
[
u
]
ε
:
ψ
0
(
b
⊗
b
)
ε
+
k
X
i
=1
ψ
i
(
˜
b
⊗
˜
b
)
ε
i
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
∇
ψ
0
: (
b
⊗
b
)
ε
+
k
X
i
=1
∇
ψ
i
: (
˜
b
⊗
˜
b
)
ε
i
d
x
d
t
:=
I
41
+
I
42
+
I
43
.
DOI:10.12677/pm.2021.115089746
n
Ø
ê
Æ
<
k
w
I
41
I
41
=
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
⊗
[
u
]
ε
:
ψ
0
(
b
⊗
b
)
ε
−
(
b
⊗
b
)
+
k
X
i
=1
ψ
i
(
˜
b
⊗
˜
b
)
ε
i
−
(
b
⊗
b
)
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
⊗
([
u
]
ε
−
u
) : (
b
⊗
b
)d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
⊗
u
: (
b
⊗
b
)d
x
d
t.
^
H¨older
Ø
ª
9
(1.4),(1.9),(1.10),(2.4),
Ú
n
2.1
9
·
K
2.1,
·
K
2.2,
·
K
2.3,
lim
δ
→
0
lim
ε
→
0
I
41
= 0
.
(3.6)
2
w
I
42
I
42
=
−
Z
T
0
Z
Ω
ξ
τ
η
δ
∇
[
u
]
ε
:
ψ
0
(
b
⊗
b
)
ε
−
(
b
⊗
b
)
+
k
X
i
=1
ψ
i
(
˜
b
⊗
˜
b
)
ε
i
−
(
b
⊗
b
)
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
(
∇
[
u
]
ε
−∇
u
) : (
b
⊗
b
)d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
∇
u
: (
b
⊗
b
)d
x
d
t.
d
H¨older
Ø
ª
9
(1.4),(1.10),(2.4),
·
K
2.1,
·
K
2.2
Ú
·
K
2.3,
lim
δ
→
0
lim
ε
→
0
I
42
=
−
Z
T
0
Z
Ω
ξ
τ
∇
u
: (
b
⊗
b
)d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
b
·
div(
b
⊗
u
)d
x
d
t.
(3.7)
•
w
I
43
I
43
=
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
∇
ψ
0
:
(
b
⊗
b
)
ε
−
b
⊗
b
+
k
X
i
=1
∇
ψ
i
:
(
˜
b
⊗
˜
b
)
ε
i
−
b
⊗
b
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
k
X
i
=0
∇
ψ
i
: (
b
⊗
b
)d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
∇
ψ
0
:
(
b
⊗
b
)
ε
−
b
⊗
b
+
k
X
i
=1
∇
ψ
i
:
(
˜
b
⊗
˜
b
)
ε
i
−
b
⊗
b
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
∇
k
X
i
=0
ψ
i
: (
b
⊗
b
)d
x
d
t
d
H¨older
Ø
ª
9
(1.4),(1.10),(2.4),
·
K
2.1,
·
K
2.2
Ú
·
K
2.3,
lim
δ
→
0
lim
ε
→
0
I
43
= 0
.
(3.8)
DOI:10.12677/pm.2021.115089747
n
Ø
ê
Æ
<
Š
â
(3.6),(3.7),(3.8)
k
lim
δ
→
0
lim
ε
→
0
(
I
41
+
I
42
+
I
43
) =
−
Z
T
0
Z
Ω
ξ
τ
b
·
div(
b
⊗
u
)d
x
d
t.
Ú
n
3.2.
y
.
.
Ú
n
3.3.
(3.2)
¥
1
Ê
‘
÷
v
lim
δ
→
0
lim
ε
→
0
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
∇
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
= 0
.
y
²
k
©
Ü
È
©
,
k
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
ψ
0
∇
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
·
[
u
]
ε
ψ
0
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
div[
u
]
ε
ψ
0
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∇
ψ
0
∇
(
|
b
|
2
2
)
ε
+
k
X
i
=1
∇
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
:=
I
51
+
I
52
+
I
53
.
k
w
I
51
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
·
[
u
]
ε
ψ
0
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
·
[
u
]
ε
ψ
0
(
|
b
|
2
2
)
ε
−
|
b
|
2
2
+
k
X
i
=1
ψ
i
(
|
˜
b
|
2
2
)
ε
i
−
|
b
|
2
2
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
·
([
u
]
ε
−
u
)
ψ
0
|
b
|
2
2
+
k
X
i
=1
ψ
i
|
b
|
2
2
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
∇
η
δ
·
u
|
b
|
2
2
d
x
d
t.
^
H¨older
Ø
ª
9
(1.4),(1.9),(1.10),(2.4),
Ú
n
2.1
9
·
K
2.1,
·
K
2.2,
·
K
2.3,
lim
δ
→
0
lim
ε
→
0
I
51
= 0
.
(3.9)
DOI:10.12677/pm.2021.115089748
n
Ø
ê
Æ
<
2
w
I
52
,
−
Z
T
0
Z
Ω
ξ
τ
η
δ
div[
u
]
ε
ψ
0
(
|
b
|
2
2
)
ε
+
k
X
i
=1
ψ
i
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
η
δ
div[
u
]
ε
ψ
0
(
|
b
|
2
2
)
ε
−
|
b
|
2
2
+
k
X
i
=1
ψ
i
(
|
˜
b
|
2
2
)
ε
i
−
|
b
|
2
2
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
(div[
u
]
ε
−
div
u
)(
|
b
|
2
2
)d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
div
u
(
|
b
|
2
2
)d
x
d
t.
d
H¨older
Ø
ª
9
(1.4),(1.10),(2.4),
·
K
2.1,
·
K
2.2
Ú
·
K
2.3,
lim
δ
→
0
lim
ε
→
0
I
52
=
−
Z
T
0
Z
Ω
ξ
τ
η
δ
div
u
(
|
b
|
2
2
)d
x
d
t
= 0
.
(3.10)
•
w
I
53
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∇
ψ
0
∇
(
|
b
|
2
2
)
ε
+
k
X
i
=1
∇
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∇
ψ
0
∇
(
|
b
|
2
2
)
ε
−
|
b
|
2
2
+
k
X
i
=1
∇
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
−
|
b
|
2
2
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
k
X
i
=0
∇
ψ
i
|
b
|
2
2
d
x
d
t
=
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·
∇
ψ
0
∇
(
|
b
|
2
2
)
ε
−
|
b
|
2
2
+
k
X
i
=1
∇
ψ
i
∇
(
|
˜
b
|
2
2
)
ε
i
−
|
b
|
2
2
d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
η
δ
[
u
]
ε
·∇
(
k
X
i
=0
ψ
i
)
|
b
|
2
2
d
x
d
t.
d
H¨older
Ø
ª
9
(1.4),(1.10),(2.4),
·
K
2.1,
·
K
2.2
Ú
·
K
2.3,
lim
δ
→
0
lim
ε
→
0
I
53
= 0
.
(3.11)
Š
â
(3.9),(3.10),(3.11)
k
lim
δ
→
0
lim
ε
→
0
(
I
51
+
I
52
+
I
53
) = 0
.
Ú
n
3.3.
y
.
.
d
(3.3),(3.4),(3.5),
Ú
n
3.1,
Ú
n
3.2
9
Ú
n
3.3
•
,
-
•
§
(3.2)
¥
ε
→
0
,δ
→
0
−
1
2
Z
T
0
Z
Ω
ξ
0
τ
|
u
|
2
d
x
d
t
+
Z
T
0
Z
Ω
ξ
τ
b
·
div(
b
⊗
u
)d
x
d
t
+
µ
Z
T
0
Z
Ω
ξ
τ
|∇
u
|
2
d
x
d
t
= 0
.
(3.12)
DOI:10.12677/pm.2021.115089749
n
Ø
ê
Æ
<
(
Ü
•
§
(1.1)
3
,
Ù
¥
Z
T
0
Z
Ω
ξ
τ
b
·
div(
b
⊗
u
)d
x
d
t
=
Z
T
0
Z
Ω
ξ
τ
b
·
div(
b
⊗
u
)d
x
d
t
−
Z
T
0
Z
Ω
ξ
τ
b
·
div(
u
⊗
b
)d
x
d
t
=
Z
T
0
Z
Ω
ξ
τ
b
·
div(
b
⊗
u
)
−
div(
u
⊗
b
)
d
x
d
t
=
Z
T
0
Z
Ω
ξ
τ
b
·
∂
t
b
d
x
d
t
=
−
1
2
Z
T
0
Z
Ω
ξ
0
τ
|
b
|
2
d
x
d
t,
l
,(3.12)
z
{
¤
−
1
2
Z
T
0
Z
Ω
ξ
0
τ
(
|
u
|
2
+
|
b
|
2
)d
x
d
t
+
µ
Z
T
0
Z
Ω
ξ
τ
|∇
u
|
2
d
x
d
t
= 0
.
(3.13)
du
ξ
τ
(
t
)
∈C
1
0
((
τ,T
−
τ
)),
é
u
?
¿
t
0
>
0 ,
?
¿
τ,α
,
¦
τ
+
α<t
0
,
ξ
τ
(
t
) =
0
,
0
≤
t
≤
τ,
t
−
τ
α
,τ<t
≤
τ
+
α,
1
,τ
+
α<t
≤
t
0
,
t
0
+
α
−
t
α
,t
0
<t
≤
t
0
+
α,
0
,t
≥
t
0
+
α.
(3.14)
ò
(3.14)
“
\
(3.13),
−
1
2
α
Z
τ
+
α
τ
Z
Ω
(
|
b
|
2
+
|
u
|
2
)d
x
d
t
+
1
2
α
Z
t
0
+
α
t
0
Z
Ω
(
|
b
|
2
+
|
u
|
2
)d
x
d
t
+
µ
Z
τ
+
α
τ
t
−
τ
α
Z
Ω
|∇
u
|
2
d
x
d
t
+
µ
Z
t
0
τ
+
α
Z
Ω
|∇
u
|
2
d
x
d
t
+
µ
Z
t
0
+
α
t
0
−
t
+
t
0
+
α
α
Z
Ω
|∇
u
|
2
d
x
d
t
= 0
.
(3.15)
d
È
©
ý
é
ë
Y5
,
lim
α
→
0
Z
τ
+
α
τ
t
−
τ
α
Z
Ω
|∇
u
|
2
d
x
d
t
= 0;
lim
α
→
0
Z
t
0
+
α
t
0
−
t
+
t
0
+
α
α
Z
Ω
|∇
u
|
2
d
x
d
t
= 0
.
DOI:10.12677/pm.2021.115089750
n
Ø
ê
Æ
<
P
E
(
t
) =
1
2
Z
Ω
(
|
b
|
2
+
|
u
|
2
)d
x,
F
(
t
) =
µ
Z
t
0
Z
Ω
|∇
u
|
2
d
x
d
s.
ä
ó
:
u
,
b
∈C
0
[0
,T
];
L
2
(Ω)
.
(3.16)
¯¢
þ
,
d
(1.1)
2
,(1.4),(1.8),(1.9),(1.10),
•
u
t
∈
L
2
0
,T
;
H
−
1
(Ω)
.
3
Ú
n
2.4.
¥
,
X
=
L
2
(Ω)
,Y
=
H
−
1
(Ω),
l
u
∈C
0
[0
,T
];
L
2
w
(Ω)
.
Ó
n
,
d
(1.1)
3
,(1.4),(1.9),(1.10)
Ú
Ú
n
2.4.,
b
∈C
0
[0
,T
];
L
2
w
(Ω)
.
(
Ü
U
þ
Ø
ª
(1.6)
Ú
Ð
Š
^
‡
,
u
,
b
∈C
0
[0
,T
];
L
2
(Ω)
.
3
(3.15)
¥
,
du
(3.16)
Ú
V
‚
‡
©
½
n
,
-
α
→
0,
(
E
+
F
)(
t
0
) = (
E
+
F
)(
τ
)
.
(3.17)
3
(3.17)
¥
,
du
(3.16),
-
τ
→
0
+
,
(
E
+
F
)(
t
0
) = (
E
+
F
)(0)
,t
0
∈
(0
,T
)
,
d
=
(1.6).
d
t
0
?
¿
5
,
®
¤
½
n
1.1.
y
²
.
ë
•
©
z
[1]Chen,R.M.,Liang,Z.L.,Wang,D.H.andXu,R.Z.(2020)EnergyEqualityinCompressibleFluidswith
PhysicalBoundaries.
SIAMJournalonMathematicalAnalysis
,
52
,1363-1385.
https://doi.org/10.1137/19M1287213
[2]Kufner,A.,John,O.andFuk,S.(1977)FunctionSpaces.Academia,Prague.
[3]Evans,L.C.(1998)PartialDifferentialEquations.AmericanMathematicalSociety,Providence,RI.
[4]Lions, P.L.(1996)Mathematical Topicsin FluidMechanics.Vol.1,Incompressible Models.OxfordUniver-
sityPress,NewYork.
DOI:10.12677/pm.2021.115089751
n
Ø
ê
Æ