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AdvancesinAppliedMathematics
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,2023,12(5),2049-2066
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.125209
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TheWell-PosednessofaDelayed
Non-AutonomousMicropolarFluid
on
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BoundedDomains
QilingWang
SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing
Received:Apr.7
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,2023;accepted:Apr.29
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,2023
Abstract
Inthispaper,westudythewell-posenessofanon-autonomousdelayedmicropolar
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,2023,12(5):2049-2066.
DOI:10.12677/aam.2023.125209
é
fluidon
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boundeddomains.Weprovetheexistenceofsolutionsbythemethodof
theGarlerinapproximation.Then we usetheenergymethodtoprovetheuniqueness
andthestabilityofsolutions.
Keywords
MicropolarFluid,Delay,Well-Posedness
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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3
L
2
(Ω)
×
L
2
(Ω)
4
•
§
…
‰
ê
•
k·k
H
=
k·k
…
é
ó
˜
m
•
H
∗
=
H,
DOI:10.12677/aam.2023.1252092052
A^
ê
Æ
?
Ð
é
V
=
V
3
H
1
(Ω)
×
H
1
(Ω)
4
•
§
…
‰
ê
•
k·k
V
=
k·k
2
,
2
…
é
ó
˜
m
•
V
∗
,
b
H
=
H
×
L
2
(Ω)
‰
ê
•
k·k
b
H
…
é
ó
˜
m
•
b
H
∗
,
b
V
=
V
×
H
1
0
(Ω)
‰
ê
•
k·k
b
V
…
é
ó
˜
m
•
b
V
∗
,
Ù
¥
k·k
b
H
Ú
k·k
b
V
/
ª
•
k
(
u,v
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k
b
H
= (
k
u
k
2
H
+
k
v
k
2
)
1
2
,
k
(
u,v
)
k
b
V
= (
k
u
k
2
V
+
k
v
k
2
H
1
)
1
2
.
3
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¥
§
X
J
v
k
·
u
)
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·
‚
ò
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H
Ú
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b
H
{
z
•
^
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Ò
k·k
.
(
·
,
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L
2
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½
b
H
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§
h·
,
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V
Ú
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b
V
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b
V
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ó
È
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,
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n
‡
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Ž
f
"
1
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f
A
½
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é
?
¿
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=(
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b
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1
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2
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φ
1
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2
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á
h
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i
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r
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∇
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φ
3
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+
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r
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i
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i
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j
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i
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2
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i
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3
∂x
i
d
x,
D
(
A
)=
b
V
∩
(
H
2
(Ω))
3
.
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g
§
½
Â
Ž
f
B
(
·
,
·
)
•
µ
é
?
¿
u
=(
u
1
,u
2
)
∈
V,ψ
=(
ψ
1
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2
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3
)
∈
b
V,φ
= (
φ
1
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2
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3
)
∈
b
V,
¤
á
h
B
(
u,w
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i
= ((
u
·∇
)
w,φ
) =
3
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j
=1
2
X
i
=1
Z
Ω
u
i
∂ψ
j
∂x
i
φ
j
d
x,
•
§
½
Â
Ž
f
N
(
·
)
•
N
(
w
) = (
−
2
ν
r
∇×
w,
−
2
ν
r
∇×
u,
+4
ν
r
ω
)
,
∀
w
= (
u,ω
)
∈
b
V.
•
§
·
‚
‰
Ñ
Ž
f
A
,
B
(
·
,
·
)
Ú
N
(
·
)
˜
5
Ÿ
§
ë
„
©
z
[3,11].
Ú
n
2.1.
(
ë
„
©
z
[3,11])
(1)
•
3
ü
‡
~
ê
c
1
Ú
c
2
¦
c
1
h
Aw,w
i
6
k
w
k
2
b
V
6
c
2
h
Aw,w
i
,
∀
w
∈
b
V.
(12)
d
§
é
u
?
¿
w
∈
D
(
A
)
k
min
ν
+
ν
r
,α
k∇
w
k
2
≤h
Aw,w
i
6
k
w
kk
Aw
k
6
λ
−
1
2
1
k∇
w
kk
Aw
k
(13)
(2)
•
3
˜
‡
~
ê
λ,
§
•
•
6
u
Ω
,
ù
é
u
?
¿
(
u,ψ,ϕ
)
∈
V
×
b
V
×
b
V
¦
DOI:10.12677/aam.2023.1252092053
A^
ê
Æ
?
Ð
é
|h
B
(
u,ψ
)
,ϕ
i|
6
λ
k
u
k
1
/
2
k∇
u
k
1
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2
k∇
ψ
kk
ϕ
k
1
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2
k∇
ϕ
k
1
/
2
,
λ
k
u
k
1
/
2
k∇
u
k
1
/
2
k∇
ϕ
kk
ψ
k
1
/
2
k∇
ψ
k
1
/
2
.
(14)
d
§
X
J
(
u,ψ,ϕ
)
∈
V
×
D
(
A
)
×
D
(
A
)
,
K
|h
B
(
u,ψ
)
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i|
6
λ
k
u
k
1
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2
k∇
u
k
1
/
2
k∇
ϕ
kk∇
ψ
k
1
/
2
k
Aψ
k
1
/
2
.
(15)
(3)
ù
p
•
3
˜
‡
~
ê
c
¦
N
(
ψ
)
6
c
k
ψ
k
b
V
,
∀
ψ
∈
b
V.
(16)
d
§
−h
N
(
ψ
)
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i
6
1
4
k
Aψ
k
2
+
c
2
(
ν
r
)
k
ψ
k
2
b
V
,
∀
ψ
∈
D
(
A
)
,
(17)
δ
1
k
ψ
k
2
b
V
6
h
Aψ,ψ
i
+
h
Nψ,ψ
i
,
∀
ψ
∈
D
(
A
)
.
(18)
ù
p
δ
1
=
min
{
ν,α
}
.
3
Ú
n
2.1
Ä
:
þ
§
·
‚
?
˜
Ú
k
±
e
Ú
n
"
Ú
n
2.2.
(
ë
„
©
z
[22])
(1)
A
´
˜
‡
l
V
V
∗
Ú
D
(
A
)
b
H
‚
5
ë
Y
Ž
f
,
N
(
·
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´
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‡
l
b
V
b
H
‚
5
ë
Y
Ž
f
.
(2)
B
(
·
,
·
)
´
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‡
l
V
×
b
V
b
V
∗
‚
5
ë
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Ž
f
§
d
§
é
∀
u
∈
V,
∀
w
∈
b
V
k
h
B
(
u,ψ
)
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i
=
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B
(
u,ϕ
)
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i
,
∀
u
∈
V,
∀
ψ
∈
b
V,
∀
ϕ
∈
b
V.
(19)
3.
N
·
½
5
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(4)-(9)
f
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k
§
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1
2
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0
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Ž
f
§
·
‚
(4)-(9)
f
/
ª
(
ë
„
©
z
[22])
∂w
∂t
+
Aw
+
B
(
u,w
)+
N
(
w
) =
F
(
t
)+
G
(
t,W
t
)
, t>
0
,
(20)
w
t
=0
=
w
0
=
w
(
s
) = (
u
(
s
)
,ω
(
s
)) =
φ
(
s
)
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∈
(
−∞
,
0]
,
(21)
Ù
¥
§
w
= (
u,ω
)
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(
t
) =
F
(
t
+
x
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f
(
t
+
x
)
,
e
f
(
t
+
x
))
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(
t,W
t
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g
(
t,u
t
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,
e
g
(
t,ω
t
))
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0
,
ž
¢
¼
ê
u
t
Ú
ω
t
d
ª
(11)
½
Â
"
•
?
n
Ã
•
ž
¢
§
·
‚
Ú
\
˜
m
BCL
−∞
(
b
H
),
½
Â
X
e
µ
DOI:10.12677/aam.2023.1252092054
A^
ê
Æ
?
Ð
é
BCL
−∞
(
b
H
) =
{
ϕ
∈
C
((
−∞
,
0];
b
H
) :lim
s
→−∞
ϕ
(
s
)
3
b
H
¥
•
3
}
BCL
−∞
(
b
H
)
´
˜
‡
Banach
˜
m
§
…
‰
ê
•
µ
k
ϕ
k
BCL
−∞
(
b
H
)
:=sup
s
∈
(
−∞
,
0]
k
ϕ
(
s
)
k
.
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Š
¯
K
(20)-(21)
f
)
½
Â
X
e
"
½
Â
3.1.
é
∀
T>
0
,
X
J¼
ê
w
∈
C
((
−∞
,T
];
b
H
)
∩
L
2
(0
,T
;
b
V
)
…
w
0
=
φ
(
s
)
∈
BCL
−∞
(
b
H
)
Ú
u
(
t,x
) =
φ,t
∈
(
τ
−
h,τ
)
,
¦
é
∀
t
∈
(0
.T
)
,
∀
ϕ
∈
b
V
•
§
d
dt
(
w,ϕ
)+
h
Aw,ϕ
i
+
h
B
(
u,w
)
,ϕ
i
+
h
N
(
w
)
,ϕ
i
=
h
F
(
t
)
,ϕ
i
+(
G
(
t,w
t
)
,ϕ
)
3
©
Ù
D
0
(0
,T
)
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e
¤
á
§
K
w
´
•
§
(20)
-
(21)
3
«
m
(
−∞
,T
]
þ
˜
‡
f
)
.
•
¼
Ð
Š
¯
K
(20)-(21)
)
N
·
½
5
§
·
‚
I
‡
é
¼
ê
F
Ú
G
J
Ñ
˜
b
"
(I)
b
é
?
¿
T>
0
,
F
(
·
,x
) = (
f
(
·
,x
)
,
e
f
(
·
,x
))
∈
L
2
(0
,T
;
b
V
∗
)
.
(II)
b
G
: [0
,T
]
×
BCL
−∞
(
b
H
)
7→
G
(
t,ξ
)
∈
(
L
2
(Ω))
3
÷
v
µ
(i)
é
∀
ξ
∈
BCL
−∞
(
b
H
)
,
N
[0
,T
]
3
t
7→
G
(
t,ξ
)
∈
(
L
2
(Ω))
3
´
Œ
ÿ
¶
(ii)
G
(
·
,
0) = (0
,
0
,
0);
(iii)
•
3
˜
‡
~
ê
L
G
>
0
¦
é
u
?
¿
t
∈
[0
,T
]
,ξ,η
∈
BCL
−∞
(
b
H
)
,
k
G
(
t,ξ
)
−
G
(
t,η
)
k
6
L
G
k
ξ
−
η
k
BCL
−∞
(
b
H
)
.
^
‡
(ii)
Ú
(iii)
L
²
k
G
(
t,ξ
)
k
6
L
G
k
ξ
k
BCL
−∞
(
b
H
)
,
∀
ξ
∈
BCL
−∞
(
b
H
)
.
(22)
e
5
y
²
•
§
(20)-(21)
f
)
•
3
5
!
•
˜
5
!
-
½
5
"
½
n
3.1.
(
f
)
•
3
5
)
é
?
¿
T>
0
,
b
φ
∈
BCL
−∞
(
b
H
)
´
‰
½
§
¿
…
b
(
I
)
Ú
(
II
)
¤
á
§
K
¯
K
(20)
-
(21)
3
«
m
(
−∞
,T
]
¥–
•
3
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‡
f
)
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y
·
‚
ò
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n
‡
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½
5
y
²
½
n
3.1.
Ú
½
1.
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q
)
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Ü
•
3
5
·
‚
Ä
k
£
½
Â
Ž
f
A
˜
5
Ÿ
"
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þ
§
Š
â
ý
Ž
f
²
;
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n
Ø
(
„
©
z
[24])
§
•
3
˜
‡
A
Š
S
{
λ
n
}
∞
n
=1
÷
v
0
<λ
1
6
λ
2
6
···
6
λ
n
6
···
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n
→
+
∞
n
→∞
,
Ú
Ù
é
A
A
•
þ
x
{
v
n
}
∞
n
=1
⊆
D
(
A
),
§
´
b
H
˜
m
F
Ë
A
Ä
§
…
d
{
v
1
,v
2
,
···
,v
n
,
···}
Ü
DOI:10.12677/aam.2023.1252092055
A^
ê
Æ
?
Ð
é
¤
§
3
b
V
˜
m
¥
È
—
§
¦
Av
n
=
λ
n
v
n
,
∀
n
∈
N
.
b
8
Ü
V
m
:= span
{
v
1
,
···
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m
}
,
¿
•
Ä
N
P
m
w
:=
m
P
j
=1
(
w,v
j
)
v
j
,
w
∈
b
H
½
w
∈
b
V
§
é
u
z
‡
T>
0
,
½
Â
¯
K
(20)-(21)
Galerkin
C
q
)
•
w
m
(
t
) :=
m
X
j
=1
γ
m,j
(
t
)
v
j
,
ù
p
X
ê
γ
m,j
(
t
)
÷
v
e
~
‡
©
…
ܯ
K
d
d
t
w
m
(
t
)
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j
+
h
Aw
m
(
t
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j
i
+
h
B
(
u
m
(
t
))
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m
(
t
))
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j
i
+
h
N
(
w
m
(
t
))
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j
i
=
h
F
(
t
)
,v
j
i
+(
G
(
t,w
m
t
)
,v
j
)
,
1
6
j
6
m,t
∈
(0
,T
)
,
(23)
w
m
(
s
) =
P
m
φ
(
s
)
, s
∈
(
−∞
,
0]
,
(24)
3
D
0
(0
,T
)
¥
•
Ä
•
§
(23).
þ
ã
Ã
¡
ž
¢
~
•
¼
‡
©•
§
|
÷
v
Û
Ü
)
•
3
•
˜
5
^
‡
(
ë
•
©
z
[25])
"
Ï
d
§
·
‚
Ñ
(23)-(24)
k
˜
‡
½
Â
3
[0
,t
m
)
…
0
≤
t
m
≤
T
•
˜
Û
Ü
)
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e
5
§
·
‚
ò
¼
˜
‡
k
O
§
¿
(
)
w
m
(
¢
•
3
u
‡
«
m
[0
,T
].
Ú
½
2.
k
O
ò
(23)
z
‡
•
§¦
±
γ
m,j
(
t
)
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= 1
,
···
,m
¿
¦
Ú
§
|
^
(18)
Ú
h
B
(
u,ψ
)
,ψ
i
= 0(
„
©
z
[22]
)
k
1
2
d
d
t
k
w
m
(
t
)
k
2
+
δ
1
k
w
m
(
t
)
k
2
b
V
6
h
F
(
t
)
,w
m
(
t
)
i
+(
G
(
t,w
m
)
,w
m
(
t
))
, t
∈
(0
,T
)
.
(25)
q
Ï
•
k
w
m
t
k
BCL
−∞
(
b
H
)
= sup
θ
6
0
k
w
m
(
t
+
θ
)
k
>
k
w
m
(
t
)
k
,
0
<t<T,
(26)
·
‚
(
Ü
(22),(25),(26)
Ú
Cauchy
Ø
ª
k
1
2
d
d
t
k
w
m
(
t
)
k
2
+
δ
1
k
w
m
(
t
)
k
2
b
V
6
h
F
(
t
)
,w
m
(
t
)
i
+(
G
(
t,w
m
)
,w
m
(
t
))
6
k
F
(
t
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2
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w
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2
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(
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(27)
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DOI:10.12677/aam.2023.1252092056
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d
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(
b
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k
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t
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2
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2
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o
,
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Ú
sup
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k
w
m
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t
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2
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3
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DOI:10.12677/aam.2023.1252092057
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DOI:10.12677/aam.2023.1252092058
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DOI:10.12677/aam.2023.1252092059
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u
m
(
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L
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t
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3
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b
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l
b
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±
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g
(
t,u
m
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→
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(
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t
)
3
L
2
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;
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ñ
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DOI:10.12677/aam.2023.1252092060
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DOI:10.12677/aam.2023.1252092061
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DOI:10.12677/aam.2023.1252092062
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t
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.
(66)
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q
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(60)
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4
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2
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t
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2
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V
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(67)
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u
s
∈
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],
·
‚
k
k
w
s
k
BCL
−∞
(
b
H
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= sup
θ
6
0
k
w
(
s
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θ
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= max
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θ
∈
(
−∞
,
−
s
]
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φ
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s
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θ
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,
sup
θ
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0]
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w
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6
max
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d
b
(
II
)(iii)
Ú
(68)
t
Z
0
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(
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(1)
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k
w
(
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k·
max
θ
∈
[0
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w
(
θ
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k
d
s.
(69)
n
Ü
(18),(67)
Ú
(69),
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(66)
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©
1
2
k
w
(
t
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k
2
+
δ
1
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0
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(
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2
b
V
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s
6
1
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d
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Z
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k
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max
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∈
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]
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w
(
θ
)
k
d
s.
DOI:10.12677/aam.2023.1252092063
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=
k
w
(
t
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2
6
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1
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t
0
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2
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2
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k
w
(
s
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k·
max
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∈
[0
,s
]
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w
(
θ
)
k
d
s.
(70)
du
2
L
G
k
φ
(
s
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k
BCL
−∞
(
b
H
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t
Z
0
k
w
(
s
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k
d
s
6
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2
k
φ
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2
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t
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w
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k
d
s
2
6
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2
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t
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2
d
s
6
c
k
φ
(
s
)
k
2
BCL
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(
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H
)
+
c
t
Z
0
max
r
∈
[0
,s
]
k
w
(
r
)
k
2
d
s.
(71)
ò
t
O
†
•
r
∈
[0
,t
],
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(70)
¥
r
(
r
∈
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(71)
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2
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[1]Eringen,A.C.(1966)TheoryofMicropolarFluids.
JournalofMathematicsandMechanics
,
16
,1-18.https://doi.org/10.1512/iumj.1967.16.16001
[2]Lukaszewicz,G. (1999)Micropolar Fluids:Theory and Applications. SpringerScience & Busi-
nessMedia,Berlin.
[3]Lukaszewicz,G. (2001)Long TimeBehavior of2D MicropolarFluidFlows.
Mathematicaland
ComputerModelling
,
34
,487-509.https://doi.org/10.1016/S0895-7177(01)00078-4
[4]Dong,B.Q.andZhang,Z.(2010)GlobalRegularityofthe2DMicropolarFluidFlowswith
ZeroAngularViscosity.
JournalofDifferentialEquations
,
249
,200-213.
https://doi.org/10.1016/j.jde.2010.03.016
DOI:10.12677/aam.2023.1252092064
A^
ê
Æ
?
Ð
é
[5]Ferreira,L.C.F.andPrecioso,J.C.(2013)ExistenceofSolutionsforthe3D-MicropolarFluid
System with Initial Data in Besov-Morrey Spaces.
Zeitschriftf¨urAngewandteMathematikund
Physik
,
64
,1699-1710.https://doi.org/10.1007/s00033-013-0310-8
[6]Galdi,G.P.andRionero,S.(1977)ANoteontheExistenceandUniquenessofSolutionsof
theMicropolarFluidEquations.
InternationalJournalofEngineeringScience
,
15
,105-108.
https://doi.org/10.1016/0020-7225(77)90025-8
[7]Lukaszewicz,G. (1990) On Non-Stationary Flows of Incompressible Asymmetric Fluids.
Math-
ematicalMethodsintheAppliedSciences
,
13
,219-232.
https://doi.org/10.1002/mma.1670130304
[8]Lukaszewicz, G. (2003) Asymptotic Behavior of Micropolar Fluid Flows.
InternationalJournal
ofEngineeringScience
,
41
,259-269.https://doi.org/10.1016/S0020-7225(02)00208-2
[9]Chen,J.,Chen,Z.M.andDong,B.Q.(2006)Existenceof
H
2
-GlobalAttractorsoftwo-
DimensionalMicropolar FluidFlows.
JournalofMathematical AnalysisandApplications
,
322
,
512-522.https://doi.org/10.1016/j.jmaa.2005.09.011
[10]Dong,B.Q.andChen,Z.M.(2006)GlobalAttractorsofTwo-DimensionalMicropolarFluid
FlowsinSomeUnboundedDomains.
AppliedMathematicsandComputation
,
182
,610-620.
https://doi.org/10.1016/j.amc.2006.04.024
[11]Lukaszewicz, G. andSadowski, W.(2004)Uniform Attractor for2D Magneto-Micropolar Fluid
FlowinSomeUnboundedDomains.
Zeitschriftf¨urAngewandteMathematikundPhysik
,
55
,
247-257.https://doi.org/10.1007/s00033-003-1127-7
[12]Nowakowski,B.(2013)Long-TimeBehaviorofMicropolarFluidEquationsinCylindrical
Domains.
NonlinearAnalysis:RealWorldApplications
,
14
,2166-2179.
https://doi.org/10.1016/j.nonrwa.2013.04.005
[13]Zhao,C.,Zhou,S.andLian,X.(2008)
H
1
-UniformAttractorandAsymptoticSmoothing
EffectofSolutionsforaNonautonomousMicropolarFluidFlowin2DUnboundedDomains.
NonlinearAnalysis:RealWorldApplications
,
9
,608-627.
https://doi.org/10.1016/j.nonrwa.2006.12.005
[14]Chen,J.,Dong,B.Q.andChen,Z.M.(2007)UniformAttractorsofNon-HomogeneousMi-
cropolarFluidFlowsinNon-SmoothDomains.
Nonlinearity
,
20
,1619-1635.
https://doi.org/10.1088/0951-7715/20/7/005
[15]Chen,J.,Dong, B.Q. and Chen,Z.M. (2007) Pullback Attractors of Non-Autonomous Microp-
olarFluidFlows.
JournalofMathematicalAnalysisandApplications
,
336
,1384-1394.
https://doi.org/10.1016/j.jmaa.2007.03.044
[16]Lukaszewicz,G.andTarasi´nska,A.(2009)On
H
1
-PullbackAttractorsforNonautonomous
MicropolarFluidEquationsinaBoundedDomain.
NonlinearAnalysis:Theory,Methods&
Applications
,
71
,782-788.https://doi.org/10.1016/j.na.2008.10.124
DOI:10.12677/aam.2023.1252092065
A^
ê
Æ
?
Ð
é
[17]Chen,G.(2009)PullbackAttractorforNon-HomogeneousMicropolarFluidFlowsinNon-
SmoothDomains.
NonlinearAnalysis:RealWorldApplications
,
10
,3018-3027.
https://doi.org/10.1016/j.nonrwa.2008.10.005
[18]Zhao,C.and Zhou,S. (2007) Pullback Attractors for a Non-AutonomousIncompressibleNon-
NewtonianFluid.
JournalofDifferentialEquations
,
238
,394-425.
https://doi.org/10.1016/j.jde.2007.04.001
[19]Caraballo,T.andReal,J.(2003)AsymptoticBehaviourofTwo-DimensionalNavier-Stokes
EquationswithDelays.
ProceedingsoftheRoyalSocietyofLondon.SeriesA:Mathematical,
PhysicalandEngineeringSciences
,
459
,3181-3194.https://doi.org/10.1098/rspa.2003.1166
[20]Caraballo,T.andReal,J.(2004) Attractors 2D-Navier-StokesModeswithDelays.
Journalof
DifferentialEquations
,
205
,271-497.https://doi.org/10.1016/j.jde.2004.04.012
[21]Marin-Rubio, P. and Real, J.(2007) Attractors for 2D-Navier-StokesEquations withDelayson
SomeUnboundedDomains.
NonlinearAnalysis:Theory,Methods&Applications
,
67
,2784-
2799.https://doi.org/10.1016/j.na.2006.09.035
[22]Zhao,C.andSun,W.(2017)GlobalWell-PosednessandPullbackAttractorsforaTwo-
DimensionalNon-AutonomousMicropolarFluidFlowswithInfiniteDelays.
Communications
inMathematicalSciences
,
15
,97-121.https://doi.org/10.4310/CMS.2017.v15.n1.a5
[23]Liu,L.,Caraballo,T.andMar´ın-Rubio,P.(2018)StabilityResultsfor2DNavier-Stokes
EquationswithUnboundedDelay.
JournalofDifferentialEquations
,
265
,5685-5708.
https://doi.org/10.1016/j.jde.2018.07.008
[24]M´alek,J.,Neˇcas,J.,Rokyta,M.andRuˇziˇcka,M.(1996) Weak andMeasure-Valued Solutions
toEvolutionaryPDE.CRCPress,NewYork.
[25]Hino,Y.,Kato,J.andNaito,T.(1991)FunctionalDifferentialEquationswithInfiniteDelay.
Springer-Verlag,Berlin.
[26]Chepyzhov,V.V.andVishik,M.I.(2002)AttractorsforEquationsofMathematicalPhysics.
AmericanMathematicalSociety,Providence.https://doi.org/10.1090/coll/049
[27]Mar´ın-Rubio,P.,Real,J.andValero,J.(2011)PullbackAttractorsforaTwo-Dimensional
Navier-StokesModelinanInfiniteDelayCase.
NonlinearAnalysis:Theory,MethodsAppli-
cations
,
74
,2012-2030.https://doi.org/10.1016/j.na.2010.11.008
DOI:10.12677/aam.2023.1252092066
A^
ê
Æ
?
Ð