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AdvancesinAppliedMathematics
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,2021,10(7),2529-2552
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.107264
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ANoteonthe
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Regularityof3D
IncompressibleNavier-StokesEquation
ChengmingYang
∗
,ZhenqiongCui
ShanghaiNormalUniversity,Shanghai
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[J].
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2552.DOI:10.12677/aam.2021.107264
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Received:Jun.21
st
,2021;accepted:Jul.11
th
,2021;published:Jul.23
rd
,2021
Abstract
Inthispaper,wemainlystudythe
H
1
regularityofthesolutionof3Dincompressible
Navier-Stokesequations.Firstly,thelocalwell-fitlemmaforsolutionsof3Dincom-
pressible Navier-Stokesequationsisgiven andprovedindetail.Secondly,by applying
thelocalwell-fitlemmaofthesolutionmentionedabove,firstly,theglobalregularity
ofthesolutioninthecaseofsmallinitialdatacanbeprovedstrictly.Second,the
H
1
regularityofthesolutionof3DincompressibleNavier-Stokesequationsisprovedfor
allpossible
U
0
andallpossible
F
.Inthispaper,itisemphasizedthatthesolution
is
H
1
regularnotonlyforthemaximumpossible
U
0
andafixed
F
,butalsoforthe
maximumpossible
U
0
andthemaximumpossible
F
.
Keywords
H
1
Regularity, TheLocalWell-FitLemma,3DIncompressibleNavier-StokesEquations
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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≤
NR
2
0
§
@
o
•
3
T
N
0
>
0
§
¦
||
A
1
2
u
0
||
2
+
2
ν
||
P
f
||
2
∞
T
N
0
≤
NR
2
0
,
•
,
T
N
= min
{
T
N
0
,T
∗∗∗
}
,
¤
±
•
3
T
N
,
¦
||
A
1
2
u
(
t
)
||
2
≤
NR
2
0
,
[0
,T
N
]
.
½
n
4.1
(
ê
â
K
5
)
3
?
¿˜
‡
k
.
«
•
Ω
⊂
R
3
þ
•
Ä
e
ã
¯
K
U
0
+
νAU
+
B
(
U,U
) =
PF,
(4.3)
DOI:10.12677/aam.2021.1072642533
A^
ê
Æ
?
Ð
¤
²
§
w
Ù
¥
P
L
«
L
2
(Ω)
"
Ñ
Ý
˜
m
Ý
K
Ž
f
,
(4
.
3)
÷
v
Direchlet
>
.
^
‡
,
…
F
Ø
•
6
t,
X
J
(4
.
17)
Ð
©ê
â
¿
©
,
@
o
(4
.
17)
k
˜
‡
Û
K
)
.
y
²
.
(4
.
3)
ª
†
AU
Š
S
È
,
¦
^
-
‡
O
ª
[5]
Ú
Young
Ø
ª
Œ
±
1
2
d
d
t
||
A
1
2
U
||
2
L
2
(Ω)
+
ν
||
AU
||
2
L
2
(Ω)
≤|
(
PF,AU
)
|
+
|
(
B
(
U,U
)
,AU
)
|
≤||
PF
||
L
2
(Ω)
||
AU
||
L
2
(Ω)
+
C
8
(Ω)
||
A
1
2
U
||
3
2
L
2
(Ω)
||
AU
||
1
2
L
2
(Ω)
≤
ν
4
||
AU
||
2
L
2
(Ω)
+
1
ν
||
PF
||
L
2
(Ω)
+
27
4
ν
3
C
4
8
||
A
1
2
U
||
6
L
2
(Ω)
+
ν
4
||
AU
||
2
L
2
(Ω)
,
=
k
d
d
t
||
A
1
2
U
||
2
L
2
(Ω)
+
λ
1
ν
||
A
1
2
U
||
2
L
2
(Ω)
≤
d
d
t
||
A
1
2
U
||
2
L
2
(Ω)
+
ν
||
AU
||
2
L
2
(Ω)
≤
2
ν
||
PF
||
L
2
(Ω)
+
27
2
ν
3
C
4
8
||
A
1
2
U
||
6
L
2
(Ω)
.
-
R
2
0
=
||
A
1
2
U
0
||
2
L
2
(Ω)
+
||
PF
||
2
L
2
(Ω)
,
½
N >
max(1
,
4
λ
1
ν
2
)
,
é
þ
ã
N>
1
,
k
||
A
1
2
U
0
||
2
L
2
(Ω)
+
||
PF
||
2
L
2
(Ω)
≤
NR
2
0
,
|
^
Lamma(4
.
1),
•
3
T
N
¦
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
≤
NR
2
0
,
0
≤
t
≤
T
N
.
(4.4)
Ø
”
b
[0
,T
∞
)
´¦
(4
.
4)
ª
¤
á
•
Œ
ž
m
«
m
,
e
ã
?
Ø
·
‚
3
[0
,T
∞
)
þ
?
1
d
d
t
||
A
1
2
U
||
2
L
2
(Ω)
+
λ
1
ν
||
A
1
2
U
||
2
L
2
(Ω)
≤
2
ν
||
PF
||
2
L
2
(Ω)
+
27
2
ν
3
C
4
8
||
A
1
2
U
||
6
L
2
(Ω)
,
du
(4
.
3)
Ð
©ê
â
¿
©
,
·
‚
Œ
±
b
27
2
ν
3
C
4
8
NR
2
0
≤
λ
1
ν
2
,
d
d
t
||
A
1
2
U
||
2
L
2
(Ω)
+
λ
1
ν
2
||
A
1
2
U
||
2
L
2
(Ω)
≤
2
ν
||
PF
||
2
L
2
(Ω)
.
(4.5)
A^
Gronwall
Ø
ª
,
||
A
1
2
U
||
2
L
2
(Ω)
≤
e
−
λ
1
νt
2
(
||
A
1
2
U
0
||
2
L
2
(Ω)
+
Z
t
0
2
ν
||
PF
||
2
L
2
(Ω)
ds
)
≤
max(1
,
4
λ
1
ν
2
)(
||
A
1
2
U
0
||
2
L
2
(Ω)
+
||
PF
||
2
L
2
(Ω)
)
<NR
2
0
,
(4.6)
•
,
A^
‡
y
{
5
y
²
T
max
=
∞
.
DOI:10.12677/aam.2021.1072642534
A^
ê
Æ
?
Ð
¤
²
§
w
Ä
k
,
Ä
½1
˜
«
œ
/
µ
T
∞
=
T
max
<
∞
,
d
ž
,[0
,T
∞
) = [0
,T
max
)
,
Ï
•
T
max
<
∞
,
¤
±
lim
t
→
T
−
max
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
=
∞
.
=
M
=
NR
2
0
ž
,
∃
δ>
0
,
∀
t
∈
(0
,T
max
)
,
•
‡
T
max
−
δ<t<T
max
,
Ò
k
||
A
1
2
U
(
t
)
||
2
>NR
2
0
,
=
∃
t
0
∈
(
T
max
−
δ,T
max
) = (
T
∞
−
δ,T
∞
)
,
¦
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
>NR
2
0
,
ù
†
[0
,T
∞
)
½
Â
g
ñ
.
Ù
g
,
Ä
½1
«
œ
/
µ
T
∞
<T
max
<
∞
,
du
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
,
3
[0
,T
max
)
þ
'
u
t
ë
Y
,
¤
±
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
•
3
[0
,T
∞
)
þ
ë
Y
¿
…
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
≤
NR
2
0
,
[0
,T
∞
)
.
du
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
3
T
∞
ë
Y
,
¤
±
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
≤
NR
2
0
,
[0
,T
∞
]
.
l
(4
.
6)
•
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
<NR
2
0
,
[0
,T
∞
]
.
(4.7)
é
(4
.
7)
ª
|
^
3
T
∞
ë
Y5
,
||
A
1
2
U
(
t
)
||
2
L
2
(Ω)
≤
NR
2
0
,t
∈
(
T
∞
,T
∞
+
δ
)
.
ù
†
[0
,T
∞
)
½
Â
g
ñ
.
n
þ
¤
ã
,
T
max
=
∞
,
y
.
.
Ú
n
4.2
^
‡
H1
Ú
(3
.
1)
¤
á
,
@
o
•
3
k
1
,k
2
,
1
>
0
,
é
u
0
<
≤
1
,
•
3
T
1
=
T
1
(
)
>
0
,
¦
u
(
t
)
∈
D
(
A
1
2
)
,
0
≤
t
≤
T
1
…
¤
á
||
A
1
2
v
(
T
1
)
||
2
≤
4
η
−
2
1
+
k
2
1
η
−
4
3
,
||
A
1
2
w
(
T
1
)
||
2
≤
k
2
2
2+
r
3
η
−
2
4
.
(4.8)
DOI:10.12677/aam.2021.1072642535
A^
ê
Æ
?
Ð
¤
²
§
w
y
²
.
½
Â
R
2
0
=
η
−
2
1
+
p
η
−
2
3
+
η
−
4
3
+2
η
−
2
2
+2
r
3
η
−
2
4
,
Ï
•
||
A
1
2
u
0
||
2
+
||
P
f
||
2
∞
=
||
A
1
2
v
0
||
2
+
||
A
1
2
ω
0
||
2
+
||
(
M
+
I
−
M
)
P
f
||
2
∞
≤||
A
1
2
v
0
||
2
+
||
A
1
2
ω
0
||
2
+2
||
MP
f
||
2
∞
+2
||
(
I
−
M
)
P
f
||
2
∞
≤
η
−
2
1
+
p
η
−
2
3
+2
η
−
2
2
+2
r
3
η
−
2
4
≤
η
−
2
1
+
p
η
−
2
3
+2
η
−
2
2
+2
r
3
η
−
2
4
+
η
−
4
3
=
R
2
0
,
À
N
= max
{
4
,
7
2
D
2
}
+1
>
1
,
A^
Lemma4
.
1,
•
3
T
N
>
0,
¦
||
A
1
2
u
(
t
)
||
2
≤
NR
2
0
,
[0
,T
N
]
.
(4.9)
P
[0
,T
∞
)
L
«¦
(4
.
23)
¤
á
•
Œ
ž
m
«
m
.
X
J
T
∞
<
∞
,
@
o
||
A
1
2
u
(
t
)
||
2
=
NR
2
0
.
(4.10)
e
5
,
r
t
•
›
3
[0
,T
∞
)
þ
?
1
?
Ø
Ï
•
(
I
−
M
)
B
(
v,v
) =
B
(
v,v
)
−
MB
(
v,v
) =
B
(
v,v
)
−
B
(
v,v
) = 0
,
@
o
'
u
ω
©
þ
•
§
•
d
ω
dt
+
νA
ω
= (
I
−
M
)
P
f
−
(
I
−
M
)(
B
(
ω,v
)+
B
(
v,ω
)+
B
(
ω,ω
))
.
(4.11)
(4
.
11)
ª
†
A
ω
Š
S
È
,
1
2
d
ω
dt
||
A
1
2
ω
||
2
+
ν
||
A
ω
||
2
≤|
((
I
−
M
)
P
f,A
ω
)
|
+
|
(
b
(
ω,v,A
w
))
|
+
|
(
b
(
v,ω,A
w
))
|
+
|
(
b
(
ω,ω,A
w
))
|
,
DOI:10.12677/aam.2021.1072642536
A^
ê
Æ
?
Ð
¤
²
§
w
|
^
Young
Ø
ª
1
2
d
ω
dt
||
A
1
2
ω
||
2
+
ν
||
A
ω
||
2
≤|
((
I
−
M
)
P
f,A
ω
)
|
+
|
(
b
(
ω,v,A
w
))
|
+
|
(
b
(
v,ω,A
w
))
|
+
|
(
b
(
ω,ω,A
w
))
|
≤
ν
2
||
A
ω
||
2
+
1
2
ν
||
(
I
−
M
)
P
f
||
2
∞
+
C
3
5
32
||
A
1
2
ω
||
15
32
||
A
1
2
v
||||
A
1
2
ω
||
49
32
+
C
4
1
4
||
A
1
2
v
||||
A
1
2
ω
||
1
2
||
A
ω
||
3
2
+
C
2
1
2
||
A
1
2
ω
||
3
2
||
A
ω
||
3
2
,
du
Mω
= 0
,
¦
^
(3
.
4)
d
dt
||
A
1
2
ω
||
2
+
ν
||
A
ω
||
2
≤
1
ν
||
(
I
−
M
)
P
f
||∞
2
+2
C
15
32
5
C
3
5
8
||
A
1
2
v
||||
A
ω
||
2
+2
C
1
2
5
C
4
3
4
||
A
1
2
v
||||
A
ω
||
2
+2
C
1
2
5
C
2
||
A
1
2
ω
||||
A
ω
||
2
,
5
¿
||
A
1
2
ω
||≤||
A
1
2
u
||
,
||
A
1
2
v
||≤||
A
1
2
u
||
,
¤
±
·
‚
d
dt
||
A
1
2
ω
||
2
+(
ν
−
D
1
5
8
||
A
1
2
u
||
)
||
A
ω
||
2
≤
1
ν
||
(
I
−
M
)
P
f
||
2
∞
,
(4.12)
Ù
¥
D
1
= 2
C
15
32
5
C
3
++2
C
1
2
5
C
4
+2
C
1
2
5
C
2
.
é
u
0
≤
t<T
∞
,
|
^
b
^
‡
H
1
D
1
5
8
||
A
1
2
u
||≤
D
1
5
8
N
1
2
R
0
=
D
1
5
8
N
1
2
(
η
−
1
1
+
p
2
η
−
1
3
+
η
−
2
3
+
√
2
η
−
1
2
+
√
2
r
6
η
−
1
4
)
→
0
,
→
0
.
Ï
d
,
∃
2
>
0
,
¦
D
1
5
8
N
1
2
R
0
≤
ν
2
,
0
<
≤
2
.
(4.13)
(
Ü
(4
.
12)
Ú
(4
.
13)
d
dt
||
A
1
2
ω
||
2
+
ν
2
||
A
ω
||
2
≤
1
ν
||
(
I
−
M
)
P
f
||
2
∞
,
(4.14)
|
^
Ø
ª
(3
.
4)
d
dt
||
A
1
2
ω
||
2
+
νC
−
2
5
−
2
2
||
A
1
2
ω
||
2
≤
1
ν
||
(
I
−
M
)
P
f
||
2
∞
,
(4.15)
DOI:10.12677/aam.2021.1072642537
A^
ê
Æ
?
Ð
¤
²
§
w
^
Gronwall
Ø
ª
,
||
A
1
2
ω
||
2
≤
e
−
R
t
0
νC
−
2
5
−
2
2
ds
||
A
1
2
ω
0
||
2
+
Z
t
0
1
ν
||
(
I
−
M
)
P
f
||
2
∞
≤
e
νC
−
2
5
−
2
2
t
||
A
1
2
ω
0
||
2
+
2
C
2
5
2
ν
2
||
(
I
−
M
)
P
f
||
2
∞
,
(4.16)
•
¦
e
νC
−
2
5
−
2
2
t
||
A
1
2
ω
0
||
2
=
2
C
2
5
2
ν
2
||
(
I
−
M
)
P
f
||
2
∞
,
=
e
νC
−
2
5
−
2
2
t
p
η
−
2
3
=
2
C
2
5
2
ν
2
r
3
η
−
2
4
,
(4.17)
Œ
±
é
T
1
>
0
,
÷
v
(4
.
17)
,
Ï
d
À
T
1
=
T
1
(
)
>
0
,
T
1
def
=2
C
2
5
2
ν
−
1
Q
(
)
,
Ù
¥
Q
(
) =
|
ln(2
C
2
5
ν
−
2
2+
r
3
−
p
η
−
2
4
η
2
3
)
|
.
(4.18)
•
y
(4
,
18)
¤
á
,
A
÷
v
e
¡
‡
¦
2
C
2
5
2+
r
3
−
p
η
2
3
η
−
2
4
≤
1
,
(4.19)
~
X
§
•
IÀ
η
4
=
−
ln
,
=
Œ
÷
v
‡
¦
.
Ï
d
,
∃
3
>
0
,
é
u
0
<
≤
3
,
•
3
T
1
=
T
1
(
)
>
0
÷
v
(4
.
19)
.
e
5
,
ä
ó
T
1
<T
∞
.
é
u
T
1
≤
t<T
∞
,
k
||
A
1
2
ω
||
2
≤
e
νC
−
2
5
−
2
2
T
1
||
A
1
2
ω
0
||
2
+
2
C
2
5
2
ν
2
||
(
I
−
M
)
P
f
||
2
∞
=
4
C
−
2
5
2
ν
2
r
3
η
−
2
4
=
k
2
2
2+
r
3
η
−
2
4
,
Ù
¥
k
2
2
= 4
C
−
2
5
ν
−
2
.
'
u
v
O
,
·
‚
•
›
t
3
[0
,T
1
]
þ
dv
dt
+
νA
v
=
MP
f
−
MB
(
v,v
)
−
MB
(
ω,v
)
−
B
(
v,ω
)
−
B
(
ω,ω
)
,
(4.20)
DOI:10.12677/aam.2021.1072642538
A^
ê
Æ
?
Ð
¤
²
§
w
(4
.
20)
ª
†
A
v
Š
S
È
,
1
2
d
dt
||
A
1
2
v
||
2
+
ν
||
A
v
||
2
≤|
(
MP
f,A
v
)
|
+
|
(
MB
(
v,v
)
,A
v
)
|
+
|
(
MB
(
ω,ω
)
,A
ω
)
|
≤|
(
MP
f,A
v
)
|
+
|
b
(
v,v,A
v
)
|
+
|
(
w,w,A
v
)
|
,
du
b
(
v,ω,A
v
) =
b
(
ω,v,A
v
) = 0
ä
N
(
Ø
•
„
[1]
é
þ
ª
|
^
Young
Ø
ª
,
1
2
d
dt
||
A
1
2
v
||
2
+
ν
||
A
v
||
2
≤||
MP
f
||
2
∞
||
A
v
||
+
C
1
||
v
||
1
2
||
A
1
2
v
||||
A
v
||
3
2
+
C
2
1
2
||
A
1
2
ω
||
3
2
||
A
ω
||
1
2
||
A
v
||
≤
ν
2
||
A
v
||
2
+
1
2
ν
||
MP
f
||
2
∞
||
2
+
C
1
||
v
||
1
2
||
A
1
2
v
||||
A
v
||
3
2
+
C
2
1
2
||
A
1
2
ω
||
3
2
||
A
ω
||
1
2
||
A
v
||
,
=
k
d
dt
||
A
1
2
v
||
2
+
ν
||
A
v
||
2
≤
1
ν
||
MP
f
||
2
∞
+2(
ν
4
||
A
v
||
2
+
27
4
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
4
+
ν
4
||
A
v
||
2
+
1
ν
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
)
≤
1
ν
||
MP
f
||
2
∞
+
ν
2
||
A
v
||
2
+
27
2
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
4
+
ν
2
||
A
v
||
2
+
2
ν
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
,
X
d
dt
||
A
1
2
v
||
2
≤
1
ν
||
MP
f
||
2
∞
+
27
2
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
4
+
2
ν
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
= (
27
2
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
2
)
||
A
1
2
v
||
2
+
1
ν
||
MP
f
||
2
∞
+
2
ν
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
,
é
þ
ª
A^
Gronwall
Ø
ª
,
||
A
1
2
v
||
2
≤
e
R
t
0
27
2
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
2
ds
[
||
A
1
2
v
0
||
2
+
Z
t
0
1
ν
||
MP
f
||
2
∞
+
2
ν
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
ds
]
≤
e
G
(
t
)
(
||
A
1
2
v
0
||
2
+
H
(
t
))
,
(4.21)
Ù
¥
H
(
t
) =
Z
t
0
1
ν
||
MP
f
||
2
∞
+
2
ν
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
ds,
G
(
t
) =
Z
t
0
27
2
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
2
ds.
DOI:10.12677/aam.2021.1072642539
A^
ê
Æ
?
Ð
¤
²
§
w
y
3
§
O
H
(
t
)
,
0
≤
t
≤
T
1
.
é
(4
.
14)
ª
È
©
ν
2
Z
t
0
||
A
ω
||
2
ds
≤
t
ν
||
(
I
−
M
)
P
f
||
2
∞
+
||
A
1
2
w
0
||
2
,
@
o
Z
t
0
||
A
ω
||
2
ds
≤
2
t
ν
2
||
(
I
−
M
)
P
f
||
2
∞
+
2
ν
||
A
1
2
w
0
||
2
,
(4
.
16)
ü
>
n
g
•
§
Ø
ª
˜
Œ
•
||
A
1
2
ω
||
6
≤
(
e
−
νC
−
2
5
−
2
2
t
||
A
1
2
ω
0
||
2
+
2
C
2
5
2
ν
2
||
(
I
−
M
)
P
f
||
2
∞
)
3
≤
4(
e
−
3
νC
−
2
5
−
2
2
t
||
A
1
2
ω
0
||
6
+
8
C
6
5
6
ν
6
||
(
I
−
M
)
P
f
||
6
∞
)
,
(4.22)
é
(4
.
22)
ª
È
©
Z
t
0
||
A
1
2
ω
||
6
ds
≤
4
Z
t
0
(
e
−
3
νC
−
2
5
−
2
2
t
||
A
1
2
ω
0
||
6
+
8
C
6
5
6
ν
6
||
(
I
−
M
)
P
f
||
2
∞
)
6
ds
≤
4(
2
C
2
5
2
3
ν
||
A
1
2
ω
0
||
6
+
8
C
6
5
6
ν
6
t
||
(
I
−
M
)
P
f
||
6
∞
)
,
¦
^
H?lder
Ø
ª
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
ds
≤
(
Z
t
0
||
A
ω
||
ds
)
1
2
(
Z
t
0
||
A
1
2
ω
||
6
ds
)
1
2
≤
2(
2
t
ν
2
||
(
I
−
M
)
P
f
||
2
∞
+
2
ν
||
A
1
2
w
0
||
2
)
1
2
(
2
C
2
5
2
3
ν
||
A
1
2
ω
0
||
6
+
8
C
6
5
6
ν
6
t
||
(
I
−
M
)
P
f
||
6
∞
)
1
2
≤
4
C
5
ν
(
t
1
2
ν
1
2
||
(
I
−
M
)
P
f
||
2
∞
+
||
A
1
2
w
0
||
)
(
1
√
3
||
A
1
2
ω
0
||
3
+
2
C
2
5
2
ν
5
2
t
1
2
||
(
I
−
M
)
P
f
||
3
∞
)
,
@
o
2
C
2
2
ν
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
ds
≤
8
C
2
2
C
5
ν
−
2
2
(
t
1
2
ν
1
2
||
(
I
−
M
)
P
f
||
2
∞
+
||
A
1
2
w
0
||
)
(
1
√
3
||
A
1
2
ω
0
||
3
+
2
C
2
5
2
ν
5
2
t
1
2
||
(
I
−
M
)
P
f
||
3
∞
)
≤
D
2
(
t
1
2
||
(
I
−
M
)
P
f
||
2
∞
+
||
A
1
2
w
0
||
)
(
||
A
1
2
ω
0
||
3
+
2
t
1
2
||
(
I
−
M
)
P
f
||
3
∞
)
,
DOI:10.12677/aam.2021.1072642540
A^
ê
Æ
?
Ð
¤
²
§
w
Ù
¥
D
2
=
8
C
2
2
C
5
ν
−
2
max(
1
√
3
,
2
C
2
5
ν
5
2
)max(1
,
1
ν
1
2
)
.
•
H
(
t
)
≤
1
ν
T
1
η
−
2
2
+
D
2
2
(
T
1
2
1
r
6
η
−
1
4
+
p
2
η
−
1
3
)(
3
p
2
η
−
3
3
+
T
1
2
1
2+
r
3
η
−
3
4
)
≤
2
C
2
5
ν
2
2
Q
(
)
η
−
2
2
+
D
2
η
−
4
3
+
2
D
2
C
2
5
ν
2
Q
(
)
η
−
4
4
+
3
D
2
4
η
−
4
3
+
D
2
C
4
5
ν
2
2
Q
(
)
2
η
−
4
4
+
√
2
D
2
C
5
ν
1
2
3
2
Q
(
)
1
2
η
−
1
3
η
−
3
4
≤
E
1
(
)+
7
4
D
2
η
−
4
3
,
Ù
¥
E
1
(
) =
D
3
(
2
Qη
−
2
2
+
2
Qη
−
4
4
+
Q
1
2
3
2
η
−
1
3
η
−
3
4
+
2
Qη
−
4
4
)
,
…
D
3
= max(
2
C
2
5
ν
−
2
,
2
D
2
C
2
5
ν
,
√
2
D
2
C
5
ν
1
2
,
D
2
C
4
5
ν
2
)
,
d
b
^
‡
H
1
,
Ø
J
y
→
0
+
ž
k
E
1
(
)
→
0
,
Ï
d
,
||
A
1
2
v
(
t
)
||
2
≤
e
G
(
t
)
(
η
−
2
1
+
E
1
(
)+
7
4
D
2
η
−
4
3
)
,
0
≤
t
≤
T
1
.
e
5
,
O
G
(
t
)
§
¿
`
²
G
(
t
)
¿
©
.
(2
.
2)
ª
†
u
Š
S
È
,
|
^
b
(
u,u,u
) = 0
,
Ú
©
z
[8]
1
2
d
dt
||
u
||
2
+
ν
||
A
1
2
u
||
2
≤|
(
P
f,u
)
|
≤|
(
A
−
1
2
P
f,A
1
2
u
)
|
≤||
A
1
2
u
||||
A
−
1
2
P
f
||
∞
≤
ν
2
||
A
1
2
u
||
2
+
1
2
ν
||
A
−
1
2
P
f
||
2
∞
,
=
k
d
dt
||
u
||
2
+
ν
||
A
1
2
u
0
||
2
≤
1
ν
||
A
−
1
2
P
f
||
2
∞
,
(4.23)
DOI:10.12677/aam.2021.1072642541
A^
ê
Æ
?
Ð
¤
²
§
w
È
©
||
u
||
2
−||
u
0
||
2
+
ν
Z
t
0
||
A
1
2
u
0
||
2
ds
≤
t
ν
||
A
−
1
2
P
f
||
2
∞
≤
2
t
ν
(
||
A
−
1
2
MP
f
||
2
∞
+
||
A
−
1
2
(
I
−
M
)
P
f
||
2
∞
)
≤
2
T
1
ν
(
||
A
−
1
2
MP
f
||
2
∞
+
||
A
−
1
2
(
I
−
M
)
P
f
||
2
∞
)
≤
4
C
2
5
2
Q
ν
2
(
||
A
−
1
2
MP
f
||
2
∞
+
||
A
−
1
2
(
I
−
M
)
P
f
||
2
∞
)
,
¦
^
(3
.
3)
†
(3
.
4)
||
u
||
2
≤||
u
0
||
2
+
4
C
2
5
2
Q
ν
2
λ
1
||
MP
f
||
2
∞
+
4
C
4
5
4
Q
ν
2
||
(
I
−
M
)
P
f
||
2
∞
=
||
v
0
||
2
+
||
ω
0
||
2
+
4
C
2
5
2
Q
ν
2
λ
1
||
MP
f
||
2
∞
+
4
C
4
5
4
Q
ν
2
||
(
I
−
M
)
P
f
||
2
∞
≤
1
λ
1
||
A
1
2
v
0
||
2
+
C
2
5
2
||
A
1
2
ω
0
||
2
+
4
C
2
5
2
Q
ν
2
λ
1
||
MP
f
||
2
∞
+
4
C
4
5
4
Q
ν
2
||
(
I
−
M
)
P
f
||
2
∞
,
=
k
||
v
||
2
≤||
u
||
2
≤
1
λ
1
||
A
1
2
v
0
||
2
+
C
2
5
2
||
A
1
2
ω
0
||
2
+
4
C
2
5
2
Q
ν
2
λ
1
||
MP
f
||
2
∞
+
4
C
4
5
4
Q
ν
2
||
(
I
−
M
)
P
f
||
2
∞
≤
D
4
(
||
A
1
2
v
0
||
2
+
2
||
A
1
2
ω
0
||
2
+
2
Q
||
MP
f
||
2
∞
+
4
Q
||
(
I
−
M
)
P
f
||
2
∞
)
,
Ù
¥
D
4
= max(
1
λ
1
,C
2
5
,
4
C
4
5
ν
2
,
4
C
2
5
ν
2
λ
1
)
.
|
^
e
ã
(
Ø
||
A
1
2
v
(
t
)
||
2
≤||
A
1
2
u
(
t
)
||
2
≤
NR
2
0
,
·
‚
Œ
±
G
(
t
) =
Z
t
0
27
2
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
2
ds
≤
27
2
ν
3
C
4
1
D
4
T
1
(
η
−
2
1
+
2+
p
η
−
2
3
+
2
Qη
−
4
2
+
4+
r
3
Qη
−
2
4
)
NR
2
0
≤
27
C
2
5
C
4
1
D
4
ν
4
2
Q
(
)(
η
−
2
1
+
2+
p
η
−
2
3
+
2
Qη
−
4
2
+
4+
r
3
Qη
−
2
4
)(
η
−
2
1
+
p
η
−
2
3
+
η
−
4
3
+2
η
−
2
2
+2
r
3
η
−
2
4
)
≤
E
2
(
)
,
Ù
¥
E
2
(
) =
D
5
2
Q
(
)(
η
−
2
1
+
2+
p
η
−
2
3
+
2
Qη
−
4
2
+
4+
r
3
Qη
−
2
4
)(
η
−
2
1
+
p
η
−
2
3
+
η
−
4
3
+2
η
−
2
2
+2
r
3
η
−
2
4
)
,
DOI:10.12677/aam.2021.1072642542
A^
ê
Æ
?
Ð
¤
²
§
w
…
D
5
=
27
C
2
5
C
4
1
D
4
ν
4
.
a
q
/
§
|
^
b
^
‡
H
1
,
Ø
J
y
E
2
(
)
→
0
as
→
0
.
•
,
3
N
= 1+max(4
,
7
2
D
2
)
e
,
·
‚
À
4
>
0,
¦
e
E
2
(
)
≤
2
,E
1
(
)
≤
η
−
2
1
,
2
C
2
5
2
3
≤
ν
2
,
0
<
≤
4
•
y
²
T
1
<T
∞
,
,
·
‚
¦
^
‡
y
{
b
T
∞
=
∞
,
@
o
w
,
k
T
1
<T
∞
=
∞
.
Ï
d
·
‚
•
I
b
T
∞
≤
T
1
<
∞
,
˜
•
¡
,
k
||
A
1
2
ω
(
T
∞
)
||
2
≤||
A
1
2
ω
0
||
2
+
1
2
k
2
2
2
||
(
I
−
M
)
P
f
||
2
∞
≤
p
η
−
2
3
+
1
2
k
2
2
2+
r
3
η
−
2
4
≤
p
η
−
2
3
+
1
2
k
2
2
2
3
+
r
3
η
−
2
4
≤
p
η
−
2
3
+
r
3
η
−
2
4
,
…
||
A
1
2
v
(
T
∞
)
||
2
≤
e
E
2
(
)
(
η
−
2
1
+
E
1
(
)+
7
4
D
2
η
−
4
3
)
≤
4
η
−
2
1
+
7
2
D
2
η
−
4
3
.
•
,
·
‚
||
A
1
2
u
(
T
∞
)
||
2
= 4
η
−
2
1
+
7
2
D
2
η
−
4
3
+
p
η
−
2
3
+
r
3
η
−
2
4
<
(1+max(4
,
7
2
D
2
))
R
2
0
=
NR
2
0
.
(4.24)
,
˜
•
¡
,
X
J
T
∞
<
∞
,
@
o
||
A
1
2
u
(
T
∞
)
||
2
=
NR
2
0
,
ù
†
(4
.
23)
ª
g
ñ
.
Ï
d
,
T
1
<T
∞
.
-
k
2
1
=
7
2
D
2
,k
2
2
= 4
C
2
5
ν
−
2
,
1
def
=
4
,
·
‚
k
||
A
1
2
v
(
T
1
)
||
2
≤
4
η
−
2
1
+
k
2
1
η
−
4
3
,
||
A
1
2
ω
(
T
1
)
||
2
≤
k
2
2
2+
r
3
η
−
2
4
.
DOI:10.12677/aam.2021.1072642543
A^
ê
Æ
?
Ð
¤
²
§
w
y
.
.
Ú
n
4.3
^
‡
H
1
Ú
H
(
a,b
)
¤
á
,
Ù
¥
a
Ú
b
¿
©
Œ
,
@
o
•
3
0
>
0
¦
é
z
‡
,
0
<
≤
0
,
e
Ð
©ê
â
÷
v
b
^
‡
(3
.
1)
§
K
•
§
(2
.
2)
)
u
(
t
)
÷
v
µ
0
≤
t
≤
2
T
0
ž
k
u
(
t
)
∈
D
(
A
1
2
)
,
…
T
0
≤
t
≤
2
T
0
ž
k
||
A
1
2
v
(
t
)
||
2
≤
1
2
(4
η
−
2
1
+
k
2
1
η
−
4
3
)
,
||
A
1
2
w
(
t
)
||
2
≤
k
2
2
2+
r
3
η
−
2
4
.
(4.25)
y
²
.
·
‚
3
Ú
n
4
.
2
Ä
:
þ
y
²
Ú
n
4
.
3.
½
Â
R
2
0
= 1+(
η
−
2
+2
η
−
2
2
+
d
1
)[1+
e
2
D
19
η
−
4
e
4
D
19
η
−
4
2
]+2
r
3
η
−
2
4
.
du
||
A
1
2
u
0
||
2
+
||
P
f
||
2
∞
≤||
A
1
2
v
0
||
2
+
||
A
1
2
ω
0
||
2
+2
||
MP
f
||
2
∞
+2
||
(
I
−
M
)
P
f
||
2
∞
≤
4
η
−
2
1
+
k
2
1
η
−
4
3
+
k
2
2
2+
r
3
η
−
2
4
+2
η
−
2
2
+2
r
3
η
−
2
4
≤
R
2
0
,
½
N
= max(1
,D
21
,D
22
)
>
1
,
A^
Lemma4
.
1,
•
3
T
N
>
0
,
¦
||
A
1
2
u
(
t
)
||
2
≤
NR
2
0
t
∈
[0
,T
N
]
.
P
[0
,T
∞
)
L
«¦
þ
¡
Ø
ª
¤
á
•
Œ
ž
m
«
m
.
e
T
∞
<
∞
,
K
k
||
A
1
2
u
(
t
)
||
2
=
NR
2
0
.
(4.26)
(4
.
11)
ª
†
A
ω
,
(4
.
13)
ª
Ú
D
1
.
é
u
0
≤
t<T
∞
k
D
2
1
5
4
||
A
1
2
u
||
2
≤
D
2
1
N
5
4
R
2
0
→
0
,as
→
0
b
a
≥
2
D
19
,b
≥
4
D
19
,
l
b
^
‡
H
1
Ú
H
(
a,b
)
µ
→
0
ª
,
e
ª
ª
u
0
5
4
R
2
0
=
5
4
(1+
η
−
2
+2
η
−
2
2
+
d
1
+
η
−
2
e
2
D
1
9
η
−
4
e
4
D
19
η
−
4
2
+2
η
−
2
2
e
2
D
19
η
−
4
e
4
D
19
η
−
4
2
+
d
1
e
2
D
19
η
−
4
e
4
D
19
η
−
4
2
+2
r
3
η
−
2
4
)
.
@
o
,
•
3
5
>
0
,
¦
D
1
N
1
2
5
8
R
0
≤
ν
2
,
0
<
≤
5
l
(4
.
16)
ª
||
A
1
2
ω
(
t
)
||
2
≤
e
−
νC
−
2
5
−
2
2
t
||
A
1
2
ω
0
||
2
+
2
C
2
5
2
ν
2
||
(
I
−
M
)
P
f
||
2
∞
≤
3
2
k
2
2
2+
r
3
η
−
2
4
,
(4.27)
DOI:10.12677/aam.2021.1072642544
A^
ê
Æ
?
Ð
¤
²
§
w
é
þ
ª
l
0
t
È
©
Z
t
0
||
A
1
2
ω
||
2
ds
≤
2
ν
||
A
1
2
ω
0
||
2
+
2
t
ν
2
||
(
I
−
M
)
P
f
||
2
∞
≤
max(
2
k
2
2
ν
,
2
ν
2
)[
2
+
t
]
r
3
η
−
2
4
≤
D
2
9
[
2
+
t
]
r
3
η
−
2
4
,
Ù
¥
0
≤
t<
1
.
é
þ
ª
l
t
−
1
t
È
©
Z
t
t
−
1
||
A
1
2
ω
||
2
ds
≤
2
ν
||
A
1
2
ω
(
t
−
1)
||
2
+
2
ν
2
||
(
I
−
M
)
P
f
||
2
∞
≤
max(
3
ν
k
2
2
,
2
ν
2
)
r
3
η
−
2
4
≤
D
2
10
r
3
η
−
2
4
,
Ù
¥
1
≤
t<T
∞
.
Ó
§
È
©
Z
t
0
||
A
1
2
ω
(
s
)
||
6
ds
≤
4
Z
t
0
(
e
−
3
νC
−
2
5
2
2
k
2
2
6+
r
η
−
6
4
+
8
C
6
5
2
ν
6
r
η
−
6
4
ds
)
≤
4
Z
t
0
(
e
−
3
νC
−
2
5
2
2
k
2
2
+
1
8
k
6
2
)
ds
6+
r
η
−
6
4
≤
4
k
6
2
max(
2
C
2
5
3
ν
,
1
8
)[
2
+
t
]
6+
r
η
−
6
4
≤
D
2
11
[
2
+
t
]
6+
r
η
−
6
4
,
Ù
¥
0
≤
t<
1
.
d
0
<
≤
1
k
,
a
q
Z
t
t
−
1
||
A
1
2
ω
(
s
)
||
6
ds
≤
2
D
2
11
6+
r
η
−
6
4
,
Ù
¥
1
≤
t<T
∞
.
¦
^
H?lder
Ø
ª
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||≤
(
Z
t
0
||
A
1
2
ω
||
6
)
1
2
(
Z
t
0
||
A
ω
||
2
)
1
2
≤
D
9
[
2
+
t
]
1
2
r
6
η
−
1
4
D
11
[
2
+
t
]
1
2
3+
r
2
η
−
3
4
=
D
9
D
11
[
2
+
t
]
3+
2
r
3
η
−
4
4
,
(4.28)
DOI:10.12677/aam.2021.1072642545
A^
ê
Æ
?
Ð
¤
²
§
w
Ù
¥
0
≤
t<
1
.
Ó
¦
^
H?lder
Ø
ª
Z
t
t
−
1
||
A
1
2
ω
||
3
||
A
ω
||≤
(
Z
t
t
−
1
||
A
1
2
ω
||
6
)
1
2
(
Z
t
t
−
1
||
A
ω
||
2
)
1
2
≤
D
10
r
6
η
−
1
4
√
2
D
11
3+
r
2
η
−
3
4
≤
2
D
10
D
11
3+
2
r
3
η
−
4
4
,
(4.29)
Ù
¥
1
≤
t<T
∞
.
e
5
,(4
.
20)
ª
†
v
Š
S
È
,
(
Ü
b
(
v,v,v
) = 0
!
(3
.
3)
ª
Ú
(3
.
4)
ª
1
2
d
dt
||
v
||
2
+
λ
1
ν
||
v
||
2
≤
1
2
d
dt
||
v
||
2
+
ν
||
A
v
||
2
≤||
MPf
||
∞
||
v
||
+
C
2
1
2
||
A
1
2
ω
||
3
2
||
A
ω
||
1
2
||
v
||
≤
1
λ
1
2
1
(
||
MPf
||
∞
+
C
2
1
2
||
A
1
2
ω
||
3
2
||
A
ω
||
1
2
)
,
|
^
Young
Ø
ª
1
2
d
dt
||
v
||
2
+
ν
||
A
1
2
v
||
2
≤
ν
4
||
A
1
2
v
||
2
+
1
νλ
1
||
MPf
||
2
∞
+
ν
4
||
A
1
2
v
||
2
+
1
νλ
1
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
,
Ø
J
y
e
ã
(
Ø
d
dt
||
v
||
2
+
ν
||
A
1
2
v
||
2
≤
2
νλ
1
(
||
MPf
||
2
∞
+
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
)
,
d
dt
||
v
||
2
+
λ
1
ν
||
A
1
2
v
||
2
≤
2
νλ
1
(
||
MPf
||
2
∞
+
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
)
,
¦
^
Gronwall
Ø
ª
||
v
||
2
≤
e
−
λ
1
t
(
||
v
0
||
2
+
Z
t
0
2
νλ
1
(
||
MPf
||
2
∞
+
C
2
2
||
A
1
2
ω
||
3
||
A
ω
||
))
ds
≤
e
−
λ
1
t
||
v
0
||
2
+
2
νλ
1
||
MPf
||
2
∞
te
−
λ
1
t
+
2
νλ
1
C
2
2
e
−
λ
1
t
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
ds
≤
e
−
λ
1
t
||
v
0
||
2
+
2
(
νλ
1
)
2
||
MPf
||
2
∞
+
2
νλ
1
C
2
2
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
ds
≤
e
−
λ
1
t
1
λ
1
||
A
1
2
v
0
||
2
+
2
(
νλ
1
)
2
||
MPf
||
2
∞
+
2
νλ
1
C
2
2
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
ds
≤
e
−
λ
1
t
1
λ
1
(4
η
−
2
1
+
k
2
1
η
−
4
3
)+
2
(
νλ
1
)
2
η
−
2
2
+
2
(
νλ
1
)
2
C
2
2
D
9
D
11
[
2
+
t
]
3+
2
r
3
η
−
4
4
≤
max(
1
λ
1
,
2
(
νλ
1
)
2
,
2
(
νλ
1
)
2
C
2
2
D
9
D
11
)(
e
−
νλ
1
t
(4
η
−
2
1
+
k
2
1
η
−
4
3
)+
η
−
2
2
+[
2
+
t
]
4+
2
r
3
η
−
4
4
)
=
D
12
γ
(
,t
)
,
DOI:10.12677/aam.2021.1072642546
A^
ê
Æ
?
Ð
¤
²
§
w
Ù
¥
D
12
def
=max(
1
λ
1
,
2
(
νλ
1
)
2
,
2
(
νλ
1
)
2
C
2
2
D
9
D
11
)
,
γ
(
,t
)
def
=
e
−
νλ
1
t
η
−
2
+
η
−
2
2
+[
2
+
t
]
4+
2
r
3
η
−
4
4
.
a
q
/
,
||
v
||
2
−||
v
0
||
2
−
ν
Z
t
0
||
A
1
2
v
||
2
ds
≤
2
νλ
1
(
||
MP
f
||
2
∞
t
+
C
2
2
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
ds
)
,
d
b
^
‡
(3
.
1)
Ú
(4
.
28)
Z
t
0
||
A
1
2
v
||
2
ds
≤
2
t
λ
1
ν
2
||
MP
f
||
2
∞
+
1
ν
||
v
0
||
2
+
2
λ
1
ν
2
C
2
2
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
ds
≤
2
t
λ
1
ν
2
η
−
2
2
+
1
νλ
1
(4
η
−
2
1
+
k
2
1
η
−
4
3
)+
2
λ
1
ν
2
C
2
2
D
9
D
11
[
2
+
t
]
3+
2
r
3
η
−
4
4
≤
max(
e
νλ
1
νλ
1
,
2
λ
1
ν
2
,
2
λ
1
ν
2
C
2
2
D
9
D
11
)(
e
−
νλ
1
(4
η
−
2
1
+
k
2
1
η
−
4
3
)+
η
−
2
2
+[
2
+
t
]
3+
2
r
3
η
−
4
4
)
≤
D
13
γ
(
,t
)
,
Ù
¥
D
13
def
=max(
e
νλ
1
νλ
1
,
2
λ
1
ν
2
,
2
λ
1
ν
2
C
2
2
D
9
D
11
)
.
a
q
/
Z
t
t
−
1
||
A
1
2
v
||
2
ds
≤
2
λ
1
ν
2
||
MP
f
||
2
∞
+
1
ν
||
v
(
t
−
1)
||
2
+
2
λ
1
ν
2
C
2
2
Z
t
t
−
1
||
A
1
2
ω
||
3
||
A
ω
||
ds
≤
1
ν
D
12
γ
(
,t
)+
2
λ
1
ν
2
η
−
2
2
+
2
λ
1
ν
2
C
2
2
2
D
9
D
11
[
2
+
t
]
3+
2
r
3
η
−
4
4
≤
1
ν
(
D
12
γ
(
,t
)+
2
λ
1
ν
η
−
2
2
+
2
λ
1
ν
C
2
2
2
D
10
D
11
3+
2
r
3
η
−
4
4
)
≤
1
ν
max(
D
12
,
2
λ
1
ν
+
D
12
,
2
D
12
+
4
λ
1
ν
C
2
2
10
D
11
)
γ
(
,t
)
≤
D
14
γ
(
,t
)
,
|
^
±
þ
O
§
k
Z
t
0
||
v
||
2
||
A
1
2
v
||
2
ds
≤
sup
0
≤
s
≤
t
||
v
||
2
Z
t
0
||
A
1
2
v
||
2
ds
≤
D
12
D
13
γ
2
(
,t
)
≤
e
νλ
1
D
12
D
13
γ
2
(
,t
)
.
DOI:10.12677/aam.2021.1072642547
A^
ê
Æ
?
Ð
¤
²
§
w
a
q
/
,
Z
t
t
−
1
||
v
||
2
||
A
1
2
v
||
2
ds
≤
e
νλ
1
D
12
D
14
γ
2
(
,t
)
.
l
(4
.
21)
ª
•
||
A
1
2
v
||
2
≤
e
G
(
t
)
(
||
A
1
2
v
0
||
2
+
H
(
t
))
,
Ù
¥
G
(
t
) =
Z
t
0
27
2
ν
3
C
4
1
||
A
1
2
v
||
2
||
v
||
2
ds
=
27
2
ν
3
C
4
1
e
νλ
1
D
12
D
13
γ
2
(
,t
)
=
D
17
γ
2
(
,t
)
,
…
H
(
t
) =
1
ν
Z
t
0
||
MP
f
||
2
∞
+
2
C
2
2
ν
Z
t
0
||
A
1
2
ω
||
3
||
A
ω
||
2
ds
≤
1
ν
η
−
2
2
+
2
C
2
2
D
9
D
11
ν
[
2
+
t
]
4+
2
r
3
η
−
4
4
.
(
Ü
þ
ã
O
§
||
A
1
2
v
(
t
)
||
2
≤
e
D
17
γ
2
(
,t
)(
1
ν
η
−
2
2
+
2
C
2
2
D
9
D
11
ν
[
2
+
t
]
4+
2
r
3
η
−
4
4
+4
η
−
2
1
+
k
2
1
η
−
4
3
)
≤
max(
e
νλ
1
,
1
ν
,
2
C
2
2
D
9
D
11
ν
)
=
D
16
γ
(
,t
)
e
D
17
γ
2
(
,t
)
,
5
¿
d
dt
||
A
1
2
v
||
2
≤
(
27
2
ν
3
C
4
1
||
v
||
2
||
A
1
2
v
||
2
)
||
A
1
2
v
||
2
+
1
ν
||
MP
f
||
2
∞
+
2
C
2
2
ν
||
A
1
2
ω
||
3
||
A
ω
||
2
,
é
þ
ª
3
t
∈
[1
,
∞
)
þ
¦
^
˜
—
Gronwall
Ø
ª
§
||
A
1
2
v
||
2
≤
[
Z
t
t
−
1
||
A
1
2
v
||
2
ds
+
Z
t
t
−
1
(
1
ν
||
MP
f
||
2
∞
+
2
C
2
2
ν
||
A
1
2
ω
||
3
||
A
ω
||
2
)
ds
]
e
27
2
ν
3
C
4
1
e
νλ
1
D
12
D
14
γ
2
(
,t
)
≤
[
D
14
ν
2
γ
(
,t
)+
1
ν
η
−
2
2
+
4
C
2
2
D
10
D
11
ν
3+
2
r
3
η
−
4
4
]
e
27
2
ν
3
C
4
1
e
νλ
1
D
12
D
14
γ
2
(
,t
)
≤
[
D
14
ν
2
γ
(
,t
)+
1
ν
η
−
2
2
+
4
C
2
2
D
10
D
11
ν
4+
2
r
3
η
−
4
4
]
e
27
2
ν
3
C
4
1
e
νλ
1
D
12
D
14
γ
2
(
,t
)
≤
(
D
14
+max(
1
ν
,
4
C
2
2
D
10
D
11
ν
))
γ
(
,t
)
e
27
2
ν
3
C
4
1
e
νλ
1
D
12
D
14
γ
2
(
,t
)
,
DOI:10.12677/aam.2021.1072642548
A^
ê
Æ
?
Ð
¤
²
§
w
@
o
·
‚
||
A
1
2
v
(
t
)
||
2
≤
D
16
γ
(
,t
)
e
D
17
γ
2
(
,t
)
||
A
1
2
v
(
t
)
||
2
≤
(
D
14
+max(
1
ν
,
4
C
2
2
D
10
D
11
ν
))
γ
(
,t
)
e
27
2
ν
3
C
4
1
e
νλ
1
D
12
D
14
γ
2
(
,t
)
,
…
||
A
1
2
v
(
t
)
||
2
≤
D
18
γ
(
,t
)
e
D
19
γ
2
(
,t
)
,
Ù
¥
D
18
= max(
D
16
,D
14
+max(
1
ν
,
4
C
2
2
D
10
D
11
ν
))
,D
19
=
27
C
4
1
2
ν
3
e
νλ
1
max(
D
13
,D
14
)
D
12
.
·
‚
Ž
‡
é
T
0
=
T
0
(
).
Ï
•
η
→∞
§
→
0
+
,
Œ
±
b
2
D
19
η
−
4
>
1
,
‡
¦
2
D
19
e
−
2
νλ
1
t
η
−
4
≤
1
.
-
T
0
def
=
1
2
νλ
1
ln(2
D
19
η
−
4
)
>
0
,
…
E
3
(
)
def
=(
2
+2
T
0
(
))
4+
2
r
3
η
−
4
4
,
Ø
J
O
E
3
(
)
Ú
3
2
k
2
2
2+
r
3
η
−
2
4
E
3
(
) = (
2
+2
T
0
(
))
4+
2
r
3
η
−
4
4
= (1+
1
νλ
1
ln(2
D
19
)
η
−
4
)
4+
2
r
3
η
−
4
4
= (1+
1
νλ
1
(ln(2
D
19
)+ln
η
−
4
))
4+
2
r
3
η
−
4
4
= (1+
1
νλ
1
max(ln(2
D
19
,
1)))(1+ln
η
−
4
)
4+
2
r
3
η
−
4
4
≤
D
20
,
…
3
2
k
2
2
2+
r
3
η
−
2
4
≤
3
2
k
2
2
(
4+
2
r
3
η
−
4
4
)
1
2
≤
3
2
k
2
2
(1+ln
η
−
4
)
1
2
(
4+
2
r
3
η
−
4
4
)
1
2
≤
3
2
k
2
2
(1+ln
η
−
4
)(
4+
2
r
3
η
−
4
4
)
1
2
≤
D
21
,
DOI:10.12677/aam.2021.1072642549
A^
ê
Æ
?
Ð
¤
²
§
w
…
k
||
A
1
2
v
(
t
)
||
2
≤
D
18
γ
(
,t
)
e
D
19
γ
2
(
,t
)
≤
D
18
(
e
−
νλ
1
t
η
−
2
+
η
−
2
2
+[
+
t
]
4+
2
r
3
η
−
4
4
)
e
D
19
(2
e
−
2
νλ
1
η
−
4
+4
η
−
2
2
+4
D
2
19
)
≤
D
18
e
4
D
19
D
2
20
(
e
−
νλ
1
t
η
−
2
+
η
−
2
2
+
D
20
)
e
(
4
D
19
η
−
4
2
+1)
≤
D
22
(
e
−
νλ
1
t
η
−
2
+
η
−
2
2
+
D
20
)
e
(
4
D
19
η
−
4
2
+1)
,
Ï
d
,
||
A
1
2
u
(
t
)
||
2
=
||
A
1
2
v
(
t
)
||
2
+
||
A
1
2
ω
(
t
)
||
2
≤
D
22
(
e
−
νλ
1
t
η
−
2
+
η
−
2
2
+
D
20
)
e
2
D
19
η
−
4
e
4
D
19
η
−
4
2
+
D
21
e
0
≤
max(1
,D
21
,D
22
)((
η
−
2
+
η
−
2
2
+
D
20
)
e
2
D
19
η
−
4
e
4
D
19
η
−
4
2
)
<NR
2
0
.
(4.30)
e
5
,
·
‚
¦
^
‡
y
{
y
²
2
T
0
≤
T
∞
.
X
J
T
∞
=
∞
,
@
o
2
T
0
≤
T
∞
.
Ï
d
·
‚
b
T
∞
<
2
T
0
<
∞
,
l
(4
.
30)
ª
•
||
A
1
2
u
(
T
∞
)
||
2
<NR
2
0
,
ù
†
(4
.
26)
ª
g
ñ
.
Ï
d
2
T
0
≤
T
∞
.
•
,
·
‚
5
y
²
(4
.
25)
ª
.
•
›
t
3
[
T
0
,
2
T
0
]
þ
l
(4
.
27)
ª
||
A
1
2
ω
(
t
)
||
2
≤
[
k
2
2
e
−
2
νC
−
2
5
−
2
2
T
0
+
1
2
k
2
2
]
2+
r
3
η
−
2
4
,
•
‡
À
T
0
≥
2ln2
C
2
5
2
ν
,
@
o
∃
8
,
0
<
≤
8
,
¦
e
−
νC
−
2
5
−
2
2
T
0
≤
1
2
,
0
<
≤
8
.
Ï
d
,
||
A
1
2
ω
(
t
)
||
2
≤
[
k
2
2
e
−
νC
−
2
5
−
2
2
T
0
+
1
2
k
2
2
]
2+
r
3
η
−
2
4
≤
[
1
2
k
2
2
+
1
2
k
2
2
]
2+
r
3
η
−
2
4
=
k
2
2
2+
r
3
η
−
2
4
,
DOI:10.12677/aam.2021.1072642550
A^
ê
Æ
?
Ð
¤
²
§
w
d
®
•
O
ª
Œ
||
A
1
2
v
(
t
)
||
2
≤
D
22
(
e
−
νλ
1
t
η
−
2
+
η
−
2
2
+
D
20
)
e
4
D
19
η
−
2
2
+1
≤
D
22
(
η
−
2
2
+
1
√
2
D
19
+
D
20
)
e
4
D
19
η
−
2
2
+1
≤
D
22
(
η
−
2
2
+
D
24
)
e
4
D
19
η
−
2
2
+1
= Γ(
η
−
2
2
)
,
@
o
k
Γ(
η
−
2
2
) =
D
22
η
−
2
2
e
4
D
19
η
−
4
2
+1
+
D
22
√
2
D
19
e
4
D
19
η
−
4
2
+1
+
D
20
D
22
e
4
D
19
η
−
4
2
+1
.
(4.31)
d
b
^
‡
H
(
a,b
)
,
•
∃
10
,
f
u
0
<
≤
10
D
22
η
−
2
2
e
4
D
19
η
−
4
2
+1
=
D
22
η
−
2
2
e
4
D
19
η
−
4
2
e
≤
1
6
(4
η
−
2
1
+
k
2
1
η
−
4
3
)
.
a
q
/
,
∃
11
,
é
u
0
<
≤
11
,
D
22
√
2
D
19
e
4
D
19
η
−
4
2
+1
≤
1
6
(4
η
−
2
1
+
k
2
1
η
−
4
3
)
.
a
q
/
,
∃
12
,
é
u
0
<
≤
12
,
D
20
D
22
e
4
D
19
η
−
4
2
+1
≤
1
6
(4
η
−
2
1
+
k
2
1
η
−
4
3
)
.
13
= min(
10
,
11
,
12
)
,
¦
Γ(
η
−
2
2
)
≤
1
2
(4
η
−
2
1
+
k
2
1
η
−
4
3
)
,
0
<
≤
13
.
é
u
0
<
≤
13
,
k
||
A
1
2
v
(
t
)
||
2
≤
1
2
(4
η
−
2
1
+
k
2
1
η
−
4
3
)
,T
0
≤
t
≤
2
T
0
.
0
=
13
.
y
.
.
½
n
4.2
(
H
1
K
5
)
η
i
,i
=1
,
2
,
3
,
4
,r,
Ú
p
÷
v
^
‡
H
1
Ú
H
(
a,b
)
,
Ù
¥
a
Ú
b
¿
©
Œ
§
@
o
•
3
0
>
0
,k
2
>
0
,
Ú
˜
‡
ë
Y
¼
ê
Γ
∈
C
([0
,
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