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AdvancesinAppliedMathematics
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,2022,11(11),7512-7523
PublishedOnlineNovemb er2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.1111796
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Low-RegularityIntegratorforthe
BiharmonicNLSEquation
CuiNing
SchoolofFinancialMathematicsandStatistics,GuangdongUniversityofFinance,Guangzhou
Guangdong
Received:Sep.28
th
,2022;accepted:Oct.21
st
,2022;published:Nov.1
st
,2022
Abstract
Inthispaper,weintroduceafirstorderlow-regularityintegratorforthebiharmonic
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[J].
A^
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,2022,11(11):7512-7523.
DOI:10.12677/aam.2022.1111796
w
}
nonlinearSchr¨odingerequation.Itonlyrequirestheboundednessofthreeadditional
derivativesof thesolutionto be thefirst orderconvergent.Byrigorous erroranalysis,
we show thatthe scheme provides firstorderaccuracyin
H
γ
(
T
d
)
forroughinitial data
in
H
γ
+3
(
T
d
)
with
γ>
d
2
.
Keywords
BiharmonicNonlinearSchr¨odingerEquation,Low-RegularityIntegrator,FirstOrder
Convergent
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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x
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C
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(2
π
)
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Z
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d
e
−
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x
f
(
x
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d
x
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DOI:10.12677/aam.2022.11117967514
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2
(
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2
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1
∈
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d
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ξ
−
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1
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ξ
1
.
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0
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•
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2
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γ
(
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d
)
=
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ξ
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d
(1+
|
ξ
|
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2
γ
|
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f
ξ
|
2
.
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½
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f
(
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−
1
•
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(
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−
1
f
=
(
|
ξ
|
−
2
ˆ
f
ξ
,
ξ
6
= 0
,
0
,
ξ
= 0
.
d
,
·
‚
„
I
‡
^
X
e
¼
ê
µ
ϕ
(
z
) =
e
z
−
1
z
,z
6
= 0
,
1
,z
= 0
.
(4)
y
3
,
·
‚
‰
Ñ
‡
^
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-
‡
O
.
Ú
n
2.
(
å
5
Ÿ
)
é
f
∈
H
γ
,t
∈
R
k
k
e
−
i
(
−
∆)
2
t
f
k
H
γ
=
k
f
k
H
γ
.
Proof.
Š
â
H
γ
‰
ê
½
Â
,
Œ
•
k
e
−
i
(
−
∆)
2
t
f
k
H
γ
=
X
ξ
∈
Z
d
(1+
|
ξ
|
)
2
γ
|
\
e
−
i
(
−
∆)
2
t
f
ξ
|
2
=
X
ξ
∈
Z
d
(1+
|
ξ
|
)
2
γ
|
e
−
i
|
ξ
|
4
t
ˆ
f
ξ
|
2
=
X
ξ
∈
Z
d
(1+
|
ξ
|
)
2
γ
|
ˆ
f
ξ
|
2
=
k
f
k
H
γ
.
DOI:10.12677/aam.2022.11117967515
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Ú
n
3.
(Kato-Ponce
Ø
ª
)
é
?
¿
γ>
d
2
,
f,g
∈
H
γ
,
k
e
¡
Ø
ª
¤
á
k
fg
k
H
γ
≤k
f
k
H
γ
k
g
k
H
γ
.
Proof.
T
Ú
n
y
²
Œ
ë
„
©
z
[27].
3.
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3
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0
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ò
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Ñ
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9
ž
˜
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'¼
ê
¥
˜
m
C
þ
x
,
X
u
(
t
) =
u
(
t,
x
).
,
,
·
‚
^
τ>
0
L
«ž
m
Ú
•
,
t
n
=
nτ
L
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m
‚
.
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5
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•
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(1),
Ï
L
Duhamel
ú
ª
,
Œ
u
(
t,
x
) =
e
−
it
(
−
∆)
2
u
0
−
iµ
Z
t
0
e
−
i
(
t
−
s
)(
−
∆)
2
(
|
u
(
s,x
)
|
2
u
(
s,
x
))
ds.
Ú
\
Û
=
C
þ
v
(
t,
x
) =
e
it
(
−
∆)
2
u
(
t,
x
)
,
(5)
k
v
(
t
n
+
τ
) =
v
(
t
n
)
−
iµ
Z
τ
0
e
i
(
t
n
+
s
)(
−
∆)
2
h
|
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
+
s
)
|
2
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
+
s
)
i
ds.
(6)
d
,
Û
=
C
þ
÷
v
∂
t
v
(
t
) =
−
iµe
it
(
−
∆)
2
h
|
e
−
it
(
−
∆)
2
v
(
t
)
|
2
e
−
it
(
−
∆)
2
v
(
t
)
i
,v
(0) =
u
0
.
(7)
Ï
•·
‚
•
I
‡
˜
Â
ñ
,
¤
±
Œ
±
^
v
(
t
n
+
s
)
≈
v
(
t
n
)
{
z
þ
ã
‚
ª
,
v
(
t
n
+
τ
) =
v
(
t
n
)+Φ(
v
(
t
n
))+
R
n
1
,
Ù
¥
Φ(
v
(
t
n
)) =
−
iµ
Z
τ
0
e
i
(
t
n
+
s
)(
−
∆)
2
h
|
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
)
|
2
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
)
i
ds,
Ú
R
n
1
=
−
iµ
Z
τ
0
e
i
(
t
n
+
s
)(
−
∆)
2
|
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
+
s
)
|
2
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
+
s
)
−|
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
)
|
2
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
)
ds.
(8)
DOI:10.12677/aam.2022.11117967516
A^
ê
Æ
?
Ð
w
}
ù
p
R
n
1
Œ
±
w
Š
´
˜
‡
Ø
›
”
K
5
p
‘
,
Œ
±
3
Ž
{
E
¥
ï
.
e
5
,
·
‚
•
I
‡
é
Φ(
v
(
t
n
))
?
1
©
Û
,
|
^
Fourier
Ð
mŒ
Φ(
v
(
t
n
)) =
−
iµ
X
ξ
∈
Z
d
X
ξ
1
,
ξ
2
,
ξ
3
∈
Z
d
ξ
=
ξ
1
+
ξ
2
+
ξ
3
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
x
e
it
n
α
Z
τ
0
e
isα
ds,
Ù
¥
·
‚
{P
¯
ˆ
v
1
=
¯
ˆ
v
ξ
1
(
t
n
),ˆ
v
2
=ˆ
v
ξ
2
(
t
n
)
Ú
ˆ
v
3
=ˆ
v
ξ
3
(
t
n
),
…
α
=
|
ξ
|
4
+
|
ξ
1
|
4
−|
ξ
2
|
4
−|
ξ
3
|
4
.
|
^
ξ
=
ξ
1
+
ξ
2
+
ξ
2
+
ξ
3
,
K
α
Œ
±
©
)
¤
α
=2
|
ξ
1
|
4
+
3
X
j,k
=1
j
6
=
k
|
ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
|
ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
ξ
2
j
¯
ξ
k
¯
ξ
h
=2
|
ξ
1
|
4
+
β,
Ù
¥
β
=
3
X
j,k
=1
j
6
=
k
|
ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
|
ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
ξ
2
j
¯
ξ
k
¯
ξ
h
=
O
(
3
X
j,k
=1
j
6
=
k
|
ξ
j
|
3
|
ξ
k
|
)
.
Ï
d
,
·
‚
Œ
±
ò
Φ(
v
(
t
n
))
¤
Φ(
v
(
t
n
)) =
−
iµ
X
ξ
∈
Z
d
X
ξ
1
,
ξ
2
,
ξ
3
∈
Z
d
ξ
=
ξ
1
+
ξ
2
+
ξ
3
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
x
e
it
n
α
Z
τ
0
e
is
2
|
ξ
1
|
4
ds
+
R
n
2
=
−
iµ
X
ξ
∈
Z
d
X
ξ
1
,
ξ
2
,
ξ
3
∈
Z
d
ξ
=
ξ
1
+
ξ
2
+
ξ
3
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
x
e
it
n
α
e
iτ
2
|
ξ
1
|
4
−
1
iτ
2
|
ξ
1
|
4
+
R
n
2
=
−
iµτe
it
n
(
−
∆)
2
(
e
−
it
n
(
−
∆)
2
v
(
t
n
))
2
··
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
(
t
n
)
+
R
n
2
,
Ù
¥
R
n
2
=
−
iµ
X
ξ
∈
Z
d
X
ξ
1
,
ξ
2
,
ξ
3
∈
Z
d
ξ
=
ξ
1
+
ξ
2
+
ξ
3
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
x
e
it
n
α
Z
τ
0
e
is
2
|
ξ
1
|
4
(
e
isβ
−
1)
ds.
(9)
ù
p
R
n
2
X
J
w
¤
´
˜
‡p
‘
Ø
?
Ž
{
,
K
‡
›
”
n
ê
Š
•
“
d
.
Ï
d
,
D
K
p
‘
R
n
1
Ú
R
n
2
,
·
‚
v
(
t
n
+1
)
≈
v
(
t
n
)+Ψ
n
(
v
(
t
n
))
,
DOI:10.12677/aam.2022.11117967517
A^
ê
Æ
?
Ð
w
}
Ù
¥
Ψ
n
(
f
) =
−
iµτe
it
n
(
−
∆)
2
(
e
−
it
n
(
−
∆)
2
f
)
2
·
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
f
.
(10)
•
,
·
‚
¤
ê
Š
Ž
{
E
µ
v
n
+1
=
v
n
−
iµτe
it
n
(
−
∆)
2
(
e
−
it
n
(
−
∆)
2
v
n
)
2
·
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
n
,
(11)
ù
p
n
≥
0,
v
0
=
u
0
,
¿
…
v
n
=
v
n
(
x
).
ò
Û
=
C
þ
(5)
‡
L
5
“
\
(11),
=
Œ
V
N
Ú
š
‚
5
Schr¨odinger
•
§
(1)
˜
Â
ñ
ê
Š
Ž
{
(2).
4.
½
n
1
y
²
!
·
‚
Ï
L
Û
Ü
Ø
©
Û
Ú
-
½
5
©
Û
é
˜
Â
ñ
(
J
‰
Ñ
˜
‡
î
‚
y
²
.
d
Ú
n
2
Œ
•
,
Û
=
C
þ
(5)
3
Sobolev
˜
m
´
å
,
K
k
k
u
(
t
n
)
−
u
n
k
H
γ
=
k
e
−
it
n
(
−
∆)
2
v
(
t
n
)
−
e
−
it
n
(
−
∆)
2
v
n
k
H
γ
=
k
v
(
t
n
)
−
v
n
k
H
γ
.
Ï
d
,
·
‚
•
I
‡
y
²
é
v
(
t
n
)
Ú
v
n
˜
Â
ñ
½
n
.
£
1
3
!
¥
°
(
)
(6)
v
(
t
n
+1
) =
v
(
t
n
)+Ψ
n
(
v
(
t
n
))+
R
n
1
+
R
n
2
,
Ú
ê
Š
)
(11)
v
n
+1
=
v
n
+Ψ
n
(
v
n
)
.
Ï
d
,
Œ
ü
‡
)
Ø
v
(
t
n
+1
)
−
v
n
+1
=
L
n
+
S
n
,
Ù
¥
L
n
=
R
n
1
+
R
n
2
,
Ú
S
n
=
v
(
t
n
)
−
v
n
+Ψ
n
(
v
(
t
n
))
−
Ψ
n
(
v
n
)
.
e
¡
,
·
‚
ò
O
Û
Ü
Ø
L
n
Ú
-
½
5
S
n
.
Ú
n
4.
(
Û
Ü
Ø
)
γ>
d
2
.
b
u
0
∈
H
γ
+3
,
K
•
3
~
ê
τ
0
Ú
C>
0
,
¦
é
u
?
¿
0
<τ
≤
τ
0
,
k
e
Ø
ª
¤
á
kL
n
k
H
γ
≤
Cτ
2
,
(12)
DOI:10.12677/aam.2022.11117967518
A^
ê
Æ
?
Ð
w
}
Ù
¥
τ
0
Ú
C>
0
=
•
6
u
T
Ú
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
.
Proof.
Š
â
L
n
½
Â
,
k
kL
n
k
H
γ
≤kR
n
1
k
H
γ
+
kR
n
2
k
H
γ
.
d
R
n
1
½
Â
(8),
|
^
Ú
n
2
Ú
Ú
n
3
Œ
kR
n
1
k
H
γ
.
Z
τ
0
k
(
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
+
s
)
−
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
))
|
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
+
s
)
|
2
+(
|
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
+
s
)
|
2
−|
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
)
|
2
)
e
−
i
(
t
n
+
s
)(
−
∆)
2
v
(
t
n
)
k
H
γ
ds
.
Z
τ
0
k
v
(
t
n
+
s
)
−
v
(
t
n
)
k
H
γ
k
v
(
t
n
+1
)
k
2
Hγ
+
k
v
(
t
n
)
k
Hγ
k
v
(
t
n
+
s
)
−
v
(
t
n
)
k
H
γ
(
k
v
(
t
n
+
s
)
k
Hγ
+
k
v
(
t
n
)
k
Hγ
)
ds.
2
d
ª
(7),
Œ
•
k
v
(
t
n
+
s
)
−
v
(
t
n
)
k
H
γ
.
Z
s
0
k
∂
t
v
(
t
n
+
t
)
k
H
γ
dt
.
Z
s
0
k|
e
−
i
(
t
n
+
t
)(
−
∆)
2
v
(
t
n
+
t,x
)
|
2
e
−
i
(
t
n
+
t
)(
−
∆)
2
v
(
t
n
+
t,x
)
k
H
γ
dt
.
Z
s
0
k
v
k
3
L
∞
((0
,T
);
H
γ
+3
)
dt
.
Cs
k
v
k
3
L
∞
((0
,T
);
H
γ
)
.
Ï
d
,
·
‚
Œ
kR
n
1
k
H
γ
.
Cτ
2
k
v
k
5
L
∞
((0
,T
);
H
γ
)
.
Ï
•
|
e
isβ
−
1
|∼|
sβ
|
,
Š
â
R
n
2
½
Â
(9)
Œ
kR
n
2
k
2
H
γ
.
X
ξ
∈
R
d
(1+
|
ξ
|
2
)
2
γ
|
X
ξ
1
,
ξ
2
,
ξ
3
∈
Z
d
ξ
=
ξ
1
+
ξ
2
+
ξ
3
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
x
e
it
n
α
Z
τ
0
e
is
2
|
ξ
1
|
4
(
e
isβ
−
1)
ds
|
2
.
C
X
ξ
∈
R
d
(1+
|
ξ
|
2
)
2
γ
X
ξ
1
,
ξ
2
,
ξ
3
∈
Z
d
ξ
=
ξ
1
+
ξ
2
+
ξ
3
|
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
Z
τ
0
sβds
|
2
.
qd
|
β
|
.
O
(
P
3
j,k
=1
j
6
=
k
|
ξ
j
|
3
|
ξ
k
|
),
Œ
•
kR
n
2
k
2
H
γ
.
Cτ
4
X
ξ
∈
R
d
(1+
|
ξ
|
2
)
2
γ
|
3
X
j,k
=1
j
6
=
k
(1+
|
ξ
j
|
)
3
(1+
|
ξ
k
|
)
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
|
2
.
Cτ
4
k
v
k
6
L
∞
((0
,T
);
H
γ
+3
)
.
DOI:10.12677/aam.2022.11117967519
A^
ê
Æ
?
Ð
w
}
n
þ
Œ
•
,
kL
n
k
H
γ
≤kR
n
1
k
H
γ
+
kR
n
2
k
H
γ
≤
Cτ
2
,
Ù
¥
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
.
–
d
,
·
‚
¤
Ú
n
4
y
²
.
Ú
n
5.
(
-
½
5
)
γ>
d
2
.
b
u
0
∈
H
γ
+3
,
K
•
3
~
ê
τ
0
Ú
C>
0
,
¦
é
u
?
¿
0
<τ
≤
τ
0
,
k
e
Ø
ª
¤
á
kS
n
k
H
γ
≤
(1+
Cτ
)
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
3
H
γ
,
Ù
¥
τ
0
Ú
C>
0
=
•
6
u
T
Ú
k
v
k
L
∞
((0
,T
);
H
γ
)
.
Proof.
d
S
n
½
Â
Œ
•
,
kS
n
k
H
γ
≤k
v
(
t
n
)
−
v
n
k
H
γ
+
k
Ψ
n
(
v
(
t
n
))
−
Ψ
n
(
v
n
)
k
H
γ
.
£
Ψ
n
½
Â
(10),
Œ
k
Ψ(
v
(
t
n
))
−
Ψ(
v
n
)
k
H
γ
.
Cτ
k
(
e
−
it
n
(
−
∆)
2
v
(
t
n
))
2
·
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
(
t
n
)
−
(
e
−
it
n
(
−
∆)
2
v
n
)
2
·
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
n
k
Hγ
.
(
Ü
Ú
n
2,
Œ
k
Ψ(
v
(
t
n
))
−
Ψ(
v
n
)
k
H
γ
.
Cτ
k
(
v
(
t
n
)
−
v
n
)(
v
(
t
n
)+
v
n
)
·
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
(
t
n
)
k
Hγ
+
Cτ
k
(
e
−
it
n
(
−
∆)
2
v
n
)
2
·
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
(
t
n
)
−
e
−
it
n
(
−
∆)
2
v
n
k
Hγ
.
d
Ú
n
3,
?
˜
Ú
Œ
k
Ψ(
v
(
t
n
))
−
Ψ(
v
n
)
k
H
γ
.
Cτ
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
2
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
3
H
γ
.
Cτ
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
3
H
γ
,
Ù
¥
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
)
.
Ï
d
,
·
‚
Œ
±
Ú
n
5
y
²
.
e
¡
,
(
Ü
Û
Ü
Ø
O
Ú
-
½
5
(
J
,
·
‚
‰
Ñ
½
n
1
y
²
.
Proof.
Š
â
Ú
n
4
Ú
Ú
n
5,
Œ
k
v
(
t
n
+1
)
−
v
n
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