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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(11),7512-7523
PublishedOnlineNovemb er2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.1111796
VNÚš‚5Schr¨odinger•§$KŽ{
www}}}
2À7KÆ7KêƆÚOÆ§2À2²
ÂvFϵ2022c928F¶¹^Fϵ2022c1021F¶uÙFϵ2022c111F
Á‡
©ïÄVNÚš‚5Schr¨odinger•§äk˜Âñ˜«$KŽ{,Ž{3
›”nêcJeŒ±ˆ˜Âñ.Óž,·‚ÏLî‚Ø©Û,y²Њá
uH
γ+3
(T
d
)ž,VNÚš‚5Schr¨odinger•§3H
γ
(T
d
)þäk˜Âñ,Ù¥γ>
d
2
"
'…c
VNÚš‚5Schr¨odinger•§§$KŽ{§˜Âñ
Low-RegularityIntegratorforthe
BiharmonicNLSEquation
CuiNing
SchoolofFinancialMathematicsandStatistics,GuangdongUniversityofFinance,Guangzhou
Guangdong
Received:Sep.28
th
,2022;accepted:Oct.21
st
,2022;published:Nov.1
st
,2022
Abstract
Inthispaper,weintroduceafirstorderlow-regularityintegratorforthebiharmonic
©ÙÚ^:w}.VNÚš‚5Schr¨odinger•§$KŽ{[J].A^êÆ?Ð,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 firstorderaccuracyinH
γ
(T
d
)forroughinitial data
inH
γ+3
(T
d
)withγ>
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/
1.Úó
©•ÄeãVNÚš‚5Schr¨odinger•§





iu
t
= (−∆)
2
u+µ|u|
2
u,(t,x) ∈(R
+
,T
d
),
u(0,x) = u
0
(x),
(1)
Ù¥T
d
=(0,2π)
d
,u(t,x)´(R
+
,T
d
)þEŠ¼ê,u
0
(x)∈H
γ
(T
d
)(γ≥0)•‰½ÐŠ,µ=±1,
¿…d≥2.
VNÚš‚5Schr¨odinger•§•oš‚5Schr¨odinger•§,kŒþÆöéÙ?1ï
Ä.Zhu[1]y²à.oš‚5Schr¨odinger•§Ÿþ.žÄ)C©(±9)
»5.Guo [2]y²à.oš‚5Schr¨odinger•§Ÿþ‡.žÛ·½5±9»•Ð
Š)Ñ5.Ù¦a.Schr¨odinger•§·½5!-½5!»5•¡•2•ïÄ,Œë
„[3–11].
·‚Ø=Œ±lêÆnØþïÄš‚5Schr¨odinger•§,Ù¢,·‚„Œ±^ꊕ{
&?š‚5Schr¨odinger•§,~Xk•©•{[12,13],k••{[14,15],©•{[16,17],Ì•
{[18,19],ØëYGalerkin•{[20,21],•êÈ©ì{[22,23].
3¢S)¹¥,d uD(6Ä!ÿþØ5!‘Å5Ï,ЩêâÏ~Ã{÷vp1
w5‡¦.ŒÜ©•{‡¦ÐŠäkép1w5,•§)Ø1wž, •k3T)þN\˜
DOI:10.12677/aam.2022.11117967513A^êÆ?Ð
w}
½K5b,âU¢y,˜Âñ.Ïd,éu°()u(t
n
),·‚8I´é˜«$
KêŠ)u
n
,¦eª¤á:
ku(t
n
)−u
n
k
H
γ
≤Cτ
α
.
Ù¥,τL«žmÚ•,αL«Âñê,Њu
0
(x)=u(0,x)áu,‡H
γ+s
˜m.·‚F"s¦þ
,=ЩŠ¦ŒUk$K5,¢yαÂñ,ù•´8cŒ[é'5$K5Ž{.
éu²;š‚5Schr¨odinger•§,Lubich [24]|^StrangeSplittingŽ{,ЊáuH
γ+2
ž,
T•§3H
γ
¥˜Âñ.5,OstermannÚSchratz [25]|^#.•ê.Ž{,Њá
uH
γ+1
ž,3H
γ
¥˜Âñ.•C,WuÚYao [26]‰Ñ˜«FourierÈ©Ž{,Њá
uH
γ
ž,y²3H
γ
¥˜Âñ,TŽ{ØI‡›”?Ûê.
©ïÄVNÚš‚5Schr¨odinger•§(1),(Üþã•{,·‚E˜«#$KŽ{,
½Â•µ
u
n
= e
−iτ(−∆)
2
u
n−1
−iµτe
−iτ(−∆)
2

(u
n−1
)
2
·ϕ(2iτ(−∆)
2
)u
n−1

.(2)
·‚̇½n•
½n1.u
n
´VNÚš‚5Schr¨odinger•§(1)÷v(2)‚ªêŠ),½T>0.é?¿
γ>
d
2
,bu
0
(x) ∈H
γ+3
(T
d
),•3~êτ
0
,C>0¦é?¿0 <τ≤τ
0
,k
ku(t
n
)−u
n
k
H
γ+3
≤Cτ,n= 0,1,2,...,
T
τ
,(3)
Ù¥τ
0
ÚC>0=•6uTÚkuk
L
∞
((0,T);H
γ+3
)
.
©SüXe.312 !¥,·‚‰Ñ˜ÎÒÚ˜k^Ún.313 !¥,·‚‰Ñ˜
ꊂª̇EL§.314!¥,·‚y²½n1.
2.ý•£
3!¥,·‚0˜½Â!5ŸÚ-‡O.••BPÒ,·‚¦^A.B½öB&A
5L«Xe¹Âµ•3,ýé~êC>0,¦A≤CB,¿¦^A∼B5L«A.B.A.
éu•þξ:=(ξ
1
,...,ξ
d
) ∈Z
d
,ξ
1
:= (ξ
11
,...,ξ
1d
) ∈Z
d
,x:=(x
1
,...,x
d
) ∈T
d
,SÈÚ½ÂX
e:
ξ·x= ξ
1
x
1
+...+ξ
d
x
d
,|ξ|
2
= |ξ
1
|
2
+...+|ξ
d
|
2
.
3±ÏT
d
þ,·‚½Â¼êfFourierC†•
ˆ
f
ξ
=
1
(2π)
d
Z
T
d
e
−iξ·x
f(x)dx,
ÙFourier_C†•f(x) =
P
ξ∈Z
d
e
−iξ·x
ˆ
f
ξ
.
DOI:10.12677/aam.2022.11117967514A^êÆ?Ð
w}
é¼êf∈L
2
(T
d
),·‚½Âf(x)FourierÐmª•
f(x) =
X
ξ∈Z
d
ˆ
f
ξ
e
iξ·x
.
FourierC†~^5Ÿ
kfk
2
L
2
= 2π
X
ξ∈Z
d
|
ˆ
f
ξ
|
2
,
Ú
c
fg(ξ) =
X
ξ,ξ
1
∈Z
d
ˆ
f
ξ−ξ
1
ˆg
ξ
1
.
½ÂSobolev˜mH
γ
,γ>0‰ê•


f


2
H
γ
(T
d
)
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|
ˆ
f
ξ
|
2
.
·‚½ÂŽf(−∆)
−1
•
\
(−∆)
−1
f=
(
|ξ|
−2
ˆ
f
ξ
, ξ6= 0,
0, ξ= 0.
d,·‚„I‡^Xe¼êµ
ϕ(z) =



e
z
−1
z
,z6= 0,
1,z= 0.
(4)
y3,·‚‰Ñ‡^˜-‡O.
Ún2.(å5Ÿ)éf∈H
γ
,t∈Rk
ke
−i(−∆)
2
t
fk
H
γ
= kfk
H
γ
.
Proof.ŠâH
γ
‰ê½Â,Œ•
ke
−i(−∆)
2
t
fk
H
γ
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|
\
e
−i(−∆)
2
t
f
ξ
|
2
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|e
−i|ξ|
4
t
ˆ
f
ξ
|
2
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|
ˆ
f
ξ
|
2
= kfk
H
γ
.
DOI:10.12677/aam.2022.11117967515A^êÆ?Ð
w}
Ún3.(Kato-PonceØª)é?¿γ>
d
2
,f,g∈H
γ
,ke¡Øª¤á
kfgk
H
γ
≤kfk
H
γ
kgk
H
γ
.
Proof.TÚny²Œë„©z[27].
3.ꊂªE
3!¥,·‚0ꊂªEL§.•{zÎÒ,·‚òŽÑ¤9ž˜ƒ'¼ê¥
˜mCþx,Xu(t) = u(t,x).,,·‚^τ>0L«žmÚ•,t
n
= nτL«žm‚.
VNÚš‚5Schr¨odinger•§(1),ÏLDuhamelúª,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=Cþ
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=Cþ÷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‡˜Âñ,¤±Œ±^v(t
n
+s) ≈v(t
n
){zþオ,
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.2022.11117967516A^êÆ?Ð
w}
ùpR
n
1
Œ±wŠ´˜‡Ø›”K5p‘,Œ±3Ž{E¥ï.
e5,·‚•I‡éΦ(v(t
n
))?1©Û,|^FourierÐmŒ
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
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
e
it
n
α
Z
τ
0
e
isα
ds,
Ù¥·‚{P
¯
ˆv
1
=
¯
ˆv
ξ
1
(t
n
),ˆv
2
=ˆv
ξ
2
(t
n
)Úˆv
3
=ˆv
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
,KαŒ±©)¤
α=2|ξ
1
|
4
+
3
X
j,k=1
j6=k
|ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h=1
j6=k6=h
|ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h=1
j6=k6=h
ξ
2
j
¯
ξ
k
¯
ξ
h
=2|ξ
1
|
4
+β,
Ù¥
β=
3
X
j,k=1
j6=k
|ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h=1
j6=k6=h
|ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h=1
j6=k6=h
ξ
2
j
¯
ξ
k
¯
ξ
h
= O(
3
X
j,k=1
j6=k
|ξ
j
|
3
|ξ
k
|).
Ïd,·‚Œ±òΦ(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
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
e
it
n
α
Z
τ
0
e
is2|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
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
e
it
n
α
e
iτ2|ξ
1
|
4
−1
iτ2|ξ
1
|
4
+R
n
2
=−iµτe
it
n
(−∆)
2

(e
−it
n
(−∆)
2
v(t
n
))
2
··ϕ(2iτ(−∆)
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
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
e
it
n
α
Z
τ
0
e
is2|ξ
1
|
4
(e
isβ
−1)ds.(9)
ùpR
n
2
XJw¤´˜‡p‘Ø?Ž{,K‡›”nꊕ“d.Ïd,DKp‘R
n
1
ÚR
n
2
,·‚
v(t
n+1
) ≈v(t
n
)+Ψ
n
(v(t
n
)),
DOI:10.12677/aam.2022.11117967517A^êÆ?Ð
w}
Ù¥
Ψ
n
(f) = −iµτe
it
n
(−∆)
2

(e
−it
n
(−∆)
2
f)
2
·ϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
f

.(10)
•,·‚¤ꊎ{Eµ
v
n+1
= v
n
−iµτe
it
n
(−∆)
2

(e
−it
n
(−∆)
2
v
n
)
2
·ϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
v
n

,(11)
ùpn≥0,v
0
= u
0
,¿…v
n
= v
n
(x).
òÛ=Cþ(5)‡L5“\(11),=ŒVNÚš‚5Schr¨odinger•§(1)˜ÂñêŠ
Ž{(2).
4.½n1y²
!·‚ÏLÛÜØ©ÛÚ-½5©Ûé˜Âñ(J‰Ñ˜‡î‚y².dÚn2Œ•,
Û=Cþ(5)3Sobolev˜m´å,Kk
ku(t
n
)−u
n
k
H
γ
= ke
−it
n
(−∆)
2
v(t
n
)−e
−it
n
(−∆)
2
v
n
k
H
γ
= kv(t
n
)−v
n
k
H
γ
.
Ïd,·‚•I‡y²év(t
n
)Úv
n
˜Âñ½n.
£13!¥°()(6)
v(t
n+1
) = v(t
n
)+Ψ
n
(v(t
n
))+R
n
1
+R
n
2
,
ÚêŠ)(11)
v
n+1
= v
n
+Ψ
n
(v
n
).
Ïd,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OÛÜØL
n
Ú-½5S
n
.
Ún4.(ÛÜØ)γ>
d
2
.bu
0
∈H
γ+3
,K• 3~êτ
0
ÚC>0,¦éu?¿0 <τ≤τ
0
,
keØª¤á
kL
n
k
H
γ
≤Cτ
2
,(12)
DOI:10.12677/aam.2022.11117967518A^êÆ?Ð
w}
Ù¥τ
0
ÚC>0=•6uTÚkvk
L
∞
((0,T);H
γ+3
)
.
Proof.ŠâL
n
½Â,k
kL
n
k
H
γ
≤kR
n
1
k
H
γ
+kR
n
2
k
H
γ
.
dR
n
1
½Â(8),|^Ún2ÚÚn3Œ
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
kv(t
n
+s)−v(t
n
)k
H
γ
kv(t
n+1
)k
2
Hγ
+kv(t
n
)k
Hγ
kv(t
n
+s)−v(t
n
)k
H
γ
(kv(t
n
+s)k
Hγ
+kv(t
n
)k
Hγ
)ds.
2dª(7),Œ•
kv(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
kvk
3
L
∞
((0,T);H
γ+3
)
dt
.Cskvk
3
L
∞
((0,T);H
γ
)
.
Ïd,·‚Œ
kR
n
1
k
H
γ
.Cτ
2
kvk
5
L
∞
((0,T);H
γ
)
.
Ï•|e
isβ
−1|∼|sβ|,ŠâR
n
2
½Â(9)Œ
kR
n
2
k
2
H
γ
.
X
ξ∈R
d
(1+|ξ|
2
)
2γ
|
X
ξ
1
,ξ
2
,ξ
3
∈Z
d
ξ=ξ
1
+ξ
2
+ξ
3
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
e
it
n
α
Z
τ
0
e
is2|ξ
1
|
4
(e
isβ
−1)ds|
2
.C
X
ξ∈R
d
(1+|ξ|
2
)
2γ
X
ξ
1
,ξ
2
,ξ
3
∈Z
d
ξ=ξ
1
+ξ
2
+ξ
3
|
¯
ˆv
1
ˆv
2
ˆv
3
Z
τ
0
sβds|
2
.
qd|β|.O(
P
3
j,k=1
j6=k
|ξ
j
|
3
|ξ
k
|),Υ
kR
n
2
k
2
H
γ
.Cτ
4
X
ξ∈R
d
(1+|ξ|
2
)
2γ
|
3
X
j,k=1
j6=k
(1+|ξ
j
|)
3
(1+|ξ
k
|)
¯
ˆv
1
ˆv
2
ˆv
3
|
2
.Cτ
4
kvk
6
L
∞
((0,T);H
γ+3
)
.
DOI:10.12677/aam.2022.11117967519A^êÆ?Ð
w}
nþŒ•,
kL
n
k
H
γ
≤kR
n
1
k
H
γ
+kR
n
2
k
H
γ
≤Cτ
2
,
Ù¥C=•6ukvk
L
∞
((0,T);H
γ+3
)
.–d,·‚¤Ún4y².
Ún5.(-½5)γ>
d
2
.bu
0
∈H
γ+3
,K•3~êτ
0
ÚC>0,¦éu?¿0 <τ≤τ
0
,k
eØª¤á
kS
n
k
H
γ
≤(1+Cτ)kv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
3
H
γ
,
Ù¥τ
0
ÚC>0=•6uTÚkvk
L
∞
((0,T);H
γ
)
.
Proof.dS
n
½ÂŒ•,
kS
n
k
H
γ
≤kv(t
n
)−v
n
k
H
γ
+kΨ
n
(v(t
n
))−Ψ
n
(v
n
)k
H
γ
.
£Ψ
n
½Â(10),Œ
kΨ(v(t
n
))−Ψ(v
n
)k
H
γ
.Cτk(e
−it
n
(−∆)
2
v(t
n
))
2
·ϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
v(t
n
)
−(e
−it
n
(−∆)
2
v
n
)
2
·ϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
v
n
k
Hγ
.
(ÜÚn2,Œ
kΨ(v(t
n
))−Ψ(v
n
)k
H
γ
.Cτk(v(t
n
)−v
n
)(v(t
n
)+v
n
)·ϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
v(t
n
)k
Hγ
+Cτk(e
−it
n
(−∆)
2
v
n
)
2
·ϕ(2iτ(−∆)
2
)

e
−it
n
(−∆)
2
v(t
n
)−e
−it
n
(−∆)
2
v
n

k
Hγ
.
dÚn3,?˜ÚŒ
kΨ(v(t
n
))−Ψ(v
n
)k
H
γ
.Cτkv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
2
H
γ
+Cτkv(t
n
)−v
n
k
3
H
γ
.Cτkv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
3
H
γ
,
Ù¥C=•6ukvk
L
∞
((0,T);H
γ
)
.Ïd,·‚Œ±Ún5y².
e¡,(ÜÛÜØOÚ-½5(J,·‚‰Ñ½n1y².
Proof.ŠâÚn4ÚÚn5,Œ
kv(t
n+1
)−v
n+1
k
H
γ
≤Cτ
2
+(1+Cτ)kv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
3
H
γ
,
DOI:10.12677/aam.2022.11117967520A^êÆ?Ð
w}
Ù¥n= 0,1,2,...,
T
τ
−1ÚC=•6uTÚkvk
L
∞
((0,T);H
γ+3
)
.
|^S“{ÚGronwall’sØª,·‚Œ
kv(t
n+1
)−v
n+1
k
H
γ
≤Cτ
2
n
X
j=0
(1+Cτ)
j
≤Cτ,n= 0,1,2,...,
T
τ
−1.
ùÒ¤˜Âñ½ny².
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11901120)"
ë•©z
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