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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(6),3011-3020
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126302
˜afYÅ•§±Ïš˜—ëY5
éééÜÜÜ
-Ÿ“‰ŒÆêƉÆÆ§-Ÿ
ÂvFϵ2023c528F¶¹^Fϵ2023c623F¶uÙFϵ2023c630F
Á‡
©ïÄ¥ÌfYÅ.±Ï…ܯK§·‚ÄkEü‡)S§¦‚3H
s
(T),s>
3/2¥´k.§¿…3Њž•«mÂñ"§´ùü‡Sƒmåle.3?¿ž
•T´˜‡š"~ê§ù¿›X•§)N3Sobolev˜m¥´š˜—ëY"
'…c
…ܯK§š˜—ëY5§fYÅ.
NonuniformContinuityofaClass
ofShallowWaterWaveEquation
SenlinCai
SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing
Received:May28
th
,2023;accepted:Jun.23
rd
,2023;published:Jun.30
th
,2023
Abstract
Inthispaper,theperiodicCauchyproblemofshallowwaterwavemodelwithmoderate
amplitudeisstudied,wefirstconstructtwosolutionsequences,theyareboundedin
©ÙÚ^:éÜ.˜afYÅ•§±Ïš˜—ëY5[J].A^êÆ?Ð,2023,12(6):3011-3020.
DOI:10.12677/aam.2023.126302
éÜ
H
s
(T),s>3/2,andconvergetozeroattheinitialtimeinterval,buttheboundofthe
distancebetweenthesetwosequenceisanon-zeroconstanceatany time.Thismeans
thatthesolutionmappingoftheequationisnon-uniformlycontinuousinSobolev
space.
Keywords
CauchyProblem,NonuniformContinuity,ShallowWaterWave
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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
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








η
t
+η
x
+
3
2
ηη
x
+δ
2
(aη
xxx
−µη
xxt
)−
3
8
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2
η
2
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+
3
16
ε
3
η
3
η
x
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15
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ε
4
η
4
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−
1
24
εδ
2
(cη
x
η
xx
+dηη
xxx
)−
21
256
ε
5
η
5
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+
27
512
ε
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η
6
η
x
−
1
32
ε
2
δ
2
(eηη
x
η
xx
+fη
2
η
xxx
+gη
3
x
),x∈R,t>0,
η(x,0) = η
0
(x),x∈R,t= 0.
(1.1)
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Ñ~fÒ´¥ÌfYÅCamassa-Holm•§µ
u
t
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u
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+3uu
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= 0.(1.2)
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5!» y–(„[5–9])§d3©z[10]¥§Šö$^KatoŒ+½ny²§·½5"´
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DOI:10.12677/aam.2023.1263023012A^êÆ?Ð
éÜ
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Y5‰ïÄ¿…É©z[13]éu§Š5¿´§•§(1.1)!Constantin-Lannes•§ÚCH•
§A©O´a
1
u+ a
2
u
2
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−µθ
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γuÚu§w,©¤ïÄ•§•3²•‘"Ïd§·‚
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½n1.1.e•3Њu
0
(x)∈H
s
(T)…s>3/2,@o•§(1.1)Њ)Nu
0
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k.8H
s
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s
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s−1
(T))´š˜—ëY"
½n1.1wŠ·‚•§(1.1))3¢ËÅH
s
(T)˜m¥´š˜—ëY"©(Xe"e
˜ Ü©§·‚£I‡^ØªÚÄ½Â§31nÜ©§·‚½ÂCq)¿OØ§3
1oÜ©§·‚OCq)†ý¢)ƒmØ¿$^¢ËŘmŠØª5Ÿ§y²
(J"
2.ý•£
©-ŽfΛ=(1 −∂
2
x
)
1/2
,KŠ^uL
2
(R)ŽfΛ
−2
Œ±^†§ƒ'‚¼êG(x)=
1
2
e
−|x|
L«•
Λ
−2
f(x) = (G∗f)(x) =
1
2
Z
R
e
−|x−y|
f(y)dy,f∈L
2
(R).
éu?¿s∈R,|^ŽfΛ
s
= (1−∂
2
x
)
s/2
½Âe¡$Ž
d
Λ
s
f(ξ) = (1+ξ
2
)
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ˆ
f(ξ),
Ù¥
ˆ
f(ξ)L«Fp“C†
ˆ
f(ξ)
.
=
Z
R
e
−ixξ
f(x)dx,ξ∈R.
@oéu?¿f∈H
s
(R)k
kfk
H
s
.
= kfk
H
s
(R)
=

Z
R
(1+ξ
2
)
s
|
ˆ
f(ξ)|
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dξ

1/2
<∞.
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¯
b, Ù¥θ=
1
√
µδ
, Ïd•§(1.1)Œ
=z•
(
u
t
+(a
1
u+a
2
u
2
)u
x
= f(u,u
x
),x∈R, t>0,
u(x,0) = u
0
(x),x∈R, t= 0,
(2.1)
Ù¥
f(u,u
x
) =(1−∂
2
x
)
−1
∂
x
(b
1
u
2
+b
2
u
3
+b
3
u
4
+b
4
u
5
+b
5
u
6
+b
6
u
7
+b
7
u
2
x
+b
8
uu
2
x
)+b
9
(1−∂
2
x
)
−1
u
3
x
.
(2.2)
3©¥,kfk
H
s
=kfk
H
s
(R)
,ÎÒ.Ú&^5L«ƒA•¹˜‡~êØª.~
DOI:10.12677/aam.2023.1263023013A^êÆ?Ð
éÜ
Xf(x) .g(x)L«éu,~êc>0,f(x) ≤cg(x).·‚„I‡^±eÚn.
Ún2.1.XJr>0,KH
r
∩L
∞
´˜‡“ê.d
(i)er>0,K
kfgk
H
r
≤c
r
(kfk
L
∞
kgk
H
r
+ kgk
L
∞
kfk
H
r
);
(ii)er>1/2,K
kfgk
H
r−1
≤c
r
kfk
H
r
kgk
H
r−1
;
(iii)e0 ≤r≤1,s>3/2,r+s≥2,K
kfgk
H
r−1
≤c
r,s
kfk
H
s−1
kgk
H
r−1
.
ùp1^´Calderon−Coifman−Meyer.†fO,„©z[14].
Ún2.2.(„[15]¥Ún1)XJ[Λ
r
,f]g= Λ
r
(fg)−fΛ
r
g…Λ = (1−∂
2
x
)
1
2
,Ke¡†fO
¤á
(i)er>0,K
k[Λ
r
,f]gk
L
2
≤c
r
(k∂
x
fk
L
∞
kΛ
r−1
gk
L
2
+ kΛ
r
fk
L
2
kgk
L
∞
);
(ii)er+1 ≥0,s>3/2,r+1 ≤s,K
k[Λ
r
∂
x
,f]gk
L
2
≤c
r,s
kfk
H
s
kgk
H
r
.
Ún2.3.(„[16])bσ
1
<σ<σ
2
…f∈H
σ
1
.K
kfk
H
σ
≤kfk
σ
2
−σ
σ
2
−σ
1
H
σ
1
kfk
σ−σ
1
σ
2
−σ
1
H
σ
2
.
3.ECq)
3ù˜!¥§·‚•Ä•§Cq)¿…-•§(2.1)Cq)/ª•
u
ω,n
(x,t) = ωn
−1
−n
−s
cosA,A= nx−(a
1
ω+a
1
ω
2
λ
−1
)t,
(3.1)
Ù¥ωu1½ö-1¿…nê"y3·‚OùCq)H
σ
‰êØ§òCq
)u
ω,n
(x,t)“\•§(2.1),·‚Œ±±eØµ
F= u
ω,n
t
(x,t)+a
1
u
ω,n
+a
2
(u
ω,n
)
2
u
ω,n
x
−f(u
ω,n
,u
ω,n
x
)
(3.2)
w,u
ω,n
t
+a
1
u
ω,n
+a
2
(u
ω,n
)
2
u
ω,n
x
Œ±¤Xeª
u
ω,n
t
+a
1
u
ω,n
+a
2
(u
ω,n
)
2
u
ω,n
x
= −1/2a
1
λ
−2s+1
sin2A−a
1
ωλ
−2s
sin2A+1/2a
2
λ
−3s
sin2Acos
2
A,
(3.3)
Ù¥·‚^5Ÿsin2A= 2sinAcosA.
DOI:10.12677/aam.2023.1263023014A^êÆ?Ð
éÜ
Ún3.1.-σ,α,βáu¢ê§náuê…n1,K±eª¤áµ
kcos(n(x+α)−β) k
H
σ
=
√
2π(1+n
2
)
σ/2
≈n
σ
.(3.4)
XJ£3.4¤¥cos†¤sinE,¤á"Kéu?¿s≥0,·‚k
ku
ω,n
(x,t) k
H
σ
=kωn
−1
−n
−s
cosA,k
H
σ
.n
−1
+n
−s+σ
,n1.(3.5)
Proof.ù‡Úny²Ú©z[17]¥y²aq§©òØ2?1y²"
e¡y²(3.3)H
σ
§¦^Ún3.1§·‚Œ±
k−1/2a
1
λ
−2s+1
sin2A−a
1
ωλ
−2s
sin2A+1/2a
2
λ
−3s
sin2Acos
2
Ak
H
σ
.λ
−2s+1+σ
+λ
−2s+σ
+λ
−3s+σ
.
(3.6)
e5y²f(u,u
x
)H
σ
§Ù¥·‚I‡^Ún2.1(ii)!Cauchy-SchwarzØªÚÚn3.1,
kf(u,u
x
)k
H
σ
= k(1−∂
x
)
−1
∂
x
(b
1
u
2
+b
2
u
3
+b
2
u
4
+b
3
u
4
+b
4
u
5
+b
5
u
6
+b
6
u
7
+b
7
u
2
x
+b
8
uu
2
x
)k
H
σ
+kb
9
(1−∂
x
)
−1
u
3
x
k
H
σ
.λ
−2s+1+σ
+λ
−s−1+σ
.
(3.7)
Šâ±þO§·‚Œ±±e·K"
·K3.1.éus>3/2,1/2 <σ≤1,n1,·‚k
kFk
H
σ
.n
−r
s
,(3.8)
Ù¥r
s
>0…
r
s
=
(
2s−1−σ,3/2 <s<2,
s+1−σ,s≥2.
(3.9)
4.Cq)†ý¢)ƒmØ
-u
ω,n
(t,x)´•§(2.1)±Ï)§Ku
ω,n
(t,x)÷v
(
∂
x
u
ω,n
+(a
1
u
ω,n
+a
2
u
2
ω,n
)∂
x
u
ω,n
−f(u
ω,n
,∂
x
u
ω,n
) = 0,
u
ω,n
(0,x) = u
ω,n
(0,x) = ωn
−1
−n
−s
cos(nx).
(4.1)
•OCq)†ý¢)Ø§·‚
v= u
ω,n
−u
ω,n
.
DOI:10.12677/aam.2023.1263023015A^êÆ?Ð
éÜ
w,§v÷v±eª







∂
t
v= F−a
1
u
ω,n
∂
x
v−a
1
v∂
x
u
ω,n
−a
2
(u
ω,n
)
2
∂
x
u
ω,n
+a
2
(u
ω,n
)
2
∂
x
u
ω,n
+f(u
ω,n
,∂
x
u
ω,n
)−f(u
ω,n
,∂
x
u
ω,n
),
V(0,x) = 0.
(4.2)
Ù¥F(3.2)½Â…÷vØª(3.8).
·K4.1.en1,s>3/2,1/2 <σ≤min1,s−1,K
kv(t) k
H
σ
.
=ku
ω,n
(t)−u
ω,n
(t) k
H
σ
.n
−r
s
,n1,0 ≤t≤T,(4.3)
Ù¥r
s
>0…d(3.9)‰Ñ"
Proof.^ŽfΛ
σ
¦±•§(4.2)ü>§,^Λ
σ
v¦±(J§2é±Ï˜mSx?1È©·‚
Œ±
1
2
d
dt
kv(t)k
2
H
σ
=
Z
T
Λ
σ
FΛ
σ
vdx−a
1
Z
T
Λ
σ
u
ω,n
∂
x
vΛ
σ
vdx−a
1
Z
T
Λ
σ
v∂
x
u
ω,n
Λ
σ
vdx
−a
2
Z
T
Λ
σ
(u
ω,n
)
2
∂
x
u
ω,n
Λ
σ
vdx+a
2
Z
T
Λ
σ
(u
ω,n
)
2
∂
x
u
ω,n
Λ
σ
vdx
+
Z
T
Λ
σ
(f(u
ω,n
,∂
x
u
ω,n
−f(u
ω,n
,∂
x
u
ω,n
))Λ
σ
vdx
= E
1
+E
2
++E
3
+E
4
+E
5
+E
6
.
(4.4)
e5·‚éØFH
ρ
?1O§Äk1˜‘OXeµ
|E
1
|=




Z
T
Λ
σ
FΛ
σ
vdx




≤kFk
H
σ
+ kvk
H
σ
.n
−r
s
||v(t)||
H
σ
(4.5)
1‘OXeµ
|E
2
|.




Z
T
Λ
σ
∂
x
(u
ω,n
,v)Λ
σ
vdx




+




Z
T
Λ
σ
v∂
x
u
ω,n
Λ
σ
vdx




≤||[Λ
σ
∂
x
,u
ω,n
]v||
L
2
||v||
H
σ
+
1
2




Z
T
∂
x
u
ω,n
(Λ
σ
v)
2
dx




+||u
ω,n
||
H
s
||v||
2
H
σ
.||u
ω,n
||
H
s
||v||
2
H
σ
,
(4.6)
Ù¥^©ÜÈ©ÚÚn2.2(ii).e5·‚Oª(4.4)1n‘Xeµ
|E
3
|=




Z
T
Λ
σ
v∂
x
u
ω,n
Λ
σ
vdx




≤||v∂
x
u
ω,n
||
H
σ
||v||
H
σ
≤C(||v||
H
σ
||∂
x
u
ω,n
||
L
∞
+||∂
x
u
ω,n
||
H
σ
||v||
H
σ
)||v||
H
σ
.||u
ω,n
||
H
s
||v||
2
H
σ
,
(4.7)
DOI:10.12677/aam.2023.1263023016A^êÆ?Ð
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Ù¥$^Ún2.1(i)ÚCauchy-SchwarØª"
aqc¡y²L§§·‚U•{ØH
σ
O§¿ò(J(Üå5Œ±
1
2
||v||
2
H
σ
.B||v||
2
H
σ
+n
−r
s
||v||
H
σ
.
(4.8)
=
d
dt
||v||
H
σ
.B||v||
H
σ
+n
−r
s
,
(4.9)
Ù¥B´•{‘N"·‚A^Øª
||u
ω,n
(t)||
H
s
.||u
ω,n
(0)||
H
s
= ||u
ω,n
(0)||
H
s
Œ±
||v(t)||
H
σ
.n
−r
s
,n10 ≤t≤T,
(4.10)
ùÒy²·K4.1.
e5·‚y²½n1.1
Proof.4·‚Äkbs>3/2,¿…u
1,n
(x,t)Úu
−i,n
(x,t)©O´•§(2.1)3Њ^‡•u
1,n
(x,0)Ú
u
−1,n
(x,0)),…ùü‡)áuC([0,T];H
s
(T)).dª(3.1)Ú||u(t)||
H
s
≤2||u(0)||
H
s
,·‚U
||u
1,n
(x,t)||
H
s
+||u
−1,n
(x,t)||
H
s
≤2C||u
1,n
(0)||
H
s
+||u
−1,n
(0)||
H
s
.1.(4.11)
t= o…λ−→∞žk
||u
1,n
(0)−u
−1,n
(0)||
H
s
−→0.(4.12)
t<0ž§$^nØªŒ±
||u
1,n
(x,t)||
H
s
+||u
−1,n
(x,t)||
H
s
≤2C||u
1,n
(0)||
H
s
+||u
−1,n
(0)||
H
s
.1.(4.13)
t= o…λ−→∞žk
||u
1,n
(t)−u
−1,n
(t)||
H
s
≥||u
1,n
(t)−u
−1,n
(t)||
H
s
−||u
1,n
(t)−u
1,n
(t)||
H
s
−||u
−1,n
(t)−u
1,n
(t)||
H
s
.
(4.14)
$^cosα−cosβ= −2sin
α+β
2
sin
α−β
2
Œ
||u
1,n
(t)−u
−1,n
(t)||
H
s
= ||2n
−1
−2n
−s
sin(nx−a
1
ωt−a
2
ω
2
λ
−1
t)sin(a
1
ωt+a
2
ω
2
λ
−1
t)||
H
s
.
(4.15)
DOI:10.12677/aam.2023.1263023017A^êÆ?Ð
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k= [s]+2,$^Ún3.1Ú||u(t)||
H
s
≤2||u(0)||
H
s
Œ
||u
ω,n
(t)−u
ω,n
(t)||
H
k
≤||u
ω,n
(t)||
H
k
+2||u
ω,n
(0)||
H
k
.n
k−s
,0 <t≤T.
(4.16)
¦^Ún2.3…-s
1
= σ,s
2
= [s]+2k
||u
ω,n
(t)−u
ω,n
(t)||
H
s
≤||u
ω,n
(t)−u
ω,n
(t)||
k−s
k−σ
H
σ
||u
ω,n
(t)−u
ω,n
(t)||
s−σ
k−σ
H
k
.n
−r
s
(k−s)
k−σ
n
(k−s)(s−σ)
k−σ
.n
−(r
s
−s+σ)
k−σ
,
(4.17)
òƒcr
s
“\þªŒ||u
ω,n
(t)−u
ω,n
(t)||
H
s
.n
−ε
s
,Ù¥
ε
s
=
(
(s−1)(k−s)
k−σ
,3/2 <s<2,
(k−s)
k−σ
,s≥2,
(4.18)
ùpε
s
Œu"§·‚(4.14)ü>4•e(.k
liminf
n→∞
ku
1,n
(t)−u
−1,n
(t) k
H
s
≥liminf
n→∞
(ku
1,n
(t)−u
−1,n
(t) k
H
s
−||u
1,n
(t)−u
1,n
(t) k
H
s
−||u
−1,n
(t)−u
−1,n
(t) k
H
s
)
&




sin
a
1
t+a
2
λ
−1
t
2




>0,
Ù¥0 <t<min
n
T,
2π
|a
1
+a
2
λ
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|
o
.ùÒy²½n1.1.
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5"
ë•©z
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