MR2976041
Takamori KATO
Takamori KATO
Well-Posedness for the Fifth Order KdV Equation
Funkcialaj Ekvacioj. Serio Internacia
55
2012
17--53
http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-1/55_17.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2976041
We consider the Cauchy problem of the fifth order KdV equation with low regularity data. We cannot apply the iteration argument to this problem when initial data is given in the Sobolev space $H^s$ for any $s\in\mathbb{R}$. So we give initial data in $H^{s,a}= H^s\cap\dot{H}^a$ with $a\leq\min\{s,0\}$. Then we recover more derivatives of the nonlinear term to be able to use the iteration method. Therefore we obtain the local well-posedness in $H^{s,a}$ in the case $s\geq\max\{-1/4,-2a-2\}$, $-3/2<a\leq-1/4$ and $(s,a)\neq(-1/4,-7/8)$. Moreover, we obtain the ill-posedness in some sense when $s<\max\{-1/4,-2a-2\}$, $a\leq-3/2$ or $a>-1/4$. The main tool is a variant of the Fourier restriction norm method, which is based on Kishimoto's work (2009).
Fifth order KdV equation, Well-posedness, Cauchy problem, Fourier restriction norm method, Low regularity.
35Q55.
55-17
2012
Well-Posedness for the Fifth Order KdV Equation
Takamori KATO
Takamori KATO
1
Bejenaru, J.; Tao, T.
Sharp well-posedness and ill-posedness results for a quadratic nonlinear Schrödinger equation
J. Funct. Anal.
233
2006
228-259
MR2204680
2
Benjamin, T. B.
Impulse, flow force and variational principles
IMA J. Appl. Math.
32
1984
3-68
MR0740456
3
Benney, D. J.
A general theory for interactions between short and long waves
Studies in Appl. Math.
56
1977
81-94
MR0463715
4
Bourgain, J.
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I: Schrödinger equations
Geom. Funct. Anal.
3
1993
107-156
MR1209299
5
Bourgain, J.
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part II: The KdV-equation
Geom. Funct. Anal.
3
1993
209-262
MR1215780
6
Bourgain, J.
Periodic Korteweg de Vries equation with measures as initial data
Selecta Math. (N.S.)
3
1997
115-159
MR1466164
7
Chen, W.; Li, J.; Miao, C.; Wu, J.
Low regularity solution of two fifth-order KdV type equations
J. Anal. Math.
107
2009
221-238
MR2496405
8
Ginibre, J.; Tsutsumi, Y.; Velo, G.
On the Cauchy problem for the Zakharov system
J. Funct. Anal.
151
1997
384-436
MR1491547
9
Grünrock, A.
On the hierarchies of higher order mKdV and KdV equaions
Cent. Eur. J. Math.
8
2010
500-536
MR2653659
10
Holmer, J.
Local ill-posedness of the 1D Zakharov system
Electron. J. Differential Equations
2007
2007
no. 24, 22pp
MR2299578
11
Kato, T. K.
Remark on well-posedness and ill-posedness for the KdV equation
Electron. J. Differential Equations
2010
2010
no. 142, 15pp
MR2729463
12
Kenig, C. E.; Ponce, G.; Vega, L.
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle
Comm. Pure Appl. Math.
46
1993
527-620
MR1211741
13
Kenig, C. E.; Ponce, G.; Vega, L.
Higher-order nonlinear dispersive equations
Proc. Amer. Math. Soc.
122
1994
157-166
MR1195480
14
Kishimoto, N.
Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity
Differential Integral Equations
22
2009
447-464
MR2501679
15
Kishimoto, N.; Tsugawa, K.
Local well-posedness for quadratic Schrödinger equations and "good" Boussinesq equation
Differential Integral Equations
23
2010
463-493
MR2654246
16
Koch, H.; Tzvetkov, N.
Nonlinear wave interactions for the Benjamin-Ono equation
Int. Math. Res. Not.
2005
2005
1833-1847
MR2172940
17
Koch, H.; Tzvetkov, N.
On finite energy solutions of the KP-I equation
Math. Z.
258
2008
55-68
MR2350033
18
Kwon, S.
On the fifth order KdV equation: Local well-posedness and lack of uniform continuity of the solution map
J. Differential Equations
245
2008
2627-2659
MR2455780
19
Kwon, S.
Well-posedness and ill-posedness of the fifth order modified KdV equation
Electron. J. Differential Equations
2008
2008
no. 1, 15pp
MR2368888
20
Molinet, L.; Saut, J. C.; Tzvetkov, N.
Ill-posedness issue for the Benjamin-Ono and related equations
SIAM J. Math. Anal.
33
2001
982-988
MR1885293
21
Molinet, L.; Saut, J. C.; Tzvetkov, N.
Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation
Duke Math. J.
115
2002
353-384
MR1944575
22
Muramatu, T.; Taoka, S.
The initial value problem for the 1-D semilinear Schrödinger equation in Besov space
J. Math. Soc. Japan
56
2004
853-888
MR2071676
23
Oliver, P. L.
Hamiltonian and non-Hamiltonian models for water waves
Lecture Notes in Phys., 195, Springer-Verlag, New York
1984
pp. 273--290
MR0755731
24
Pilod, D.
On the Cauchy problem for higher-order nonlinear dispersive equations
J. Differential Equations
245
2008
2055-2077
MR2446185
25
Ponce, G.
Lax pairs and higher order models for water waves
J. Differential Equations
102
1993
360-381
MR1216734
26
Tao, T.
Multilinear weighted convolution of $L^2$ functions, and application to nonlinear dispersive equations
Amer. J. Math.
123
2001
839-908
MR1854113
27
Triebel, H.
Theory of Function Spaces
Monographs in Mathematics, 78. Birkhäuser Verleg, Basel
1983
MR0781540
28
Tsugawa, K.
Well-posedness and weak rotation limit for the Ostrovsky equation
J. Differential Equations
247
2009
3163-3180
MR2571572