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March  2014, 13(2): 527-542. doi: 10.3934/cpaa.2014.13.527

Well-posedness for the supercritical gKdV equation

Nils Strunk 1,

1. 

Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld, Germany

Received  September 2012 Revised  July 2013 Published  October 2013

In this paper we consider the supercritical generalized Korteweg-de~Vries equation $\partial_t\psi + \partial_{x x x}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5 \leq p \in R$. We prove a local well-posedness result in the homogeneous Besov space $\dot B_\infty^{s_p,2}(R)$, where $s_p=\frac12-\frac{2}{p-1}$ is the scaling critical index. In particular local well-posedness in the smaller inhomogeneous Sobolev space $H^{s_p}(R)$ can be proved similarly. As a byproduct a global well-posedness result for small initial data is also obtained.
Keywords: supercritical, Korteweg-de~Vries equation, KdV-like equations, Besov space., Cauchy problem, well-posedness.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B3.
Citation: Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
References:
[1]

Jöran Bergh and Jörgen Löfström, "Interpolation Spaces. An Introduction",, Springer-Verlag, (1976).  doi: 10.1007/978-3-642-66451-9.  Google Scholar

[2]

Michael Christ, James E. Colliander and Terence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations,, Amer. J. Math., 125 (2003), 1235.  doi: 10.1353/ajm.2003.0040.  Google Scholar

[3]

Luiz G. Farah, Felipe Linares and Ademir Pastor, The supercritical generalized KdV equation: global well-posedness in the energy space and below,, Math. Res. Lett., 18 (2011), 357.  doi: 10.4310/MRL.2011.v18.n2.a13.  Google Scholar

[4]

Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3,, Differential Integral Equations, 18 (2005), 1333.   Google Scholar

[5]

Martin Hadac, Sebastian Herr and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 26 (2009), 917.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[6]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Comm. Pure Appl. Math., 46 (1993), 527.  doi: 10.1002/cpa.3160460405.  Google Scholar

[7]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[8]

Herbert Koch and Jeremy L. Marzuola, Small data scattering and soliton stability in $\dot H^{-\frac16}$ for the quartic KdV Equation,, Anal. PDE, 5 (2012), 145.  doi: 10.2140/apde.2012.5.145.  Google Scholar

[9]

Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators,, Comm. Pure Appl. Math., 58 (2005), 217.  doi: 10.1002/cpa.20067.  Google Scholar

[10]

Herbert Koch and Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces,, Int. Math. Res. Not. IMRN, 2007 (2007).  doi: 10.1093/imrn/rnm053.  Google Scholar

[11]

Luc Molinet and Francis Ribaud, On the Cauchy problem for the generalized Korteweg-de Vries equation,, Comm. Partial Differential Equations, 28 (2003), 2065.  doi: 10.1081/PDE-120025496.  Google Scholar

[12]

Terence Tao, Scattering for the quartic generalised Korteweg-de Vries equation,, J. Differential Equations, 232 (2007), 623.  doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[13]

Norbert Wiener, The quadratic variation of a function and its fourier coefficients.,, Journ. Math. Phys., 3 (1924), 72.   Google Scholar

show all references

References:
[1]

Jöran Bergh and Jörgen Löfström, "Interpolation Spaces. An Introduction",, Springer-Verlag, (1976).  doi: 10.1007/978-3-642-66451-9.  Google Scholar

[2]

Michael Christ, James E. Colliander and Terence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations,, Amer. J. Math., 125 (2003), 1235.  doi: 10.1353/ajm.2003.0040.  Google Scholar

[3]

Luiz G. Farah, Felipe Linares and Ademir Pastor, The supercritical generalized KdV equation: global well-posedness in the energy space and below,, Math. Res. Lett., 18 (2011), 357.  doi: 10.4310/MRL.2011.v18.n2.a13.  Google Scholar

[4]

Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3,, Differential Integral Equations, 18 (2005), 1333.   Google Scholar

[5]

Martin Hadac, Sebastian Herr and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 26 (2009), 917.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[6]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Comm. Pure Appl. Math., 46 (1993), 527.  doi: 10.1002/cpa.3160460405.  Google Scholar

[7]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573.  doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[8]

Herbert Koch and Jeremy L. Marzuola, Small data scattering and soliton stability in $\dot H^{-\frac16}$ for the quartic KdV Equation,, Anal. PDE, 5 (2012), 145.  doi: 10.2140/apde.2012.5.145.  Google Scholar

[9]

Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators,, Comm. Pure Appl. Math., 58 (2005), 217.  doi: 10.1002/cpa.20067.  Google Scholar

[10]

Herbert Koch and Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces,, Int. Math. Res. Not. IMRN, 2007 (2007).  doi: 10.1093/imrn/rnm053.  Google Scholar

[11]

Luc Molinet and Francis Ribaud, On the Cauchy problem for the generalized Korteweg-de Vries equation,, Comm. Partial Differential Equations, 28 (2003), 2065.  doi: 10.1081/PDE-120025496.  Google Scholar

[12]

Terence Tao, Scattering for the quartic generalised Korteweg-de Vries equation,, J. Differential Equations, 232 (2007), 623.  doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[13]

Norbert Wiener, The quadratic variation of a function and its fourier coefficients.,, Journ. Math. Phys., 3 (1924), 72.   Google Scholar

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