Well-posedness for the supercritical gKdV equation

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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

Abstract
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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 and 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, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. doi: 10.1007/978-3-642-66451-9.
[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), 1235-1293. doi: 10.1353/ajm.2003.0040.
[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), 357-377. doi: 10.4310/MRL.2011.v18.n2.a13.
[4]	Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.
[5]	Martin Hadac, Sebastian Herr and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941, Erratum: ibid., 3 (2010), 971-972. doi: 10.1016/j.anihpc.2008.04.002.
[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), 527-620. doi: 10.1002/cpa.3160460405.
[7]	Carlos E. Kenig, Gustavo Ponce and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[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), 145-198. doi: 10.2140/apde.2012.5.145.
[9]	Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa.20067.
[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), Art. ID rnm053, 36. doi: 10.1093/imrn/rnm053.
[11]	Luc Molinet and Francis Ribaud, On the Cauchy problem for the generalized Korteweg-de Vries equation, Comm. Partial Differential Equations, 28 (2003), 2065-2091. doi: 10.1081/PDE-120025496.
[12]	Terence Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019.
[13]	Norbert Wiener, The quadratic variation of a function and its fourier coefficients., Journ. Math. Phys., 3 (1924), 72-94.

show all references

References:

[1]	Jöran Bergh and Jörgen Löfström, "Interpolation Spaces. An Introduction", Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. doi: 10.1007/978-3-642-66451-9.
[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), 1235-1293. doi: 10.1353/ajm.2003.0040.
[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), 357-377. doi: 10.4310/MRL.2011.v18.n2.a13.
[4]	Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.
[5]	Martin Hadac, Sebastian Herr and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941, Erratum: ibid., 3 (2010), 971-972. doi: 10.1016/j.anihpc.2008.04.002.
[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), 527-620. doi: 10.1002/cpa.3160460405.
[7]	Carlos E. Kenig, Gustavo Ponce and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[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), 145-198. doi: 10.2140/apde.2012.5.145.
[9]	Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa.20067.
[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), Art. ID rnm053, 36. doi: 10.1093/imrn/rnm053.
[11]	Luc Molinet and Francis Ribaud, On the Cauchy problem for the generalized Korteweg-de Vries equation, Comm. Partial Differential Equations, 28 (2003), 2065-2091. doi: 10.1081/PDE-120025496.
[12]	Terence Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019.
[13]	Norbert Wiener, The quadratic variation of a function and its fourier coefficients., Journ. Math. Phys., 3 (1924), 72-94.

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