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

Well-posedness for the supercritical gKdV equation

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.
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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