The low regularity global solutions for the critical generalized KdV equation

Abstract

We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\mathbb{R})$ for $s>6/13$. Of course, we require that the mass is strictly less than that of the ground state in the focusing case. The main approach is the "I-method" together with the multilinear correction analysis. Moreover, we use some "partially refined" argument to lower the upper control of the multiplier in the resonant interactions. The result improves the previous works of Fonseca, Linares, Ponce (2003) and Farah (2009).

27pages, the mistake in the previous version is corrected; using I-method with the resonant decomposition gives an improvement over our previous results

References in corpus (3)

• Global rough solutions to the critical generalized KdV equation
• Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^2$
• A remark on global well-posedness below L^2 for the gKdV-3 equation

Cited by in corpus (7)

• Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$
• Global rough solutions to the critical generalized KdV equation
• Large data scattering for the defocusing supercritical generalized KdV equation
• Nonlinear Profile Decomposition and the Concentration Phenomenon for Supercritical Generalized KdV Equations
• On the mass-critical generalized KdV equation
• Random data Cauchy problem for a generalized KdV equation in the supercritical case
• Global well-posedness for periodic generalized KdV equation