The low regularity global solutions for the critical generalized KdV equation

TL;DR

This paper establishes global well-posedness for the critical generalized KdV equation in Sobolev spaces with low regularity (s > 6/13) using the I-method and multilinear correction techniques, advancing previous results.

Contribution

It introduces refined analytical methods to lower the regularity threshold for global solutions of the critical generalized KdV equation.

Findings

Global well-posedness for s > 6/13 in Sobolev spaces

Improved bounds on resonant interaction multipliers

Extension of previous results by Fonseca, Linares, Ponce, and Farah

Abstract

We prove that the Cauchy problem of the mass-critical generalized KdV equation is globally well-posed in Sobolev spaces $H^s(\R)$ 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).