Global well-posedness for periodic generalized KdV equation

TL;DR

This paper proves the global well-posedness of periodic generalized KdV equations in certain Sobolev spaces, improving previous results and establishing sharp conditions for the quintic case using advanced analytical methods.

Contribution

It advances the understanding of gKdV equations by establishing sharper global well-posedness results in periodic settings, especially for the quintic case.

Findings

Global well-posedness for s ≥ 1/2 in quartic case

Global well-posedness for s > 5/9 in quintic case

Improved results over previous work from 2004

Abstract

In this paper, we show the global well-posedness for periodic gKdV equations in the space $H^s(\mathbb{T})$, $s\ge \frac12$ for quartic case, and $s> \frac59$ for quintic case. These improve the previous results of I-team in 2004. In particular, the result is sharp for quintic case. The main approaches are the I-method combining with the resonance decomposition developed by Miao et al in 2010, and a bilinear Strichartz estimate in periodic setting.