Global well-posedness for the Kawahara equation with low regularity data

TL;DR

This paper proves the global well-posedness of the Kawahara equation with low regularity initial data by combining the Fourier restriction norm method and the I-method in a modified Bourgain space.

Contribution

It introduces a novel approach by applying the I-method to the modified Bourgain space for the Kawahara equation, extending local solutions globally.

Findings

Established local well-posedness in a suitable function space.

Extended solutions globally in time using the I-method.

Applied the I-method to a modified Bourgain space for the first time in this context.

Abstract

We consider the global well-posedness for the Cauchy probelem of the Kawahara equation which is one of the fifth order KdV type equations. We first establish the local well-posedness in a more suitable function space for the global well-posedness by a variant of the Fourier restriction norm method. Next, we extend local solutions globally in time by the I-method. In this paper, we apply the I-method to the modified Bourgain space.