Low regularity data for the periodic Kawahara equation

TL;DR

This paper investigates the well-posedness of the periodic Kawahara equation with low regularity initial data, establishing local and global results for certain Sobolev spaces and contrasting with non-periodic cases.

Contribution

It extends the understanding of the Kawahara equation's well-posedness to low regularity data in the periodic setting, using Fourier restriction and I-method techniques.

Findings

Local well-posedness for s ≥ -3/2

Ill-posedness for s < -3/2

Global solutions extend for s ≥ -1

Abstract

In this paper, we consider the well-posedness for the Cauchy problem of the Kawahara equation with low regularity data in the periodic case. We obtain the local well-posedness for $s \geq -3/2$ by a variant of the Fourier restriction norm method. On the other hand, we show the ill-posedness for $s <-3/2$ in weak sense. Moreover, the local solutions can be extended globally in time for $s \geq -1$ by the I-method. This is a shape contrast to the results in the non-periodic setting, where the critical exponent is equal to -2.