Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation

TL;DR

This paper establishes local and global well-posedness results for the Kawahara equation in low regularity Sobolev spaces, extending techniques used for KdV and addressing challenges due to fewer symmetries.

Contribution

It introduces a novel application of the I-method to the Kawahara equation, overcoming difficulties from its lack of symmetries and invariances.

Findings

Proves local well-posedness in H^{-7/4}

Extends to global well-posedness in H^s for s ≥ -7/4

Addresses challenges from reduced symmetries of the Kawahara equation

Abstract

We prove that the Kawahara equation is locally well-posed in $H^{-7/4}$ by using the ideas of $\bar{F}^s$-type space \cite{GuoKdV}. Next we show it is globally well-posed in $H^s$ for $s\geq -7/4$ by using the ideas of "I-method" \cite{I-method}. Compared to the KdV equation, Kawahara equation has less symmetries, such as no invariant scaling transform and not completely integrable. The new ingredient is that we need to deal with some new difficulties that are caused by the lack symmetries of this equation.