Low regularity solutions of two fifth-order KdV type equations

TL;DR

This paper proves local well-posedness for Kawahara and modified Kawahara equations in low regularity Sobolev spaces using dyadic block estimates and Bourgain spaces, advancing understanding of these fifth-order KdV type equations.

Contribution

It establishes the first local well-posedness results for Kawahara and modified Kawahara equations at low regularity levels in Sobolev spaces.

Findings

Well-posedness for Kawahara in $H^s$ with $s>-rac74$

Well-posedness for modified Kawahara in $H^s$ with $s\\ge -\frac14$

Development of dyadic block estimates using Tao's $[k; Z]$ multiplier norm method

Abstract

The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in $H^s({\mathbf R})$ with $s>-\frac74$ and the local well-posedness for the modified Kawahara equation in $H^s({\mathbf R})$ with $s\ge-\frac14$. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the $[k; Z]$ multiplier norm method of Tao \cite{Tao2001} and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.