A liouville property with application to asymptotic stability for the camassa-holm equation

TL;DR

This paper proves a Liouville property for certain solutions of the Camassa-Holm equation, showing they must be peakons, and establishes their asymptotic stability, including for trains of peakons.

Contribution

It introduces a Liouville property for almost localized solutions and demonstrates asymptotic stability of peakons and their trains in the Camassa-Holm equation.

Findings

Almost localized solutions are necessarily peakons.

Peakons are asymptotically stable in H 1 with measure-valued momentum.

Train of peakons also exhibit asymptotic stability.

Abstract

We prove a Liouville property for uniformly almost localized (up to translations) H 1-global solutions of the Camassa-Holm equation with a momentum density that is a non negative finite measure. More precisely, we show that such solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of H 1-functions with a momentum density that belongs to M + (R). Finally, we also get an asymptotic stability result for train of peakons.