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

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.