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A Liouville Property with Application to Asymptotic Stability for the Camassa–Holm Equation

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Abstract

We prove a Liouville property for uniformly almost localized (up to translations) H1-global solutions of the Camassa–Holm equation with a momentum density that is a non-negative finite measure. More precisely, we show that such a solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of H1-functions with a momentum density that belongs to \({\mathcal{M}_+(\mathbb{R})}\). Finally, we also get an asymptotic stability result for a train of peakons.

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Authors and Affiliations

  1. Institut Denis Poisson, Université de Tours, Université d’Orléans, CNRS, Parc Grandmont, 37200, Tours, France

    Luc Molinet

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  1. Luc Molinet
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Correspondence to Luc Molinet.

Additional information

Communicated by A. Bressan

L.M. was partially supported by the French ANR Project GEODISP.

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Molinet, L. A Liouville Property with Application to Asymptotic Stability for the Camassa–Holm Equation. Arch Rational Mech Anal 230, 185–230 (2018). https://doi.org/10.1007/s00205-018-1243-3

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  • DOI: https://doi.org/10.1007/s00205-018-1243-3

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