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A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  21 May 2020

JOSÉ A. CARRILLO Open the ORCID record for JOSÉ A. CARRILLO [Opens in a new window] ,
KATRIN GRUNERT Open the ORCID record for KATRIN GRUNERT [Opens in a new window]  and
HELGE HOLDEN Open the ORCID record for HELGE HOLDEN [Opens in a new window]
JOSÉ A. CARRILLO
Affiliation:
Mathematical Institute, University of Oxford, OxfordOX2 6GG, UK; [email protected]
KATRIN GRUNERT
Affiliation:
Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491Trondheim, Norway; [email protected], [email protected]
HELGE HOLDEN
Affiliation:
Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491Trondheim, Norway; [email protected], [email protected]

Abstract

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We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) 35B35: Stability
Secondary: 35B60: Continuation and prolongation of solutions
Type
Differential Equations
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e27
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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