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A CONTINUOUS INTERPOLATION BETWEEN CONSERVATIVE AND DISSIPATIVE SOLUTIONS FOR THE TWO-COMPONENT CAMASSA–HOLM SYSTEM

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

Published online by Cambridge University Press:  07 January 2015

KATRIN GRUNERT ,
HELGE HOLDEN  and
XAVIER RAYNAUD
KATRIN GRUNERT
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; [email protected]
HELGE HOLDEN
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; [email protected] Centre of Mathematics for Applications, University of Oslo, NO-0316 Oslo, Norway; [email protected]
XAVIER RAYNAUD
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway; [email protected] SINTEF ICT, Applied Mathematics, P.O. Box 124, NO-0314 Oslo, Norway; [email protected]

Abstract

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We introduce a novel solution concept, denoted ${\it\alpha}$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa–Holm system on the line with vanishing asymptotics. All the ${\it\alpha}$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) 35B35: Stability
Secondary: 35Q20: Boltzmann equations
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e1
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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/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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