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

KATRIN GRUNERT; HELGE HOLDEN; XAVIER RAYNAUD

doi:10.1017/fms.2014.29

Forum of Mathematics, Sigma (Jan 2015)

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

KATRIN GRUNERT,
HELGE HOLDEN,
XAVIER RAYNAUD

Affiliations

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

DOI: https://doi.org/10.1017/fms.2014.29
Journal volume & issue: Vol. 3

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.

Published in Forum of Mathematics, Sigma

ISSN: 2050-5094 (Online)
Publisher: Cambridge University Press
Country of publisher: United Kingdom
LCC subjects: Science: Mathematics
Website: https://www.cambridge.org/core/journals/forum-of-mathematics-sigma

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