Global conservative and dissipative solutions of the generalized Camassa-Holm equation

Shouming Zhou; Chunlai Mu

doi:10.3934/dcds.2013.33.1713

Article Contents

2013, Volume 33, Issue 4: 1713-1739. Doi: 10.3934/dcds.2013.33.1713

This issue Previous Article Well-posedness for a modified two-component Camassa-Holm system in critical spaces Next Article

Global conservative and dissipative solutions of the generalized Camassa-Holm equation

Shouming Zhou^1, and
Chunlai Mu^1,

1.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331

Received: November 2011

Revised: February 2012

Published: April 2013

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Abstract

This paper is devoted to the continuation of solutions to the generalized Camassa-Holm equation beyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.

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Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35B10, 35Q53.

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