Local well-posedness of the Camassa-Holm equation on the real line

Jae Min Lee; Stephen C. Preston

doi:10.3934/dcds.2017139

Article Contents

2017, Volume 37, Issue 6: 3285-3299. Doi: 10.3934/dcds.2017139

This issue Previous Article Two-phase incompressible flows with variable density: An energetic variational approach Next Article The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities

Local well-posedness of the Camassa-Holm equation on the real line

Jae Min Lee^1, and
Stephen C. Preston^1,2,

1.
Department of Mathematics, The Graduate Center, City University of New York, New York, NY 10016, USA
2.
Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210, USA

Received: November 30, 2016

Revised: December 31, 2016

Published: May 2017

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In this paper we prove the local well-posedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms with an appropriate decaying condition. This work was motivated by G. Misiolek who proved the same result for the Camassa-Holm equation on the periodic domain. We use the Lagrangian approach and rewrite the equation as an ODE on the Banach space. Then by using the standard ODE technique, we prove existence and uniqueness. Finally, we show the continuous dependence of the solution on the initial data by using the topological group property of the diffeomorphism group.

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Mathematics Subject Classification: 35Q35, 53D25.

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