We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation

COCLITE, Giuseppe Maria;
2006

Abstract

We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11586/48082
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