Preprint 2005-017

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

G. M. Coclite and K. H. Karlsen

Abstract: We consider a shallow water equation of Camassa–Holm type, containing 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 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 Lp setting.

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Giuseppe Maria Coclite, <>
Kenneth Hvistendahl Karlsen, <>
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<> April 16 2005.

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