Preprint 2005-017

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

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 L^p setting.

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Author(s):

Giuseppe Maria Coclite, <giusepc@math.uio.no>

Kenneth Hvistendahl Karlsen, <kennethk@math.uio.no>

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<giusepc@math.uio.no> April 16 2005.