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; Karlsen, K. H.. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - 31:8(2006), pp. 1253-1272. [10.1080/03605300600781600]

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

COCLITE, Giuseppe Maria;
2006-01-01

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.
2006
A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation / Coclite, Giuseppe Maria; Karlsen, K. H.. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - 31:8(2006), pp. 1253-1272. [10.1080/03605300600781600]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/93879
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