A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation

Abstract

We consider a shallow water equation of the 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.

Keywords:

• Burgers equation
• Compensated compactness
• Conservation law
• Entropy condition
• Shallow water equation
• Singular limit

2000 Mathematics Subject Classification:

• 35G25
• 35L65
• 35B25

Acknowledgments

The research of K. H. Karlsen is supported by the Research Council of Norway through the BeMatA program and an Outstanding Young Investigators Award and by the European network HYKE, contract HPRN-CT-2002-00282.