A NOTE ON THE CONVERGENCE OF THE SOLUTIONS OF THE CAMASSA-HOLM EQUATION TO THE ENTROPY ONES OF A SCALAR CONSERVATION LAW

We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution 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.