A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law

We consider the high order Camassa-Holm equation, which is a non linear dispersive equation of the fifth order. We prove that as the diffusion and dispersion parameters tends to zero, the solutions converge to the entropy ones 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.