Singular limit problem of the Camassa–Holm type equation

We consider a shallow water equation of Camassa–Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove the strong convergence and establish the condition under which, as diffusion and dispersion parameters tend to zero, smooth solutions of the shallow water equation converge to the entropy solution of a scalar conservation law using methodology developed by Hwang and Tzavaras [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion–dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229–1254]. The proof relies on the kinetic formulation of conservation laws and the averaging lemma.