A singular limit problem for the Rosenau-Korteweg-de Vries-regularized long wave and Rosenau-regularized long wave equations

We consider the Rosenau-Korteweg-de Vries-regularized long wave and Rosenau-Korteweg-de Vries equations, which contain nonlinear dispersive effects. We prove that, as the diffusion parameter tends to zero, the solutions of the dispersive equations converge 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.