On the convergence of the modified Rosenau and the modified Benjamin-Bona-Mahony equations

We consider the modified Rosenau and the modified Benjamin-Bona-Mahony equations, which contain nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equations converge to entropy solutions of a scalar conservation laws. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting