  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.
