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We consider the Rosenau–Korteweg-de Vries-regularized long wave and Rosenau-regularized long wave equations, which contain nonlinear dispersive effects. We prove that by adding small diffusion to the equations, as the diffusion and dispersion parameters tends to zero, the solutions of the duffusive/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 theL  p${L^{p}}$    setting.

A Singular Limit Problem for the Rosenau–Korteweg-de Vries-Regularized Long Wave and Rosenau-regularized Long Wave Equations