A note on convergence of the solutions of Benjamin–Bona–Mahony type equations

We consider the Benjamin-Bona-Mahony and the modified Benjamin-Bona-Mahony equations, which contains nonlinear dispersive effects. We prove that as the diffusion and dispersion parameters tend to zero, the solutions of these dispersive equations converge to the entropy ones of a scalar conservation law. The argument relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.