Well-posedness and Dispersive/Diffusive Limit of a Generalized Ostrovsky–Hunter Equation

Abstract

We consider a generalized Ostrovsky–Hunter equation and the corresponding generalized Ostrovsky one with nonlinear dispersive effects. For the first equation, we study the well-posedness of entropy solutions for the Cauchy problem. For the second equation, we prove that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the first one. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.