A singular limit problem for the Rosenau-Korteweg-de Vries-regulared long wave and Rosenau-korteweg-de Viers equation

TL;DR

This paper investigates the limit behavior of solutions to Rosenau-Korteweg-de Vries equations with nonlinear dispersive effects, showing convergence to entropy solutions of scalar conservation laws as diffusion vanishes.

Contribution

It establishes the convergence of solutions to the Rosenau-Korteweg-de Vries equations to entropy solutions of scalar conservation laws using a priori estimates and compensated compactness.

Findings

Solutions converge to entropy solutions as diffusion parameter approaches zero

The proof employs a priori estimates and compensated compactness in L^p spaces

Unique entropy solutions are characterized as the limit of dispersive equations

Abstract

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.