Zero diffusion-dispersion limits for scalar conservation laws

TL;DR

This paper proves the convergence of solutions to hyperbolic conservation laws with vanishing diffusion and dispersion, improving previous results and establishing optimal conditions for convergence in multiple dimensions.

Contribution

It extends Schonbek's work by providing an optimal balance condition between diffusion and dispersion, and generalizes convergence results to multi-dimensional laws.

Findings

Convergence of regularized solutions to discontinuous solutions established

Optimal conditions on diffusion and dispersion parameters derived

Multi-dimensional convergence proven using DiPerna's theorem

Abstract

We consider solutions of hyperbolic conservation laws regularized with vanishing diffusion and dispersion terms. Following a pioneering work by Schonbek, we establish the convergence of the regularized solutions toward discontinuous solutions of the hyperbolic conservation law. The proof relies on the method of compensated compactness in the $L^2$ setting. Our result improves upon Schonbek's earlier results and provides an optimal condition on the balance between the relative sizes of the diffusion and the dispersion parameters. A convergence result is also established for multi-dimensional conservation laws by relying on DiPerna's uniqueness theorem for entropy measure-valued solutions.