Conservation laws with vanishing nonlinear diffusion and dispersion

TL;DR

This paper investigates the limiting behavior of solutions to conservation laws with diminishing nonlinear diffusion and dispersion, establishing convergence to entropy solutions under specific conditions, inspired by classical pseudo-viscosity methods.

Contribution

It provides a rigorous analysis of the convergence of solutions with vanishing nonlinear diffusion and dispersion to entropy solutions, extending classical pseudo-viscosity approaches.

Findings

Proves convergence to entropy solutions under certain conditions.

Establishes the relative size condition between diffusion and dispersion.

Connects classical pseudo-viscosity methods to modern analysis.

Abstract

We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This work is motivated by the pseudo-viscosity approximation introduced by Von Neumann in the 50's.