On the convergence of quasilinear viscous approximations with BV initial data

TL;DR

This paper proves that quasilinear viscous approximations converge to the unique entropy solution of scalar conservation laws with BV initial data on bounded domains, ensuring well-posedness under these conditions.

Contribution

It establishes the convergence of viscous approximations to the entropy solution for scalar conservation laws with BV initial data, extending previous results to bounded domains.

Findings

Almost everywhere convergence of viscous approximations

Uniqueness of the entropy solution in the BV setting

Convergence holds on bounded domains in 

Abstract

We show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it Bardos-Leroux-Nedelec}) of the corresponding scalar conservation laws on a bounded domain in $\mathbb{R}^{d}$ whenever the initial data is essentially bounded and a function of bounded variation.