On the Convergence of Quasilinear Viscous Approximations Using Compensated Compactness

TL;DR

This paper proves that quasilinear viscous approximations converge almost everywhere to the unique entropy solution of scalar conservation laws in bounded domains, using the method of compensated compactness.

Contribution

It demonstrates the convergence of quasilinear viscous approximations to entropy solutions employing compensated compactness, extending previous results to bounded domains.

Findings

Almost everywhere convergence of viscous approximations.

Uniqueness of the entropy solution in bounded domains.

Application of compensated compactness method.

Abstract

Method of compensated compactness is used to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it Bardos et.al}\cite{MR542510}) of the corresponding scalar conservation laws in a bounded domain in $\mathbb{R}^{d}$, where the viscous term is of the form $\varepsilon div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$.