On the Convergence of Quasilinear Viscous Approximations with Degenerate Viscosity

TL;DR

This paper proves that quasilinear viscous approximations with degenerate viscosity converge to the unique entropy solution of scalar conservation laws, using velocity averaging techniques on bounded domains.

Contribution

It establishes convergence of viscous approximations with degenerate viscosity to entropy solutions, extending previous results to a broader class of viscous terms.

Findings

Almost everywhere convergence of viscous approximations

Uniqueness of entropy solutions in the degenerate viscosity setting

Application of velocity averaging lemma in convergence proof

Abstract

We use Velocity Averaging lemma to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it F. Otto}) of the corresponding scalar conservation laws on a bounded domain in $\mathbb{R}^{d}$, where the viscous term is of the form $\varepsilon\,div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$ and $B\geq 0$.