On the convergence of quasilinear viscous approximations using compensated compactness and Kinetic Formulation

TL;DR

This paper proves that quasilinear viscous approximations converge to the unique entropy solution of scalar conservation laws using compensated compactness and kinetic formulation methods.

Contribution

It introduces a novel approach combining compensated compactness and kinetic formulation to establish convergence of viscous approximations.

Findings

Almost everywhere limit of viscous approximations equals the entropy solution.

Convergence holds for scalar conservation laws with quasilinear viscous terms.

The method ensures uniqueness and stability of the limit solution.

Abstract

We use the method of Compensated Compactness and Kinteic Formulation 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 $\varepsilon\,div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$.