Scalar Green Function Bounds for Instantaneous Shock Location and One-dimensional Stability of Viscous Shock Waves

TL;DR

This paper establishes the nonlinear stability of scalar viscous shock waves by extending existing methods from systems of conservation laws, providing a simplified approach for scalar equations and small-amplitude cases.

Contribution

It adapts and simplifies the methods for nonlinear stability analysis of viscous shocks specifically for scalar conservation laws, extending prior work from Burgers equation to general scalar cases.

Findings

Proved nonlinear stability of scalar viscous shock waves.

Extended methods from Burgers to general scalar equations.

Provided a simple rescaling argument for small-amplitude shocks.

Abstract

In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun ["Instantaneous shock location ..."] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give by a rescaling argument a simple treatment of nonlinear stability in the small-amplitude case.