Instantaneous shock location and one-dimensional nonlinear stability of viscous shock wave

TL;DR

This paper demonstrates an approach for tracking the instantaneous location of viscous shocks in conservation laws, showing its asymptotic equivalence to other established methods and providing insights into nonlinear stability analysis.

Contribution

It introduces a simple, unified framework for instantaneous shock location that aligns with existing methods and enhances understanding of viscous shock stability.

Findings

The shock location choice is asymptotically equivalent to least-squares fit.

The shock tracking approach applies broadly to various localized projections.

The method simplifies stability analysis of viscous shocks.

Abstract

We illustrate in a simple setting the instantaneous shock tracking approach to stability of viscous conservation laws introduced by Howard, Mascia, and Zumbrun. This involves a choice of the definition of instanteous location of a viscous shock-- we show that this choice is time-asymptotically equivalent both to the natural choice of least-squares fit pointed out by Goodman and to a simple phase condition used by Gu\`es, M\'etivier, Williams, and Zumbrun in other contexts. More generally, we show that it is asymptotically equivalent to any location defined by a localized projection