On the problem of stability of viscous shocks

TL;DR

This paper investigates the spectral stability of viscous shock solutions in conservation laws, introducing a piece-wise linear model to simplify stability analysis and demonstrating stability loss examples that can be smoothed.

Contribution

It proposes a novel piece-wise linear model for viscous conservation laws that simplifies the spectral stability problem to linear algebra, enabling explicit stability analysis.

Findings

Stability can be analyzed using linear algebra in the piece-wise linear model.

Examples of stability loss are constructed within this model.

Stability loss examples can be smoothed to smooth models.

Abstract

We consider the problem of spectral stability of traveling wave solutions $u=\gamma(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories $\gamma$ of a system of ODE. In general conditions of stability can be obtained only numerically. We propose a model class of piece-wise linear (discontinuous) vector fields $F$ for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.