On the Problem of Stability of Viscous Shocks

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 ODEs. In general conditions of stability can be obtained only numerically. We propose a model class of piecewise 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.