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 ${\partial }_{t}u+{\partial }_{x}F(u)={\partial }_x^{2}u$. 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.