Nonlinear Stability and Instability of Overcompressive Shock Waves

Overcompressive shock waves arise in nonlinear elasticity, magnetohydrodynamics, multiphase flows and other physical situations. The inviscid models are non-strictly hyperbolic and there have been controversies on the admissibility of overcompressive shocks. We study the nonlinear stability of these waves for the viscous models and show that there are two distinct types. The first type is a combination of classical shock waves and is nonlinear stable with respect to the strength of dissipations. For sufficiently small dissipation a fixed perturbation will give rise to waves of distinct speeds. The second type is stable, but not uniformly with respect to the strength of dissipations. For sufficiently small dissipation a fixed perturbation will give rise to waves of distinct speeds.