Zero dissipation and stability of shocks (Q1270876)

The author studies the nonlinear stability of viscous shock waves in the process of vanishing viscosity parameters. In the time asymptotic states the two non-classical shocks, overcompressive and undercompressive shocks, are nonlinearly stable. It is proved that the overcompressive shocks are nonlinearly stable, however not uniformly with respect to the viscosity parameter. This is also showed for a rotationally invariant system, which is derived from magneto-hydrodynamics and nonlinear elasticity. The existence and speed of undercompressive shocks depend on the relative strength of the dissipation parameters, and the shocks are uniformly stable with respect to the viscosity parameters. This situation is demonstrated on a combustion model. It is important to note that these phenomena for overcompressive and undercompressive shocks hold for general hyperbolic systems, and they are in contrast to the classical shocks for gas dynamics.