On stability of fast shock waves in classical and relativistic MHD

We are concerned with the multidimensional structural stability of fast shock waves in classical (nonrelativistic) and relativistic MHD. Stability is studied for the special case of parallel shocks, i.e. the magnetic field is supposed to be parallel to the normal to the discontinuity surface. The main aim of the work is to find the domains of linearized uniform stability for planar shock waves, where the stability problem satisfies the uniform Kreiss-Lopatinski condition. For the special case when only one characteristic mode of a linearized hyperbolic system is incoming, and the others are outgoing, we deduce an equivalent form of this condition which plays the crucial role in the stability analysis. We show that fast parallel MHD shock waves in a polytropic gas, with an arbitrary adiabat index gamma, are weakly linearly stable, in the sense of the general (not necessarily uniform) Lopatinski condition. This refines the old results of Gardner and Kruskal, with gamma < 3. Moreover, we exactly describe the domain of uniform (structural) stability. For fast parallel shock waves in relativistic MHD, it is proved that the instability and weak stability domains coincide with those of shock waves in relativistic gas dynamics. We find likewise the domain of uniform stability that turns out to be smaller than that of relativistic gas dynamic shock waves.