The structure of MFD shock waves in a model of plasma

Germain, P. described the question of the existence of structure of MFD shock waves in the case of rectilinear motion, for purely transverse magnetic fields in a model of plasma, and gave a proof of existence for the case of neglecting viscosity and heat conduction. The mathematical question is stated in terms of a four-dimensional system of ordinary differential equations which depends on four viscosity parameters. These equations admit two rest points, independent of the viscosity parameters. The problem considered in this article is to show that for all values of the viscosities, there is an orbit running from one rest point to the other one. Moreover the shock is stable or has the usual type of profile. In order to solve the above problem we show that our system is gradient-like (except for the very special case η=κ=σ-1=0). This result enables us to use the Conley theory and solve the problem. Moreover, some limiting cases for singular viscosities are investigated; in particular, we show how the general results in the classical one fluid M.H.D. theory are obtained when 'the plasma viscosity' β tends to zero. (author). 12 refs., 10 figs