Existence and Bifurcation of Viscous Profiles for All Intermediate Magnetohydrodynamic Shock Waves

Abstract

A viscous profile for a magnetohydrodynamic shock wave is given by a heteroclinic orbit of a six-dimensional gradient-like system of ordinary differential equations. This system, and tjius possibly the existence of a viscous profile, vary with an array $\delta $ of four positive dissipation coefficients. It is known that for each choice of $\delta $, all “classical” and “degenerate intermediate” shocks as well as some “nondegenerate intermediate” shocks have viscous profiles, and that, vice versa, each given nondegenerate intermediate shock has no viscous profile for some range of $\delta $. Complementing this picture, it is shown that (i) each nondegenerate intermediate shock does have a (family of) viscous profile(s) for a certain other range of $\delta $, and (ii) such profiles, for all intermediate shocks sharing the same relative flux, are generated in a global heteroclinic bifurcation. Both (i) and (ii) are proved in a regime of $\delta $ in which the dissipative effects due to electrical resistivity and longitudinal viscosity dominate those associated with transverse viscosity and heat conduction: The constructive proof is based on a recently formulated method in geometric singular perturbation theory.