Existence, Bifurcation, and Stability of Profiles for Classical and Non-Classical Shock Waves

Abstract

This paper surveys the authors’ recent results on viscous shock waves in PDE systems of conservation laws with non-convexity and non-strict hyperbolicity. Particular attention is paid to the physical model of magnetohydrodynamics. The plan of the paper is as follows. Sections 1 and 2 introduce the classes of systems and the classes of shock waves we consider and recall how profiles for small-amplitude shocks are constructed via center manifold analyses of a corresponding system of ODEs. Section 3 describes the global picture, i. e., large-amplitude shock waves, for the case of magnetohydrodynamics, first the solution set of the Rankine-Hugoniot jump conditions, then a heteroclinic bifurcation occurring in the ODE system for the profiles. Section 4 presents a method for the numerical identification of heteroclinic manifolds, which is applied in Sections 5 and 6 to the case of magnetohydrodynamics. The numerical treatment confirms and details the analytical findings and, more notably, extends them considerably; in particular, it allows to study the existence / non-existence of profiles and the aforementioned heteroclinic bifurcation globally. Section 7 discusses the stability of viscous shock waves; the important nonuniformity of the vanishing viscosity limit for, in particular, non-classical MHD shock waves is not addressed in this paper.