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MiS Preprint

Existence, bifurcation, and stability of profiles for classical and non-classical shock waves

Heinrich Freistühler, Christian Fries and Christian Rohde


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 dicusses 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.


Related publications

2001 Repository Open Access
Heinrich Freistühler, Christian Fries and Christian Rohde

Existence, bifurcation, and stability of profiles for classical and non-classical shock waves

In: Ergodic theory, analysis, and efficient simulation of dynamical systems / Bernd Fiedler (ed.)
Berlin [u. a.] : Springer, 2001. - pp. 287-309