Existence and Bifurcaton of MHD-Profiles

Weak solutions of nonlinear hyperbolic systems of partial differential equations can develop discontinuities within finite time. Additionally, this loss of regularity leads to nonuniqueness of the weak solution. To identify the physically relevant solution one way of reasoning is provided by the so-called viscosity method. A weak solution is accepted to be the physically relevant one if it is the limit of classical solutions of an associated parabolically-regularized problem for vanishing viscosity parameters. Here we consider the viscosity method for MHD intermediate shock waves. The viscous solutions turn out to be solutions of an ordinary boundary value problem on the real line that depends on various parameters (corresponding to fluid viscosity, resistivity, for instance). In the language of dynamical systems they constitute connecting orbits. The MHD system is of particular interest since the orbits arise in multiparameter families and undergo various types of bifurcation.

Global Bifurcation for the Full Orbital Structure

Starting from analytical work by Freistühler and collaborators the project is concerned with numerical investigations for existence and bifurcation of viscous profiles. A typical configuration of viscous profiles under the variation of the resistivity parameter can be seen above.

This project is joint work with H. Freistühler (MPI Leipzig) and was supported by the DFG priority research program Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme.

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