One-Dimensional Discontinuous Flows in Relativistic Magnetohydrodynamics

We consider discontinuous flows of relativistic magnetic fluid with a general equation of state that is not supposed to be normal in the sense of Bethe and Weyl. The criteria of admissibility of the shock waves without a supposition of the relativistic version of the convexity condition are obtained. The results are used to analyze the cases of perpendicular and parallel shock waves. 1. It is well known that the uniqueness of solutions of an initial value problem for quasilinear equations such as equations of fluid dynamics may be lost if discontinuities of initial data are present [1]. In this case some additional requirements are needed to single out a unique discontinuous solution. In hydrodynamics with a normal equation of state (in the sense of Bethe and Weyl [1]) the role of such a requirement is fulfilled by the entropy criterion. In case of anomalous fluid with a nonconvex equation of state some more stringent criteria are needed that may be represented as requirements to the form of shock adiabate (see, e.g., [2], [3]). The generalization to the relativistic shock waves is worked out in [4], [5]. In this paper the analogous problem in relativistic magnetohydrodynamics is considered. In order to formulate the conditions for existence of admissible shock waves in the fluid with a nonconvex (in the relativistic sense) equation of state we modify the approach of [4], [5] based on the small viscosity considerations. The resulting criterion is expressed in terms of shock adiabats, defined by the equation of state, and some pattern curves in the plane of thermodynamic coordinates pressure-conserving charge density. This criterion is applied to study parallel and perpendicular shocks in a one-dimensional magnetohydrodynamic flow. We start from the equations of motion of a viscous relativistic fluid with perfect electrical conductivity, permeated by a magnetic field [6], [7] @µ (T µ + µ ) = 0, (1)