Introduction

The simplest model for a relativistic medium is that of a relativistic fluid. When the medium interacts electromagnetically and is electrically highly conducting the simplest description is in terms of relativistic magneto-fluid dynamics.

From the mathematical viewpoint relativistic fluid dynamics (RFD) and magneto-fluid dynamics (RMFD) have mainly been treated in the framework of general relativity, that is, as describing possible sources of the gravitational field. This means that both the RFD and RMFD equations have been studied in conjunction with Einstein's equations.

In this framework Lichnerowicz (1967) has made a thorough and deep investigation of the initial value problem, and by using the theory of Leray systems, has obtained a local existence and uniqueness theorem in a suitable function class.

In many applications (particularly in plasma physics) one can neglect the gravitational field generated by the medium in comparison with the background gravitational field, or, in many cases, one can simply assume special relativity.

Mathematically this amounts to taking into account only the conservation equations for the matter, neglecting Einstein's equations. The resulting theory can be called test relativistic fluid dynamics or magneto-fluid dynamics. These theories are mathematically much simpler than the full general relativistic ones, and, consequently, stronger and more detailed results can be obtained.

In Section 2.1, following ideas originally introduced by Friedrichs (1974) and developed by Ruggeri and Strumia (1981a), we give a covariant definition of a quasi-linear hyperbolic system. The concept of systems of conservation laws is also introduced in this section.