The Galerkin Method for Global Solutions to the Maxwell-Boltzmann System

We prove an existence and uniqueness of solution to the Maxwell- Boltzmann coupled system globally in time. We first of all describe the background space-time and the unknown functions, making some hypotheses concerning the potentials of gravitation a and b; which determine the gravitational field g, the distribution function f which is unknown and is subject to the Boltzmann equation, as well as the collision kernel sigma which appears in the collision operator. We after clarify the choice of the function spaces and we establish step by step, using Sobolev theorems, all the essential energy estimations leading to the global existence theorem. The method used for the investigation of the global existence combine the Galerkin method which is applied in a particular separable Hilbert space which is a Sobolev space with weight, and the standard theory on the first order differential systems. We then give at the end, the physical significance of our work.