The Galerkin Approach for the Einstein-Boltzmann Equation

The Einstein-Boltzmann equation has already been considered by some authors. But in this work we investigate, using a new method the global in time existence, uniqueness and regularity of solutions to the Magnetized Einstein-Boltzmann equation on a Locally Rotationally Symmetric Bianchi type I model, where we have assumed the LRS both for the matter and the geometry. We use for the study of the relativistic Boltzmann equation, the Galerkin approach in some weighted Sobolev spaces that are also separable. Next we study the Einstein system: we follow Rendall and Uggla in (1). Then, we couple these equations and we investigate, using an original method, their time global existence. In the present work,all lemmas, propositions and theorems leading to the global existence theorem are original. We have also given all their proofs.