THE EINSTEIN-BOLTZMANN SYSTEM FOR WEIGHTED SOBOLEV SEPARABLE SPACES ON BIANCHI TYPE I COSMOLOGICAL MODELS : GLOBAL EXISTENCE, UNIQUENESS AND REGULARITY

The present article discusses the global dynamics of a particular Bianchi I cosmological model with a cosmological constant, where the matter elds consist of a collisional, relativistic kinetic gas and pressure-less dust. Because of the symmetries of spacetime, the evolution of the dust component is completely determined by its initial density and the evolution of the metric eld. One is then left with solving the Einstein-Boltzmann system of equations for a Bianchi type I conguration. within this framework, is discussed the global existence and uniqueness results. To this purpose, based on a Galerkin method, we rst show local uniqueness and existence results for the Boltzmann equation, assuming the metric coecients are given and xed. In a next step, assuming that the distribution function is given, we show local existence of solutions to the Einstein evolution equations. Finally, the two results are put together and local and then global future existence of solutions to the coupled Einstein-Boltzmann system is shown, assuming initial expansion in all spatial directions and non-negative cosmological constant.

Also is explained in the paper the choice of function spaces : the weighted Sobolev and separable spaces for the Boltzmann equation; some special spaces for the Einstein equations, and also are clearly displayed all the proofs leading to the main theorems. The problem of the initial constraints is also highly studied.