Homogénéisation déterministe d’équations cinétiques

Abstract

In this thesis, we study the homogenization of Vlasov type equations. Two cases are treated here. In the first part of the work, we carry out the homogenization of Vlasov equation under some structural hypotheses on the electromagnetic field. Two main results were proved in this case. The first one (Theorem 17) shows the equivalence between the duality in the sense of distributions and the duality defined by the mean value, while the second (Theorem 19) is nothing but the homogenization result obtained under general deterministic setting. Few concrete examples are presented here as illustration. The second part of the work deals with the homogenization of linear Boltzmann equation. Once again, the study leads us to two important results. The first one (Proposition 5) gives the solution to the cellular problem under more general setting, and the second (Theorem 22) is the homogenization result stated here, as in the first part, under general deterministic setting. This result shows that, under appropriate structural hypotheses, the particles density converges to the solution of a drift-diffusion equation. Some illustrations of this result are presented in the document.