On the Boltzmann equation with long-range interactions

Most of the theory of the Boltzmann equation has been built upon 'Grad's cut-off' assumption, i.e that the kinetic cross-section appearing in Boltzmann's collision operator should be locally integrable. This assumption is however true only for particles interacting via finite-range forces. The case of long-range interactions has been the object of a lot of interest in the past few years, due to the complete change in the structure of the equation, and in the qualitative behavior of solutions. In this talk, I present the extension of the DiPerna-Lions theory to the case of long-range interactions, together with the proof of a conjecture by Lions, about regularizing properties of the Boltzmann equation without cut-off. Parts of the work were done in collaboration with Desvillettes, Alexandre, Wennberg.