On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations

This paper deals with the spatially homogeneous Boltzmann equation when grazing collisions are involved. We study in a unified setting the Boltzmann equation without cut-off, the Fokker-Planck-Landau equation, and the asymptotics of grazing collisions for a very broad class of potentials; in particular, we are able to derive rigorously the Landau equation for the Coulomb potential. In order to do so, we introduce a new definition of weak solutions, based on entropy production.