On the Boltzmann equation for long-range interactions

We study the Boltzmann equation without Grad's angular cutoff assumption. We introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables. Angular singularities occur as soon as one is interested in long-range interactions, while singularities in the relative velocity variable Occur in the study of soft potentials, in particular, Coulomb interaction. Together with several new estimates. this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions. (C) 2002 John Wiley & Sons, Inc.