ON A NEW CLASS OF WEAK SOLUTIONS TO THE SPATIALLY HOMOGENEOUS BOLTZMANN AND

ON A NEW CLASS OF WEAK SOLUTIONS TO THE SPATIALLY HOMOGENEOUS BOLTZMANN AND LANDAU EQUATIONS C. VILLANI Abstract. This paper deals with the spatially homogeneous Boltz- mann equation when grazing collisions are involved. We study in a unified frame 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 spatially homogeneous Landau equation with a Coulomb potential. In order to do so, we introduce a new defi- nition of weak solutions, based on the entropy production. Contents 1. Introduction 1 2. Basic identities and formulation of the problem 6 3. Definitions and main results 12 4. Hard and moderately soft potentials 20 5. Lack of a priori estimates for very soft potentials 25 6. H-solutions 27 7. The Coulomb potential 33 Appendix A. Uniform gain of moments 35 References 36 1. Introduction We are concerned in this paper with equations arising from kinetic theory, describing the evolution of a gas (or a plasma) which is not at equilibrium. They can be written as (1) ∂f∂t = Q(f, f), t ≥ 0, v ? R 3 where the unknown function f(t, v) is assumed to be nonnegative, and stands for the density of particles at time t with velocity v; Q(f, f) is a quadratic non-local operator describing the collisions within the gas, 1

• grazing


• weak solution


• collision


• differences between hard


• soft potentials


• cut


• fokker- planck-landau equation


• without cut


• boltzmann equation