Archive for Rational Mechanics and Analysis

, Volume 143, Issue 3, pp 273–307

On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations

  • Cédric Villani

DOI: 10.1007/s002050050106

Cite this article as:
Villani, C. Arch Rational Mech Anal (1998) 143: 273. doi:10.1007/s002050050106


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.

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Cédric Villani
    • 1
  1. 1.Ecole Normale SupérieureParis Cedex 05France