Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Convergence from Boltzmann to Landau processes with soft potential and particle approximations


Code(s) de Classification MSC:

Résumé: In the present paper, we firstly extend the probabilistic interpretation of spatially homogeneous Boltzmann equations without angular cutoff due firstly to Tanaka and generalized by Fournier-M\'el\'eard, to some soft potential cases for a large class of initial data. We relate a measure solution of the Boltzmann equation to the solution of a Poisson-driven stochastic differential equation. Then we consider renormalized such equations which make prevail the grazing collisions, and we prove the convergence of the associated Boltzmann processes to a process related to the Landau equation initially introduced by Gu\'erin. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. An approximation of a solution of the Landau equation with soft potential via colliding stochastic particle systems is derived from this result. We then deduce a Monte-Carlo algorithm of simulation by a conservative particle method following the asymptotics of grazing collisions. Numerical results are given.

Mots Clés: Boltzmann equations without cutoff and soft potential ; Landau equation with soft potential ; Nonlinear stochastic differential equations ; Interacting particle systems ; Monte-Carlo algorithm

Date: 2001-11-15

Prépublication numéro: PMA-698

Postscript file : PMA-698.ps