Propagation of chaos for the Landau equation with moderately soft potentials

Abstract : We consider the 3D Landau equation for moderately soft potentials ($\gamma\in(-2,0)$ with the usual notation) as well as a stochastic system of $N$ particles approximating it. We first establish some strong/weak stability estimates for the Landau equation, which are satisfying only when $\gamma \in [-1,0)$. We next prove, under some appropriate conditions on the initial data, the so-called propagation of molecular chaos, i.e. that the empirical measure of the particle system converges to the unique solution of the Landau equation. The main difficulty is the presence of a singularity in the equation. When $\gamma \in (-1,0)$, the strong-weak uniqueness estimate allows us to use a coupling argument and to obtain a rate of convergence. When $\gamma \in (-2,-1]$, we use the classical martingale method introduced by McKean. To control the singularity, we have to take advantage of the regularity provided by the entropy dissipation. Unfortunately, this dissipation is too weak for some (very rare) aligned configurations. We thus introduce a perturbed system with an additional noise, show the propagation of chaos for that perturbed system and finally prove that the additional noise is almost never used in the limit.
Type de document :
Article dans une revue
Annals of Probability, Institute of Mathematical Statistics, 2016, 44 (6), pp.3581 - 3660. <10.1214/15-AOP1056>
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01257022
Contributeur : Aigle I2m <>
Soumis le : vendredi 15 janvier 2016 - 16:22:36
Dernière modification le : lundi 29 mai 2017 - 14:27:22

Identifiants

Collections

Citation

Nicolas Fournier, Maxime Hauray. Propagation of chaos for the Landau equation with moderately soft potentials. Annals of Probability, Institute of Mathematical Statistics, 2016, 44 (6), pp.3581 - 3660. <10.1214/15-AOP1056>. <hal-01257022>

Partager

Métriques

Consultations de la notice

57