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 fully satisfactory only when $\gamma\in[-1,0)$. We next prove, under some appropriate conditions on the initial data, the so-called propagation of molecular chaos, that is, 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 this system and finally prove that the additional noise is almost never used in the limit $N\to\infty$.