Propagation of chaos for the Boltzmann equation with very soft potentials

TL;DR

This paper proves propagation of chaos for the Boltzmann equation with very soft potentials by constructing solutions to Kac's particle system and establishing convergence of empirical measures.

Contribution

It introduces new estimates on Fisher information dissipation and a novel inequality related to fractional heat flow, solving an open problem for very soft potentials.

Findings

Empirical measures of Kac's system converge to the Boltzmann solution.

New estimates control system singularities.

A novel inequality on fractional heat flow is developed.

Abstract

We build solutions to Kac's particle system and show that their empirical measures converge to the solution of the space-homogeneous Boltzmann equation in the regime of very soft potentials. This proves propagation of chaos for the last class of kernels for which it was still open. The proof relies on new estimates on the dissipation of the Fisher information along the Boltzmann equation, which allow us to control the strong singularities of the system. These estimates are obtained thanks to a new inequality related to the fractional heat flow on the sphere, that might be of independent interest.