Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials

TL;DR

This paper establishes stability and uniqueness results for the 3D homogeneous Boltzmann equation with moderately soft potentials using Wasserstein distances, and analyzes the convergence of a stochastic particle system to the solution.

Contribution

It provides the first strong/weak stability estimates for this class of Boltzmann equations and quantifies the convergence rate of the Nanbu particle system to the solution.

Findings

Proves stability and uniqueness of weak solutions under finite entropy and moments.

Derives convergence rates for the Nanbu particle system approximating the Boltzmann solution.

Uses probabilistic coupling methods for analysis.

Abstract

We prove a strong/weak stability estimate for the 3D homogeneous Boltzmann equation with moderately soft potentials ($\gamma\in(-1,0)$) using the Wasserstein distance with quadratic cost. This in particular implies the uniqueness in the class of all weak solutions, assuming only that the initial condition has a finite entropy and a finite moment of sufficiently high order. We also consider the Nanbu $N$-stochastic particle system which approximates the weak solution. We use a probabilistic coupling method and give, under suitable assumptions on the initial condition, a rate of convergence of the empirical measure of the particle system to the solution of the Boltzmann equation for this singular interaction.