The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 28, Number 2 (2018), 1136-1189.
Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials
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.
Ann. Appl. Probab., Volume 28, Number 2 (2018), 1136-1189.
Received: May 2016
Revised: February 2017
First available in Project Euclid: 11 April 2018
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Xu, Liping. Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials. Ann. Appl. Probab. 28 (2018), no. 2, 1136--1189. doi:10.1214/17-AAP1327. https://projecteuclid.org/euclid.aoap/1523433633