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.
"Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials." Ann. Appl. Probab. 28 (2) 1136 - 1189, April 2018. https://doi.org/10.1214/17-AAP1327