On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

TL;DR

This paper establishes uniqueness and well-posedness results for the spatially homogeneous Boltzmann equation with strong angular singularities, using Wasserstein distance inequalities and probabilistic coupling methods.

Contribution

It provides the first uniqueness results for strong angular singularities beyond Maxwell molecules, employing a probabilistic approach and Wasserstein distance analysis.

Findings

Proves a Wasserstein distance inequality for solutions.

Establishes local and global well-posedness for certain potentials.

Demonstrates uniqueness for strong angular singularities.

Abstract

We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ideas of Tanaka: we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other.