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

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.

Cited by in corpus (4)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Well-posedness of the spatially homogeneous Landau equation for soft potentials
• A new regularization possibility for the Boltzmann equation with soft potentials
• A probabilistic view on the general relativistic Boltzmann equation