On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity
Nicolas Fournier, Clément Mouhot
TL;DR
The paper addresses the well-posedness of the spatially homogeneous Boltzmann equation with a non-cutoff kernel featuring a moderate angular singularity. It develops a Kantorovich-Rubinstein distance-based inequality between two weak solutions, derived via a Tanaka-inspired coupling and a precise post-collision velocity parameterization, to obtain quantitative stability. This framework yields new uniqueness and stability results: for hard potentials (with exponential tail initial data) and for moderately soft potentials (finite energy and entropy with possible extra moments), improving upon prior weighted Sobolev approaches. It also establishes propagation of exponential moments for hard potentials and extends the analysis to soft potentials, broadening the class of physically relevant kernels that admit unique, stable solutions.
Abstract
We prove an inequality on the Kantorovich-Rubinstein distance --which can be seen as a particular case of a Wasserstein metric-- between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity. Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the physically relevant cases of hard and moderately soft potentials. In the case of hard potentials, we relax the regularity assumption of [6], but we need stronger assumptions on the tail of the distribution (namely some exponential decay). We thus obtain the first uniqueness result for measure initial data. In the case of moderately soft potentials, we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy (for very moderately soft potentials), plus sometimes an additionnal moment condition. We thus improve significantly on all previous results, where weighted Sobolev spaces were involved.