On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity
Mouhot, Clément; Fournier, Nicolas (2009), On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity, Communications in Mathematical Physics, 289, 3, p. 803-824. http://dx.doi.org/10.1007/s00220-009-0807-3
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00135991/en/
Journal nameCommunications in Mathematical Physics
MetadataShow full item record
Abstract (EN)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 . 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 , 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.
Subjects / KeywordsKantorovich-Rubinstein distance; Boltzmann equation without cutoff; uniqueness; Wasserstein distance; long-range interaction
Showing items related by title and author.
Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials. Desvillettes, Laurent; Mouhot, Clément (2007) Article accepté pour publication ou publié