On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity - Laboratoire d'Analyse et de Mathématiques Appliquées Accéder directement au contenu
Article Dans Une Revue Journal of Statistical Physics Année : 2008

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

Résumé

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.

Dates et versions

hal-00794176 , version 1 (25-02-2013)

Identifiants

Citer

Nicolas Fournier, Hélène Guérin. On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. Journal of Statistical Physics, 2008, 131 (4), pp.749-781. ⟨10.1007/s10955-008-9511-5⟩. ⟨hal-00794176⟩
146 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More