HAL : hal-00135991, version 1

arXiv : math.AP/0703283

DOI : 10.1007/s00220-009-0807-3

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Communications in Mathematical Physics 289, 3 (2009) 803-824

On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity

Nicolas Fournier 1, Clément Mouhot 2

(2009)

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.

1 :	Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
	CNRS : UMR8050 – Université Paris XII - Paris Est Créteil Val-de-Marne – Université Paris XII - Paris Est Créteil Val-de-Marne
2 :	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine

Domaine

:

Mathématiques/Equations aux dérivées partielles Mathématiques/Physique mathématique

Mots Clés : Boltzmann equation without cutoff – long-range interaction – uniqueness – Wasserstein distance – Kantorovich-Rubinstein distance

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Contributeur : Clément Mouhot <>

Soumis le : Samedi 10 Mars 2007, 11:33:44

Dernière modification le : Mardi 2 Février 2010, 15:56:21