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| HAL: hal-00289384, version 1 |
| arXiv: 0806.3379 |
| DOI: 10.1016/j.jfa.2008.11.008 |
| Detailed view |  Export this paper |
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| Journal of Functional Analysis 256, 8 (2009) 2542-2560 |
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| Well-posedness of the spatially homogeneous Landau equation for soft potentials |
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| Nicolas Fournier 1Hélène Guérin 2 |
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| (2009-04-15) |
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| We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderatley soft potentials, we assume additionnally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka. |
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| 1: | Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA) |
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Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout |
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| 2: | Institut de Recherche Mathématique de Rennes (IRMAR) |
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CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne |
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| Subject | : | Mathematics/Probability Mathematics/Analysis of PDEs |
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| Fokker-Planck-Landau equation – soft potentials – plasma physics – uniqueness – Wasserstein distance – quadratic cost. |
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| hal-00289384, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00289384 | |
| oai:hal.archives-ouvertes.fr:hal-00289384 | |
| From: Hélène Guérin | |
| Submitted on: Friday, 20 June 2008 14:39:44 | |
| Updated on: Tuesday, 16 February 2010 11:09:03 | |
