Particle approximation of Vlasov equations with singular forces: Propagation of chaos

Hauray, Maxime; Jabin, Pierre-Emmanuel

doi:10.24033/asens.2261

Particle approximation of Vlasov equations with singular forces: Propagation of chaos
[Approximation particulaire des équations de Vlasov avec noyaux de force singuliers : la propagation du chaos]

Hauray, Maxime ; Jabin, Pierre-Emmanuel

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 891-940.

Résumé
Abstract

Nous montrons la validité de l'approximation par champ moyen et prouvons la propagation du chaos pour un système de particules en interaction par le biais d'une force avec singularité $1 / {| x |}^{α}$ , avec $α < 1$ en dimension $d \geq 3$ . Nous traitons également le cas de forces avec troncature et des singularités pouvant aller jusqu'à $α < d - 1$ . Ce dernier résultat permet presque d'atteindre les cas d'interaction coulombiennes ou gravitationnelles et requiert seulement de très petits paramètres de troncature lorsque la singularité est proche de $α = 1$ .

We justify the mean field approximation and prove the propagation of chaos for a system of particles interacting with a singular interaction force of the type $1 / {| x |}^{α}$ , with $α < 1$ in dimension $d \geq 3$ . We also provide results for forces with singularity up to $α < d - 1$ but with a large enough cut-off. This last result thus almost includes the case of Coulombian or gravitational interactions, but it also allows for a very small cut-off when the strength of the singularity $α$ is larger but close to one.

Publié le : 2015-09-24

DOI : 10.24033/asens.2261

Classification : 35Q82, 35-02, 82C22; 35L65, 35Q85, 35Q70, 82C40.
Keywords: Derivation of kinetic equations, particle methods, Vlasov equation, propagation of chaos and mean field limits.
Mot clés : Dérivation des modèles cinétiques, méthodes particulaires, équation de Vlasov, propagation du chaos et limites de champ moyen.

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     title = {Particle approximation of {Vlasov} equations with singular forces: {Propagation} of chaos},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {891--940},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Hauray, Maxime; Jabin, Pierre-Emmanuel. Particle approximation of Vlasov equations with singular forces: Propagation of chaos. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 891-940. doi : 10.24033/asens.2261. http://www.numdam.org/articles/10.24033/asens.2261/

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