From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge

Julien Barré; David Chiron; Thierry Goudon; Nader Masmoudi

doi:10.5802/jep.24

From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge
[Des équations de Vlasov-Poisson et Vlasov-Poisson-Fokker-Planck aux équations d’Euler incompressibles : le cas de charge finie]

Julien Barré ¹ ; David Chiron ¹ ; Thierry Goudon ² ; Nader Masmoudi ³

¹ Université de Nice Sophia Antipolis, CNRS UMR 7351, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice, France
² Université de Nice Sophia Antipolis, CNRS UMR 7351, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice, France & Inria, Sophia Antipolis Méditerranée Research Centre, Project COFFEE
³ Courant Institute for Math. Sciences, New York University 251 Mercer St., New York, NY 10012, U.S.A.

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 247-296.

Résumé
Abstract

Nous étudions le régime asymptotique de forts champs électriques qui conduit du système de Vlasov-Poisson aux équations d’Euler incompressibles. Nous abordons aussi le système de Vlasov-Poisson-Fokker-Planck qui induit des effets dissipatifs additionnels. L’originalité de cette étude réside dans le fait qu’on suppose la charge totale finie et confinée par un fort champ extérieur. En conséquence, l’équation limite est posée dans un domaine borné dont la géométrie est déterminée par ce champ confinant. L’analyse s’étend au cas où la densité limite est inhomogène ; l’équation d’Euler est alors remplacée par l’équation des lacs (ou modèle anélastique).

We study the asymptotic regime of strong electric fields that leads from the Vlasov–Poisson system to the Incompressible Euler equations. We also deal with the Vlasov–Poisson–Fokker–Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non–homogeneous and where the Euler equation is replaced by the Lake Equation, also called Anelastic Equation.

Reçu le : 2015-02-16
Accepté le : 2015-10-15

MR Zbl

DOI : 10.5802/jep.24

Classification : 82D10, 35Q35, 82C40
Keywords: Plasma physics, Vlasov–Poisson system, Vlasov–Poisson–Fokker–Planck system, incompressible Euler equations, lake equations, quasi–neutral regime, modulated energy, relative entropy
Mot clés : Physique des plasmas, système de Vlasov-Poisson, système de Vlasov-Poisson-Fokker-Planck, équations d’Euler incompressibles, équation des lacs, régime quasi-neutre, énergie modulée, entropie relative

Affiliations des auteurs :

Julien Barré ¹ ; David Chiron ¹ ; Thierry Goudon ² ; Nader Masmoudi ³

¹ Université de Nice Sophia Antipolis, CNRS UMR 7351, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice, France
² Université de Nice Sophia Antipolis, CNRS UMR 7351, Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108 Nice, France & Inria, Sophia Antipolis Méditerranée Research Centre, Project COFFEE
³ Courant Institute for Math. Sciences, New York University 251 Mercer St., New York, NY 10012, U.S.A.

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Julien Barr\'e and David Chiron and Thierry Goudon and Nader Masmoudi},
     title = {From {Vlasov{\textendash}Poisson} and {Vlasov{\textendash}Poisson{\textendash}Fokker{\textendash}Planck} systems to incompressible {Euler} equations: the~case~with finite charge},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {247--296},
     publisher = {\'Ecole polytechnique},
     volume = {2},
     year = {2015},
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     mrnumber = {3426749},
     zbl = {1339.82010},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.24/}
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Julien Barré; David Chiron; Thierry Goudon; Nader Masmoudi. From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 247-296. doi : 10.5802/jep.24. https://jep.centre-mersenne.org/articles/10.5802/jep.24/

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