From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge
Journal de l’École polytechnique - Mathématiques, Volume 2  (2015), p. 247-296

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.

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).

DOI : https://doi.org/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
@article{JEP_2015__2__247_0,
     author = {Barr\'e, Julien and Chiron, David and Goudon, Thierry and Masmoudi, Nader},
     title = {From Vlasov--Poisson and Vlasov--Poisson--Fokker--Planck systems to incompressible Euler equations: the~case~with finite charge},
     journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
     publisher = {Ecole polytechnique},
     volume = {2},
     year = {2015},
     pages = {247-296},
     doi = {10.5802/jep.24},
     language = {en},
     url = {http://www.numdam.org/item/JEP_2015__2__247_0}
}
Barré, Julien; Chiron, David; Goudon, Thierry; Masmoudi, Nader. From Vlasov–Poisson and Vlasov–Poisson–Fokker–Planck systems to incompressible Euler equations: the case with finite charge. Journal de l’École polytechnique - Mathématiques, Volume 2 (2015) , pp. 247-296. doi : 10.5802/jep.24. http://www.numdam.org/item/JEP_2015__2__247_0/

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