Singularities of the Stationary Solutions to the Vlasov-Poisson System in a PolygonReport as inadecuate

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1 CALVI - Scientific computation and visualization IRMA - Institut de Recherche Mathématique Avancée, LSIIT - Laboratoire des Sciences de l-Image, de l-Informatique et de la Télédétection, Inria Nancy - Grand Est, IECL - Institut Élie Cartan de Lorraine 2 Ecole Supérieure de Technologie d-Essaouira 3 EDP - Equations aux dérivées partielles IECL - Institut Élie Cartan de Lorraine 4 CORIDA - Robust control of infinite dimensional systems and applications IECN - Institut Élie Cartan de Nancy, LMAM - Laboratoire de Mathématiques et Applications de Metz, Inria Nancy - Grand Est

Abstract : We present an existence result for the stationary Vlasov-Poisson system in a bounded domain of~$\R^{N}$, with more general hypotheses than considered so far in the literature. In particular, we prove the equivalence of the kinetic approach which consists in looking for the equilibrium distribution function and the potential approach where the unknown is the electrostatic potential at equilibrium. We study the dependence of the solution on parameters such as the total mass of the distribution, or those entering in the boundary conditions of the potential. Focusing on the case of a plane polygon, we study the singular behavior of the solution near the reentrant corners, and examine the dependence of the singularity coefficients on the parameters of the problem. Numerical experiments illustrate and confirm the analysis.

Keywords : Large solutions Nonlinear elliptic problem Vlasov-Poisson system Stationary solutions Corner singularity

Author: Fahd Karami - Simon Labrunie - Bruno Pinçon -



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