Singularities of the stationary solutions to the Vlasov-Poisson system in a polygon

Fahd Karami (1, 2) , Simon Labrunie (1, 3) , Bruno Pinçon (4, 3)

1 CALVI - Scientific computation and visualization
2 Ecole Supérieure de Technologie d'Essaouira
3 IECL - Institut Élie Cartan de Lorraine
4 CORIDA - Robust control of infinite dimensional systems and applications

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.

Mots clés: Corner singularity; Stationary solutions; Vlasov--Poisson system; Nonlinear elliptic problem; Large solutions

Domaines: Equations aux dérivées partielles [math.AP]

Fahd Karami, Simon Labrunie, Bruno Pinçon. Singularities of the stationary solutions to the Vlasov-Poisson system in a polygon. Mathematical Models and Methods in Applied Sciences, 2013, 23 (6), pp.1029-1066. ⟨10.1142/S0218202513500024⟩. ⟨hal-00501842v3⟩