From Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck Systems to Incompressible Euler Equations: the case with finite charge

TL;DR

This paper investigates the transition from Vlasov-Poisson systems to incompressible Euler and Lake equations under strong electric fields with finite charge, including dissipative effects and non-homogeneous densities.

Contribution

It introduces a novel analysis of finite charge confinement leading to bounded domain Euler and Lake equations, extending previous asymptotic regimes to more complex scenarios.

Findings

Derived limiting equations in bounded domains with finite charge

Extended analysis to non-homogeneous densities and Lake equations

Incorporated dissipative effects from Vlasov-Poisson-Fokker-Planck system

Abstract

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.