Existence of solutions for a bi-species kinetic model of a cylindrical Langmuir probe

TL;DR

This paper proves the existence of solutions for a collisionless bi-species plasma model near a cylindrical Langmuir probe, combining weak solutions for Vlasov equations and a strong solution for the Poisson equation.

Contribution

It introduces a new existence proof for a coupled Vlasov-Poisson system in a cylindrical geometry, with a novel reformulation of the Poisson equation.

Findings

Existence of weak solutions for Vlasov equations.

Existence of a strong solution for the Poisson equation.

Reformulation of the Poisson equation as a 1D non-linear non-local problem.

Abstract

In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists in a bi-species Vlasov-Poisson equation in a domain contained between two cylinders with prescribed boundary conditions. The interior cylinder models the probe while the exterior cylinder models the interaction with the plasma core. We prove the existence of a weak-strong solution for this model in the sense that we get a weak solution for the 2 Vlasov equations and a strong solution for the Poisson equation. The first parts of the article are devoted to explain the model and proceed to a detailed study of the Vlasov equations. This study leads to a reformulation of the Poisson equation as a 1D non-linear and non-local equation and we prove it admits a strong solution using an iterative fixed-point procedure.