Communications in Mathematical Sciences

Volume 21 (2023)

Number 4

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

Pages: 1097 – 1134



Mehdi Badsi (Laboratoire de Mathématiques Jean Leray, Nantes Université, Nantes, France)

Ludovic Godard-Cadillac (Laboratoire de Mathématiques Jean Leray, Nantes Université, Nantes, France)


In this article, we study a collisionless kinetic model for plasmas in the neighborhood of a cylindrical metallic Langmuir probe. This model consists of 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 two Vlasov equations and a strong solution for the Poisson equation. The first parts of the article are devoted to explaining the model and proceed to a detailed study of the Vlasov equations. This study then 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.


cylindrical Langmuir probe, stationary Vlasov–Poisson equations, boundary value problem, non-local semi-linear Poisson equation

2010 Mathematics Subject Classification

35Q83, 82D10

Received 31 January 2022

Received revised 9 September 2022

Accepted 12 September 2022

Published 24 March 2023