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

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 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.

Variational radial sheath solutions for a kinetic model of a cylindrical Langmuir probe

“…Using this generic phase space decomposition, we construct explicitly a solution to the Vlasov equation. For details and comments on the construction, we refer the reader to Badsi and Godard-Cadillac [7]. For the sake of conciseness, we shall often use the subscript s ∈ {i, e} to denote a quantity that corresponds to the ions when s = i or to the electrons when s = e. We therefore obtain expressions for weak solution of Vlasov equation, as detailed in the two following propositions obtained at Badsi and Godard-Cadillac [7].…”

“…Outside the sheath, trajectories of particles can be much complex depending on whether effective potential barriers exist. For a generic study of the phase space for a cylindrical Langmuir probe, we refer to our previous work [7].…”

“…Although ubiquitous in applications, Langmuir probes have not been investigated that much in the mathematical literature while it is possible to model the interaction of the probe as a stationary boundary value problem based on the Vlasov-Poisson equations. We refer to earlier studies [7][8][9][10][11] for the construction of stationary solutions in bounded domains. The main purpose of the present work is thus to pursue our previous study [7] where we have derived the model and proven the existence of weak-strong solutions.…”

“…We refer to earlier studies [7][8][9][10][11] for the construction of stationary solutions in bounded domains. The main purpose of the present work is thus to pursue our previous study [7] where we have derived the model and proven the existence of weak-strong solutions. Our previous study relies on the method of characteristics and a phase space study which yields quasi-explicit formula for the distributions functions.…”

“…We then state our main result in Section 3. In Section 4, we summarize the phase space study associated with the Vlasov equations done in Badsi and Godard-Cadillac [7]. It yields notably the reformulation of the problem as a nonlinear and nonlocal Poisson problem that is presented in Section 5.…”