On the three-dimensional finite Larmor radius approximation: the case of electrons in a fixed background of ions

TL;DR

This paper analyzes a plasma physics model with finite Larmor radius, deriving a fluid system for specific data, proving local existence of solutions, and studying the limit as the Debye length vanishes using advanced mathematical techniques.

Contribution

It extends previous work by considering electrons in a fixed ion background, deriving a new fluid system, and rigorously analyzing the limit with Cauchy-Kovalevskaya methods.

Findings

Formal derivation of a fluid system for monokinetic data.

Proof of local in time existence of analytic solutions.

Rigorous analysis of the zero Debye length limit.

Abstract

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows \cite{DHK1}, where the tri-dimensional analysis of a Vlasov-Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell-Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in \cite{DHK1}. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the Debye length vanishes) to a new anisotropic fluid system. This is achieved thanks to Cauchy-Kovalevskaya type techniques, as introduced by Caflisch \cite{Caf} and Grenier \cite{Gre1}.