Asymptotics of the three dimensional Vlasov equation in the large magnetic field limit

TL;DR

This paper rigorously derives the guiding-center approximation for the 3D Vlasov equation under strong magnetic fields, including first-order corrections and long-term behaviors in specific geometries.

Contribution

It provides a mathematically rigorous derivation of the guiding-center approximation in 3D with inhomogeneous magnetic fields, including first-order corrections and long-time analysis.

Findings

Guiding-center approximation derived rigorously in 3D

First-order correction terms computed and justified

Analysis of long-time behavior in specific geometries

Abstract

We study the asymptotic behavior of solutions to the Vlasov equation in the presence of a strong external magnetic field. In particular we provide a mathematically rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large inhomogeneous magnetic fields. First order corrections are computed and justified as well, including electric cross field, magnetic gradient and magnetic curvature drifts. We also treat long time behaviors on two specific examples, the two dimensional case in carte-sian coordinates and a poloidal axi-symmetric geometry, the former for expository purposes. Algebraic manipulations that underlie concrete computations make the most of the linearity of the stiffest part of the system of characteristics instead of relying on any particular variational structure.