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

Filbet, Francis; Rodrigues, L. Miguel

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1009-1067.

Nous étudions le comportement asymptotique des solutions de l’équation de Vlasov en présence d’un fort champ magnétique externe. En particulier, nous justifions rigoureusement l’obtention de l’approximation centre-guide dans un cadre général en dimension $3$ pour un champ magnétique inhomogène. Les corrections d’ordre $1$ sont également décrites et justifiées, y compris le terme $E\times B$, les gradients du champ magnétique et les effets de courbure. En outre, nous traitons le comportement en temps long pour deux exemples spécifiques, le cas bidimensionnel en coordonnées cartésiennes (pour ses vertus pédagogiques) et une géométrie toroïdale axi-symétrique. Notre approche est essentiellement basée sur des manipulations algébriques, plutôt que sur une structure variationnelle particulière.

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 cartesian coordinates and a toroidal 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. At last, we analyze a smoothed Vlasov-Poisson system, thus showing how our arguments may be extended to deal with the nonlinearity arising from self-consistent fields.

Classification : 35Q83, 78A35, 82D10, 35B40
Keywords: Vlasov equation, guiding center approximation, gyrokinetics, asymptotic analysis
Mot clés : Analyse asymptotique, équation de Vlasov, approximation centre-guide, gyro-cinétique

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