Averaging of highly-oscillatory transport equations

Philippe Chartier; Nicolas Crouseilles; Mohammed Lemou; Florian Méhats

doi:10.3934/krm.2020039

Article Contents

2020, Volume 13, Issue 6: 1107-1133. Doi: 10.3934/krm.2020039

This issue Previous Article A semigroup approach to the convergence rate of a collisionless gas Next Article Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder

Averaging of highly-oscillatory transport equations

1.
Univ Rennes, Inria, Irmar, Mingus Team, Campus de Beaulieu, F-35042 Rennes, France
2.
Univ Rennes, CNRS, Irmar, Mingus Team, Campus de Beaulieu, F-35042 Rennes, France
3.
Univ Rennes, Irmar, Mingus Team, Campus de Beaulieu, F-35042 Rennes, France

^* Corresponding author: Philippe Chartier
^* Corresponding author: Philippe Chartier

Received: February 2020

Revised: June 2020

Early access: September 2020

Published: November 2020

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Abstract

In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature are re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.

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Mathematics Subject Classification: Primary:34C29, 82B40, 35Q83.

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