Expansion of a singularly perturbed equation with a two-scale converging convection term

Alexandre  Mouton

doi:10.3934/dcdss.2016058

Article Contents

2016, Volume 9, Issue 5: 1447-1473. Doi: 10.3934/dcdss.2016058

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Expansion of a singularly perturbed equation with a two-scale converging convection term

Alexandre Mouton^1,

1.
Laboratoire Paul Painlevé, CNRS & Universit é de Sciences et Technologies Lille 1, Cit é Scientifique, F-59655 Villeneuve d'Ascq

Received: February 2015

Revised: September 2015

Published: October 2016

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Abstract

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $\epsilon^{-1}$, $\epsilon \ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $\epsilon$ by using some two-scale convergence tools. Following Frénod & Sonnendrücker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $\epsilon^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $\epsilon$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams.

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Mathematics Subject Classification: Primary: 35Q83, 76M40, 78A35, 82D10; Secondary: 76M40.

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