A minimization formulation of a bi-kinetic sheath

Mehdi  Badsi; Martin Campos  Pinto; Bruno  Després

doi:10.3934/krm.2016010

Article Contents

2016, Volume 9, Issue 4: 621-656. Doi: 10.3934/krm.2016010

This issue Previous Article Next Article A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror

A minimization formulation of a bi-kinetic sheath

1.
UPMC-Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, 4, pl. Jussieu F75252 Paris cedex 05

Received: September 30, 2014

Revised: June 30, 2015

Published: November 2016

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Abstract

The mathematical description of the interaction between a plasma and a solid surface is a major issue that still remains challenging. In this paper, we model this interaction as a stationary and bi-kinetic Vlasov-Poisson-Ampère boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be determined from the ampibolarity of the particle flows as the unique solution of a non linear equation. Based on variational techniques, our analysis establishes the well-posedness of the model, provided that the incoming ion distribution satisfies a moment condition that generalizes the historical Bohm criterion of plasma physics. Quantitative estimates are also given, together with numerical illustrations that validate the robustness of our approach.

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Mathematics Subject Classification: 35B35, 35L60, 82D10, 35B30.

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