A general way to confined stationary Vlasov-Poisson plasma configurations

Yulia O. Belyaeva; Björn Gebhard; Alexander L. Skubachevskii

doi:10.3934/krm.2021004

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2021, Volume 14, Issue 2: 257-282. Doi: 10.3934/krm.2021004

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A general way to confined stationary Vlasov-Poisson plasma configurations

1.
RUDN University, 6, Mikluhko–Maklaya str., 117198 Moscow, Russia
2.
Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

^* corresponding author
^* corresponding author

Received: September 30, 2020

Early access: December 2020

Published: April 2021

The first and third author are supported by the Russian Foundation for Basic Research grant 20-01-00288 (section 5) and "RUDN University Program 5-100" (section 6). The second author is supported by the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD) (Project numbers: M-2018b-2, A-2019b-5_d)

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Abstract

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain $ \Omega\subset\mathbb{R}^3 $ describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of $ \Omega $. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.

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Mathematics Subject Classification: Primary: 35Q83; Secondary: 82D10.

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