Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

Jin Woo Jang; Robert M. Strain; Tak Kwong Wong

doi:10.3934/krm.2021039

Article Contents

2022, Volume 15, Issue 4: 569-604. Doi: 10.3934/krm.2021039

This issue Previous Article Lagrangian dual framework for conservative neural network solutions of kinetic equations Next Article Kinetic description of stable white dwarfs

Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

1.
Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea
2.
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA
3.
Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

^* Corresponding author: Robert M. Strain
^* Corresponding author: Robert M. Strain

†Supported by the German DFG grant CRC 1060, the Korean Basic Science Research Institute Fund NRF-2021R1A6A1A10042944 and the Korean IBS grant IBS-R003-D1.
*Partially supported by the NSF grants DMS-1764177 and DMS-2055271 of the USA.
‡Partially supported by the HKU Seed Fund for Basic Research under the project code 201702159009, the Start-up Allowance for Croucher Award Recipients, Hong Kong General Research Fund (GRF) grant "Solving Generic Mean Field Type Problems: Interplay between Partial Differential Equations and Stochastic Analysis" with project number 17306420, and Hong Kong GRF grant "Controlling the Growth of Classical Solutions of a Class of Parabolic Differential Equations with Singular Coefficients: Resolutions for Some Lasting Problems from Economics" with project number 17302521.

Received: July 2021

Revised: November 2021

Early access: November 2021

Published: August 2022

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Abstract

Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system.

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Mathematics Subject Classification: Primary: 35Q83, 35Q61, 82C40, 82D10.

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Figure 1. The domain $ \Delta $

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