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

Abstract

<p style='text-indent:20px;'>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 <i>Vlasov-Maxwell</i> system in a two-dimensional annulus when a huge (<i>but finite-in-time</i>) 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 <i>within a finite time interval</i> 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 <i>Vlasov-Maxwell</i> system.</p>

Lipschitz continuous solutions of the Vlasov-Maxwell systems with a conductor boundary condition

“…In other words, the acceleration of particles due to this external electromagnetic field is repellent to the boundary, which is the exact opposite effect of gravitation/polarization electric field satisfying the Pannekoek-Rosseland condition. In some sense, one can reduce the initial-boundary value problem to the Cauchy problem when the confining magnetic field is dominant ( [63,33]).…”

“…As far as the authors know, Theorem (1)-( 3) provide the first unique solvability of the RVM system when the physical boundary has a global effect (cf. [63,33]). The time span T of existence depends on the size of the initial data f 0 , E 0 , B 0 and their derivatives, and c 1 in (0.16).…”