The Relativistic Vlasov-Maxwell System with External Electromagnetic Fields

Abstract

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov – Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma consists of several particle species, the particles are located in a container $\Omega\subset\mathbb R^3$, and are subject to boundary conditions on $\partial\Omega$. In the first two parts of this work, we deal with the situation that there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. In order to allow interaction between the exterior and the interior of the container, we do not impose perfect conductor boundary conditions for the electromagnetic fields – in contrast to other papers dealing with a similar setting, but without external currents – but consider the fields as functions on whole space $\mathbb R^3$ and model objects that are placed in space via given matrix-valued functions $\varepsilon$ (the permittivity) and $\mu$ (the permeability). Firstly, a weak solution concept is introduced and existence of global-in-time solutions is proved, as well as the redundancy of the divergence part of the Maxwell equations in this weak solution concept. Secondly, since a typical aim in fusion plasma physics is to keep the amount of particles hitting $\partial\Omega$ as small as possible (since they damage the reactor wall), while the control costs should not be too exhaustive (to ensure efficiency), we consider a suitable minimization problem with the Vlasov – Maxwell system as a constraint. This problem is analyzed in detail. In particular, we prove existence of minimizers and establish an approach to derive first order optimality conditions. In the third part of this work, we consider the case that the plasma is located in an infinitely long cylinder and is influenced by an external magnetic field. We prove existence of stationary solutions (extending in the third space direction infinitely) and give conditions on the external magnetic field under which the plasma is confined inside the cylinder, that is, it stays away from the boundary of the cylinder.%%%%Die zeitliche Entwicklung eines kollisionsfreien Plasmas wird durch das relativistische Vlasov-Maxwell-System modelliert, das die Vlasov-Gleichung (die Transportgleichung) mit den Maxwell-Gleichungen der Elektrodynamik koppelt. Es wird der Fall betrachtet, dass das Plasma aus mehreren Teilchenspezies besteht, die Teilchen sich in einem Behalter $\Omega\subset\mathbb R^3$ befinden und auf $\partial\Omega$ Randbedingungen genugen. In den ersten beiden Teilen dieser Arbeit wird die Situation behandelt, dass externe Strome vorhanden sind, typischerweise auserhalb des Behalters, die bei entsprechender Justierung als Steuerung des Plasmas dienen konnen. Um eine Interaktion zwischen dem Auseren und dem Inneren des Behalters zu ermoglichen, werden keine Randbedingungen eines perfekten Leiters fur die elektromagnetischen Felder verlangt – im…