The Relativistic Vlasov–Maxwell System with External Electromagnetic Fields
Speaker: Dr. Jörg Weber
Affiliation: Universität Bayreuth, Germany
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 situation that the plasma is located in a bounded container and that there are external currents, typically in the exterior of the container, which may serve as a control of the plasma if adjusted suitably. Firstly, a weak solution concept is introduced and existence of global-in-time solutions is discussed. Secondly, since a typical aim in fusion plasma physics is to keep the amount of particles hitting the boundary 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. In particular, we address the following questions: Does a minimizer of this problem exist? How can first order optimality conditions be established rigorously? Here, the difficulty lies in the very weak solution concept and the possible non-uniqueness of solutions. A natural question is also whether steady states of this system exist which are confined inside the container. In the last part of the talk, this problem is investigated in a simplified setup, where the container is an infinitely long cylinder and an external magnetic field is present.