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 N particle species, the particles are located in a bounded container , and are subject to boundary conditions on ∂Ω. Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We do not impose perfect conductor boundary conditions for the electromagnetic fields but consider the fields as functions on whole space and model objects, that are placed in space, via given matrix‐valued functions ε (the permittivity) and μ (the permeability). 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.
Highlights
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system
We consider the following setting: there are N species of particles, all of which are located in a container Ω ⊂ R3, which is a bounded domain, for example, a fusion reactor
In the exterior of Ω, there are external currents, for example, in electric coils, that may serve as a control of the plasma if adjusted suitably
Summary
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system. The more important motivation of our paper is the following: the papers concerning plasma in a domain we are aware of deal with perfect conductor boundary conditions for the electromagnetic fields Such a setup can model no interaction between this domain and the exterior. The main aim of fusion plasma research is to adjust these external currents “suitably.” we impose Maxwell's equations globally in space and model objects like the reactor wall, electric coils, and almost perfect superconductors via ε and μ. For a more detailed overview, we refer to Rein[11] and to the book of Glassey,[12] which deals with other PDE systems in kinetic theory
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