Weak Solutions of the Relativistic Vlasov-Maxwell System with External Currents

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 $Ω\subset\mathbb R^3$, and are subject to boundary conditions on $\partialΩ$. 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 $\mathbb R^3$ and model objects, that are placed in space, via given matrix-valued functions $\varepsilon$ (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.