Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder

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. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. 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 and give conditions on the external magnetic field under which the plasma is confined inside the cylinder, i.e., it stays away from the boundary of the cylinder.

On the two and one-half dimensional Vlasov-Poisson system with an external magnetic field: global well-posedness and stability of confined steady states

“…In [22], confined steady states for the two-dimensional analogue of (1.1) (meaning that x, p ∈ R 2 ) were constructed. The same has been done in [39] for the two and one-half dimensional relativistic Vlasov-Maxwell system. In both papers an additional symmetry, namely rotational symmetry about the z-axis, is assumed, and the ansatz…”

A general way to confined stationary Vlasov-Poisson plasma configurations

“…Stationary solutions of the Vlasov-Poisson equations have been studied in various settings [5,6,7,16,22,30,35,31,38,39,41]. Let us focus on the ones addressing the confinement problem.…”

“…On Ω = R 3 stationary solutions confined to an infinite cylinder and with the Newtonian electric potential have been constructed in [22]. In [41] stationary confined solutions in an infinite cylinder have also been constructed for the relativistic Vlasov-Maxwell system. The proofs in [22,41] rely, after a suitable ansatz and reduction, on a fixed point argument.…”

“…In [41] stationary confined solutions in an infinite cylinder have also been constructed for the relativistic Vlasov-Maxwell system. The proofs in [22,41] rely, after a suitable ansatz and reduction, on a fixed point argument.…”

“…The strategy to find stationary solutions also here is based on the well-known method, see [5,6,30,35,38,39,41], of exploiting first integrals of the characteristic systemẋ…”