Confined steady states of a Vlasov‐Poisson plasma in an infinitely long cylinder

Abstract

We consider the two-dimensional Vlasov-Poisson system to model a two-component plasma whose distribution function is constant with respect to the third space dimension. First, we show how this two-dimensional Vlasov-Poisson system can be derived from the full three-dimensional model.The existence of compactly supported steady states with vanishing electric potential in a three-dimensional setting has already been investigated in the literature. We show that these results can easily be adapted to the two-dimensional system. However, our main result is to prove the existence of compactly supported steady states even with a nontrivial self-consistent electric potential.KEYWORDS magnetic confinement, nonlinear partial differential equations, stationary solutions, Vlasov-Poisson equation MSC CLASSIFICATION35Q83; 82D10

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

“…The ansatz (12) in turn can be inserted into the definition of ρ and j to derive representations of these densities in terms of the potentials.…”

“…The question about existence of confined steady states for a Vlasov-Poisson plasma (that is, B = 0) by means of an external magnetic field was considered in [18] and [12]. The approach of the latter work is similar to ours but needs some smallness assumption on the ansatz functions, which we can avoid, and is restricted to homogeneous external magnetic fields parallel to the symmetry axis.…”

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.…”

“…The existence of stationary solutions to (1), (2), (3) with vanishing potential and density distribution functions supported away from the considered boundary, as well as compactly supported distribution functions have first been shown to exist for Ω being an infinite cylinder and a half-space in [35,7]. 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.…”

“…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.…”