Stationary solutions of the Vlasov–Poisson system for two-component plasma under an external magnetic field in a half-space

Abstract

We consider the first mixed problem for the Vlasov-Poisson equations with an external magnetic field in a half-space. This problem describes the evolution of the density distributions of ions and electrons in a high temperature plasma with a fixed potential of electric field on a boundary. For a sufficiently large induction of external magnetic field there are constructed stationary solutions supported on some distance from the hyperplane x1 = 0.

A general way to confined stationary Vlasov-Poisson plasma configurations

“…Since by our general assumption in this section I β λb is a first integral to the Vlasov equation (19) considered with parameter λb, we directly see that f β λ solves the Vlasov equation considered with potential ϕ λ and it remains to check the generalized Poisson equation (7).…”

“…In general, stationary solutions of the two-component Vlasov-Poisson equation with trivial potential can be found by relating the cutoff functions ψ + , ψ − on an algebraic level similar to (26). In that case the use of the sub-/supersolution method or a fixed point argument is not needed, see for example [22,35,7]. Moreover, in [7] it has been shown that the trivial potential case allows additional first integrals of the characteristic equations to exist.…”

“…In that case the use of the sub-/supersolution method or a fixed point argument is not needed, see for example [22,35,7]. Moreover, in [7] it has been shown that the trivial potential case allows additional first integrals of the characteristic equations to exist. 4.…”

“…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.…”

“…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].…”