Debye screening for the stationary Vlasov-Poisson equation in interaction with a point charge

Stability of a Point Charge for the Vlasov–Poisson System: The Radial Case

“…The existence and stability of other equilibriums has been considered for the Vlasov-Poisson system with repulsive interactions, most notably in connection to Landau damping [1,4,12,18,30]. In the case of Vlasov-Poisson with attractive interactions, there are many more equilibriums and their linear and nonlinear (in)stability have been studied [16,17,22,29,32], but the analysis of asymptotic stability is very challenging.…”

“…This contains the main term. Integrating by parts, we observe that [1] (ϑ, α, t)γ 2 (ϑ, α)dϑdα [1] (ϑ, α, t)γ 2 (ϑ, α)dϑdα = 1 { R(ϑ,α)≤r } ∂ α χ [1] (ϑ, α, t) [1] (ϑ, α, t) [1] (ϑ, α, t) [1] (ϑ, α, t)…”

“…It turns out that we will also need some moments of first derivatives. Given a weight function ω(θ, a), we define ω (1) (θ, a) := (a + a −1 )ω(θ, a), ω (2) (θ, a) := aω(θ, a), and we compute that ∂ t (ω (1) γ θ ) − λ{ , ω (1) γ θ } − λ∂ θ ∂ a • ω (1) (1) γ θ − λ∂ a • ∂ θ ω (1) γ θ , ∂ t (ω (2) γ a ) − λ{ , ω (2) γ a } + λ∂ θ ∂ a • ω (2) γ a − λ (2) γ a − λ∂ a • ∂ θ ω (2) γ a .…”

“…Writing ω = ω p,q for simplicity of notation, using (3.5) we find that (1) γ θ • ω (1) |∂ a | • |∂ θ ω (1) p,q γ θ • ω (1) p,q γ θ |dθ da…”

“…We consider initial data 1 of the form M = q c δ + q g μ 2 0 dxdv, where q c > 0 is the charge of the Dirac mass and q g > 0 is the charge per particle of the gas, which results in purely repulsive interactions. We track the singular and the absolutely continuous parts of a solution as 1 Here the initial continuous density f 0 = μ 2 0 is assumed to be non-negative, a condition which is then propagated by the flow and allows us to work with functions μ in an L 2 framework rather than a general non-negative function f in L 1 -see also our previous work [21] for more on this.…”