Linear stability of the Vlasov–Poisson system with boundary conditions

We study the linear stability of the Vlasov–Poisson system in one dimension on a finite interval and half line with the in-flow and specular reflection boundary conditions. Conditions for the existence and non-existence of stationary solutions are obtained. The behavior of the perturbed solutions is investigated in different contexts. We show that, if the equilibrium distribution μμ is a strictly decreasing function of local energy density, then the Vlasov–Poisson system on a finite interval is stable in the (weighted) L2L2 sense for the in-flow and specular reflection boundary conditions. For the case μμ is strictly increasing, we prove a similar result on a finite interval under some extra assumptions. Finally, in the last part of the article, the spectral stability of the Vlasov–Poisson system on an unbounded interval is studied with similar monotonicity assumptions on μμ.