The Kinetic and Hydrodynamic Bohm Criterions for Plasma Sheath Formation

The purpose of this paper is to mathematically investigate the formation of a plasma sheath, and to analyze the Bohm criterions which are required for the formation. Bohm derived originally the (hydrodynamic) Bohm criterion from the Euler--Poisson system. Boyd and Thompson proposed the (kinetic) Bohm criterion from kinetic point of view, and then Riemann derived it from the Vlasov--Poisson system. In this paper, we prove the solvability of boundary value problems of the Vlasov--Poisson system. On the process, we see that the kinetic Bohm criterion is a necessary condition for the solvability. The argument gives a simpler derivation of the criterion. Furthermore, the hydrodynamic criterion can be derived from the kinetic criterion. It is of great interest to find the relation between the solutions of the Vlasov--Poisson and Euler--Poisson systems. To clarify the relation, we also study the hydrodynamic limit of solutions of the Vlasov--Poisson system.