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.Comment: 24 pages, 2 figure

Traveling Waves of the Vlasov--Poisson System

We consider the Vlasov--Poisson system describing a two-species plasma with spatial dimension $1$ and the velocity variable in $\mathbb{R}^n$. We find the necessary and sufficient conditions for the existence of solitary waves, shock waves, and wave trains of the system, respectively. To this end, we need to investigate the distribution of ions trapped by the electrostatic potential. Furthermore, we classify completely in all possible cases whether or not the traveling wave is unique. The uniqueness varies according to each traveling wave when we exclude the variant caused by translation. For the solitary wave, there are both cases that it is unique and nonunique. The shock wave is always unique. No wave train is unique.Comment: 56 pages, 9 figure

A priori estimates of solutions to the motion of an inextensible hanging string

We consider the initial boundary value problem to the motion of an inextensible hanging string of finite length under the action of the gravity. In this problem, the tension of the string is also an unknown quantity. It is determined as a unique solution to a two-point boundary value problem, which is derived from the inextensibility of the string together with the equation of motion, and degenerates linearly at the free end. We derive a priori estimates of solutions to the initial boundary value problem in weighted Sobolev spaces under a natural stability condition. The necessity for the weights results from the degeneracy of the tension. Uniqueness of solution is also proved.Comment: 40 pages, 3 figure

Nonlinear Stability and Instability of Plasma Boundary Layers

We investigate the formation of a plasma boundary layer (sheath) by considering the Vlasov--Poisson system on a half-line with the completely absorbing boundary condition. In an earlier paper by the first two authors, the solvability of the stationary problem is studied. In this paper, we study the nonlinear stability and instability of these stationary solutions of the Vlasov--Poisson system

Gravitino Dark Matter without R-parity

Cosmological issues are examined when gravitino is the lightest superparticle (LSP) and R-parity is broken. Decays of the next lightest superparticles occur rapidly via R-parity violating interaction, and thus they do not upset the big-bang nucleosynthesis, unlike the R-parity conserving case. The gravitino LSP becomes unstable, but its lifetime is typically much longer than the age of the Universe. It turns out that observations of diffuse photon background coming from radiative decays of the gravitino do not severely constrain the gravitino abundance, and thus the gravitino weighing less than around 1 GeV can be dark matter of the Universe when bilinear R-parity violation generates a neutrino mass which accounts for the atmospheric neutrino anomaly.Comment: 11 pages, 1 figur