Stability and existence of stationary solutions to the Euler-Poisson equations in a domain with a curved boundary

TL;DR

This paper investigates the existence and stability of stationary solutions to the Euler-Poisson equations in domains with curved boundaries, extending plasma sheath formation analysis beyond planar walls under Bohm's criterion.

Contribution

It establishes existence and stability results for stationary solutions in nonplanar geometries, generalizing sheath formation criteria to curved boundary domains.

Findings

Existence of stationary solutions under Bohm's criterion.

Asymptotic stability of these solutions.

Shape of walls influences sheath formation.

Abstract

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of walls immersed in a plasma, and to analyze qualitative information of such a sheath layer. In the case of planar wall, Bohm proposed a criterion on the velocity of the positive ion for the formation of sheath, and several works gave its mathematical validation. It is of more interest to analyze the criterion for the nonplanar wall. In this paper, we study the existence and asymptotic stability of stationary solutions for the Euler-Poisson equations in a domain of which boundary is drawn by a graph. The existence and stability theorems are shown by assuming that the velocity of the positive ion satisfies the Bohm criterion at infinite distance. What most interests us in these theorems is that the criterion together with a suitable necessary condition guarantees the formation of sheaths as long as the shape of walls is drawn by a graph.