A boundary‐value problem for the stationary vlasov‐poisson equations: The plane diode

Abstract

The stationary Vlasov-Poisson boundary value problem in a spatially one-dimensional domain is studied. The equations describe the flow of electrons in a plane diode. Existence is proved when the boundary condition (the cathode emission distribution) is a bounded function which decays superlinearly or a Dirac mass. Uniqueness is proved for (physically realistic) boundary conditions which are decreasing functions of the velocity variable. It is shown that uniqueness does not always hold for the Dirac mass boundary conditions. IntroductionThe construction of particle accelerators and free electron lasers requires electron guns which produce relativistic electron beams of high quality (low emittance and high current). Numerical simulations of increasingly sophisticated models are used in the design of electron guns: see Among these are the study of boundary conditions (which give rise to problems well known to physicists: the existence of a boundary layer near the cathode due to space-charge-limited currents, existence of wakes, etc.) and the study of the stationary equations.In this paper, we shall consider the most basic model-the stationary Vlasov-Poisson equations for classical electronic conduction in a plane diode. Although physically elementary, even this model contains difficulties due to space-charge limitation of the current, and so our study may be of value as a background for a study of the full multi-dimensional, time-dependent problem. In addition, the current paper is one of the few to include boundaries in a study of the Vlasov-Poisson equations (another discussion can be found in [S]).

Уравнения Власова - Пуассона Для Двукомпонентной Плазмы В Однородном Магнитном Поле

“…Исследованию стационарных решений уравнений Власова-Пуассона посвя-щены работы В. В. Веденяпина [114], [115], Ю. Батта и К. Фабиана [12], Ю. Бат-та, В. Фальтенбахера и Э. Хорста [13], С. И. Похожаева [91], К. Грингарда и П. А. Равьяра [48] и Г. Райна [95]. Стационарные решения уравнений Вла-сова-Максвелла и бифуркация решений рассматривалась в статьях [79], [80], [100], [101], [105]- [108].…”

Scientific Description of Summer Semiconductor Program

“…They have given an analysis of the Vlasov-Poisson system in a onedimensional geometry. The techniques developed by the author [3,4] to construct stationary solutions for kinetic models in any kind of geometry are quiet different of those of [1,2]. They are based on the use of upper solutions of the Vlasov equation.…”

“…Week 3 -Quantum Effects (I week), July 31 -Aug. 2,1991 As the dimensions of modem devices decrease to ultrasmall levels, quantum effects begin to play a major role. Examples are the quantization of states in confined regions -thin channels, quantum-wires etc.…”

Permanent regimes for the Vlasov–Maxwell equations with specular boundary conditions

“…the boundary conditions need to be considered [3], [16]. The stationary problems were studied by Greengard and Raviart [15], Poupaud [23]. Results for the time periodic case can be found in [4], [6].…”