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The Child–Langmuir Asymptotics of the Vlasov–Poisson Equation for Cylindrically or Spherically Symmetric Diodes Part 1: Statement of the Problem and Basic Estimates
Author(s) -
Degond P.,
Jaffard S.,
Poupaud F.,
Raviart P. A.
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19960310)19:4<287::aid-mma774>3.0.co;2-h
Subject(s) - mathematics , poisson's equation , statement (logic) , mathematical analysis , vlasov equation , poisson distribution , plasma , mathematical physics , physics , quantum mechanics , statistics , law , political science
Abstract The Child–Langmuir asymptotics of the Vlasov–Poisson system provides a model for vacuum diodes which operate under large biases. In these conditions the energy of the injected particles at the cathode is very small compared with the applied external bias. From the mathematical view point, this leads to an interesting and non‐standard asymptotic problem for the Vlasov–Poisson equation, which has already been investigated in the one‐dimensional Cartesian case, in [7]. The purpose of this paper is to extend the analysis to the cylindrically or spherically symmetric case. Surprisingly, the behaviour of the solutions of the model is somehow different than in the Cartesian case. This feature had not been noticed by the physicists before. Furthermore, the mathematical analysis is much more involved than in [7] because of the geometrical effects, and the techniques that are used are quite different. They mainly rely on the use of supersolutions in the spirit of [18, 19]. This work is divided in two parts. In this first part, we state the problem and establish the basic estimates which are needed for the asymptotic analysis.