THE CHILD-LANGMUIR ASYMPTOTICS OF THE VLASOV-POISSON EQUATION FOR CYLINDRICALLY OR SPHERICALLY SYMMETRICAL DIODES .1. STATEMENT OF THE PROBLEM AND BASIC ESTIMATES

The Child-Langmuir asymptotics of the Vlasov-Poisson system provides a model for vacuum diodes which operate under large biases. In these co nditions the energy of the injected particles at the cathode is very s mall compared with the applied external bias. From the mathematical vi ew point, this leads to an interesting and non-standard asymptotic pro blem for the Vlasov-Poisson equation, which has already been investiga ted in the one-dimensional Cartesian case, in [7]. The purpose of this paper is to extend the analysis to the cylindrically or spherically s ymmetric case. Surprisingly, the behaviour of the solutions of the mod el is somehow different than in the Cartesian case. This feature had n ot been noticed by the physicists before. Furthermore, the mathematica l analysis is much more involved than in [7] because of the geometrica l effects, and the techniques that are used are quite different. They mainly rely on the use of supersolutions in the spirit of [18, 19]. Th is work is divided in two parts. In this first part, we state the prob lem and establish the basic estimates which are needed for the asympto tic analysis.