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The Child-Langmuir asymptotics of the Vlasov-Poisson equation for cylindrically or spherically symmetric diodes. I: Statement of the problem and basic estimates. (English) Zbl 0844.35086

Summary: 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.
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 because of the geometrical effects, and the techniques that are used are quite different. They mainly rely on the use of supersolutions.
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. [For part II, see ibid., 313-340 (1996; reviewed below)].

MSC:

35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
76X05 Ionized gas flow in electromagnetic fields; plasmic flow

Citations:

Zbl 0844.35087
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References:

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