Degond, P.; Jaffard, S.; Poupaud, F.; Raviart, P. A. 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 Math. Methods Appl. Sci. 19, No. 4, 287-312 (1996). 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)]. Cited in 1 ReviewCited in 6 Documents 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 Keywords:Child-Langmuir asymptotics; Vlasov-Poisson system; cylindrically or spherically symmetric case; supersolutions Citations:Zbl 0844.35087 PDFBibTeX XMLCite \textit{P. Degond} et al., Math. Methods Appl. Sci. 19, No. 4, 287--312 (1996; Zbl 0844.35086) Full Text: DOI References: [1] Bardos, Ann. Scient. Ec. Norm. Sup., 4e série 3 pp 185– (1970) [2] Analyse fonctionnelle, théorie et applications, Masson, Paris, 1983. [3] Degond, C.R. Acad. Sci. Paris 310 pp 607– (1990) [4] ’The Child-Langmuir law in the kinetic theory of charged-particles. Part 1. electron flows in vacuum’, in: Advances in Kinetic Theory, Theory and Computation (ed.), World Scientific, Singapore, 1994. · Zbl 0863.76091 [5] Degond, Math. Methods in the Appl. Sci. 19 pp 313– (1996) [6] and , ’The multidimensional Child-Langmuir problem in the kinetic theory of charged-particles’, in preparation. [7] Degond, Asymptotic Anal. 4 pp 187– (1991) [8] Degond, Asymptotic Anal. 6 pp 1– (1992) [9] Di Capua, IEEE Trans. Plasma Sci. PS11 pp 205– (1983) [10] Large Ion Beams, Fundamentals of Generation and Propagation, Wiley, New York, 1988. [11] Greengard, Comm. Pure Appl. Math. 43 pp 473– (1990) [12] ’Electron trajectory program’, SLAC Technical Report 166, September 1973. [13] Charged Particle Beams, Wiley, New-York, 1990. [14] Langmuir, Rev. Mod. Phys. 3 pp 191– (1931) [15] Lawconnell, Phys. Fluids B2 pp 629– (1990) · doi:10.1063/1.859297 [16] The physics of charged-particle beams, International Series of monographs on physics N{\(\deg\)} 75, Oxford University Press, Oxford, 1988. [17] Lovelace, Phys. Fluids 17 pp 1263– (1974) [18] Poupaud, C. R. Acad. Sci. Paris 311 pp 307– (1990) [19] Poupaud, Forum Mathematicum 4 pp 499– (1992) [20] Ron, IEEE Trans. Plasma Sci. PS1 pp 85– (1973) [21] Weisterman, Nucl. Instruments Methods Phys. Res. A260 pp 271– (1988) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.