An asymptotic analysis of the one-dimensional Vlasov–Poisson system: the Child–Langmuir law

Article type: Research Article

Authors: Degond, P. | Raviart, P.A.

Affiliations: Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Abstract: We perform the asymptotic analysis of the one-dimensional Vlasov–Poisson system when singular boundary data are prescribed. Such a singular perturbation problem arises in the modelling of vacuum diodes under very large applied bias, and gives rise to the well-known “Child-Langmuir law”. In this paper, we provide a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter ε is set equal to zero) and the boundary layer problem, which gives a sharp qualitative information.

DOI: 10.3233/ASY-1991-4301

Journal: Asymptotic Analysis, vol. 4, no. 3, pp. 187-214, 1991