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

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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.

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