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The Child–Langmuir Asymptotics of the Vlasov–Poisson Equation for Cylindrically or Spherically Symmetric Diodes Part 2: Analysis of the Reduced Problem and Determination of the Child–Langmuir Current
Author(s) -
Degond P.,
Jaffard S.,
Poupaud F.,
Raviart P. A.
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19960310)19:4<313::aid-mma773>3.0.co;2-#
Subject(s) - poisson's equation , langmuir , current (fluid) , mathematics , diode , plasma , langmuir probe , poisson distribution , mathematical analysis , physics , quantum mechanics , plasma diagnostics , chemistry , statistics , thermodynamics , aqueous solution
This paper analyses the Child–Langmuir asymptotics of the Vlasov–Poisson problem in cylindrical or spherical symmetry. The problem was stated in the first part of this paper [2]. We recall that the Child–Langmuir asymptotics concerns the boundary value problem for the Vlasov–Poisson system in the situation where the thermal energy of the injected particles at the boundary is small compared with the external applied bias. In the first part, we derived the set of estimates which allow us to pass to the limit in the asymptotic problem. In the present part, we analyse the limit (or ‘reduced’) problem, which leads us to a characterization of the limit or ‘Child–Langmuir’ current which flows through the system.