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Half-space problem of the Boltzmann equation for charged particles. (English) Zbl 0939.82040
Summary: For two particular collision kernels, we explicitly solve the one-dimensional stationary half-space boundary value problem of the linear Boltzmann equation including a constant external field via an extension of Case’s eigenfunction technique. In the first collision model we reproduce a solution recently obtained by Cercignani; in the second model the solution of the stationary boundary value problem is presented for the first time.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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[1] C. Cercignani,The Boltzmann Equation and Its Application, Springer, New York, 1986. · Zbl 0594.76068
[2] K. M. Case, Elementary solutions of the transport equation and their applications,Annals of Physics 9:1 (1960). · Zbl 0087.42301
[3] P. F. Zweifel, A generalized transport equation,Transport Theory Stat. Phys. 11:183 (1983). · Zbl 0527.47033
[4] W. Greenberg and C. V. M. van der Mee, Abstract kinetic equations relevant to supercritical media,J. Func. Anal. 57:111 (1984). · Zbl 0584.47050
[5] C. Dalitz, Exact solutions of the semiconductor Boltzmann equation,Physica A 203:125 (1994).
[6] C. Dalitz, Initial-boundary value problems of the Boltzmann equation at zero temperature.Transport Theory Stat. Phys. (1997), in press. · Zbl 0903.35075
[7] C. Cercignani, An explicitly solvable kinetic model for semiconductors,J. Stat. Phys. 77:1039 (1994). · Zbl 0839.76075
[8] J. Piasecki, Approach to field-induced stationary state in a gas of hard rods,J. Stat. Phys. 30:185 (1983).
[9] A. Gervois and J. Piasecki, Stationary velocity distribution in an external field: a onedimensional model,J. Stat. Phys. 42:1091 (1986).
[10] O. J. Eder and M. Posch, Solution of the one-dimensional linear Boltzmann equation for charged Maxwellian particles in an external field,J. Stat. Phys. 52:1031 (1988). · Zbl 1084.82544
[11] L. Hörmander,The analysis of linear partial differential operators, Part 1, Springer, Berlin (1983).
[12] G. Doetsch,Handbuch der Laplacetransformation, Teil 1, Birkhäuser, Basel (1970). · Zbl 0202.12301
[13] P. C. Stichel and D. Strothmann, Asymptotic analysis of the high field semiconductor equation,Physica A 202:553 (1994).
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