Kinetic and Related Models (KRM)

The Spherical Harmonics Expansion model coupled to the Poisson equation

Pages: 1063 - 1079, Volume 4, Issue 4, December 2011      doi:10.3934/krm.2011.4.1063

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Jan Haskovec - RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, Linz, A-4040, Austria (email)
Nader Masmoudi - Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York, 10012, United States (email)
Christian Schmeiser - Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, Vienna, A-1090, Austria (email)
Mohamed Lazhar Tayeb - Department of Mathematics, University of Tunis ElManar, Faculty of Sciences of Tunis, 2092 El-Manar, Tunisia (email)

Abstract: The Spherical Harmonics Expansion (SHE) assumes a momentum distribution function only depending on the microscopic kinetic energy. The SHE-Poisson system describes carrier transport in semiconductors with self-induced electrostatic potential. Existence of weak solutions to the SHE-Poisson system subject to periodic boundary conditions is established, based on appropriate a priori estimates and a Schauder fixed point procedure. The long time behavior of the one-dimensional Dirichlet problem with well prepared boundary data is studied by an entropy-entropy dissipation method. Strong convergence to equilibrium is proven. In contrast to earlier work, the analysis is carried out without the use of the derivation from a kinetic problem.

Keywords:  Spherical Harmonics Expansion model, Poisson equation, degenerate PDE, entropy, long time behavior.
Mathematics Subject Classification:  Primary: 35K65, 35Q82; Secondary: 58Z05.

Received: June 2011;      Revised: October 2011;      Available Online: November 2011.