# American Institute of Mathematical Sciences

December  2011, 4(4): 1063-1079. doi: 10.3934/krm.2011.4.1063

## The Spherical Harmonics Expansion model coupled to the Poisson equation

 1 RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, Linz, A-4040, Austria 2 Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York, 10012, United States 3 Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, Vienna, A-1090, Austria 4 Department of Mathematics, University of Tunis ElManar, Faculty of Sciences of Tunis, 2092 El-Manar, Tunisia

Received  June 2011 Revised  October 2011 Published  November 2011

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.
Citation: Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic & Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063
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