# American Institute of Mathematical Sciences

November  2004, 4(4): 1129-1142. doi: 10.3934/dcdsb.2004.4.1129

## Diffusion approximation for the one dimensional Boltzmann-Poisson system

 1 Mathématiques pour l'Industrie et la Physique, UMR, CNRS 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France 2 Laboratoire d'Ingéniere Mathématique, Ecole Polytechnique de Tunisie, La Marsa, Tunisia

Received  January 2003 Revised  February 2004 Published  August 2004

The diffusion limit of the initial-boundary value problem for the Boltzmann-Poisson system is studied in one dimension. By carefully analyzing entropy production terms due to the boundary, $L^p$ estimates are established for the solution of the scaled Boltzmann equation (coupled to Poisson) with well prepared initial and boundary conditions. A hybrid Hilbert expansion taking advantage of the regularity of the limiting system allows to prove the convergence of the solution towards the solution of the Drift-Diffusion-Poisson system and to exhibit a convergence rate.
Citation: N. Ben Abdallah, M. Lazhar Tayeb. Diffusion approximation for the one dimensional Boltzmann-Poisson system. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1129-1142. doi: 10.3934/dcdsb.2004.4.1129
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