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

Diffusion approximation for the one dimensional Boltzmann-Poisson system

N. Ben Abdallah 1, and  M. Lazhar Tayeb 2,

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.
Keywords: Transport equations, Boltzmann-Poisson system, Hilbert expansion., entropy inequality, Drift-Diffusion equations.
Mathematics Subject Classification: 35B45, 35B25, 82B21, 54C7.
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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