Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Diffusion approximation for the one dimensional Boltzmann-Poisson system

Pages: 1129 - 1142, Volume 4, Issue 4, November 2004      doi:10.3934/dcdsb.2004.4.1129

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N. Ben Abdallah - Mathématiques pour l'Industrie et la Physique, UMR, CNRS 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France (email)
M. Lazhar Tayeb - Laboratoire d'Ingéniere Mathématique, Ecole Polytechnique de Tunisie, La Marsa, Tunisia (email)

Abstract: 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, Drift-Diffusion equations, entropy inequality, Hilbert expansion.
Mathematics Subject Classification:  35B45, 35B25, 82B21, 54C70.

Received: January 2003;      Revised: February 2004;      Published: August 2004.