Diffusion approximation for the one dimensional Boltzmann-Poisson system
N. Ben Abdallah - Mathématiques pour l'Industrie et la Physique, UMR, CNRS 5640, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France (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.
Received: January 2003; Revised: February 2004; Available Online: August 2004.
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