Abstract

We study the approximation by diffusion of a Boltzmann equation, considering a linear time relaxation model and an inflow boundary data with a general profile. A corrected Hilbert expansion and the contraction property of the collision operator are used to establish a uniform $L^1$-estimate. We introduce a correction of the boundary layer at the first order in order to prove strong convergence and to exhibit a rate of convergence. The limit fluid model is a drift-diffusion model associated with effective boundary data obtained as the decay at infinity of a half-space problem. The analysis is performed, in the first step, for the linear case (prescribed potential). In the second step, the analysis is extended to the case of a self-consistent potential (Poisson coupling) in one dimension by carefully combining the relative entropy method and a perturbation of the Hilbert expansion; giving the convergence and rate of convergence.

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