Diffusion limit of a Boltzmann-Poisson system: case of general inflow boundary data profile

We study the approximation by diffusion of a Boltzmann equation, considering alinear time relaxation model and an inflow boundary data with a general profile. Acorrected Hilbert expansion and the contraction property of the collision operatorare used to establish a uniformL1-estimate. We introduce a correction of theboundary layer at the first order in order to prove strong convergence and toexhibit a rate of convergence. The limit fluid model is a drift-diffusion modelassociated with effective boundary data obtained as the decay at infinity of ahalf-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 caseof a self-consistent potential (Poisson coupling) in one dimension by carefullycombining the relative entropy method and a perturbation of the Hilbert expansion;giving the convergence and rate of convergence.