FROM BOLTZMANN EQUATION TO SPHERICAL HARMONICS EXPANSION MODEL: DIFFUSION LIMIT AND POISSON COUPLING

The diffusion approximation of an initial-boundary value problem for a Boltzmann Poisson system is studied. An elastic operator modeling electron-impurity collision is considered. A relative entropy is used to control the terms coming from the boundary and to prove useful L-2-estimates for the renormalized solutions of the scaled Boltzmann equation (coupled to Poisson). A careful analysis of a relative entropy for high velocity allows us to show uniform bounds for the total mass and the kinetic energy which gives the compactness of the self-consistent electrostatic potential. Then, the moment method is used to prove the convergence of the renormalized solutions to a weak solution of a Spherical Harmonics Expansion (or SHE-) model coupled to the Poisson equation.