On fluid limit for the semiconductors Boltzmann equation

This paper is devoted to the derivation of (non-linear) drift-diffusion equations from the semiconductor Boltzmann equation. Collisions are taken into account through the non-linear Pauli operator, but we do not assume relation on the cross section such as the so-called detailed balance principle. In turn, equilibrium states are implicitly defined. This article follows and completes the contribution of Mellet (Monatsh. Math. 134 (4) (2002) 305–329) where the electric field is given and does not depend on time. Here, we treat the self-consistent problem, the electric potential satisfying the Poisson equation. By means of a Hilbert expansion, we shall formally derive the asymptotic model in the general case. We shall then rigorously prove the convergence in the one-dimensional case by using a modified Hilbert expansion.