The quasineutral limit in the quantum drift-diffusion equations

Abstract

The quasineutral limit in the transient quantum drift-diffusion equations in one space dimension is rigorously proved. The model consists of a fourth-order parabolic equation for the electron density, including the quantum Bohm potential, coupled to the Poisson equation for the electrostatic potential. The equations are supplemented with Dirichlet–Neumann boundary conditions. For the proof uniform a priori bounds for the solutions of the semi-discretized equations are derived from so-called entropy functionals. The drift term involving the electrostatic potential is estimated by proving a new bound for the electric energy. Since the electrostatic potential is not an admissible test function, an auxiliary test function has been carefully constructed.