The quasineutral limit in the quantum drift-diffusion equations

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 socalled 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.