Convergence of Nonlinear Schrödinger-Poisson Systems to the Compressible Euler equations

The combined semi-classical and quasineutral limit in the bipolar defocusing nonlinear Schrödinger-Poisson system in the whole space is proven. The electron and current densities, defined by the solution of the Schrödinger-Poisson system, converge to the solution of the compressible Euler equation with nonlinear pressure. The corresponding Wigner function of the Schrödinger-Poisson system converges to a solution of a nonlinear Vlasov equation. The proof of these results is based on estimates of a modulated energy functional and on the Wigner measure method