Existence and some limits of stationary solutions to a one-dimensional bipolar Euler–Poisson system

In this paper, we study the stationary flow for a one-dimensional isentropic bipolar Euler–Poisson system (hydrodynamic model) for semiconductor devices. This model consists of the continuous equations for the electron and hole densities, and their current densities, coupled the Poisson equation of the electrostatic potential. In a bounded interval supplemented by the proper boundary conditions, we first show the unique existence of stationary solutions of the one-dimensional isentropic hydrodynamic model, based on the Schauder fixed-point principle and the careful energy estimates. Next, we investigate the zero-electron-mass limit, combined zero-electron mass and zero-hole mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit, respectively. We also show the strong convergence of the sequence of solutions and give the associated convergence rates.