Relaxation limit of the one-dimensional bipolar Euler–Poisson system in the bound domain

Abstract

In this paper, we present a one-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas, which takes the form of bipolar isothermal Euler–Poisson with electric field and frictional damping added to the momentum equations. From proper scaling, when the relaxation time in the bipolar Euler–Poisson system tends to zero, we can obtain the bipolar drift-diffusion equation. First, we show that the solutions to the initial boundary value problems of the bipolar Euler–Poisson system and the corresponding drift-diffusion equation converge to their stationary solutions as time tends to infinity, respectively. Then, it is shown that the solution for the bipolar Euler–Poisson equation converges to that of the corresponding bipolar drift-diffusion equations as the relaxation time tends to zero with the initial layer.