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

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.