Relaxation Limit of the One-Dimensional Bipolar Isentropic Euler-Poisson System in the Bound Domain

In this paper, we investigate a physically relevant hydrodynamic model for a bipolar semiconductor device considering Ohmic conductor boundary conditions and a non-flat doping profile. From proper scaling, when the relaxation time in the bipolar isentropic Euler-Poisson system tends to zero, we can obtain the bipolar drift-diffusion equations. First, we show that the solutions to the initial boundary value problems of the bipolar isentropic Euler-Poisson system and the corresponding drift-diffusion equations converge to their corresponding stationary solutions as the time tends to infinity, respectively. Then, it is shown that the solution for the bipolar isentropic Euler-Poisson equations converges to that of the corresponding bipolar drift-diffusion equations as the relaxation time tends to zero with the initial layer. These results are proven by the decay estimates of solutions, which are derived by energy methods.