Combined Semi - Classical And Relaxation Limits Of One - Dimensional Quantum Euler - Poisson Equation

In this paper, we discuss the combined semi-classical and relaxation limit of a one-dimensional isentropic quantum hydrodynamical model for semiconductors. The quantum hy-drodynamic equations consist of the isentropic Euler-Poisson equations for the particle density and current density, including the quantum potential and a momentum relaxation term. The mo-mentum equation is highly nonlinear and contains a dispersive term with third-order derivatives.I show the relaxation limit as the relaxation time e tends to zero. First of all, we simplify the original equation and take the relaxation time c goes to zero, then initial layer phenomena appears. After that, we construct formal approximations of initial layer solution to the nonlinear problem. Next, assuming some regularity of the solution to the reduced problem, we prove the existence of classical solution. At last, we justify the validity of the formal approximations in any fixed compact subset of the uniform time interval.