The combined semi-classical and relaxation limit in a quantum hydrodynamic semiconductor model

We discuss the combined semi-classical and relaxation limit of a one-dimensional isentropic quantum hydrodynamical model for semiconductors. The quantum hydrodynamic equations consist of the iserrtropic Eider equations for the particle density and current density; including the quantum potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. With the help of the Maxwell-type iteration; we prove that; as the relaxation tine and Planck constant tend to zero, periodic initial-value problems of a. scaled one-dimensional isentropic quantum hydrodynamic model have unique smooth solutions existing in the time interval where the classical drift; diffusion model has smooth solutions. Meanwhile, we justify a, formal derivation of the classical drift-diffusion model from the quantum hydrodynamic; model.