Steady-State Solutions And Asymptotic Limits Of Quantum Hydrodynamic Model For Semiconductors

In this paper,the steady-state quantum hydrodynamic model for semiconductors is analyzed on the several dimensional space in first part. We show the existence of solutions for sufficiently small relaxation time by applying a truncation method, Leray-Schauder's fixed point theorem and estimates of the elliptic equation.For the bipolar quantum hydrodynamic model, the solutions is unique if Planck constant is large sufficiently. The uniqueness is investigated in thermal equilibrium for any Planck constant. In the last of this part,the relaxation time,dispersive limit are shown for the bipolar and unipolar equations,respectively.In the second part,the steady-state viscous quantum hydrodynamic model in the one-dimensional space is studied. The existence and uniqueness of strong solutions with general boundary conditions is obtained under a " subsonic " condition.Viscosity vanishing limit,dispersion limit and neutral charge limit are carried out in the last.A-prior estimate is based on constructing an auxiliary function and the proofs rely on the truncation technique, the Leray-Schauder fixed point theorem and an exponential variable transformation.