Diffusion relaxation limit of a nonisentropic hydrodynamic model for semiconductors

In this paper, we discussed a general multidimensional nonisentropic hydrodynamical model for semiconductors with small momentum relaxation time. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. With the help of the Maxwell-type iteration, we prove that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled multidimensional nonisentropic hydrodynamic model has a unique smooth solution existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the drift-diffusion models from the nonisentropic hydrodynamic models.