On Mathematic Analysis Of The Viscous Quantum Hydrodynamic Model In Semiconductor

The viscous quantum hydrodynamic model arises from physics. It has higcr value not, only in the research fields but also in practice, therefore, it has been a very important part in semiconductor field. The viscous quantum hydrodynamic model is composed of two conservation equations with particle density n, the current density ,1 and the Poisson equation. Two conservation equations are derived from a Wigner-Fokker-Planck model, they contain a third-order quantum correction term and second-order viscous term. Our object in this paper is the steady-state visous quantum hydrodynamic model in one space dimension, we study mainly existence and uniqueness of solutions in the physically motivated the Dirichlet and Neumann boundary conditions, two important limits as well.First of all, in this paper, we reformulate the problem as a fourth-order elliptic equation by using an exponential variable transformation. Second, we build up a priori estimates of solutions by using the truncated method in the case of "weakly'1 subsonic. And then, we get existence of solutions which is based on a priori estimates to the fourth-order equation by the Leray-Schauder fixed point theorem. In the following, we prove the uniqueness of solution when the physical parameter is enough small. In the end, we prove inviscid limit and zero-space-charge limit.