Quasineutral limit for the full quantum hydrodynamic system for semiconductors

In this paper, we consider a three-dimensional full quantum hydrodynamic system for particle density, current density, energy density and electrostatic potential, coupled with a Poisson equation. Formally, as the Debye length tends to zero (i.e., the quasineutral limit), we see that the full quantum hydrodynamic system can be reduced to the incompressible nonisentropic Euler type equation. Based on the existence of the solutions for the full quantum hydrodynamic system and the incompressible nonisentropic Euler type equation, the quasineutral limit of the full quantum hydrodynamic system with the well-prepared initial data is rigorously showed by the method of asymptotic expansion and energy methods. The main ingredient of the proof is some rigorous uniform estimates on the error functions with respect to the Debye length. It is demonstrated that the quantum effect and the electrostatic potential do play important roles in the estimates, and we need to choose the appropriate Sobolev spaces and use the curl-div decomposition of the gradient.