The macroscopic equations of quantum hydrodynamical (QHD) type are used to simulate the motion of charge transport in ultra submicron semiconductor devices, where quantum effects, depending on particle resonant tunnelling through potential barriers and charge density built-up in quantum wells, take place. Therefore, QHD models are important and dominative in the description of the motion of electrons or holes transport under the self-consistent electric field. For last two decades, there have been many mathematical studies about the QHD model for semiconductors in different settings. However,...