This paper is devoted to staudy the existence and the time-asymptotic of multidimensional quantum hydrodynamic equations for the electron particle density, the current density and the electrostatic potential in spatial periodic domain. The equations are formally analogous of the classical hydrodynamics but differ in the momentum equation, which is forced by an additional nonlinear dispersion term, (due to the quantum Bohm potential,) and are used in the modelling of quantum effects on semiconductors devices.
We prove the local-in-time existence of the solutions, in the case of general, nonconvex pressure-density relation and large and regular initial data. Furthermore we propose a “subsonic” type stability condition related to that one of the classical hydrodynamical equations. When this condition is satisfied, the local-in-time solutions exist globally in-time and converge time exponentially toward the corresponding steady-state. Since for this problem classical methods like, for instance, the Friedrichs theory for symmetric hyperbolic systems cannot be used, we investigate via an iterative procedure an extended system, which incorporates the one under investigation as a special case. In particular the dispersive terms appears in the form of a fourth-order wave type equation.
Quantum hydrodynamic models become important and necessary to model and simulate electron transport, affected by extremely high electric fields, in ultra-small sub-micron semiconductor devices, such as resonant tunnelling diodes, where quantum effects (like particle tunnelling through potential barriers and built-up in quantum wells [10, 21]) take place and dominate the process. Such kinds of quantum mechanical phenomena cannot be simulated by classical hydrodynamical models. The advantage of the macroscopic quantum hydrodynamical models relies in the facts that they are not only able to describe directly the dynamics of physical observable and simulate the main characters of quantum effects but also numerically less expensive than those microscopic models like Schrödinger and Wigner-Boltzmann equations. Moreover, even in the process of semiclassical (or zero dispersion) limit, the macroscopic quantum quantities like density, momentum, and temperature converge in some sense to these of Newtonian fluid-dynamical quantities . Similar macroscopic quantum models are also used in other physical area such as superfluid  and superconductivity .