The full viscous quantum hydrodynamic system in one dimensional space

Abstract

A viscous quantum hydrodynamic system for particle density, current density, energy density, and electrostatic potential, coupled with a Poisson equation, is studied in spatial one dimensional real line. The system is self-consistent in the sense that the electric field, which forms a forcing term in the momentum and energy equations, is determined by the coupled Poisson equation. First, the existence and uniqueness of the stationary solution is proved in an appropriate Sobolev space. Then, exponential stability of the stationary solution is established by constructing an a priori estimate. Since the techniques for classical hydrodynamic equations are not applicable here due to the quantum term, the existence of a local-in-time solution is obtained by showing the existence of local-in-time solutions of a reformulated system via the iteration method.