The transient equations of viscous quantum hydrodynamics

We study the viscous model of quantum hydrodynamics in a bounded domain of space dimension 1, 2, or 3, and in the full onedimensional space. This model is a mixed-order partial differential system with nonlocal and nonlinear terms for the particle density, current density, and electric potential. By a viscous regularization approach, we show existence and uniqueness of local in time solutions. We propose a reformulation as an equation of Schrodinger type, and we prove the inviscid limit.