On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics

In this paper we consider the global existence of weak solutions to a class of Quantum Hydrodynamics (QHD) systems with initial data, arbitrarily large in the energy norm. These type of models, initially proposed by Madelung [44], have been extensively used in Physics to investigate Superfluidity and Superconductivity phenomena [19,38] and more recently in the modeling of semiconductor devices [20] . Our approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method which iterates a Schrödinger Madelung picture with a suitable wave function updating mechanism. Therefore several a priori bounds of energy, dispersive and local smoothing type, allow us to prove the compactness of the approximating sequences. No uniqueness result is provided.