On the finite 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 hydrodynamic systems with initial data that are arbitrarily large in the energy norm. This type of model, initially proposed by Madelung, has been extensively used in physics to investigate superfluidity and superconductivity phenomena and more recently in the modeling of semiconductor devices. Our approach is based on various tools, namely the wave functions polar decomposition and the construction of an 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