  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, have
been extensively used in Physics to investigate Supefluidity 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, the construction of approximate solution via a fractional
steps method, which iterates a Schr\"odinger Madelung picture with a suitable
wave function updating mechanism. Therefore several \emph{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.
