Some Results on Systems for Quantum Fluids

The purpose of this paper is first to review, in a self-consistent way, some results concerning a class of systems in Quantum Hydrodynamics. We show existence of global in time finite energy weak solutions for such systems. These are obtained by means of a polar factorization technique, which allows to define the hydrodynamic quantities without using the velocity field, which can not be well-defined in the nodal regions, i.e. where the mass density vanishes. We present then some new results by applying similar techniques to the Quantum Magnetohydrodynamics system. Here electromagnetic fields, prescribed by the Maxwell's equation, are also taken into account. Such models arise in the description of quantum plasmas.