Global, finite energy, weak solutions to a quantum magneto-hydrodynamic system with a pure-power pressure term

We prove the existence of global in time, finite energy, weak solutions to a quantum magneto--hydrodynamic system (QMHD), modeling a charged quantum fluid interacting with it self-generated electromagnetic field, with the additional presence of a pure-power, quantum statistical pressure which comes from the electron degeneracy, due to Heisenberg's uncertainty principle and Pauli's exclusion principle. No smallness and/or higher regularity is necessary. The QMHD system formally is strictly connected with the wave function dynamics, given by the related nonlinear Maxwell-Schr\"odinger system, via the so called Madelung transformations. To make rigorous this connection we actually need to be able to approximate non smooth solutions. Unfortunately the nonlinear Maxwell-Schr\"odinger system lacks of a well-posedness theory in the energy space, suitable to approximate the QMHD nonlinear terms, in particular the Lorentz force. To circumvent these difficulties here we perform a derivation by using the integral formulation for weak solutions. More precisely, we derive suitable a priori smoothing estimates for the Maxwell-Schr\"odinger system, so we can rigorously justify the derivation for a class of ``almost'' finite energy initial data. In the same regime of regularity we also prove a local smoothing effect, which guarantees the weak-stability of both the hydrodynamic variables and the Lorentz force associated to the electromagnetic field.