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We prove the existence of global in time, finite energy, weak solutions to aquantum magnetohydrodynamic system (QMHD) with large data, modeling a chargedquantum fluid interacting with a self-generated electromagnetic field. Theanalysis of QMHD relies upon the use of Madelung transformations. The rigorousderivation requires non-trivial smoothing estimates, which are obtained byassuming slightly higher regularity for the electromagnetic potential. Theseassumptions are motivated by the nonlinear dependence of the hydrodynamicsystem in terms of the underlying wave function dynamics, which issupercritical with respect to the bare energy bounds. Due to quantum effects onthe dispersive properties of QMHD, our approach requires neither smallness norhigh regularity, unlike a large amount of existing literature forEuler-Maxwell's classical system. For quantum MHD system the irrotationalityand the presence of a highly nonlinear quantum stress tensor induce muchstronger dispersive properties, as a byproduct of a close relationship with theclassical Maxwell-Schr\"odinger system. Therefore the core argument is shiftedto the analysis of the nonlinearities related to the formulation of thehydrodynamic variables through the Madelung transformations. The analysiscarried out in section 4 shows that it is necessary to go through non-trivialsmoothing estimates and these require us to assume regularity conditions, justabove the energy norms, for the initial data of the Maxwellian electromagneticpotential. In the same regime of regularity, with the help of suitable localsmoothing estimates, we also prove stability of both the hydrodynamic variablesand the Lorentz force associated with the electromagnetic field. (Abbreviatedversion, see full abstract in the paper).

Existence and Stability of almost finite energy weak solutions to the Quantum Euler-Maxwell system