Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrodinger system with a power-type nonlinearity. We show the local well-posedness in $H^2(mathbb{R}^3)times H^{3/2}(mathbb{R}^3)$ and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.
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