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This work is concerned with the minimization of quantum entropies under localconstraints of density, current, and energy. The problem arises in the work ofDegond and Ringhofer about the derivation of quantum hydrodynamical models fromfirst principles, and is an adaptation to the quantum setting of the momentclosure strategy by entropy minimization encountered in kinetic equations. Themain mathematical difficulty is the lack of compactness needed to recover theenergy constraint. We circumvent this issue by a monotonicity argumentinvolving energy, temperature and entropy, that is inspired by somethermodynamical considerations.

On the minimization of quantum entropies under local constraints