  This work is devoted to the analysis of the quantum Liouville-BGK equation.
This equation arises in the work of Degond and Ringhofer on the derivation of
quantum hydrodynamical models from first principles. Their theory consists in
transposing to the quantum setting the closure strategy by entropy minimization
used for kinetic equations. The starting point is the quantum Liouville-BGK
equation, where the collision term is defined via a so-called quantum local
equilibrium, defined as a minimizer of the quantum free energy under a local
density constraint. We then address three related problems: we prove new
results about the regularity of these quantum equilibria; we prove that the
quantum Liouville-BGK equation admits a classical solution; and we investigate
the long-time behavior of the solutions. The core of the proofs is based on a
fine analysis of the properties of the minimizers of the free energy.
