  The problem considered here is motivated by a work by B. Nachtergaele and
H.T. Yau where the Euler equations of fluid dynamics are derived from manybody
quantum mechanics, see [10]. A crucial concept in their work is that of local
quantum Gibbs states, which are quantum statistical equilibria with prescribed
particle, current, and energy densities at each point of space (here R d , d
$\ge$ 1). They assume that such local Gibbs states exist, and show that if the
quantum system is initially in a local Gibbs state, then the system stays, in
an appropriate asymptotic limit, in a Gibbs state with particle, current, and
energy densities now solutions to the Euler equations. Our main contribution in
this work is to prove that such local quantum Gibbs states can be constructed
from prescribed densities under mild hypotheses, in both the fermionic and
bosonic cases. The problem consists in minimizing the von Neumann entropy in
the quantum grand canonical picture under constraints of local particle,
current, and energy densities. The main mathematical difficulty is the lack of
compactness of the minimizing sequences to pass to the limit in the
constraints. The issue is solved by defining auxiliary constrained optimization
problems, and by using some monotonicity properties of equilibrium entropies.
