  In this paper, we consider the existence problem for a stationary
relaxational models of the quantum Boltzmann equation. More precisely, we
establish the existence of mild solution to the fermionic or bosonic quantum
BGK model in a slab with inflow boundary data. Unlike the classical case, it is
necessary to verify that the quantum local equilibrium state is well-defined,
and the transition from the non-condensed state to the condensated state
(Bosons), or from the non-saturated state to the saturated state (Fermions)
does not arise in our solution space.
