Stationary Quantum BGK Model for Bosons and Fermions in a Bounded Interval

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.