  We consider the non-relativistic quantum Boltzmann equation for fermions and
bosons. Using the nonlinear energy method and mild formulation, we justify the
global well-posedness when the density function is near the global Maxwellian
and vacuum. This work is a generalization and adaptation of the classical
Boltzmann theory. Our main contribution is a detailed analysis of the nonlinear
operator $Q$ in the quantum context. This is the first piece of a long-term
project on the quantum kinetic equations.
