  Multi-species BGK models describe the dynamics of rarefied gases with
constituent particles of different elements or compounds with potentially
non-trivial velocity distributions. In this paper, moment equations for the
bulk velocities, energies, and temperatures of a spatially homogeneous
multi-species BGK model are examined. A key challenge in analyzing these
equations is the fact that the collision frequencies are allowed to depend on
the species temperatures, which allows for more realistic simulations of dilute
gas flow. Therefore, a positive lower bound is established for the species
temperatures. With this lower bound, a global existence and uniqueness of
solutions to the coupled velocity-energy ODE system is established. The lower
bound also enables a proof of exponential decay to a unique steady-state
solution. Numerical results are presented to demonstrate how the bulk
velocities and temperatures relax for large times.
