  We consider two models for a two component gas mixture with translational and
internal energy degrees of freedom described by a BGK approximation assuming
that the number of particles of each species remains constant. The two species
are allowed to have different degrees of freedom in internal energy and are
modelled by a system of kinetic BGK equations featuring two interaction terms
to account for momentum and energy transfer between the species. We consider
two models in the literature and prove that the models satisfy a different
equation of state than the ideal gas law. Moreover, we prove that for these
models there exist a unique positive solution.

Analysis of PDEs