  We consider kinetic models for a multi component gas mixture without chemical
reactions. In the literature, one can find two types of BGK models in order to
describe gas mixtures. One type has a sum of BGK type interaction terms in the
relaxation operator, for example the model described by Klingenberg, Pirner and
Puppo, 2017 which contains well-known models of physicists and engineers for
example Hamel, 1965, and Gross and Krook, 1956, as special cases. The other
type contains only one collision term on the right-hand side, for example the
well-known model of Andries, Aoki and Perthame, 2002. For each of these two
models we prove existence, uniqueness and positivity of solutions in the first
part of the paper. In the second part, we use the first model in order to
determine an unknown function in the energy exchange of the macroscopic
equations for gas mixtures described by Dellacherie.
