Interplay of elastic and inelastic scattering operators in extended kinetic models and their hydrodynamic limits - reference manual

In this paper we survey various hydrodynamic/diffusive limits which can occur in the linear kinetic equation describing elastic and inelastic collisions when the involved operators act on different time-scales. The first part of the review is devoted to the explanation to non-experts how to use the compressed Chapman-Enskog procedure to analyse the asymptotic behaviour of a kinetic equation when some physical parameters are small with respect to the others. In the second part it is shown that all the formal steps of the procedure can be mathematically justified in a natural way leading to a rigorous asymptotic theory.