Dissipative hydrodynamic models for the diffusion of impurities in a gas

Recently, linear dissipative models of the Boltzmann equation have been introduced in [B. Loods, G. Toscani, The dissipative linear Boltzmann equation for hard spheres, J. Statist. Phys. 117 (2004) 635–664] and [G. Spiga, G. Toscani, The dissipative linear Boltzmann equation, Appl. Math. Lett. 17 (2004) 255–301]. In this note, we consider the problem of constructing suitable hydrodynamic approximations for such models.