We investigate in the diffusive scaling the limit to the macroscopic description of finite-velocity Boltzmann kinetic models, where the rate coefficient in front of the collision operator is assumed to be dependent of the mass density. It is shown that in the limit the flux vanishes while the evolution of the mass density is governed by a nonlinear parabolic equation of porous medium type. In the last part of the paper we show that our method adapts to prove the so-called Rosseland approximation in radiative transfer theory.
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Pierre-Louis Lions, Giuseppe Toscani, Diffusive limit for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoam. 13 (1997), no. 3, pp. 473–513DOI 10.4171/RMI/228