Simultaneous diffusion and homogenization asymptotic for the linear Boltzmann equation
Claude Bardos, Harsha Hutridurga · Asymptotic Analysis · 2016
This article is on the simultaneous diffusion approximation and homogenization of the linear Boltzmann equation when both the mean free path ε and the heterogeneity length scale η vanish.No periodicity assumption is made on the scattering coefficient of the background material.There is an assumption made on the heterogeneity length scale η that it scales as ε β for β ∈ (0, ∞).In one space dimension, we prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation for any β 2 in the ε → 0 limit.In any arbitrary phase space dimension, under a smallness assumption of a certain quotient involving the scattering coefficient in the H -1 2 norm, we again prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation in the ε → 0 limit.