2016/05/05 by Claude Bardos, Bardos, Claude, Harsha Hutridurga +1
Computer Science · Engineering · Mathematics · #35B27 #35B40 #35C20 #82D75 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1605.01610
openalex publication_date 2016/05/05 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
This article is on the simultaneous diffusion approximation and\nhomogenization of the linear Boltzmann equation when both the mean free path\n\ε and the heterogeneity length scale \η vanish. No periodicity\nassumption is made on the scattering coefficient of the background material.\nThere is an assumption made on the heterogeneity length scale \η that it\nscales as \ε^\β for \β\∈(0,\∞). In one space dimension,\nwe prove that the solutions to the kinetic model converge to the solutions of\nan effective diffusion equation for any \β\≤2 in the \ε\→0\nlimit. In any arbitrary phase space dimension, under a smallness assumption of\na certain quotient involving the scattering coefficient in the\nH-\(1)/(2) norm, we again prove that the solutions to the kinetic model\nconverge to the solutions of an effective diffusion equation in the\n\ε\→0 limit.\n