Rigorous derivation of the Fick cross-diffusion system from the\n multi-species Boltzmann equation in the diffusive scaling

2020/03/17 by M. Briant, Briant, Marc, Bérénice Grec +1 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Lattice Boltzmann Simulation Studies #Mathematical Biology Tumor Growth #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.2003.07891

openalex publication_date 2020/03/17 · openalex created_date 2023/06/06 · openalex updated_date 2026/07/28

Abstract

We present the arising of the Fick cross-diffusion system of equations for\nfluid mixtures from the multi-species Boltzmann in a rigorous manner in Sobolev\nspaces. To this end, we formally show that, in a diffusive scaling, the\nhydrodynamical limit of the kinetic system is the Fick model supplemented with\na closure relation and we give explicit formulae for the macroscopic diffusion\ncoefficients from the Boltzmann collision operator. Then, we provide a\nperturbative Cauchy theory in Sobolev spaces for the constructed Fick system,\nwhich turns out to be a dilated parabolic equation. We finally prove the\nstability of the system in the Boltzmann equation, ensuring a rigorous\nderivation between the two models.\n

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