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

Article type: Research Article

Authors: Briant, Marc | Grec, Bérénice^{; *}

Affiliations: Université Paris Cité, CNRS, MAP5, F-75006 Paris, France

Correspondence: [*] Corresponding author. E-mail: berenice.grec@u-paris.fr.

Abstract: We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system in the Boltzmann equation, ensuring a rigorous derivation between the two models.

Keywords: Multispecies Boltzmann equation, gaseous and fluid mixture, Fick’s equation, perturbative theory, hydrodynamical limit, Knudsen number

DOI: 10.3233/ASY-231847

Journal: Asymptotic Analysis, vol. 135, no. 1-2, pp. 55-80, 2023