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Article Dans Une Revue Asymptotic Analysis Année : 2023

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

Résumé

We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann 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.
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Dates et versions

hal-02506265 , version 1 (12-03-2020)

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Marc Briant, Bérénice Grec. Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scaling. Asymptotic Analysis, 2023, pp.1-26. ⟨10.3233/ASY-231847⟩. ⟨hal-02506265⟩
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