Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

Abstract

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

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

“…Theorem 7.1 (Th. 2.4 of [9]). Under the assumptions (H1) − (H2) − (H3) − (H4) on the collision kernel, there exists an integer s 0 , some constants δ fluid , δ B , C B > 0, ε ∈ (0, 1] and a norm…”

“…This falls into the wide literature concerning the hydrodynamical limits of kinetic equations [38]. In the context of mixtures, it has been proved in [9] that the Maxwell-Stefan model is stable for the Boltzmann multi-species equation, ensuring a rigorous derivation of the Maxwell-Stefan system in a perturbative setting. In their paper, the authors choose to consider perturbative solutions around a Maxwellian whose fluid quantities solve the limit macroscopic system as in [19,23], and use hypocoercive strategy in the spirit of [37,32,16].…”

“…In this paper, we shall tackle the rigorous limit towards the Fick model in a perturbative setting as well, following the same ideas as in [9]. More precisely, we first derive formally the Fick diffusion coefficients, and show that they are naturally linked to the inverse of the linear Boltzmann operator.…”

“…After deriving formally the Fick system from the Boltzmann one in Section 4, we will prove some properties of the Fick matrix in Section 5. We will then be able to prove a perturbative Cauchy theory for the Fick equation in Section 6, which will allow to conclude the rigorous convergence in Section 7 thanks to a result established in [9].…”

“…We conclude this proof by indicating that the general result Theorem 7.1 could in fact be rewritten under a weaker form with local-in-time H s ε -norms replaced by L 2 [0,Tmax) H s ε . We refer explicitely to[9, Equation (3.36)] that one could integrate in time. In that framework we would solely have to prove the following control ε ∂ t n L 2…”