  Recently, the authors proved [2] 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 [2]) 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.
