We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard potentials, or hard spheres collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species operator L is known to possess an explicit spectral gap λ L , in the global equilibrium weighted L 2 space. We study a new operator L ε obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order ε, ε > 0. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of L ε in the global equilibrium weighted L 2 space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to L and we show that the same estimate holds for the entropy production functional associated to L ε , up to a correction of order ε.

In this article, we consider a multi‐species kinetic model which leads to the Maxwell–Stefan equations under a standard diffusive scaling (small Knudsen and Mach numbers). We propose a suitable numerical scheme which approximates both the solution of the kinetic model in rarefied regime and the one in the diffusion limit. We prove some a priori estimates (mass conservation and nonnegativity) and well‐posedness of the discrete problem. We also present numerical examples where we observe the asymptotic‐preserving behavior of the scheme.

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