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We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hardpotentials, or hard spheres collision kernels. In a perturbative regime aroundthe global Maxwellian equilibrium, the linearized Boltzmann multi-speciesoperator $\mathbf{L}$ is known to possess an explicit spectral gap$\lambda_{\mathbf{L}}$, in the global equilibrium weighted $L^2$ space. Westudy a new operator $\mathbf{L^\varepsilon}$ obtained by linearizing theBoltzmann operator for mixtures around local Maxwellian distributions, whereall the species evolve with different small macroscopic velocities of order$\varepsilon$, $\varepsilon >0$. This is a non-equilibrium state for themixture. We establish a quasi-stability property for the Dirichlet form of$\mathbf{L^\varepsilon}$ in the global equilibrium weighted $L^2$ space. Moreprecisely, we consider the explicit upper bound that has been proved for theentropy production functional associated to $\mathbf{L}$ and we show that thesame estimate holds for the entropy production functional associated to$\mathbf{L^\varepsilon}$, up to a correction of order $\varepsilon$.

Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium Maxwell distributions