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

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 , in the global equilibrium weighted L^2 space. We study a new operator L_e obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order e, e > 0. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of L_e 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_e , up to a correction of order e.