Boltzmann equation for a binary gas mixture with perturbed reflection boundary conditions

A boundary value problem for the stationary Boltzmann equation with hard sphere in a half-space is considered for a binary mixture of gases, imposing slightly perturbed reflection boundary conditions for the both species. We assume that one of the species is dominant and close to equilibrium but the density of the other is small. Under this assumption, we prove the existence and uniqueness of the solution to the coupled system, and their accurate asymptotic behavior in the far field is obtained. Our result is based on the one species result. However, the system is not symmetric which leads us to the major difficulty in this paper. Thanks to the conservation laws for the collision operators, the nonlinear problem is solved.