Milne problem for the linear and linearized Boltzmann equations relevant to a binary gas mixture

A stationary boundary-value problem for the Boltzmann equation in a half space is considered for a binary mixture of gases when the indata on the boundary are given for the both species. Under the assumption that one of the species is dominant and close to equilibrium but the density of the other is small, the problem is decomposed into two half-space problems: the so-called Milne problem for the linearized Boltzmann equation with a source term for the dominant species and that for the linear Boltzmann equation for the low-density species. The two problems are coupled in such a way that the source term in the former is specified by the solution of the latter. The existence and uniqueness of the solutions to the two problems are proved, and their accurate asymptotic behavior in the far field is obtained. In particular, the precise rate of approach of the solution to the state at infinity is expressed in terms of the decay rate in the molecular velocity of the boundary data for both species.