Exterior Problem of Boltzmann Equation with Temperature Difference

The existence of stationary solution to an exterior domain of the Boltzmann equation was first studied by S. Ukai and K. Asano in [25, 27] and was recently generalized by S. Ukai, T. Yang, and H. J. Zhao in [29] to more general boundary conditions. We note, however, that the results obtained in [25, 29] require that the temperature of the far field Maxwellian is the same as the one of the Maxwellian preserved by the boundary conditions. The main purpose of this paper is to discuss the case when these two temperatures are different. The analysis is based on some new estimates on the linearized collision operator and the method introduced in [25, 27, 29].