Exterior problem for the Boltzmann equation with temperature difference: asymptotic stability of steady solutions

The asymptotic stability of non-equilibrium steady solutions to the exterior problem for the Boltzmann equation was first proved by Ukai and Asano in [17], under the assumption that the temperature associating with the far-field Maxwellian and the one preserved by the kinetic boundary condition are the same. In this paper, we generalize Ukai and Asano's result in the sense that the two temperatures mentioned above are allowed to be different. The proof of the main theorem is based on the ideas developed in [16], [17] and [20] as well as some new observations.