The Half-Space Problem for the Boltzmann Equation with Phase Transition at the Boundary

Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space above its condensed phase. The present paper studies the existence and uniqueness of a uniformly, exponentially decaying solution in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has been studied numerically by Y. Sone, K. Aoki and their collaborators—see section 2 of (Bardos et al., J Stat Phys 124:275–300, 2006) for a detailed presentation of these works. More recently Liu and Yu (Arch Ration Mech Anal 209:869–997, 2013) have proposed a mathematical strategy to handle problems of this type. In this paper, we describe an alternative approach to one of their results obtained in collaboration with Bernhoff (Arch Ration Mech Anal 240:51–98, 2021).