On the Boundary Layer Equations with Phase Transition in the Kinetic Theory of Gases

Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, 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 extensively studied, first by Sone, Aoki and their collaborators, by means of careful numerical simulations. See section 2 of (Bardos et al. in J Stat Phys 124:275–300, 2006) for a very detailed presentation of these works. More recently, Liu and Yu (Arch Ration Mech Anal 209:869–997, 2013) proposed an extensive mathematical strategy to handle the problems studied numerically by Sone, Aoki and their group. The present paper offers an alternative, possibly simpler proof of one of the results discussed in Liu and Yu (2013).