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

TL;DR

This paper investigates the existence and uniqueness of boundary layer solutions for the steady Boltzmann equation near equilibrium in the context of phase transition, providing an alternative proof to previous numerical and analytical results.

Contribution

It offers a new, potentially simpler mathematical proof for boundary layer solutions in the kinetic theory of gases with phase transition.

Findings

Proves existence and uniqueness of boundary layer solutions

Provides an alternative proof to previous numerical results

Clarifies the mathematical structure of phase transition boundary layers

Abstract

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 Y. Sone, K. Aoki and their collaborators, by means of careful numerical simulations. See section 2 of [C. Bardos, F. Golse, Y. Sone: J. Stat. Phys. 124 (2006), 275-300] for a very detailed presentation of these works. More recently T.-P. Liu and S.-H. Yu [Arch. Rational Mech. Anal. 209 (2013), 869-997] have proposed an extensive mathematical strategy to handle the problems studied numerically by Y. Sone, K. Aoki and their group. The present paper offers an alternative, possibly simpler proof of one of the results discussed in [T.P. Liu, S.-H. Yu, loc. cit.]