Analysis of the Boundary Layer Equation in the Kinetic Theory of Gases

The present paper completes an earlier result by S. Ukai, T. Yang, and S.-H. Yu [Commun. Math. Phys. 236 (2003), 373−393] on weakly nonlinear half-space problems for the steady Boltzmann equation with hard-sphere potential. 1. Statement of the Problem & Main Result Boundary layers are an important class of flows in the kinetic theory of gases. One of the most significant problems in this context is the study of half-space problems for the Boltzmann equation matching the state of the gas on a surface immersed in the flow to some thermodynamic equilibrium far from that surface. A classical example of this situation is the case of a phase transition, where the rarefied gas is the vapor above its condensed phase. From the mathematical viewpoint, these boundary layers explain how boundary conditions for the hydrodynamic equations can be obtained from the boundary data at the mesoscopic level of description corresponding to the kinetic theory of gases. Henceforth, we consider the steady state of a rarefied gas in a half-space R+ := {(x, y, z) ∈ R | x ≥ 0}. Received March 17, 2008. AMS Subject Classification: 82C40 (76P05).