We consider the thermal transpiration problem in the kinetic theory, which can be modeled by the linearized Boltzmann equation. It is well-known through asymptotic expansions and computations that there is a logarithmic singularity for the fluid velocity around the solid boundary. The goal of this paper is to confirm this basic phenomenon in the kinetic theory through analysis for sufficiently large Knudsen number.
We use an iterated scheme, with the "gain" part of the collision operator as a source.
The scheme yields an explicit leading term. The remaining converging terms are estimated through a refined pointwise estimate and Maxwellian upper bound for the gain part. Our analysis is motivated by the previous studies of asymptotic and computational analysis. Numerical data supporting the analysis are also provided.