Singularity of the Velocity Distribution Function in Molecular Velocity Space

Abstract

We study the boundary singularity of the solutions to the Boltzmann equation in the kinetic theory. The solution has a jump discontinuity in the microscopic velocity ${\zeta}$ on the boundary and a secondary singularity of logarithmic type around the velocity tangential to the boundary, ${\zeta_{n} \sim 0_{-}}$, where ${\zeta_{n}}$ is the component of molecular velocity normal to the boundary, pointing to the gas. We demonstrate this secondary singularity by obtaining an asymptotic formula for the derivative of the solution on the boundary with respect to ${\zeta_{n}}$ that diverges logarithmically when ${\zeta_{n} \sim 0_{-}}$. Our study is for the thermal transpiration problem between two plates for the hard sphere gases with sufficiently large Knudsen number and with the diffuse reflection boundary condition. The solution is constructed and its singularity is studied by an iteration procedure.