Singular behavior of the macroscopic quantity near the boundary for a Lorentz-gas model with the infinite-range potential (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

Possibility of the diverging gradient of macroscopic quantities near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By a study of a steady one-dimensional boundary-value problem, it is numerically confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price of the regularization, the collision integral becomes no longer finite on the boundary in the direction of the molecular velocity parallel to it. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence. The present document is a reorganized summary version of the paper coauthored with M. Hattori posted in arXiv (https://arxiv.org/abs/2106.06532).