H\"older regularity of the Boltzmann equation past an obstacle

TL;DR

This paper establishes H"older continuity for solutions to the Boltzmann equation with elastic reflection on convex obstacles, contrasting with other boundary conditions where solutions are less regular.

Contribution

It proves H"older regularity in $C^{0,\frac{1}{2}-}$ for the Boltzmann equation with elastic reflection, a significant advancement over previous results for other boundary conditions.

Findings

H"older regularity in $C^{0,\frac{1}{2}-}$ for the Boltzmann equation with elastic reflection

Contrast with solutions under diffuse and in-flow boundary conditions, which are discontinuous

Solutions are more regular than BV in the elastic reflection case

Abstract

Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory (\cite{Kim11,GKTT1}). In this paper, we prove an H\"older regularity in $C^{0,\frac{1}{2}-}_{x,v}$ for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this H\"older regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in-flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (\cite{Kim11}), and therefore the best possible regularity is BV, which has been proved in \cite{GKTT2}.