Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains

TL;DR

This paper investigates how discontinuities form and move in solutions to the Boltzmann equation within non-convex domains, highlighting the role of boundary geometry in singularity development.

Contribution

It reveals the mechanism of discontinuity formation at non-convex boundary parts and its propagation along characteristics, advancing understanding of boundary effects in Boltzmann solutions.

Findings

Discontinuities originate at non-convex boundary regions.

Discontinuities propagate along forward characteristics.

Propagation continues until boundary re-encountered.

Abstract

The formation and propagation of singularities for Boltzmann equation in bounded domains has been an important question in numerical studies as well as in theoretical studies. Consider the nonlinear Boltzmann solution near Maxwellians under in-flow, diffuse, or bounce-back boundary conditions. We demonstrate that discontinuity is created at the non-convex part of the grazing boundary, then propagates only along the forward characteristics inside the domain before it hits on the boundary again.