Propagation of boundary-induced discontinuity in stationary radiative transfer

TL;DR

This paper investigates how boundary discontinuities affect solutions in stationary radiative transfer, introducing conditions for discontinuity propagation without relying on velocity averaging, and providing a counterexample in 2D.

Contribution

It establishes new boundary conditions ensuring discontinuity propagation in stationary transport equations without using velocity averaging, and presents a counterexample in two dimensions.

Findings

Discontinuity propagates along positive characteristic lines under certain boundary conditions.

Velocity averaging lemma is not necessary for analyzing discontinuity propagation.

Piecewise continuity of boundary data is insufficient for the main propagation result.

Abstract

We consider the boundary value problem of the stationary transport equation in the slab domain of general dimensions. In this paper, we discuss the relation between discontinuity of the incoming boundary data and that of the solution to the stationary transport equation. We introduce two conditions posed on the boundary data so that discontinuity of the boundary data propagates along positive characteristic lines as that of the solution to the stationary transport equation. Our analysis does not depend on the celebrated velocity averaging lemma, which is different from previous works. We also introduce an example in two dimensional case which shows that piecewise continuity of the boundary data is not a sufficient condition for the main result.