An $L^p$ theory for stationary radiative transfer

TL;DR

This paper develops an $L^p$ theoretical framework for stationary radiative transfer equations, providing sharp a-priori estimates, derivative and trace bounds, and proving convergence of source iteration without positive absorption assumptions.

Contribution

It introduces a comprehensive $L^p$ analysis for stationary radiative transfer, including sharp estimates, derivative bounds, and convergence results under minimal assumptions.

Findings

Derived uniform a-priori estimates in weighted $L^p$ spaces.

Constructed explicit example showing estimates are sharp.

Proved convergence of source iteration without positive absorption.

Abstract

We present a self-contained analysis of the stationary radiative transfer equation in weighted $L^p$ spaces. The use of weighted spaces allows us to derive uniform a-priori estimates for $1 \le p \le \infty$ under minimal assumptions on the parameters. By constructing an explicit example, we show that our estimates are sharp and cannot be improved in general. Better estimates are however derived under additional assumptions on the parameters. We also present estimates for derivatives and traces of the solution and formulate a natural energy space, for which the data-to-solution map becomes an isomorphism. As a side result, we are able to prove uniform convergence of the source iteration for all $1 \le p \le \infty$ without the assumption of positive absorption that is frequently used in the literature.