The radiative transfer equation in the forward-peaked regime

TL;DR

This paper proves well-posedness and regularization of solutions for the radiative transfer equation in the forward-peaked regime, introducing new mathematical tools and representations that facilitate numerical simulations.

Contribution

It introduces a rigorous framework for analyzing the forward-peaked radiative transfer equation, including a fractional Laplace-Beltrami operator representation and convergence results for scattering models.

Findings

Proves instantaneous regularization of weak solutions in L1.

Provides a fractional Laplace-Beltrami operator representation of the scattering operator.

Shows convergence of Henyey-Greenstein models and algebraic decay of solutions.

Abstract

In this work we study the radiative transfer equation in the forward-peaked regime in free space. Specifically, it is shown that the equation is well-posed by proving instantaneous regularization of weak solutions for arbitrary initial datum in L1. Classical techniques for hypo-elliptic operators such as averaging lemma are used in the argument. Among the interesting aspect of the proof are the use of the stereographic projection and the presentation of a rigorous expression for the scattering operator given in terms of a fractional Laplace-Beltrami operator on the sphere, or equivalently, a weighted fractional Laplacian analog in the projected plane. Such representations may be used for accurate numerical simulations of the model. As a bonus given by the methodology, we show convergence of Henyey-Greenstein scattering models and vanishing of the solution at time algebraic rate due to scattering diffusion.