Radiative transfer with long-range interactions: regularity and asymptotics

TL;DR

This paper studies radiative transfer equations with long-range interactions, demonstrating hypoellipticity, regularity of solutions, and analyzing diffusion and peaked forward limits using probabilistic methods.

Contribution

It provides new insights into the regularizing effects, diffusion limit, and asymptotic behavior of radiative transfer equations with singular, long-range collision operators.

Findings

Solutions are infinitely differentiable in all variables due to hypoellipticity.

The diffusion limit exists with a non-zero, finite diffusion coefficient.

Grazing collisions dominate in certain regimes, affecting the asymptotic behavior.

Abstract

This work is devoted to radiative transfer equations with long-range interactions. Such equations arise in the modeling of high frequency wave propagation in random media with long-range dependence. In the regime we consider, the singular collision operator modeling the interaction between the wave and the medium is conservative, and as a consequence wavenumbers take values on the unit sphere. Our goals are to investigate the regularizing effects of grazing collisions, the diffusion limit, and the peaked forward limit. As in the case where wavenumbers take values in R^{d+1}, we show that the transport operator is hypoelliptic, so that the solutions are infinitely differerentiable in all variables. Using probabilistic techniques, we show as well that the diffusion limit can be carried on as in the case of a regular collision operator, and as a consequence that the diffusion coefficient is non-zero and finite. We finally consider the regime where grazing collisions are dominant.