Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

Christophe Gomez (1) , Olivier Pinaud (2) , Lenya Ryzhik (3)

1 I2M - Institut de Mathématiques de Marseille
2 CSU - Colorado State University [Fort Collins]
3 Department of Mathematics [Stanford]

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.

Mots clés: radiative transfer; long-range interactions; hypoellipticity; diffusion limit; peaked-forward limit

Domaines: Equations aux dérivées partielles [math.AP]

Christophe Gomez, Olivier Pinaud, Lenya Ryzhik. Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2017, 15 (2), pp.1048 - 1072. ⟨10.1137/15M1047076⟩. ⟨hal-01225235⟩