Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

Abstract

International audienceThis 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

Pencil-beam approximation of fractional Fokker-Planck

“…The hypoelliptic property of the associated integro-differential operator was also analyzed. Similar results have been obtained in [17,18] for the radiative transfer equation with long-range interactions. Analyzing the limiting scaling of this equation, the authors demonstrated the emergence of a Fokker-Planck type operator in the highly forward-peaked limit, with a diffusion component consisting of another singular integral operator whose high frequency behavior equals that of the Laplace-Beltrami operator on the sphere.…”

Monte Carlo Methods for Radiative Transfer with Singular Kernels

“…When t t S := (−λ 1 ) −1 , the distribution of the angle is almost uniform and particles are in a diffusive regime. At such large times, the solution f (t, x, k) to the RTE (1) does not depend on k anymore and satisfies a diffusion equation [22] (see also [13] in the context of singular kernels). For these time scales, it is preferrable to solve a diffusion equation instead of the RTE, and we therefore consider times not significantly We represent in Figure 13 the characteristic time t S as a function of α for a(x) = a 0 = 0.1.…”

“…Regarding the regularity of f , it was established in [13,14] that f is C ∞ in all variables for all t > 0, which is in stark contrast with the SR case. The regularity is reminiscent of the hypoelliptic nature of (1) when F is not integrable.…”

“…With the hypotheses above, the operator Q can be seen the generator of a Lévy process (K k (t)) t≥0 on S 1 , with initial condition K k (0) = k, see [13]. Introducing the position…”

Existence and Regularity of Solutions: Nonlocal and Nonlinear Models

“…The RTE (1.1) has been studied in the whole domain using slightly different approaches, based on hypo-ellipticity techniques [6], in the references [1,21]. References treating problems with boundaries are scarce in the context of kinetic equations with singular scattering, however, for the classical kinetic Fokker-Planck equation with absorbing boundary we refer to [26,27].…”