The Radiative Transfer Equation in the Forward-Peaked Regime

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 L 1 . Classical techniques for hypo-elliptic operators, such as averaging lemma, are used in the argument. Among the interesting aspects 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.

Pencil-Beam Approximation of Stationary Fokker--Planck

“…, whose link to the fractional Laplacian was noticed in [3] and established by means of a stereographic transformation S :…”

“…By introducing the following version of the fractional Laplacian on the unit sphere (as in [3]), (1.5) [(−∆ θ ) s u] S :=…”

Radiative Transfer with Long-Range Interactions: Regularity and Asymptotics

“…and defining v(t) = ψ ε (t) 2 L 2 (R d+1 ×S d ) , we can differentiate v(t) and use both the transport equation (5) and item (iii) of Lemma 4.1 to arrive at…”

Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups

“…Nonlinear problems with fractional Laplacians have been receiving a lot of attention for the past 12 years. Fractional nonlocal equations appear in several areas of pure and applied mathematics, see for instance [1,4,5,19,22].…”