The Diffusion Approximation for the Linear Boltzmann Equation with Vanishing Absorption

The present paper discusses the di↵usion approximation of the linear Boltzmann equation in cases where the collision frequency is not uniformly large in the spatial domain. Our result applies for instance to the case of radiative transfer in a composite medium with optically thin inclusions in an optically thick background medium. The equation governing the evolution of the approximate particle density coincides with the limit of the di↵usion equation with infinite di↵usion coe cient in the optically thin inclusions. 1. Presentation of the Problem Consider the linear Boltzmann equation (1) (@t + v ·rx)f(t, x, v) + Lxf(t, x, v) = 0 for the unknown f ⌘ f(t, x, v) that is the distribution function for a system of identical point particles interacting with some background material. In other words, f(t, x, v) is the number density of particles located at the position x 2 ⌦, where ⌦ is a domain of RN , with velocity v ⇢ RN at time t 0. The notation Lx designates a linear integral operator acting on the v variable in f , i.e. (2) Lxf(t, x, v) = Z RN k(x, v, w)(f(t, x, v) f(t, x, w))dμ(w) where μ is a Borel probability measure on RN , while k is a nonnegative function defined μ⌦ μ-a.e. on RN ⇥RN that measures the probability of a transition from velocity v to velocity w for particles located at the position x. Henceforth we denote (3) h i = Z RN (v)dμ(v) and ⌦ ↵ = ZZ RN⇥RN (v, w)dμ(v)dμ(w) for all 2 L1(RN , dμ) and 2 L(Rv ⇥Rw ; dμ(v)dμ(w)). We assume that k satisfies the semi-detailed balance condition (4) Z RN k(x, v, w)dμ(w) = Z RN k(x,w, v)dμ(w) and introduce the notation (5) a(x, v) := Z