Existence and uniqueness for spherically symmetric linear transport

Abstract

Abstract The purpose of this note is to establish an existence and uniqueness theorem for the spherically and azimuthally symmetric steady-state linear transport equation. The methods used are similar to those of Olhoeft1 for bounded three-dimensional geometry and of Nelson2 for linear transport in a slab. Our basic result contains the corollary that, for very general scattering laws, nonmultiplying transport of linear particles in a spherically and azimuthally symmetric situation necessarily is subcritical. This should be contrasted with the example due to Nelson2 of critical nonmultiplying linear transport in a slab. Our result also, when combined with that of Olhoeft1 (see also 3 Theorem 12 of Case and Zweifel3) establishes rigorously that the linear transport equation subject to spherically and azimuthally symmetric data has a solution that also enjoys these symmetries. Finally, we note that our basic technique should have application to the extension to spherical geometry of convergence results known...