Existence and uniqueness for spherically symmetric linear transport

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 for the discrete-ordinates method in other settings.^4–10 We have in mind here both the classical S_N approach in which the differential angular redistribution operator is angularly discretized along with the scattering integral, and the more recent work^l3–15 employing angular discretization along with the method of characteristics.