On regularity of solutions for certain linear Boltzmann transport equations

TL;DR

This paper studies the regularity of solutions to a class of linear Boltzmann transport equations modeling charged particle transport, using anisotropic Sobolev spaces to verify solution properties relevant for applications like radiation therapy.

Contribution

It provides new regularity results for approximate Boltzmann equations involving hyper-singular integrals, connecting these to anisotropic Sobolev space analysis.

Findings

Regularity of solutions established in anisotropic Sobolev spaces

Approximation of hyper-singular integrals by differential and integral operators

Results applicable to radiation therapy dose calculations

Abstract

The paper considers a class of linear Boltzmann transport equations which models a charged particle transport. The equation is an approximation of the original exact transport equation which involves hyper-singular integrals in their collision terms. Hyper-singular integrals can be approximated by partial differential operators together with partial integral operators which leads to an approximation under consideration. This type of approximation is applied, for example in the dose calculation of radiation therapy. The related transport problem is a characteristic initial inflow boundary value problem. Regularity results of solutions are verified utilizing the scales of relevant anisotropic Sobolev spaces.