Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On regularity of solutions for certain linear Boltzmann transport equations

Jouko Tervo

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.

On regularity of solutions for certain linear Boltzmann transport equations

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.

Paper Structure

This paper contains 23 sections, 18 theorems, 430 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic function spaces
  4. Anisotropic Sobolev spaces
  5. Existence of solutions
  6. Assumptions for the restricted collision operator
  7. Existence of solutions for the initial inflow boundary value problem
  8. On regularity of solutions of initial inflow boundary value problem by using explicit formulas
  9. On regularity of solutions for convection-scattering equation using explicit formulas
  10. First order regularity with respect to spatial variable
  11. First order regularity with respect to angular variable
  12. Regularity with respect to energy variable
  13. On regularity of solutions for the differential part of continuous slowing down equation
  14. On the regularity with respect to spatial variable for a complete equation (in a special case)
  15. A Solution Based on Neumann Series
  16. ...and 8 more sections

Key Result

Theorem 2.1

The inflow trace operators are (well-defined) bounded surjective operators and they have bounded right inverses $L_\pm:T^2_{\tau_\pm}(\Gamma_\pm)\to { W}^2(G\times S\times I)$ that is, $\gamma_\pm\circ L_\pm=I$. The operators $L_\pm$ are called lifts.

Theorems & Definitions (54)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1: (Fubini)
  • ...and 44 more