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abstract: 'The paper considers a coupled system of linear Boltzmann transport equations (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in radiation therapy. The evolution of charged particles (e.g. electrons and positrons) are in practice often modelled using the CSDA version of BTE because of the so-called forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems in numerical methods. We shall verify, after the preliminary discussion, that CSDA-type modelling is actually necessary due to hyper-singularities in the differential cross-sections of certain interactions, that is, first-order partial derivatives with respect to energy must be included into the transport part of charged particles. The existence and uniqueness of (weak) solutions is shown, under sufficient criteria and in appropriate $L^2$-based spaces, for a single (particle) CSDA-equation by using three techniques, the Lions-Lax-Milgram Theorem (variational approach), the theory of $m$-dissipative operators and the theory evolution operators (semigroup approach). The necessary a priori estimates are derived. In addition, we prove the corresponding results and estimates for the system of coupled transport equations. The related results are given for the adjoint problem as well. We also give some computational points (e.g. certain explicit formulas), and we outline a related inverse problem at the end of the paper.'
address:
- '$^1$University of Eastern Finland, Department of Applied Physics, Kuopio, Finland'
- '$^2$Varian Medical Systems Finland Oy, Helsinki, Finland'
- '$^3$RWTH Aachen University, MATHCCES, Schinkelstrasse 2, 52062, Germany'
- '$^4$RWTH Aachen University, IGPM, Templergraben 5, 52062, Germany'
author:
- 'J. Tervo$^1$'
- 'P. Kokkonen$^2$'
- 'M. Frank$^3$'
- 'M. Herty$^4$'
title: 'On existence of $L^2$-solutions of Coupled Boltzmann Conti