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RWTH Publication No: 710251 2018 
TITLE 
On Existence of L2solutions of Coupled Boltzmann Continuous Slowing Down Transport Equation System 
AUTHORS 
J. Tervo, P. Kokkonen, Martin Frank, Michael Herty 
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 socalled forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems in numerical methods. We shall find, after the preliminary treatments, that for some interactions CSDAtype modelling is actually necessary due to hypersingularities in the differential crosssections of certain interactions, that is, first or second order partial derivatives with respect to energy and angle 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 L2based spaces, for a single (particle) CSDAequation by using three techniques, the LionsLaxMilgram Theorem (variational approach), the theory of mdissipative operators and the theory evolution operators (semigroup approach). The due a priori estimates are derived and the positivity of solutions are retrieved. In addition, we prove the corresponding results and estimates for the system of coupled transport equations. The related existence 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. 
KEYWORDS 

DOI 
10.1016/j.jmaa.2017.11.052 
PUBLICATION 
Journal of Mathematical Analysis and Applications, Volume 460, Issue 1, Pages 271–301, 2018 