  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 find, after the preliminary treatments, that for
some interactions CSDA-type modelling is actually necessary due to
hyper-singularities in the differential cross-sections 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 $L^2$-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 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.
