On Global Existence and Regularity of Solutions for a Transport Problem Related to Charged Particles

Abstract

The paper considers a class of linear Boltzmann transport equations which models charged particle transport, for example in dose calculation of radiation therapy. The equation is an approximation of the exact transport equation containing hyper-singular integrals in its collision terms. The paper confines to the global case where the spatial domain G is the whole space R 3 : Existence results of solutions for the due initial value problem are formulated by applying variational methods. In addition, some regularity results of solutions are verified in scales of relevant anisotropic mixed-norm Sobolev spaces.

On regularity of solutions for certain linear Boltzmann transport equations

“…When G = R 3 we are not able to deduce regularity results similar to the global case [41]. The principle that the solution is more and more regular in the Sobolev sense when the data is more and more regular is not valid.…”

“…In the global case G = R 3 the regularity of solutions of initial value transport problems may increase according to data (see [41]). In the case where G is not the whole space the relevant transport problems contain the inflow boundary condition.…”

“…We remark that in the global case G = R 3 the "regularity of solution may increase according to the data". Actually, we conjectured in [41] (Discussion and Remark 4.9 therein) that in the case G = R 3 under relevant assumptions the assumption…”

Effects of tumour heterogeneous properties on modelling the transport of radiative particles

“…In order to simplify models for dose calculation while increasing accuracy, a lot of studies have focused on the use of deterministic dose calculation models as an alternative method to Monte Carlo approach that is widely used in RT clinical treatment planning. 18,[45][46][47] More specifically, the studies have focused on directly modelling the transport of radiative particles (electrons, photons, etc.) through the use of BTEs 48,49 and in addition approximating their solutions as accurately as possible.…”