On existence of spatially regular strong solutions for a class of transport equations
Jouko Tervo, Petri Kokkonen
TL;DR
The paper addresses the problem of existence and spatial regularity for linear Boltzmann transport equations with inflow boundary on bounded convex domains where the boundary is characteristic with variable multiplicity. It develops an anisotropic Sobolev framework in $H^{(m,0,0)}(G\times S\times I)$, constructs the spaces $H_0^{(m,0,0)}(G\times S\times I,\Gamma_-)$, and leverages $m$-accretivity and evolution-operator theory to treat convection-attenuation, convection-scattering, and CSDA variants. Key contributions include existence in $H_0^{(m,0,0)}$, higher-order regularity under restricted data, and a coherent methodology combining trace lifts, Green's formula, and operator theory for three transport models. The results provide a rigorous basis for high-order spatial discretizations (e.g., FEM) of transport equations in bounded domains and have direct relevance to radiation dose calculation models.
Abstract
The paper considers existence of spatially regular solutions for a class of linear Boltzmann transport equations. The related transport problem is an (initial) inflow boundary value problem. This problem is characteristic with variable multiplicity, that is, the rank of the boundary matrix (here a scalar) is not constant on the boundary. It is known that for these types of (initial) boundary value problems the full higher order Sobolev regularity cannot generally be established. In this paper we present Sobolev regularity results for solutions of linear Boltzmann transport problems when the data belongs to appropriate anisotropic Sobolev spaces whose elements are zero on the inflow and characteristic parts of the boundary.