Existence and BV-regularity for Neutron transport equation in non-convex domain

TL;DR

This paper investigates the existence and BV-regularity of solutions to the neutron transport equation in non-convex domains, employing $L^2-L^{inity}$ methods and a specialized cut-off function to handle grazing set characteristics.

Contribution

It extends the analysis of neutron transport equations to non-convex domains, establishing existence and BV-regularity results with novel boundary treatment techniques.

Findings

Proved existence of solutions using $L^2-L^{inity}$ methods.

Established BV-regularity of solutions in non-convex domains.

Constructed a cut-off function to manage grazing set characteristics.

Abstract

This paper considers the neutron transport equation in bounded domain with a combination of the diffusive boundary condition and the in-flow boundary condition. We firstly study the existence of solution in any fixed time by $L^2-L^{\infty}$ method, which was established to study Boltzmann equation in \cite{[Guo2]}. Based on the uniform estimates of the solution, we also consider the BV-regularity of the solution in non-convex domain. A cut-off function, which aims to exclude all the characteristics emanating from the grazing set $\mathfrak{S}_B$, has been constructed precisely.