Validity and regularization of classical half-space equations
For researchers in neutron transport and kinetic theory, this resolves a concern about CHS validity by proving its correctness for boundary conditions of the interior Laplace equation.
The paper shows that despite the classical half-space equation (CHS) failing to capture correct boundary layer behavior on the 2D unit disk, it still yields the correct boundary condition for the interior Laplace equation, validating its use for that purpose.
Recent result [Wu and Guo, Comm. Math. Phys., 2015] has shown that over the 2D unit disk, the classical half-space equation (CHS) for the neutron transport does not capture the correct boundary layer behaviour as long believed. In this paper we develop a regularization technique for CHS to any arbitrary order and use its first-order regularization to show that in the case of the 2D unit disk, although CHS misrepresents the boundary layer behaviour, it does give the correct boundary condition for the interior macroscopic (Laplace) equation. Therefore CHS is still a valid equation to recover the correct boundary condition for the interior Laplace equation over the 2D unit disk.