The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications

Abstract

In this study, the structure of fractional spaces generated by the two-dimensional neutron transport operator $A$ defined by formula $Au=\omega_{1}\frac{\partial u}{\partial x}+\omega _{2}\frac{\partial u}{\partial y}$ is investigated. The positivity of $A$ in $C\left( \mathbb{R}^{2}\right)$ and $L_{p}\left( \mathbb{R}^{2}\right)$, $1\leq p \lt \infty$, is established. It is established that, for any $0 \lt \alpha \lt 1$ and $1\leq p \lt \infty$, the norms of spaces $E_{\alpha ,p}\left( L_{p}\left( \mathbb{R}^{2}\right), A\right)$ and $E_{\alpha }\left( C\left( \mathbb{R}^{2}\right), A\right) , W_{p}^{\alpha } \left( \mathbb{R}^{2}\right)$ and $C^{\alpha }\left( \mathbb{R}^{2}\right)$ are equivalent, respectively. The positivity of the neutron transport operator in Hölder space $C^{\alpha }\left( \mathbb{R}^{2}\right)$ and Slobodeckij space $W_{p}^{\alpha }\left( \mathbb{R}^{2}\right)$ is proved. In applications, theorems on the stability of Cauchy problem for the neutron transport equation in Hölder and Slobodeckij spaces are provided.