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

Document Type: Special issue: Trends in Operators on Banach Spaces

Authors

1 Near East University Lefkoşa(Nicosia), Mersin 10 Turkey TRNC

2 Department of Mathematics‎, ‎Private Soyak Bahcesehir Science‎ ‎and Technology College‎, ‎Umraniye‎, ‎Istanbul‎, ‎Turkey

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 $L_{p}\left( \mathbb{R}^{2}\right)‎ ,‎$ $1\leq p<\infty‎ ~‎$ is‎ ‎established. It is established that for any~$0<\alpha <1~$and $1\leq‎
‎p<\infty‎ ~‎$the norms in 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 H\"{o}lder 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 H\"{o}lder and Slobodeckij spaces are provided‎.

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