## Advances in Operator Theory

- Adv. Oper. Theory
- Volume 4, Number 1 (2019), 140-155.

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

Allaberen Ashyralyev and Abdulgafur Taskin

#### 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.

#### Article information

**Source**

Adv. Oper. Theory, Volume 4, Number 1 (2019), 140-155.

**Dates**

Received: 12 November 2017

Accepted: 18 April 2018

First available in Project Euclid: 10 May 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1525917619

**Digital Object Identifier**

doi:10.15352/aot.1711-1261

**Mathematical Reviews number (MathSciNet)**

MR3867338

**Zentralblatt MATH identifier**

06946447

**Subjects**

Primary: 47B65: Positive operators and order-bounded operators

Secondary: 35A35: Theoretical approximation to solutions {For numerical analysis, see 65Mxx, 65Nxx} 35K30: Initial value problems for higher-order parabolic equations 34B27: Green functions

**Keywords**

Neutron transport operator fractional space Slobodeckij space positive operator

#### Citation

Ashyralyev, Allaberen; Taskin, Abdulgafur. The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications. Adv. Oper. Theory 4 (2019), no. 1, 140--155. doi:10.15352/aot.1711-1261. https://projecteuclid.org/euclid.aot/1525917619