The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications. (English) Zbl 1435.47045

In this paper, a densely defined operator \(B\) in a Banach space \(E\) is called positive if its spectrum lies in a sector \(|\arg\lambda| \le \varphi,\) \(0 \le \varphi < \pi\) and \[ \| (\lambda I - B)^{-1} \| \le \frac{M}{1 + |\lambda|}, \quad | \arg \lambda| > \varphi \, . \] The operator to which this definition is applied is \(B = -A,\) where \(A\) is the neutron transport operator \[ Au(x, y, \omega_1, \omega_2) = \omega_1 \frac{\partial u}{\partial x}+ \omega_2 \frac{\partial u}{\partial y} \] in \(\mathbb{R}^2,\) with \(\omega_1, \omega_2\) direction cosines. The first result is positivity of \(B\) in the spaces \(C(\mathbb{R}^2)\) and \(L^p(\mathbb{R}^2),\) \(1 \le p < \infty.\) Then, the space \(E_\alpha(C(\mathbb{R}^2), B)\) is introduced in terms of the growth of \(\|B(\lambda I + B)f\|_{C({\mathbb{R}^2})}\) as \(\lambda \to \infty\) and \(B\) is proved positive in \(E_\alpha(C(\mathbb{R}^2), B),\) hence in a Hölder space \(C^\alpha(\mathbb{R})\) with equivalent norm. Finally, the same results are shown for the space \(E_\alpha^p (L^p(\mathbb{R}^2), B)\) defined as \(E_\alpha(C(\mathbb{R}^2), B)\) but replacing \(C(\mathbb{R}^2)\) by \(L^p(\mathbb{R}^2),\) and for a Sobolev-Slobodecki space \(W^p_\alpha(\mathbb{R}^2)\) of equivalent norm.


47B65 Positive linear operators and order-bounded operators
47F05 General theory of partial differential operators
47B38 Linear operators on function spaces (general)
47B01 Operators on Banach spaces
35A35 Theoretical approximation in context of PDEs
35K30 Initial value problems for higher-order parabolic equations
34B27 Green’s functions for ordinary differential equations
35Q49 Transport equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure Appl. Math. 16 (1963), 121–239. · Zbl 0117.10001
[2] A. Ashyralyev, A survey of results in the theory of fractional spaces generated by positive operators, TWMS J. Pure Appl. Math. 6 (2015), no. 2, 129–157. · Zbl 1369.47049
[3] A. Ashyralyev and S. Akturk, Positivity of a one-dimensional difference operator in the half-line and its applications, Appl. Comput. Math. 14 (2015), no. 2, 204–220. · Zbl 1338.47037
[4] A. Ashyralyev and A. Taskin, Structure of fractional spaces generated by the two dimensional neutron transport operator, AIP Conf. Proc. 1759 (2016), 661–665.
[5] A. Ashyralyev and F. S. Tetikoglu, A note on fractional spaces generated by the positive operator with periodic conditions and applications, Bound. Value Probl. 2015, 2015:31, 17 pp. · Zbl 1334.34135
[6] A. Ashyralyev and P. E. Sobolevskii, Well-posedness of parabolic difference equations, Operator Theory: Advances and Applications vol. 69, Birkhäuser, Verlag, Basel, Boston, Berlin, 1994.
[7] A. Ashyralyev, N. Nalbant, and Y. Sozen, Structure of fractional spaces generated by second order difference operators, J. Franklin Inst. 351 (2014), no. 2, 713–731. · Zbl 1293.47035
[8] H. O. Fattorini, Second order linear differential equations in Banach spaces, Elsevier Science Publishing Company, North-Holland, Amsterdam, 1985. · Zbl 0564.34063
[9] S. G. Krein, Linear differential equations in Banach space, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 29. American Mathematical Society, Providence, R.I., 1971.
[10] V. I. Lebedova and P. E. Sobolevskii, Spectral properties of the transfer operator with constant coefficients in \(L_{p}\left( R^{n}\right) (~1≤ p<∞) \) spaces, Voronezh. Gosud. Univ. 1983, 54 pages. Deposited VINITI, 02.06.1983, no. 2958-83, (Russian) 1983.
[11] V. I. Lebedova, Spectral properties of the transfer operator of neutron in \(C(Ω, C(R^{n}))\) spaces, Qualitative and Approximate Methods for Solving Operator Equations, Yaroslavil, (Russian) 9 (1984), 44–51.
[12] E. Lewis and W. Miller Computational methods of neutron transport, American Nuclear Society, USA, 1993.
[13] G. I. Marchuk and V. I. Lebedev, Numerical methods in the theory of neutron transport, Taylor and Francis, USA, 1986.
[14] M. Mokhtar-Kharroubi, Mathematical topics in neutron transport theory, New aspects, World Scientific, Singapore and River Edge, N.J., 1997. · Zbl 0997.82047
[15] V. Shakhmurov and H. Musaev, Maximal regular convolution-differential equations in weighted Besov spaces, Appl. Comput. Math. 16 (2017), no. 2, 190–200. · Zbl 1474.34401
[16] P. E. Sobolevskii, Some properties of the solutions of differential equations in fractional spaces, Trudy Nauchn.-Issled. Inst. Mat. Voronezh. Gos. Univ., (Russian) 74 (1975), 68–76.
[17] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.
[18] V. I. Zhukova, Spectral properties of the transfer operator, Trudy Vsyesoyuznoy Nauchno-Prakticheskoy Konferensii, Chita 5 (2000), no. 1, 170–174.
[19] V. I. Zhukova and L. N. Gamolya, Investigation of spectral properties of the transfer operator, Dalnovostochniy Matematicheskiy Zhurnal, (Russian) 5 (2004), no. 1, 158–164.
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