Geodesic
Operator approach to the problem on small motions of an ideal relaxing fluid
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 459-489.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the problem on small motions of an ideal relaxing fluid that fills a uniformly rotating or fixed container. We prove a theorem on uniform strong solvability of the corresponding initial-boundary value problem. In the case where the system does not rotate, we find an asymptotic behavior of the solution under the stress of special form. We investigate the spectral problem associated with the system under consideration. We obtain results on localization of the spectrum, on essential and discrete spectrum, and on spectral asymptotics. For nonrotating system in zero-gravity conditions we prove the multiple basis property of a special system of elements. In this case, we find an expansion of the solution of the evolution problem in the special system of elements. 
@article{CMFD_2018_64_3_a1,
     author = {D. A. Zakora},
     title = {Operator approach to the problem on small motions of an ideal relaxing fluid},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {459--489},
     publisher = {mathdoc},
     volume = {64},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a1/}
}
TY  - JOUR
AU  - D. A. Zakora
TI  - Operator approach to the problem on small motions of an ideal relaxing fluid
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2018
SP  - 459
EP  - 489
VL  - 64
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a1/
LA  - ru
ID  - CMFD_2018_64_3_a1
ER  - 
%0 Journal Article
%A D. A. Zakora
%T Operator approach to the problem on small motions of an ideal relaxing fluid
%J Contemporary Mathematics. Fundamental Directions
%D 2018
%P 459-489
%V 64
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a1/
%G ru
%F CMFD_2018_64_3_a1
D. A. Zakora. Operator approach to the problem on small motions of an ideal relaxing fluid. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 459-489. http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a1/

[1] V. A. Avakyan, “Asymptotic distribution of the spectrum of a linear pencil perturbed by an analytic operator-function”, Funct. Anal. Appl., 12:2 (1978), 66–67 (in Russian) | MR | Zbl

[2] Yu. M. Berezanskiy, Expansion in Eigenfunctions of Self-Adjoint Operators, Naukova dumka, Kiev, 1965 (in Russian)

[3] M. Sh. Birman, M. Z. Solomyak, “Asymptotics of the spectrum of differential equations”, Totals Sci. Tech. Ser. Math. Anal., 14, 1977, 5–58 (in Russian) | MR | Zbl

[4] V. V. Vlasov, N. A. Rautian, Spectral Analysis of Functional Differential Equations, MAKS Press, Moscow, 2016 (in Russian)

[5] D. A. Zakora, “Operator approach to the Ilyushin model for a viscoelastic body of parabolic type”, Contemp. Math. Fundam. Directions, 57, 2015, 31–64 (in Russian)

[6] D. A. Zakora, “Exponential stability of one semigroup and applications”, Math. Notes, 103:5 (2018), 702–719 (in Russian) | DOI | MR | Zbl

[7] V. G. Zvyagin, M. V. Turbin, “The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids”, Contemp. Math. Fundam. Directions, 31, 2009, 3–144 (in Russian) | MR

[8] T. Kato, Perturbation Theory for Linear Operators, Mir, Moscow, 1972, (Russian translation)

[9] N. D. Kopachevsky, S. G. Kreyn, Ngo Zuy Kan, Operator Methods in Linear Hydrodynamics: Evolution and Spectral Problems, Nauka, Moscow, 1989 (in Russian)

[10] S. G. Kreyn, Linear Differential Equations in Banach Space, Nauka, Moscow, 1967 (in Russian)

[11] A. S. Markus, Introduction to Spectral Theory of Polynomial Operator Pencils, Shtiintsa, Kishinev, 1986 (in Russian)

[12] A. I. Markushevich, Theory of Analytic Functions, v. 1, Nauka, Moscow, 1967 (in Russian) | MR

[13] M. B. Orazov, Some questions of spectral theory of nonself-adjoint operators and related problems of mechanics, Doctoral Thesis, Ashkhabad, 1982 (in Russian)

[14] G. V. Radzievskiy, Quadratic pencil of operators, Preprint, Kiev, 1976 (in Russian) | MR

[15] Birman M. Sh., Solomjak M. Z., Spectral theory of self-adjoint operators in Hilbert space, D. Reidel Publ. Co., Dordrecht–Boston–Lancaser–Tokyo, 1986 | MR

[16] Gohberg I., Goldberg S., Kaashoek M. A., Classes of linear operators, v. 1, Birkhäuser, Basel–Boston–Berlin, 1990 | Zbl

[17] Goldstein J. A., Semigroups of linear operators and applications, Oxford University Press, New York, 1989 | MR

[18] Kopachevsky N. D., Krein S. G., Operator approach to linear problems of hydrodynamics, v. 2, Nonself-adjoint problems for viscous fluids, Birkhäuser, Basel–Boston–Berlin, 2003 | MR | Zbl

[19] Ralston J. V., “On stationary modes in inviscid rotating fluids”, J. Math. Anal. Appl., 44 (1973), 366–383 | DOI | MR | Zbl