Geodesic
On oscillations of two connected pendulums containing cavities partially filled with incompressible fluid
Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 4, pp. 627-677.

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

We consider the linearized problem on small oscillations of two pendulums connected to each other with a spherical hinge. Each pendulum has a cavity partially filled with incompressible fluid. We study the initial-boundary value problem as well as the corresponding spectral problem on normal motions of the hydromechanic system. We prove theorems on correct solvability of the problem on an arbitrary interval of time both in the case of ideal and viscous fluids in the cavities, and we study the corresponding spectral problems as well. 
@article{CMFD_2017_63_4_a6,
     author = {N. D. Kopachevsky and V. I. Voytitsky and Z. Z. Sitshaeva},
     title = {On oscillations of two connected pendulums containing cavities partially filled with incompressible fluid},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {627--677},
     publisher = {mathdoc},
     volume = {63},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2017_63_4_a6/}
}
TY  - JOUR
AU  - N. D. Kopachevsky
AU  - V. I. Voytitsky
AU  - Z. Z. Sitshaeva
TI  - On oscillations of two connected pendulums containing cavities partially filled with incompressible fluid
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2017
SP  - 627
EP  - 677
VL  - 63
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2017_63_4_a6/
LA  - ru
ID  - CMFD_2017_63_4_a6
ER  - 
%0 Journal Article
%A N. D. Kopachevsky
%A V. I. Voytitsky
%A Z. Z. Sitshaeva
%T On oscillations of two connected pendulums containing cavities partially filled with incompressible fluid
%J Contemporary Mathematics. Fundamental Directions
%D 2017
%P 627-677
%V 63
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2017_63_4_a6/
%G ru
%F CMFD_2017_63_4_a6
N. D. Kopachevsky; V. I. Voytitsky; Z. Z. Sitshaeva. On oscillations of two connected pendulums containing cavities partially filled with incompressible fluid. Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 4, pp. 627-677. http://geodesic.mathdoc.fr/item/CMFD_2017_63_4_a6/

[1] Yu. Sh. Abramov, Variational Methods in the Theory of Operator Pencils. Spectral Optimization, LGU, Leningrad, 1983 (in Russian) | MR

[2] M. S. Agranovich, “Spectral problems for second-order strongly elliptic systems in domains with smooth and nonsmooth boundary”, Usp. mat. nauk [Progr. Math. Sci.], 57:5 (2002), 3–78 (in Russian) | DOI | MR | Zbl

[3] T. Ya. Azizov, I. S. Iokhvidov, Basics of the Theory of Linear Operators in Spaces with Indefinite Metrics, Nauka, Moscow, 1986 (in Russian) | MR

[4] E. I. Batyr, “Small motions of system of serially connected bodies with cavities containing viscous incompressible fluid”, Dinam. sist. [Dynam. Syst.], 17, 2001, 120–125 (in Russian) | Zbl

[5] E. I. Batyr, “Small motions of system of serially connected bodies with cavities containing ideal incompressible fluid”, Uch. zap. Tavr. nats. un-ta im. V. I. Vernadskogo [Sci. Notes Vernadskii Tavria Nats. Univ.], 15(54):2 (2002), 5–10 (in Russian)

[6] E. I. Batyr, “Small motions and normal oscillations of a system of three connected bodies with cavities filled with ideal incompressible fluid]”, Uch. zap. Tavr. nats. un-ta im. V. I. Vernadskogo [Sci. Notes Vernadskii Tavria Nats. Univ.], 23(62):2 (2010), 19–38 (in Russian)

[7] E. I. Batyr, O. A. Dudik, N. D. Kopachevsky, “Small motions of bodies with cavities filled with incompressible viscous fluid”, Izv. vuzov. Sev.-Kavkaz. region. Estestv. nauki [Bull. Higher Edu. North Caucas. Reg. Nat. Sci.], 49 (2009), 15–29 (in Russian)

[8] E. I. Batyr, N. D. Kopachevsky, “Small motions and normal oscillations in systems of connected gyrostats]”, Sovrem. mat. Fundam. napravl. [Contemp. Math. Fundam. Directions], 49, 2013, 5–88 (in Russian)

[9] M. Sh. Birman, M. Z. Solomyak, “Asymptotics of spectrum of differential equations”, Itogi nauki i tekhn. Mat. analiz [Totals Sci. Tech. Math. Anal.], 14, 1977, 5–58 (in Russian) | MR | Zbl

[10] V. I. Voytitskiy, N. D. Kopachevsky, “Small motions of a system of two connected bodies with cavities partially filled with heavy viscous fluid”, Tavr. vestn. inform. i mat. [Tavricheskiy Bull. Inform. Math.], 2017, no. 2(35), 7–32 (in Russian)

[11] I. L. Vulis, M. Z. Solomyak, “Spectral asymptotics of the degenerating Steklov problem”, Vestn. LGU [Bull. Leningrad State Univ.], 1973, no. 19, 148–150 (in Russian) | MR | Zbl

[12] I. L. Vulis, M. Z. Solomyak, “Spectral asymptotics of second-order degenerating elliptic operators”, Izv. AN SSSR. Ser. Mat. [Bull. Acad. Sci. USSR. Ser. Math.], 38:6 (1974), 1362–1392 (in Russian) | MR | Zbl

[13] N. E. Zhukovskiy, “On motion of a solid body with cavities filled with homogeneous dropping fluid”, Selected Works, v. 1, Gostekhizdat, Moscow–Leningrad, 1948, 31–52 (in Russian)

[14] N. A. Karazeeva, M. Z. Solomyak, “Asymptotics of spectrum of Steklov-type problems in composite domains”, Probl. mat. analiza [Probl. Math. Anal.], 8, 1981, 36–48 (in Russian) | MR | Zbl

[15] N. D. Kopachevsky, “On oscillations of a body with cavity partially filled with heavy ideal fluid: theorems of existence, uniqueness, and stability of strong solutions”, Problemi dinamiki ta stiykosti bagatovimirnikh sistem. Zb. prats' Institutu matematiki NAN Ukraï ni [Probl. Dynamics Stability Multidimen. Syst.: Digest Works Inst. Math. Acad. Sci. Ukraine], 1:1 (2005), 158–194 (in Russian)

[16] N. D. Kopachevsky, S. G. Krein, Ngo Zuy Kan, Operator Methods in Linear Hydrodynamics: Evolutional and Spectral Problems, Nauka, Moscow, 1989 (in Russian) | MR

[17] N. D. Kopachevsky, Spectral Theory of Operator Pencils: A Special Course, Forma, Simferopol', 2009 (in Russian)

[18] N. D. Kopachevsky, Integrodifferential Equations in a Hilbert Space: A Special Course, FLP Bondarenko, Simferopol', 2012

[19] N. D. Kopachevsky, “Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms”, Sovrem. mat. Fundam. napravl. [Contemp. Math. Fundam. Directions], 57, 2015, 71–107 (in Russian)

[20] N. D. Kopachevsky, Abstract Green Formula and Some Its Applications, Forma, Simferopol', 2016 (in Russian)

[21] S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow, 1967 (in Russian) | MR

[22] S. G. Krein, N. N. Moiseev, “On oscillations of solid body containing fluid with a free boundary”, Prikl. mat. mekh. [Appl. Math. Mech.], 21:2 (1957), 169–174 (in Russian)

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

[24] S. G. Mikhlin, Variational Methods in Mathematical Physics, Nauka, Moscow, 1970 (in Russian) | MR

[25] L. S. Pontryagin, “Hermit operators in spaces with indefinite metrics”, Izv. AN SSSR. Ser. Mat. [Bull. Acad. Sci. USSR. Ser. Math.], 8:6 (1944), 243–280 (in Russian) | MR | Zbl

[26] V. A. Prigorskiy, “On some classes of bases of Hilbert spaces”, Usp. mat. nauk [Progr. Math. Sci.], 20:5(125) (1965), 231–236 (in Russian) | MR | Zbl

[27] T. A. Suslina, “Asymptotics of spectrum of variational problems on solutions of homogeneous elliptic equation with relations on part of the boundary”, Probl. mat. analiza [Probl. Math. Anal.], 9, 1984, 84–97 (in Russian)

[28] T. A. Suslina, “Asymptotics of spectrum of variational problems on solutions of elliptic equation in a domain with piecewise smooth boundary”, Zap. nauch. sem. LOMI [Notes Sci. Semin. Leningrad Dept. Math. Inst. Acad. Sci.], 147, 1985, 179–183 (in Russian) | MR | Zbl

[29] T. A. Suslina, Asymptotics of spectrum of some problems related to oscillations of fluids, Dep. to VINITI 21.11.85, No. 8058-B, Leningrad Electrotech. Inst. Commun., Leningrad, 1985 (in Russian)

[30] Kopachevsky N. D., Krein S. G., Operator Approach to Linear Problems of Hydrodynamics, v. 1, Self-Adjoint Problems for an Ideal Fluid, Birkhäuser, Basel–Boston–Berlin, 2001 | MR | Zbl

[31] Kopachevsky N. D., Krein S. G., Operator Approach to Linear Problems of Hydrodynamics, v. 1, Nonself-Adjoint Problems for Viscous Fluids, Birkhäuser, Basel–Boston–Berlin, 2003 | MR | Zbl

[32] Metivier G., “Valeurs propres d'opérateurs définis par la restriction de systèmes variationnels à des sous-espaces”, J. Math. Pures Appl. (9), 57:2 (1978), 133–156 | MR | Zbl