Abstract
An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.
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References
V. A. Avakyan, “Asymptotic distribution of the spectrum of a linear sheaf perturbed by an analytic operator-valued function,” Funkts. Anal. Appl., 12, 129–131 (1978).
A. Garadzhaev, “On normal oscillations of an ideal compressible fluid in rotating elastic containers,” Sov. Math. Dokl., 27, 313–317 (1983).
A. Garadzhaev, “On a problem of oscillations of an ideal compressible fluid in an elastic container,” Dokl. AN SSSR, 286, No. 5, 1047–1049 (1986).
A. Garadzhaev, “Spectral theory of a problem of small oscillations of an ideal fluid in a rotating elastic vessel,” Differ. Uravn., 23, No. 1, 38–47 (1987).
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators. Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, Rhode Island (1969).
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin–Heidelberg–New York (1966).
N. D. Kopachevskii, S. G. Krein, and N. Z. Kan, Operator Methods in Linear Hydrodynamics. Evolutionary and Spectral Problems [in Russian], Nauka, Moscow (1989).
S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, Springer-Verlag, Berlin–New York (1972).
A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Sheaves [in Russian], Shtiintsa, Kishinev (1986).
O. A. Oleinik, G. A. Iosif’yan, and A. S. Shamaev, Mathematical Problems of Strongly Nonuniform Media [in Russian], Moscow State Univ., Moscow (1990).
M. B. Orazov, Some questions on spectral theory of non-self-adjoint operators and related mechanical problems [in Russian], Doctoral Thesis, Ashkhabad (1982).
G. V. Radzievskii, Quadratic Operator Sheaves [in Russian], preprint, Kiev (1976).
J. V. Ralston, “On stationary modes in viscid rotating fluids,” J. Math. Anal. Appl., 44, 366–383 (1973).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 34, Proceedings of KROMSH, 2009.
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Zakora, D.A. Problem of small and normal oscillations of a rotating elastic body filled with an ideal barotropic liquid. J Math Sci 170, 173–191 (2010). https://doi.org/10.1007/s10958-010-0079-7
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DOI: https://doi.org/10.1007/s10958-010-0079-7