Journal of Mathematical Sciences

, Volume 170, Issue 2, pp 173–191 | Cite as

Problem of small and normal oscillations of a rotating elastic body filled with an ideal barotropic liquid

  • D. A. Zakora


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.


Hilbert Space Cauchy Problem Strong Solution Elastic Body Spectral Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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).MATHCrossRefGoogle Scholar
  2. 2.
    A. Garadzhaev, “On normal oscillations of an ideal compressible fluid in rotating elastic containers,” Sov. Math. Dokl., 27, 313–317 (1983).MATHGoogle Scholar
  3. 3.
    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).MathSciNetGoogle Scholar
  4. 4.
    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).MATHMathSciNetGoogle Scholar
  5. 5.
    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).Google Scholar
  6. 6.
    T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin–Heidelberg–New York (1966).MATHGoogle Scholar
  7. 7.
    N. D. Kopachevskii, S. G. Krein, and N. Z. Kan, Operator Methods in Linear Hydrodynamics. Evolutionary and Spectral Problems [in Russian], Nauka, Moscow (1989).Google Scholar
  8. 8.
    S. G. Krein, Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1967).Google Scholar
  9. 9.
    O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).MATHGoogle Scholar
  10. 10.
    J.-L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, Springer-Verlag, Berlin–New York (1972).Google Scholar
  11. 11.
    A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Sheaves [in Russian], Shtiintsa, Kishinev (1986).Google Scholar
  12. 12.
    O. A. Oleinik, G. A. Iosif’yan, and A. S. Shamaev, Mathematical Problems of Strongly Nonuniform Media [in Russian], Moscow State Univ., Moscow (1990).Google Scholar
  13. 13.
    M. B. Orazov, Some questions on spectral theory of non-self-adjoint operators and related mechanical problems [in Russian], Doctoral Thesis, Ashkhabad (1982).Google Scholar
  14. 14.
    G. V. Radzievskii, Quadratic Operator Sheaves [in Russian], preprint, Kiev (1976).Google Scholar
  15. 15.
    J. V. Ralston, “On stationary modes in viscid rotating fluids,” J. Math. Anal. Appl., 44, 366–383 (1973).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Faculty of Mathematics and Informatics, Chair of Mathematical AnalysisTaurida National V. I. Vernadsky UniversitySimferopolUkraine

Personalised recommendations