To the problem on small oscillations of a system of two viscoelastic fluids filling immovable vessel: model problem

D. A. Zakora ; N. D. Kopachevsky

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 66 (2020) no. 2, pp. 182-208.

Résumé

In this paper, we study the scalar conjugation problem, which models the problem of small oscillations of two viscoelastic fluids filling a fixed vessel. An initial-boundary value problem is investigated and a theorem on its unique solvability on the positive semiaxis is proven with semigroup theory methods. The spectral problem that arises in this case for normal oscillations of the system is studied by the methods of the spectral theory of operator functions (operator pencils). The resulting operator pencil generalizes both the well-known S. G. Kreyn's operator pencil (oscillations of a viscous fluid in an open vessel) and the pencil arising in the problem of small motions of a viscoelastic fluid in a partially filled vessel. An example of a two-dimensional problem allowing separation of variables is considered, all points of the essential spectrum and branches of eigenvalues are found. Based on this two-dimensional problem, a hypothesis on the structure of the essential spectrum in the scalar conjugation problem is formulated and a theorem on the multiple basis property of the system of root elements of the main operator pencil is proved.

Bibliographie

[1] T. Ya. Azizov, I. S. Iokhvidov, Fundamentals of the Theory of Linear Operators in Spaces with Indefinite Metric, Nauka, M., 1986 (in Russian)

[2] J. Goldstein, Semigroups of Linear Operators and Applications, Russian translation, Vyshcha shkola, Kiev, 1989

[3] T. Kato, Perturbation Theory for Linear Operators, Russian translation, Mir, M., 1972

[4] N. D. Kopachevsky, Abstract Green's Formula and Applications, OOO «Forma», Simferopol', 2016 (in Russian)

[5] N. D. Kopachevsky, “To the problem on small motions of the system of two viscoelastic fluids in a fixed vessel”, Contemp. Math. Fundam. Directions, 64, no. 3, 2018, 547–572 (in Russian)

[6] N. D. Kopachevsky, S. G. Kreyn, Ngo Zuy Kan, Operator Methods in Linear Hydrodynamic. Evolutional and Spectral Problems, Nauka, M., 1989 (in Russian)

[7] C. G. Kreyn, “On oscillations of a viscous fluid in a vessel”, Rep. Acad. Sci. USSR, 159:2 (1964), 262–265 (in Russian)

[8] C. G. Kreyn, Linear Differential Equations in a Banach Space, Nauka, M., 1967 (in Russian)

[9] C. G. Kreyn, G. I. Laptev, “To the problem on motion of a viscous liquid in an open vessel”, Funct. Anal. Appl., 2:1 (1968), 40–50 (in Russian)

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

[11] A. S. Markus, V. I. Matsaev, “Comparison theorems for spectra of linear operators and spectral asymptotics”, Proc. Moscow Math. Soc., 45, 1982, 133–181 (in Russian)

[12] A. S. Markus, V. I. Matsaev, “Theorem on spectra comparison and spectral asymptotics for the Keldysh pencil”, Math. Digest, 123:3 (1984), 391–406 (in Russian)

[13] A. I. Miloslavskiy, Spectral analysis of small oscillations of viscoelastic fluid in open container, Preprint No 1221, Univ. Math. NAS Ukraine, Kiev, 1989 (in Russian)

[14] A. I. Miloslavskiy, “Spectrum of small oscillation of viscoelastic fluid in open vessel”, Progr. Math. Sci., 44:4 (1989) (in Russian)

[15] A. I. Miloslavskiy, “Spectrum of small oscillations of viscoelastic inheritance medium”, Rep. Acad. Sci. USSR, 309:3 (1989), 532–536 (in Russian)

[16] Azizov T. Ya., Kopachevskii N. D., Orlova L. D., “Evolution and spectral problems related to small motions of viscoelastic fluid”, Am. Math. Soc. Transl., 199 (2000), 1–24

[17] Birman M. Sh., Solomyak M. Z., “Asymptotic behavior of the spectrum of differential equations”, J. Soviet Math., 12:3 (1979), 247–283

[18] Engel K.-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000

[19] Gagliardo E., “Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili”, Rend. Semin. Mat. Univ. Padova, 27 (1957), 284–305

[20] Gohberg I., Goldberg S., Kaashoek M. A., Classes of Linear Operators, v. 1, Birkhäuser, Basel–Boston–Berlin, 1990

[21] Helton J. W., “Unitary operators on a space with an indefinite inner product”, J. Funct. Anal., 6:3 (1970), 412–440

[22] 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

[23] Miloslavsky A. I., “Stability of certain classes of evolution equations”, Sib. Math. J., 26:5 (1985), 723–735

[24] Miloslavskii A. I., “Stability of a viscoelastic isotropic medium”, Soviet Phys. Dokl., 33 (1988), 300