Abstract
In this paper, mathematical models of compressible viscoelastic Maxwell, Oldroyd, and Kelvin–Voigt fluids are derived. A model of rotating viscoelastic barotropic Oldroyd fluid is studied. A theorem on strong unique solvability of the corresponding initial-boundary value problem is proved. The spectral problem associated with such a system is studied. Results on the spectrum localization, essential and discrete spectra, and spectrum asymptotics are obtained. In the case where the system is in the weightlessness state and does not rotate, results on multiple completeness and basis property of a special system of elements are proved. In such a case, under the assumption the viscosity is sufficiently large, an expansion of the solution of the evolution problem with respect to a special system of elements is obtained.
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References
T.Ya. Azizov and I. S. Iokhvidov, Foundations of Theory of Linear Operators in Spaces with Indefinite Metrics [in Russian], Nauka, Moscow (1986).
T.Ya. Azizov, N. D. Kopachevskiy and L.D. Orlova, “Evolution and spectral problems generated by the problem of small movements of viscoelastic fluid,” Tr. St.-Peterbg. Mat. Obshch., 6, 5–33 (1998).
Yu. M. Berezanskiy, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
M. Sh. Birman and M. Z. Solomyak, “Asymptotic properties of the spectrum of differential equations,” Mathematical Analysis, 14, 5–58 (1977).
M. Sh. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing Company, Dordrecht–Boston–Lancaser–Tokyo (1987).
A. Freudenthal and H. Geiringer, The Mathematical Theories of the Inelastic Continuum [Russian translation], Fizmatgiz, Moscow (1962).
I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators. Vol. 1, Birkhäuser, Basel–Boston–Berlin (1990).
J. A. Goldsteyn, Semigroups of Linear Operators and Their Applications [Russian translation], Vyshcha Shkola, Kiev (1989).
G. Grubb and G. Geymonat, “The essential spectrum of elliptic systems of mixed order,” Math. Ann., 227, 247–276 (1977).
T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow, (1972).
W. Kelvin (Thomson), “On the theory of viscoelastic fluids,” Math. A. Phys. Pap., 3, 27–84 (1875).
N. D. Kopachevsky and S. G. Krein, Operator Approach to Linear Problems of Hydrodynamics.Vol. 2: Nonself-Adjoint Problems for Viscous Fluids, Birkh¨auser, Basel–Boston–Berlin (2003).
N. D. Kopachevskiy, S.G. Krein, and Ngo Zuy Kan, Operator Methods in Linear Hydrodynamics: Evolution and Spectral Problems [in Russian], Nauka, Moscow (1989).
S.G. Krein, Linear Differential Equations in Banach Space [in Russian], Nauka, Moscow (1967).
A. S. Markus, Introduction to Spectral Theory of Polynomial Operator Pencils [in Russian], Shtiintsa, Kishinev (1986).
A. S. Markus and V. I. Matsaev, “Theorem on comparison of spectra and spectral asymptotics for the M. V. Keldysh pencil,” Mat. Sb., 123, No. 3, 391–406 (1984).
J.C. Maxwell, “On the dynamical theory of gases,” Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 157, 49–88 (1867).
J.C. Maxwell, “On the dynamical theory of gases,” Philos. Mag., 35, 129–145 (1868).
A. I. Miloslavskii, “Spectral analysis of small oscillations of a viscoelastic fluid in an open vessel,” Deposited in Ukr. NIINTI, No. 1221 (1989).
A. I. Miloslavskii, “The spectrum of small oscillations of a viscoelastic hereditary medium,” Dokl. Akad. Nauk SSSR, 309, No. 3, 532–536 (1989).
S. G. Mikhlin, “Spectrum of the operator pencil from the elasticity theory,” Uspekhi Mat. Nauk, 28, No. 3, 43–82 (1973).
J. G. Oldroyd, “On the formulation of rheological equations of state,” Proc. Roy. Soc. London. Ser. A, 200, 523–541 (1950).
J.G. Oldroyd, “The elastic and viscous properties of emulsions and suspensions,” Proc. Roy. Soc. London. Ser. A, 218, 122–132 (1953).
A. P. Oskolkov, “Initial-boundary value problems for equations of motion of Kelvin-Voight fluids and Oldroyd fluids,” Trudy Mat. Inst. Steklov., 179, 126–164 (1988).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).
K. Rektoris, Variational Methods in Mathematical Physics and Technics [Russian translation], Mir, Moscow (1985).
V. A. Solonnikov, “General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg. II,” Trudy Mat. Inst. Steklov., 92, 233–297 (1966).
W. Voight, “Uber die innere Reibung der fasten Korper, inslesondere der krystalle,” Gottinden Abh., 36, No. 1, 3–47 (1889).
W. Voight, “Uber innex Reibung faster Korper, insbesondere der Metalle,” Ann. Phys. U. Chem., 47, No. 9, 671–693 (1892).
D. A. Zakora, “Operator approach to the Il’yushin model of a viscoelastic body of parabolic type,” Sovrem. Mat. Fundam. Napravl., 57, 31–64 (2015).
V.G. Zvyagin and M.V. Turbin, “Investigation of initial-boundary value problems for Kelvin–Voigt mathematical models of motion of fluids,” Sovrem. Mat. Fundam. Napravl., 31, 3–144 (2009).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 61, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2016.
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Zakora, D.A. Oldroyd Model for Compressible Fluids. J Math Sci 239, 582–607 (2019). https://doi.org/10.1007/s10958-019-04317-7
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DOI: https://doi.org/10.1007/s10958-019-04317-7