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Justification of the averaging method for a system of equations with the Navier-Stokes operator in the principal part


Author: V. B. Levenshtam
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 1.
Journal: St. Petersburg Math. J. 26 (2015), 69-90
MSC (2010): Primary 35Q30
DOI: https://doi.org/10.1090/S1061-0022-2014-01331-3
Published electronically: November 21, 2014
MathSciNet review: 3234805
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Abstract | References | Similar Articles | Additional Information

Abstract: The averaging method is justified for a system of partial differential equations with the Navier-Stokes operator in the principal part. The right-hand side of this system (an analog of a mass force) oscillates in time with frequency $ \omega \gg 1$, depends polynomially on the unknown (an analog of the flow velocity), and involves a linear summand proportional to $ \sqrt {\omega }$. An initial-boundary value problem and a problem on time-periodic solutions are considered.


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  • 1. N. N. Bogolyubov, On some statistical methods in mathematical physics, Akad. Nauk UkrSSR, Kiev, 1945. (Russian) MR 0016575 (8:37g)
  • 2. N. N. Bogolubov and Yu. A. Mitropol'skiĭ, Asymptotic methods in the theory on nonlinear oscillations, Nauka, Moscow, 1974. (Russian) MR 0374550 (51:10750)
  • 3. V. M. Volosov and B. I. Morgunov, The averaging method in the theory of nonlinear oscillatory systems, Moskov. Gos. Univ., Moscow, 1971. (Russian)
  • 4. B. M. Levitan and V. V. Zhukov, Almost-periodic functions and differential equations, Moskov. Gos. Univ., Moscow, 1978. (Russian) MR 509035 (80d:42010)
  • 5. I. B. Simonenko, The averaging method in the theory of nonlinear equations of parabolic type with applications to the problems of hydrodinamic stability, Rostov. Gos. Univ., Rostov-on-Don, 1989. (Russian)
  • 6. -, Justification of the averaging method for the convection problem in a field of rapidly oscillating forces and for other parabolic equations, Mat. Sb. 87 (1972), 236-253. (Russian) MR 0295150 (45:4218)
  • 7. V. B. Levenshtam, The averaging method in the convection problem with high-frequency oblique vibrations, Sibirsk. Mat. Zh. 37 (1996), no. 5, 1103-1116; English transl., Sib. Math. J. 37 (1996), no. 5, 970-982. MR 1643275 (99h:35163)
  • 8. -, Asymptotic integration of the Navier-Stokes system with a rapidly oscillating mass force, Differ. Uravn. 37 (2001), no. 5, 696-705; English transl., Differ. Equ. 37 (2001), no. 5, 732-742. MR 1850733 (2002h:35238)
  • 9. -, Asymptotic integration of parabolic equations with large high-frequency summands, Sibirsk. Mat. Zh. 46 (2005), no. 4, 805-821; English transl., Sib. Math. J. 46 (2005), no. 4, 637-651. MR 2169398 (2006g:35103)
  • 10. -, Justification of the averaging method for parabolic equations containing rapidly oscillating terms with large amplitudes, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 2, 25-56; English transl., Izv. Math. 70 (2006), no. 2, 233-263. MR 2223239 (2007b:35207)
  • 11. V. I. Yudovich, Linearization method in the hydrodinamic theory of stability, Rostov. Gos. Univ., Rostov-on-Don, 1984. (Russian)
  • 12. I. B. Simonenko, Basis of the averaging method for abstract parabolic equations, Mat. Sb. 81 (1970), no. 1, 53-61. (Russian) MR 0257594 (41:2244)
  • 13. O. A. Ladyzenskaya, Mathematical problems in the dynamics of a viscous incompressible fluid, 2nd ed., Fizmatgiz, Moscow, 1970; English transl., The mathematical theory of viscous incompessible flow, Gordon and Breach Sci. Publ., New York, 1963. MR 0271559 (42:6442); MR 0155093 (27:5034b)
  • 14. M. A. Krasnosel'skiĭ, P. P. Zabreiko, E. I. Pustyl'nik, and P. E. Sobolevskiĭ, Integral operators in spaces of summable functions, Nauka, Moscow, 1966. (Russian) MR 0206751 (34:6568)
  • 15. V. B. Levenshtam, On the correlation of two classes of solutions of the Navier-Stokes equations, Vladikavkaz. Math. Zh. 12 (2010), no. 3, 56-66. (Russian) MR 2779546 (2011m:35273)

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Additional Information

V. B. Levenshtam
Affiliation: Southern Federal University, ul. Mil′chakova 8-a, Rostov-on-Don 344090, Russia
Email: vleven@math.rsu.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01331-3
Keywords: Partial differential equations, Navier--Stokes operator, rapidly oscillating summands, averaging method
Received by editor(s): August 15, 2012
Published electronically: November 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society