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Journal of Mathematical Sciences

, Volume 201, Issue 6, pp 830–858 | Cite as

Topological Approximation Approach to Study of Mathematical Problems of Hydrodynamics

  • V. G. Zvyagin
Article

Abstract

We give a description of an abstract scheme of the topological approximation method and mention those fields where its application to concrete models of hydrodynamics yields results. As an illustration, we expose in detail the problem of optimal control of right-hand sides in the initialboundary value problem describing the motion of a viscoelastic incompressible fluid in the Jeffreys model with the Jaumann objective derivative.

Keywords

Weak Solution Optimal Control Problem Global Attractor Functional Space Stokes System 
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.

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References

  1. 1.
    V.T. Dmitrienko and V.G. Zvyagin, “Investigation of a regularized model of motion of a viscoelastic medium,” in: Analytical Approaches to Multidimensional Balance Laws, 119–142, Nova, New York (2006).Google Scholar
  2. 2.
    A. V. Fursikov, “Control problems and theorems concerning the unique solvability of mixed boundary-value problems for the three-dimensional Navier–Stokes and Euler equations,” Math. Sb., 115 (157), 281–306 (1981).MathSciNetGoogle Scholar
  3. 3.
    A. V. Fursikov, Optimal Control of Distributive Systems. Theory and Applications [in Russian], Nauchnaya kniga, Novosibirsk (1999).Google Scholar
  4. 4.
    C. Gori, V. Obukhovskii, P. Rubbioni, and V. Zvyagin, “Optimization of the motion of a viscoelastic fluid via multivalued topological degree method,” Dyn. Syst. Appl., 16, 89–104 (2007).MATHMathSciNetGoogle Scholar
  5. 5.
    C. Guilliope and J.-C. Saut, “Existence results for the flow of viscoelastic fluids with differential constitutive law,” Nonlinear Anal., 15, No. 9, 849–869 (1990).CrossRefMathSciNetGoogle Scholar
  6. 6.
    R. H. W. Hoppe, M. Y. Kuzmin, W.G. Litvinov, and V.G. Zvyagin, “Flow of electrorheological fluid under conditions of slip on the boundary,” Abstr. Appl. Anal. 2006, id:43560, (2006).Google Scholar
  7. 7.
    O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York–London–Paris (1969).MATHGoogle Scholar
  8. 8.
    J. Leray, “´Etude de diverses ´equations int´egrales non lin´eaires et de quelques probl`emes que pose l’hydrodynamique,” J. Math. Pures Appl., 12, 1–82 (1933).MathSciNetGoogle Scholar
  9. 9.
    S. M. Nikolskii, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).Google Scholar
  10. 10.
    V. Obukhovskii, P. Zecca, and V. Zvyagin, “Optimal feedback control in the problem of the motion of a viscoelastic fluid,” Topol. Methods Nonlinear Anal., 23, 323–337 (2004).MathSciNetGoogle Scholar
  11. 11.
    J. Simon, “Compact sets in Lp(0, T;B),Ann. Mat. Pura Appl., 4, 65–96 (1987).Google Scholar
  12. 12.
    R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam–New York–Oxford (1981).MATHGoogle Scholar
  13. 13.
    D.A. Vorotnikov, “On the existence of weak stationary solutions of a boundary-value problem in the Jeffreys model of the motion of a viscoelastic medium,” Izv. VUZ. Ser. Mat., 9, 17–21 (2004).Google Scholar
  14. 14.
    D. A. Vorotnikov, “On repeated concentration and periodic regimes with anomalous diffusion in polymers,” Math. Sb., 201, No. 1, 59–80 (2010).CrossRefMathSciNetGoogle Scholar
  15. 15.
    D.A. Vorotnikov and V.G. Zvyagin, “On the existence of weak solutions for the initial-boundary value problem in the Jeffreys model of motion of a viscoelastic medium,” Abstr. Appl. Anal., 2004, 815–829 (2004).CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    D.A. Vorotnikov and V.G. Zvyagin, “Trajectory and global attractors of the boundary-value problem for autonomous motion equations of viscoelastic medium,” Usp. Mat. Nauk, 2, 161–162 (2006).CrossRefMathSciNetGoogle Scholar
  17. 17.
    D. A. Vorotnikov and V.G. Zvyagin, “Uniform attractors for nonautomous motion equations of viscoelastic medium,” J. Math. Anal. Appl., 325, 438–458 (2007).CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    D.A. Vorotnikov and V.G. Zvyagin, “Trajectory and global attractors of the boundary-value problem for autonomous motion equations of viscoelastic medium,” J. Math. Fluid Mech., 10, 19–44 (2008).CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    D. A. Vorotnikov and V.G. Zvyagin, “A review of results and open problems in mathematical models of motion of Jeffreys type viscoelastic media,” Vestn. VGU, Ser. Fiz. Mat., 2, 30–50 (2009).Google Scholar
  20. 20.
    A. V. Zvyagin, “On well-posedness of nonlinear equations,” Spectral and Evolution Problems, Simferopol, 20, 136–140 (2006).Google Scholar
  21. 21.
    A. V. Zvyagin, “A study of solvability of a stationary model of motion of weak water liquors of polymers,” Vestn. Voronezh. Gos. Univ., Ser. Mat., 1, 103–118 (2011).MathSciNetGoogle Scholar
  22. 22.
    A. V. Zvyagin, “Solvability of a stationary model of motion of weak water liquors of polymers,” Izv. VUZ. Ser. Mat., 2, 103–105 (2011).MathSciNetGoogle Scholar
  23. 23.
    V. G. Zvyagin and V.T. Dmitrienko, Topological Approximation Approach to the Study of Hydrodynamical Problems. The Navier–Stokes System [in Russian], URSS Editorial, Moscow (2004).Google Scholar
  24. 24.
    V. G. Zvyagin and S.K. Kondratiev, Attractors for Equations of Models of Motion for Viscoelastic Media [in Russian], VGU, Voronezh (2010).Google Scholar
  25. 25.
    V. G. Zvyagin and S.K. Kondratiev, “Attractors of weak solutions to a regularized system of motion equations for fluids with memory,” Izv. VUZ. Ser. Mat., 8, 86–89 (2011).Google Scholar
  26. 26.
    V. G. Zvyagin and M.Yu. Kuzmin, “On an optimal control problem in the Voigt model of the motion of a viscoelastic fluid,” J. Math. Sci. (N.Y.), 149, No. 5, 1618–1627 (2008).CrossRefMathSciNetGoogle Scholar
  27. 27.
    V. G. Zvyagin, M.Yu. Kuzmin, and S.V. Kornev, “On an optimal control problem in the Voigt model of the motion of a viscoelastic fluid,” Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., 2, 180–197 (2011).Google Scholar
  28. 28.
    V. G. Zvyagin and A. V. Kuznetsov, “The density of the set of right-hand sides of the initialboundary value problem for the Jeffreys model of a viscoelastic fluid,” Usp. Mat. Nauk, 6, 165–166 (2008).CrossRefMathSciNetGoogle Scholar
  29. 29.
    V. G. Zvyagin and A. V. Kuznetsov, “Optimal control in a model of the motion of a viscoelastic medium with objective derivative,” Izv. VUZ. Ser. Mat., 5, 55–61 (2009).MathSciNetGoogle Scholar
  30. 30.
    V. G. Zvyagin and V. P. Orlov, “On weak solutions of the equations of motion of a viscoelastic medium with variable boundary,” Bound. Value Probl., 3, 215–245 (2005).MathSciNetGoogle Scholar
  31. 31.
    V. G. Zvyagin and M. V. Turbin, “On existence and uniqueness of weak solutions to initialboundary value problems for the Voigt model of fluid motion in domains with time-dependent boundaries,” Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat., 2, 180–197 (2007).Google Scholar
  32. 32.
    V.G. Zvyagin and M.V. Turbin, “The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids,” J. Math. Sci. (N.Y.), 168, No. 2, 157–308 (2010).CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    V. Zvyagin and M. Turbin, “Optimal feedback control in the mathematical model of low concentrated aqueous polymer solutions,” J. Optim. Theory Appl., 148, No. 1, 146–163 (2011).CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    V. G. Zvyagin and D. A. Vorotnikov, “Approximating-topological methods in some problems of hydrodynamics,” J. Fixed Point Theory Appl., 3, No. 1, 23–49 (2008).CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    V. Zvyagin and D. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodinamics, de Gruyter Series in Nonlinear Analysis and Applications, 12, Walter de Gruyter, Berlin–New York (2008).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Voronezh State UniversityVoronezhRussia

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