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Topological Approximation Approach to Study of Mathematical Problems of Hydrodynamics

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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.

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References

  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).

  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).

    MathSciNet  Google Scholar 

  3. A. V. Fursikov, Optimal Control of Distributive Systems. Theory and Applications [in Russian], Nauchnaya kniga, Novosibirsk (1999).

    Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

  7. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York–London–Paris (1969).

    MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  9. S. M. Nikolskii, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

  11. J. Simon, “Compact sets in Lp(0, T;B),” Ann. Mat. Pura Appl., 4, 65–96 (1987).

    Google Scholar 

  12. R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam–New York–Oxford (1981).

    MATH  Google Scholar 

  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. D. A. Vorotnikov, “On repeated concentration and periodic regimes with anomalous diffusion in polymers,” Math. Sb., 201, No. 1, 59–80 (2010).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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. A. V. Zvyagin, “On well-posedness of nonlinear equations,” Spectral and Evolution Problems, Simferopol, 20, 136–140 (2006).

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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. 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. 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. 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).

    Article  MathSciNet  Google Scholar 

  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. 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).

    Article  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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. 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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

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Authors and Affiliations

  1. Voronezh State University, 1, Universitetskaya pl., Voronezh, Russia

    V. G. Zvyagin

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  1. V. G. Zvyagin
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Correspondence to V. G. Zvyagin.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 46, Proceedings of the Sixth International Conference on Differential and Functional Differential Equations and International Workshop “Spatio-Temporal Dynamical Systems” (Moscow, Russia, 14–21 August, 2011). Part 2, 2012.

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Zvyagin, V.G. Topological Approximation Approach to Study of Mathematical Problems of Hydrodynamics. J Math Sci 201, 830–858 (2014). https://doi.org/10.1007/s10958-014-2028-3

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