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Approximating topological approach to the existence of attractors in fluid mechanics

  • Victor G. Zvyagin
  • Stanislav K. Kondratyev
Article

Abstract

The aim of this paper is to demonstrate how the approximating topological method can be effectively combined with the theory of attractors of trajectory spaces in problems of fluid mechanics. First we give an exposition of the theory. Then we consider the model of motion of weak aqueous polymer solutions and prove that it has the minimal trajectory attractor and the global one. Finally we prove that the attractors of approximating problem converge to the attractors of the unperturbed one.

Mathematics Subject Classification

Primary 35B41 Secondary 76M99 

Keywords

Non-Newtonian fluids approximating topological method trajectory space trajectory attractor global attractor convergence of attractors 

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References

  1. 1.
    V. V. Amfilokhiev et al., Flows of polymer solutions with convective accelerations. In: Proceedings of Leningrad Shipbuilding Institute, vol. 96, 1975, 3–9 (in Russian).Google Scholar
  2. 2.
    A. V. Babin and M. I. Vishik, Attractors of Evolution Equations. Stud. Math. Appl. 25, North–Holland, Amsterdam, 1992.Google Scholar
  3. 3.
    Babin A.V., Vishik M.I.: Attractors of partial differential evolution equations and estimates of their dimension. Russ. Math. Surv. 38, 151–213 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babin A.V., Vishik M.I.: Regular attractors of semigroups and evolution equations. J. Math. Pures Appl. 62(9), 441–491 (1983)MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. Appl. Math. Sci. 182. Springer, New York, 2012.Google Scholar
  6. 6.
    V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics. Amer. Math. Soc. Colloq. Publ. 49, Amer. Math. Soc., Providence, RI, 2002.Google Scholar
  7. 7.
    Chepyzhov V.V., Vishik M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76(9), 913–964 (1997)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chepyzhov V.V., Vishik M.I.: Trajectory attractors for evolution equations. C. R. Math. Acad. Sci. Paris 321, 1309–1314 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Foias C., Manley O., Rosa R., Temam R.: Navier-Stokes Equations and Turbulence. Encyclopedia of Mathematics and Its Applications 83. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  10. 10.
    A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications. Trans. Math. Monographs 187, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  11. 11.
    O. A. Ladyzhenskaya, On a dynamical system generated by Navier–Stokes equations. In: Boundary-Value Problems of Mathematical Physics and Related Problems of Function Theory, Part 6, Zap. Nauchn. Sem. LOMI 27, “Nauka”, Leningrad. Otdel., Leningrad, 1972, 91–115 (in Russian).Google Scholar
  12. 12.
    Ladyzhenskaya O.A.: On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations. Russ. Math. Surv. 42, 27–73 (1987)CrossRefzbMATHGoogle Scholar
  13. 13.
    V. A. Pavlovsky, To a problem on theoretical exposition of weak aqueous solutions of polymers. Dokl. Akad. Nauk SSSR 200 (1971), 809–812 (in Russian).Google Scholar
  14. 14.
    Sell G.: Global attractors for the three-dimensional Navier-Stokes equations. J. Dynam. Differrential Equations 8, 1–33 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    G. Sell and Y. You, Dynamics of Evolutionary Equations. Springer, New York, 1998.Google Scholar
  16. 16.
    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York, 1988.Google Scholar
  17. 17.
    R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. AMS Chelsea, Providence, 2000.Google Scholar
  18. 18.
    Vishik M.I., Chepyzhov V.V.: Trajectory and Global Attractors of Three-Dimensional Navier–Stokes Systems. Math. Notes 71, 177–193 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vishik M.I., Chepyzhov V.V.: Trajectory attractors of equations of mathematical physics. Russ. Math. Surv. 66, 637–731 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vorotnikov D.A., Zvyagin V.G.: On the trajectory and global attractors for the equations of motion of a visco-elastic medium. Russ. Math. Surv. 61, 368–370 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vorotnikov D.A., Zvyagin V.G.: Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium. J. Math. Fluid Mech. 10, 19–44 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vorotnikov D.A., Zvyagin V.G.: Uniform attractors for non-automous motion equations of viscoelastic medium. J. Math. Anal. Appl. 325, 438–458 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    V. G. Zvyagin and V. T. Dmitrienko, Approximating-topological approach to investigation of problems of hydrodynamics. Navier–Stokes system. Editorial URSS, Moscow, 2004 (in Russian).Google Scholar
  24. 24.
    V. G. Zvyagin and S. K. Kondratyev, Attractors for Equations of Motions of Viscoelastic Media. Voronezh, Voronezh State University Publishing House, 2010 (in Russian).Google Scholar
  25. 25.
    Zvyagin V.G., Kondrat’ev S.K.: Attractors of weak solutions to a regularized system of motion equations for fluids with memory. Russian Mathematics 55, 75–77 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    V. G. Zvyagin and S. K. Kondratyev, Attractors of weak solutions of a regularized system of motions for fluids with memory. Sb. Math. 11 (2012), 83–104 (in Russian).Google Scholar
  27. 27.
    V. G. Zvyagin and M. V. Turbin, Mathematical Problems of Hydrodynamics of Viscoelastic Media. KRASAND, Moscow, 2012 (in Russian).Google Scholar
  28. 28.
    Zvyagin V.G., Turbin M.V.: The study of initial-boundary value problems for mathematical models of the motion of Kelvin–Voigt fluids. J. Math. Sci. 168, 157–308 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zvyagin V.G., Vorotnikov D.A.: Approximating-topological methods in some problems of hydrodynamics. J. Fixed Point Theory Appl. 3, 23–49 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics. Walter de Gruyter, Berlin, 2008.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Research Institute of MathematicsVoronezh State UniversityVoronezhRussia

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