Skip to main content

Pullback Attractors for a Model of Polymer Solutions Motion with Rheological Relation Satisfying the Objectivity Principle

Abstract

On the basis of the trajectory pullback attractors theory, this paper studies the dynamics of weak solutions for a nonautonomous model of the polymer solutions motion (with the rheological relation satisfying the objectivity principle). For this model, we establish the existence of weak solutions, determine a family of trajectory spaces, introduce the concepts of trajectory and minimal pullback attractors, and prove the existence of these attractors.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    V. B. Amfilokhiev, Ya. I. Voitkunskiy, N. P. Mazaeva, and Ya. S. Khodorkovskiy, “Flows of polymer solutions under convective accelerations,” Tr. Leningr. Korablestr. Inst., 96, 3–9 (1975).

  2. 2.

    V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc., Providence (2002).

    MATH  Google Scholar 

  3. 3.

    A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Trans. Math. Monographs, Amer. Math. Soc., Providence (2000).

    MATH  Google Scholar 

  4. 4.

    V. A. Pavlovskij, “On the theoretical description of weak aqueous solutions of polymers,” Dokl. Akad. Nauk SSSR, 200, No. 4, 809–812 (1971).

    Google Scholar 

  5. 5.

    G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., Vol. 143, Springer, New York (2002).

  6. 6.

    J. Simon, “Compact sets in the space Lp(0, T;B),” Ann. Mat. Pura Appl., 146, 65–96 (1987).

  7. 7.

    R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, Amer. Math. Soc., Providence (2000).

    MATH  Google Scholar 

  8. 8.

    D. A. Vorotnikov, “Asymptotic behavior of the non-autonomous 3D Navier–Stokes problem with coercive force”, J. Differ. Equ., 251, 2209–2225 (2011).

    MathSciNet  Article  Google Scholar 

  9. 9.

    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, No. 1, 19–44 (2008).

    MathSciNet  Article  Google Scholar 

  10. 10.

    A. V. Zvyagin, “An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated aqueous polymer solutions with objective derivative,” Sib. Math. J., 54, No. 4, 640–655 (2013).

    MathSciNet  Article  Google Scholar 

  11. 11.

    A. V. Zvyagin, “Attractors for a model of polymer motion with objective derivative in the rheological relation,” Dokl. Math., 88, No. 3, 730–733 (2013).

    MathSciNet  Article  Google Scholar 

  12. 12.

    A. V. Zvyagin, “Solvability for equations of motion of weak aqueous polymer solutions with objective derivative,” Nonlinear Anal. TMA, 90, 70–85 (2013).

    MathSciNet  Article  Google Scholar 

  13. 13.

    V. Zvyagin, “Topological approximation approach to study of mathematical problems of hydrodynamics,” J. Math. Sci., 201, No. 6, 830–858 (2014).

    MathSciNet  Article  Google Scholar 

  14. 14.

    V. Zvyagin and S. Kondratyev, “Attractors of weak solutions to the regularized system of equations of motion of fluid media with memory,” Sb. Math., 203, No. 11, 1611–1630 (2012).

    MathSciNet  Article  Google Scholar 

  15. 15.

    V. G. Zvyagin and S. K. Kondratyev, “Approximating topological approach to the existence of attractors in fluid mechanics,” J. Fixed Point Theory Appl., 13, No. 2, 359–395 (2013).

    MathSciNet  Article  Google Scholar 

  16. 16.

    V. G. Zvyagin and S. K. Kondratyev, “Pullback attractors for a model of motion of weak aqueous polymer solutions,” Dokl. Math., 90, No. 3, 660–662 (2014).

    MathSciNet  Article  Google Scholar 

  17. 17.

    V. Zvyagin and S. Kondratyev, “Pullback attractors for the model of motion of dilute aqueous polymer solutions,” Izv. Math., 79, No. 4, 645–667 (2015).

    MathSciNet  Article  Google Scholar 

  18. 18.

    V. Zvyagin and S. Kondratyev, “Pullback attractors of the Jeffreys–Oldroyd equations,” J. Differ. Equ., 260, No. 6, 5026–5042 (2016).

    MathSciNet  Article  Google Scholar 

  19. 19.

    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., 168, No. 2, 157–308 (2010).

    MathSciNet  Article  Google Scholar 

  20. 20.

    V. Zvyagin and M. Turbin, Mathematical Problems of Viscoelastic Media Hydrodynamics [in Russian], URRS, Moscow (2012).

    Google Scholar 

  21. 21.

    V. G. Zvyagin and D. A. Vorotnikov, Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics, Walter de Gruyter, Berlin (2008).

    Book  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding authors

Correspondence to V. G. Zvyagin or A. V. Zvyagin.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 5, pp. 129–157, 2016.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zvyagin, V.G., Zvyagin, A.V. Pullback Attractors for a Model of Polymer Solutions Motion with Rheological Relation Satisfying the Objectivity Principle. J Math Sci 248, 600–620 (2020). https://doi.org/10.1007/s10958-020-04898-8

Download citation