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Attractors for Incompressible Bipolar and Non-Newtonian Flows: Bounded Domains and Space Periodic Problems

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Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

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Abstract

From the existence and uniqueness theorems established in Chap. 4, both for the initial-boundary value problems, as well as for the space-periodic problems associated with nonlinear, incompressible, bipolar (μ 1 > 0) and non-Newtonian flow (μ 1 = 0), it follows that under appropriate sets of conditions one may show that the solution operator \(\boldsymbol{S}(t)\) yields a nonlinear semigroup; in this chapter we examine the behavior of the orbits of such semigroups as t → ∞. Our interest here is focused on the existence of maximal compact global attractors for bounded domains and space periodic problems.

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Notes

  1. 1.

    When there is no possibility of confusion we will often write \(\mu (\boldsymbol{e})\) in lieu of \(\mu (\vert \boldsymbol{e}\vert )\).

References

  1. Babin, A.V.: The attractor of a Navier–Stokes system in an unbounded channel-like domain. J. Dynam. Differ. Equat. 4, 555–584 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bae, H.: Existence, regularity, and decay rate of solutions of non-Newtonian flow. J. Math. Anal. Appl. 231, 467–491 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellout, H., Bloom, F., Nec̆as, J.: Weak and measure-valued solutions for incompressible non-Newtonian fluids. C.R. Acad. Sci. Paris 317, 795–800 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Bellout, H., Bloom, F., Nec̆as, J.: Existence, uniqueness, and stability of solutions to the initial-boundary value problem for bipolar viscous fluids. Differ. Integr. Equat. 8, 453–464 (1995)

    Google Scholar 

  5. Bellout, H., Bloom, F., Nec̆as, J.: Bounds for the dimensions of the attractors of nonlinear bipolar viscous fluids. Asymptotic Anal. 11, 1–37 (1995)

    Google Scholar 

  6. Bloom, F.: Lower semicontinuity of the attractors of non-Newtonian fluids. Dynam. Syst. Appl. 4, 567–580 (1995)

    MATH  Google Scholar 

  7. Bloom, F.: Attractors of non-Newtonian fluids. J. Dynam. Differ. Equat. 7, 109–140 (1995)

    Article  MATH  Google Scholar 

  8. Bloom, F.: Linearized stability of the viscous incompressible bipolar equations. Nonlinear Anal. TMA 27, 1013–1030 (1996)

    Article  MATH  Google Scholar 

  9. Babin, A.V., Vishik, M.I.: Attractors of Partial Differential Evolution Equations and Estimates of Their Dimension. Russ. Math. Surv. 38(4), 151–213 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Babin, A.V., Vishik, M.I.: Attractors for the Navier–Stokes system and for parabolic equations and estimates of their dimension. J. Sov. Math. 28, 619–627 (1983)

    Article  Google Scholar 

  11. Constantin, P., Foias, C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988)

    MATH  Google Scholar 

  12. Constantin, P., Foias, C., Temam, R.: Attractors Representing Turbulent Flows. American Mathematical Society, Providence (1985)

    Google Scholar 

  13. Constantin, P., Foias, C., Temam, R.: On the dimension of the attractors in two-dimensional turbulence. Physica D 30, 284–296 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of mathematical Physics. Amer. Math. Soc. Colloq. Pub. AMS, Providence (2002)

    MATH  Google Scholar 

  15. Chepyzhov, V.V., Vishik, M.I.: Non-autonomous Navier–Stokes system with a simple global attractor and some averaging problems. E.I.J. ESAIM 8, 467–487 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chepyzhov, V.V., Vishik, M.I., Wendland, W.L.: On non-autonomous Sine-Gordon type equations with a simple global attractor and some averaging. Discrete Cont. Dynam. Syst. 12, 27–38 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Du, Q., Gunzburger, M.D.: Analysis of a Ladyzhenskaya model for incompressible viscous flow. J. Math. Anal. Appl. 155, 21–45 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Evans, L.C.: Partial Differential Equations. AMS, Providence (1998)

    MATH  Google Scholar 

  19. Efendiev, M., Zelik, S.V.: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Comm. Pure Appl. Math. LIV, 625–688 (2001)

    Google Scholar 

  20. Foias, C., Prodi, G.: Sur le comportement global des solutions non stationnaires des équations de Navier–Stokes en dimension 2. Rend. Sem. Math. Univ. Padova 39, 1–34 (1967)

    MathSciNet  Google Scholar 

  21. Foias, C., Temam, R.: The connection between the Navier–Stokes equations, dynamical systems, and turbulence theory. In: Directions in Partial Differential Equations, pp. 55–73. Academic, New York (1987)

    Google Scholar 

  22. Ghidaglia, J.M., Temam, R.: Attractors for damped nonlinear hyperbolic equations. J. Math Pures et Appl. 66, 273–319 (1987)

    MathSciNet  MATH  Google Scholar 

  23. Guillopé, C.: Comportement à l’infini des solutions des équations de Navier–Stokes et propriétée des ensembles functionnels invariants ou attracteurs. Ann. Inst. Fourier (Grenoble) 32, 1–37 (1982)

    Google Scholar 

  24. Guo, B.L., Zhu, P.C.: Partial regularity of suitable weak solutions to the system of incompressible non-Newtonian fluids. J. Differ. Equat. 178, 281–297 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (1988)

    MATH  Google Scholar 

  26. Hao, W.: Long-time behavior of the nonlinear, incompressible, bipolar equations, Ph.D. Thesis, Northern Illinois University, August 1994

    Google Scholar 

  27. Joseph, D.D.: Stability of Fluid Motions I, II. Springer, New York (1996)

    Google Scholar 

  28. Ladyzhenskaya, O.: The mathematical theory of viscous incompressible flow. Gordon and Breach, New York (1965)

    Google Scholar 

  29. Ladyzhenskaya, O.A.: A dynamical system generated by the Navier–Stokes equations. J. Soviet Math. 3, 458–479 (1975)

    Article  MATH  Google Scholar 

  30. Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  31. Lu, S., Wu, H., Zhong, C.: Attractors for nonautonomous 2D Navier–Stokes equations with normal external forces. Discrete Cont. Dynam. Syst. 13, 701–719 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Métivier, J.: Valeurs propres d’expérateurs définis sur la restriction de systémes variationnels á des sous-espaces. J. Math. Pures Appl. 57, 133–156 (1928)

    Google Scholar 

  33. Malek, J., Nec̆as, J.: A finite dimensional attractor for three-dimensional flow of incompressible fluids. J. Differ. Equat. 127, 498–518 (1996)

    Google Scholar 

  34. Málek, J., Nec̆as, J., Rüz̆ic̈ka, M.: On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains. Adv. Differ. Equat. 6, 257–302 (2001)

    Google Scholar 

  35. Nec̆as, J.: Sur les normes équivalentes dans W p (k)(Ω) et sur la coercivité des formes formellement positives. Équations aux Derivées Partielles. Presse Univ. Montréal, Montréal (1965)

    Google Scholar 

  36. Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  37. Ruelle, D.: The turbulent fluid as a dynamical system. In: Sirovich, L. (ed.) New Perspectives in Turbulence. Springer, New York (1991)

    Google Scholar 

  38. Sell, G.R., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)

    Book  MATH  Google Scholar 

  39. Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Northern Illinois University, West Lincoln Hwy. 1425, DeKalb, Illinois, USA

    Hamid Bellout & Frederick Bloom

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  1. Hamid Bellout
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  2. Frederick Bloom
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Bellout, H., Bloom, F. (2014). Attractors for Incompressible Bipolar and Non-Newtonian Flows: Bounded Domains and Space Periodic Problems. In: Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00891-2_5

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