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Exponential Attractors in Contact Problems

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Navier–Stokes Equations

Part of the book series: Advances in Mechanics and Mathematics ((AMMA))

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Abstract

In this chapter we consider two examples of contact problems. First, we study the problem of time asymptotics for a class of two-dimensional turbulent boundary driven flows subject to the Tresca friction law which naturally appears in lubrication theory. Then we analyze the problem with the generalized Tresca law, where the friction coefficient can depend on the tangential slip rate.

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References

  1. J. Bolte, A. Daniliidis, O. Ley, L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362, 3319–3363 (2010)

    Article  MATH  Google Scholar 

  2. M. Boukrouche, G. Łukaszewicz, J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows. Int. J. Eng. Sci. 44, 830–844 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics (American Mathematical Society, Providence, RI, 2002)

    MATH  Google Scholar 

  4. J.W. Cholewa, T. Dłotko, Global Attractors in Abstract Parabolic Problems. London Mathematical Society Lecture Note Series, vol. 278 (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  5. G. Duvaut, J.L. Lions, Les inéquations en mécanique et en physique (Dunod, Paris, 1972)

    MATH  Google Scholar 

  6. L.C. Evans, Partial Differential Equations, 2nd edn. (American Mathematical Society, Providence, RI, 2010)

    MATH  Google Scholar 

  7. E. Feireisl, D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics. American Institute of Mathematical Sciences (American Mathematical Society, Providence, RI, 2010)

    Google Scholar 

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

    MATH  Google Scholar 

  9. J. Haslinger, I. Hlaváček, J. Nečas, Numerical methods for unilateral problems in solid mechanics, in Handbook of Numerical Analysis, ed. by P.G. Ciarlet, J.L. Lions, vol. IV (1996), Elsevier, pp. 313–485

    Google Scholar 

  10. I.R. Ionescu, Q.L. Nguyen, Dynamic contact problems with slip dependent friction in viscoelasticity. Int. J. Appl. Math. Comput. 12, 71–80 (2002)

    MathSciNet  MATH  Google Scholar 

  11. P. Kalita, G. Łukaszewicz, On large time asymptotics for two classes of contact problems, in Advances in Variational and Hemivariational Inequalities, ed. by W. Han, S. Migórski, M. Sofonea. Theory Numerical Analysis, and Applications (Springer, Berlin, 2015), pp. 299–332

    Google Scholar 

  12. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd edn, (Gordon and Breach, New York, 1969)

    MATH  Google Scholar 

  13. O.A. Ladyzhenskaya, G. Seregin, On semigroups generated by initial-boundary problems describing two dimensional visco-plastic flows, in Nonlinear Evolution Equations. American Mathematical Society Translation Series 2, vol. 164 (American Mathematical Society, Providence, RI, 1995), pp. 99–123

    Google Scholar 

  14. G. Łukaszewicz On the existence of an exponential attractor for a planar shear flow with the Tresca friction condition. Nonlinear Anal. Real 14, 1585–1600 (2013)

    Google Scholar 

  15. J. Málek, J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids. J. Differ Equ. 127, 498–518 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Málek, D. Pražák, Large time behavior via the method of l-trajectories. J. Differ. Equ. 181, 243–279 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Migórski, A. Ochal, Boundary hemivariational inequality of parabolic type. Nonlinear Anal. Theory 57, 579–596 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Migórski, A. Ochal, Navier-Stokes models modeled by evolution hemivariational inequalities. Discrete Cont. Dyn. Syst. Supplement, 731–740 (2007)

    Google Scholar 

  19. A. Miranville, S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Evolutionary Equations. Handbook of Differential Equations, vol. IV (Elsevier, North-Holland, Amsterdam, 2008), pp. 103–200

    Google Scholar 

  20. P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications (Birkhäuser, Basel, 1985)

    Book  MATH  Google Scholar 

  21. G.J. Perrin, J.R. Rice, G. Zheng, Self-healing slip pulse on a frictional surface. J. Mech. Phys. Solids 43, 1461–1495 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  23. J.C. Robinson, Dimensions, Embeddings, and Attractors (Cambridge University Press, Cambridge, 2011)

    MATH  Google Scholar 

  24. C.H. Scholz, The Mechanics of Earthquakes and Faulting (Cambridge University Press, Cambridge, 1990)

    Google Scholar 

  25. A. Segatti, S. Zelik, Finite-dimensional global and exponential attractors for the reaction-diffusion problem with an obstacle potential. Nonlinearity 22, 2733–2760 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Seregin, On a dynamical system generated by the two-dimensional equations of the motion of a Bingham fluid. J. Math. Sci. 70(3), 1806–1816 (1994). (Translated from Zapiski Nauchnykh Seminarov LOMI, SSSR, vol. 188, 128–142, 1991)

    Google Scholar 

  27. M. Shillor, M. Sofonea, J.J. Telega, Models and Analysis of Quasistatic Contact: Variational Methods (Springer, Berlin, Heidelberg, 2004)

    Book  MATH  Google Scholar 

  28. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. (Springer, New York, 1997)

    Book  MATH  Google Scholar 

  29. C.K. Zhong, M.H. Yang, C.Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. J. Differ. Equ. 223, 367–399 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Warszawa, Poland

    Grzegorz Łukaszewicz

  2. Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, Krakow, Poland

    Piotr Kalita

Authors
  1. Grzegorz Łukaszewicz
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  2. Piotr Kalita
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Łukaszewicz, G., Kalita, P. (2016). Exponential Attractors in Contact Problems. In: Navier–Stokes Equations. Advances in Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-27760-8_10

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