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ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO 3D CONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS WITH FINITE DELAYS

  • Le, Thi Thuy (Department of Mathematics Electric Power University)
  • Received : 2020.07.09
  • Accepted : 2020.11.23
  • Published : 2021.07.31

Abstract

In this paper we prove the existence of global weak solutions, the exponential stability of a stationary solution and the existence of a global attractor for the three-dimensional convective Brinkman-Forchheimer equations with finite delay and fast growing nonlinearity in bounded domains with homogeneous Dirichlet boundary conditions.

Keywords

References

  1. C. T. Anh and D. T. P. Thanh, Existence and long-time behavior of solutions to Navier-Stokes-Voigt equations with infinite delay, Bull. Korean Math. Soc. 55 (2018), no. 2, 379-403. https://doi.org/10.4134/BKMS.b170044
  2. J. W. Barrett and W. B. Liu, Finite element approximation of the parabolic p-Laplacian, SIAM J. Numer. Anal. 31 (1994), no. 2, 413-428. https://doi.org/10.1137/0731022
  3. L. E. J. Brouwer, Beweis der Invarianz der Dimensionenzahl, Math. Ann. 70 (1911), no. 2, 161-165. https://doi.org/10.1007/BF01461154
  4. T. Caraballo and X. Han, A survey on Navier-Stokes models with delays: existence, uniqueness and asymptotic behavior of solutions, Discrete Contin. Dyn. Syst. Ser. S 8 (2015), no. 6, 1079-1101. https://doi.org/10.3934/dcdss.2015.8.1079
  5. T. Caraballo, A. M. Marquez-Duran, and J. Real, Asymptotic behaviour of the three-dimensional α-Navier-Stokes model with delays, J. Math. Anal. Appl. 340 (2008), no. 1, 410-423. https://doi.org/10.1016/j.jmaa.2007.08.011
  6. T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2014, 2441-2453. https://doi.org/10.1098/rspa.2001.0807
  7. T. Caraballo and J. Real, Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2040, 3181-3194. https://doi.org/10.1098/rspa.2003.1166
  8. A. O. C, elebi, V. K. Kalantarov, and D. Ugurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Appl. Math. Lett. 19 (2006), no. 8, 801-807. https://doi.org/10.1016/j.aml.2005.11.002
  9. H. Chen, Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays, Proc. Indian Acad. Sci. Math. Sci. 122 (2012), no. 2, 283-295. https://doi.org/10.1007/s12044-012-0071-x
  10. P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
  11. F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2040, 3195-3202. https://doi.org/10.1098/rspa.2003.1169
  12. J. Garcia-Luengo, P. Marin-Rubio, and J. Real, Regularity of pullback attractors and attraction in H1 in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay, Discrete Contin. Dyn. Syst. 34 (2014), no. 1, 181-201. https://doi.org/10.3934/dcds.2014.34.181
  13. K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations 263 (2017), no. 11, 7141-7161. https://doi.org/10.1016/j.jde.2017.08.001
  14. V. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal. 11 (2012), no. 5, 2037-2054. https://doi.org/10.3934/cpaa.2012.11.2037
  15. J.-L. Lions, Quelques m'ethodes de r'esolution des probl'emes aux limites non lineaires, Dunod, 1969.
  16. P. Marin-Rubio, A. M. Marquez-Duran, and J. Real, Three dimensional system of globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 2, 655-673. https://doi.org/10.3934/dcdsb.2010.14.655
  17. P. Marin-Rubio, A. M. Marquez-Duran, and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. 31 (2011), no. 3, 779-796. https://doi.org/10.3934/dcds.2011.31.779
  18. P. Marin-Rubio, J. Real, and J. Valero, Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case, Nonlinear Anal. 74 (2011), no. 5, 2012-2030. https://doi.org/10.1016/j.na.2010.11.008
  19. M. T. Mohan, Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity, arXiv: 2010.12814, 2020.
  20. D. Ugurlu, On the existence of a global attractor for the Brinkman-Forchheimer equations, Nonlinear Anal. 68 (2008), no. 7, 1986-1992. https://doi.org/10.1016/j.na.2007.01.025
  21. S. Whitaker, The Forchheimer equation: a theoretical development, Transport in Porous Media 25 (1996), 272.
  22. C. Zhao, L. Kong, G. Liu, and M. Zhao, The trajectory attractor and its limiting behavior for the convective Brinkman-Forchheimer equations, Topol. Methods Nonlinear Anal. 44 (2014), no. 2, 413-433. https://doi.org/10.12775/TMNA.2014.054