Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 721673, 9 pages
http://dx.doi.org/10.1155/2015/721673
Research Article

Approximate Kelvin-Voigt Fluid Driven by an External Force Depending on Velocity with Distributed Delay

1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2School of Mathematics and Statistics, Xuchang University, Xuchang, Henan 461000, China
3School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei 430073, China

Received 23 March 2015; Accepted 2 May 2015

Academic Editor: Luca Gori

Copyright © 2015 Yantao Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. P. Oskolkov, “The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers,” Zapiski Nauchnykh Seminarov Leningrad Otdel Mathematics Institute Stekov (LOMI), vol. 38, pp. 98–136, 1973. View at Google Scholar
  2. Y. Cao, E. M. Lunasin, and E. S. Titi, “Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models,” Communications in Mathematical Sciences, vol. 4, no. 4, pp. 823–848, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. V. K. Kalantarov and E. S. Titi, “Global attractors and determining modes for the 3D Navier-Stokes-Voight equations,” Chinese Annals of Mathematics, Series B, vol. 30, no. 6, pp. 697–714, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. G.-C. Yue and C.-K. Zhong, “Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations,” Discrete and Continuous Dynamical Systems Series B, vol. 16, no. 3, pp. 985–1002, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. García-Luengo, P. Marín-Rubio, and J. Real, “Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations,” Nonlinearity, vol. 25, no. 4, pp. 905–930, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977. View at MathSciNet
  7. Y. Tang and M. Wang, “A remark on exponential stability of time-delayed Burgers equation,” Discrete and Continuous Dynamical Systems. Series B, vol. 12, no. 1, pp. 219–225, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Y. Tang and L. Zhou, “Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1290–1307, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. J. H. Wu, Theory and Applications of Partial Differential Equations, Springer, New York, NY, USA, 1996.
  10. L. Zhou, Y. B. Tang, and S. Hussein, “Stability and Hopf bifurcation for a delay competition diffusion system,” Chaos, Solitons & Fractals, vol. 14, no. 8, pp. 1201–1225, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. T. Caraballo and J. Real, “Navier-Stokes equations with delays,” The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences, vol. 457, no. 2014, pp. 2441–2453, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. T. Caraballo and J. Real, “Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays,” The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences, vol. 459, no. 2040, pp. 3181–3194, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes models with delays,” Journal of Differential Equations, vol. 205, no. 2, pp. 271–297, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. Z. Hu and Y. Wang, “Pullback attractors for a nonautonomous nonclassical diffusion equation with variable delay,” Journal of Mathematical Physics, vol. 53, no. 7, Article ID 072702, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. H. Y. Li and Y. M. Qin, “Pullback attractors for three-dimensional navier-stokes-voigt equations with delays,” Boundary Value Problems, vol. 2013, article 191, 2013. View at Publisher · View at Google Scholar · View at Scopus
  16. P. Marín-Rubio and J. Real, “Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 10, pp. 2784–2799, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. P. Marín-Rubio, J. Real, and J. Valero, “Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 2012–2030, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. P. Marín-Rubio and J. Real, “Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,” Discrete and Continuous Dynamical Systems. Series A, vol. 26, no. 3, pp. 989–1006, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A. K. Khanmamedov, “Global attractors for wave equations with nonlinear interior damping and critical exponents,” Journal of Differential Equations, vol. 230, no. 2, pp. 702–719, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. L. Yang and C.-K. Zhong, “Global attractor for plate equation with nonlinear damping,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3802–3810, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus