Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 769724, 8 pages
http://dx.doi.org/10.1155/2013/769724
Research Article

Systems of Navier-Stokes Equations on Cantor Sets

1Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China
2Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey
3Institute of Space Sciences, Magurele, Bucharest, Romania
4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
5Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida, 431, 4200-072 Porto, Portugal

Received 31 March 2013; Revised 3 June 2013; Accepted 10 June 2013

Academic Editor: Bashir Ahmad

Copyright © 2013 Xiao-Jun Yang 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. C. Foias, O. Manley, R. Rosa, and R. Temam, Navier-Stokes Equations and Turbulence, vol. 83, Cambridge University Press, Cambridge, UK, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. 1-2, Springer, New York, NY, USA, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Y. Wang, “Exact solutions of the steady-state Navier-Stokes equations,” in Annual Review of Fluid Mechanics, vol. 23, pp. 159–177, Annual Reviews, Palo Alto, CA, USA, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. B. Dubrulle, J.-P. Laval, S. Nazarenko, and O. Zaboronski, “A model for rapid stochastic distortions of small-scale turbulence,” Journal of Fluid Mechanics, vol. 520, pp. 1–21, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Alexakis, P. D. Mininni, and A. Pouquet, “Imprint of large-scale flows on turbulence,” Physical Review Letters, vol. 95, no. 26, Article ID 264503, 2005. View at Publisher · View at Google Scholar · View at Scopus
  6. P. D. Mininni, A. Alexakis, and A. Pouquet, “Large-scale flow effects, energy transfer, and self-similarity on turbulence,” Physical Review E, vol. 74, no. 1, Article ID 016303, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. R. Benzi, L. Biferale, S. Ciliberto, M. V. Struglia, and R. Tripiccione, “Scaling property of turbulent flows,” Physical Review E, vol. 53, no. 4, pp. R3025–R3027, 1996. View at Google Scholar · View at Scopus
  8. Z.-S. She and E. Leveque, “Universal scaling laws in fully developed turbulence,” Physical Review Letters, vol. 72, no. 3, pp. 336–339, 1994. View at Publisher · View at Google Scholar · View at Scopus
  9. B. I. Shraiman and E. D. Siggia, “Scalar turbulence,” Nature, vol. 405, no. 6787, pp. 639–646, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, “Extended self-similarity in turbulent flows,” Physical Review E, vol. 48, no. 1, pp. R29–R32, 1993. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Meneveau and J. Katz, “Scale-invariance and turbulence models for large-eddy simulation,” in Annual Review of Fluid Mechanics, vol. 32, pp. 1–32, Annual Reviews, Palo Alto, CA, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. P. Constantin, C. Foias, O. P. Manley, and R. Temam, “Determining modes and fractal dimension of turbulent flows,” Journal of Fluid Mechanics, vol. 150, pp. 427–440, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. V. V. Chepyzhov and A. A. Ilyin, “On the fractal dimension of invariant sets: applications to Navier-Stokes equations,” Discrete and Continuous Dynamical Systems A, vol. 10, no. 1-2, pp. 117–135, 2004. View at Google Scholar · View at MathSciNet
  14. V. Scheffer, “Turbulence and Hausdorff dimension, in Turbulence and the Navier-Stokes equations,” in Lecture Notes in Mathematics, vol. 565, pp. 94–112, Springer, Berlin, Germany, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. V. Scheffer, “Hausdorff measure and the Navier-Stokes equations,” Communications in Mathematical Physics, vol. 55, no. 2, pp. 97–112, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Boffetta, A. Mazzino, and A. Vulpiani, “Twenty-five years of multifractals in fully developed turbulence: a tribute to Giovanni Paladin,” Journal of Physics A, vol. 41, 363001, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. Cheskidov, D. D. Holm, E. Olson, and E. S. Titi, “On a Leray-α model of turbulence,” Proceedings of The Royal Society of London A, vol. 461, no. 2055, pp. 629–649, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. A. Ilyin, E. M. Lunasin, and E. S. Titi, “A modified-Leray-α subgrid scale model of turbulence,” Nonlinearity, vol. 19, no. 4, pp. 879–897, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  19. D. D. Holm, “Kármán-Howarth theorem for the Lagrangian-averaged Navier-Stokes-alpha model of turbulence,” Journal of Fluid Mechanics, vol. 467, pp. 205–214, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.
  22. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006. View at MathSciNet
  23. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  24. M. El-Shahed and A. Salem, “On the generalized Navier-Stokes equations,” Applied Mathematics and Computation, vol. 156, no. 1, pp. 287–293, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 488–494, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. Z. Z. Ganji, D. D. Ganji, A. D. Ganji, and M. Rostamian, “Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 1, pp. 117–124, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  27. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
  28. X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,” Boundary Value Problems, no. 1, pp. 131–150, 2013. View at Google Scholar
  29. X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,” Thermal Science, vol. 17, no. 2, pp. 625–628, 2013. View at Google Scholar
  30. M.-S. Hu, R. P. Agarwal, and X.-J. Yang, “Local fractional Fourier series with application to wave equation in fractal vibrating string,” Abstract and Applied Analysis, vol. 2012, Article ID 567401, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximation solution to diffusion equation on Cantor time-space,” Proceedings of the Romanian Academy A, vol. 14, no. 2, pp. 127–133, 2013. View at Google Scholar
  32. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Limited, Hong Kong, China, 2011.
  33. X. J. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011. View at Google Scholar
  34. A. Carpinteri and A. Sapora, “Diffusion problems in fractal media defined on Cantor sets,” ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik, vol. 90, no. 3, pp. 203–210, 2010. View at Publisher · View at Google Scholar · View at Scopus
  35. K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. J. Li and M. Ostoja-Starzewski, “Micropolar continuum mechanics of fractal media,” International Journal of Engineering Science, vol. 49, no. 12, pp. 1302–1310, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  37. V. E. Tarasov, “Continuous medium model for fractal media,” Physics Letters, Section A, vol. 336, no. 2-3, pp. 167–174, 2005. View at Publisher · View at Google Scholar · View at Scopus
  38. C. S. Drapaca and S. Sivaloganathan, “A fractional model of continuum mechanics,” Journal of Elasticity, vol. 107, no. 2, pp. 105–123, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  39. V. E. Tarasov, “Fractional hydrodynamic equations for fractal media,” Annals of Physics, vol. 318, no. 2, pp. 286–307, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet