Archive for Rational Mechanics and Analysis

, Volume 199, Issue 1, pp 117–144 | Cite as

On Singularity Formation of a Nonlinear Nonlocal System

  • Thomas Y. Hou
  • Congming Li
  • Zuoqiang Shi
  • Shu Wang
  • Xinwei Yu


We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94(1), 61–66 (1984)MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. 35, 771–831 (1982)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Calderon, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78(2), 289–309 (1956)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chae, D., Cordoba, A., Cordoba, D., Fontelos, M.A.: Finite time singularities in a 1D model of the quasi-geostrophic equation. Adv. Math. 194, 203–223 (2005)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Constantin, P.: Note on loss of regularity for solutions of the 3D incompressible Euler and related equations. Commun. Math. Phys. 104, 311–326 (1986)MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Constantin, P., Lax, P.D., Majda, A.J.: A simple one-dimensional model for the three-dimensional vorticity equation. Comm. Pure Appl. Math. 38(6), 715–724 (1985)MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Cordoba, A., Cordoba, D., Fontelos, M.A.: Formation of singularities for a transport equation with nonlocal velocity. Adv. Math. 162(3), 1375–1387 (2005)MathSciNetGoogle Scholar
  8. 8.
    De Gregorio, S.: On a one-dimensional model for the 3-dimensional vorticity equation. J. Stat. Phys. 59, 1251–1263 (1990)MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    De Gregorio, S.: A partial differential equation arising in a 1D model for the 3D vorticity equation. Math. Method Appl. Sci. 19(15), 1233–1255 (1996)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Duoandikoetxea, J.: The Hilbert transform and Hermite functions: a real variable proof of the L 2-isometry. J. Math. Anal. Appl. 347, 592–596 (2008)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
  12. 12.
    Hou, T.Y.: Blow-up or no blow-up? A unified computational and analytic approach to study 3-D incompressible Euler and Navier–Stokes equations. Acta Numer. 18, 277–346 (2009)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hou, T.Y., Li, R.: Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations. J. Nonlinear Sci. 16(6), 639–664 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Hou, T.Y., Lei, Z.: On partial regularity of a 3D model of Navier–Stokes equations. Commun. Math Phys. 287(2), 589–612 (2009)MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Hou, T.Y., Li, C.: Dynamic stability of the 3D axi-symmetric Navier–Stokes equations with swirl. Comm. Pure Appl. Math. 61(5), 661–697 (2008)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hou, T.Y., Lei, Z.: On the stabilizing effect of convection in 3D incompressible flows. Comm. Pure Appl. Math. 62(4), 501–564 (2009)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hou, T.Y., Shi, Z., Wang, S.: On singularity formation of a 3D model for incompressible Navier–Stokes equations. arXiv:0912.1316v1 [math.AP] (2009)Google Scholar
  18. 18.
    Li, D., Rodrigo, J.: Blow up for the generalized surface quasi-geostrophic equation with supercritical dissipation. Commun. Math. Phys. 286(1), 111–124 (2009)MATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Li, D., Rodrigo, J.: On a one-dimensional nonlocal flux with fractional dissipation. Preprint (2009)Google Scholar
  20. 20.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge, 2002Google Scholar
  21. 21.
    Okamoto, H., Ohkitani, K.: On the role of the convection term in the equations of motion of incompressible fluid. J. Phys. Soc. Jpn. 74(10), 2737–2742 (2005)MATHCrossRefADSGoogle Scholar
  22. 22.
    Shelley, M.: A study of singularity formation in vortex sheet motion by a spectrally accurate vortex method. J. Fluid Mech. 244, 493–526 (1992)MATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Temam, R.: Navier–Stokes Equations, 2nd edn. AMS Chelsea Publishing, Providence, RI, 2001Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Thomas Y. Hou
    • 1
  • Congming Li
    • 2
  • Zuoqiang Shi
    • 1
  • Shu Wang
    • 3
  • Xinwei Yu
    • 4
  1. 1.Applied and Computational MathematicsCaltech, PasadenaUSA
  2. 2.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  3. 3.College of Applied SciencesBeijing University of TechnologyBeijingChina
  4. 4.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations