Skip to main content
SpringerLink
Log in

On the Spectral Dynamics of the Deformation Tensor and New A Priori Estimates for the 3D Euler Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3D incompressible Euler equations. Using the evolution equation of the L 2 norm of spectra, we deduce new a priori estimates of the L 2 norm of vorticity. As an immediate corollary of the estimate we obtain a new sufficient condition of L 2 norm control of vorticity. We also obtain decay in time estimates of the ratios of the eigenvalues. In the remarks we discuss what these estimates suggest in the study of searching initial data leading to possible finite time singularities. We find that the dynamical behaviors of L 2 norm of vorticity are controlled completely by the second largest eigenvalue of the deformation tensor.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babin, A., Mahalov, A., Nicolaenko, B.: 3D Navier-Stokes and Euler equations with Initial Data Characterized by Uniformly Large Vorticity. Indiana Univ. Math. J. 50(1), 1–35 (2001)

    MathSciNet  Google Scholar 

  2. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Chae, D.: On the Well-Posedness of the Euler equations in the Besov and the Triebel-Lizorkin spaces. In: Proceedings of the Conference, ``Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics'', June 27–29, 2001

  4. Chae, D.: On the Well-Posedness of the Euler equations in the Triebel-Lizorkin spaces. Commun. Pure Appl. Math. 55, 654–678 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chae, D.: On the Euler equations in the critical Triebel-Lizorkin spaces. Arch. Rat. Mech. Anal. 170(3), 185–210 (2003)

    Article  MathSciNet  Google Scholar 

  6. Chae, D.: Local existence and blow-up criterion for the Euler equations in the Besov spaces. Asymp. Anal. 38(3–4), 339–358 (2004)

    Google Scholar 

  7. Chae, D.: Remarks on the blow-up criterion of the 3D Euler equations. Nonlinearity, 18, 1021–1029 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Chemin, J.-Y.: Régularité des trajectoires des particules d'un fluide incompressible remplissant l'espace. J. Math. Pures Appl. 71(5), 407–417 (1992)

    MathSciNet  Google Scholar 

  9. Chemin, J.-Y.: Perfect incompressible fluids. Oxford: Clarendon Press, 1998

  10. Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36, 73–98 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Constantin, P., Foias, C.: Navier-Stokes equations. Chicago: Univ. Chicago Press, 1988

  12. Constantin, P., Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42, 775–789 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Constantin, P., Fefferman, C., Majda, A.: Geometric constraints on potential singularity solutions for the 3D Euler equations. Commun. Partial Diff. Eqs. 21, 559–571 (1996)

    MATH  MathSciNet  Google Scholar 

  14. Cordoba, D., Fefferman, C.: On the collapse of tubes carried by 3D incompressible flows. Commun. Math. Phys. 222(2), 293–298 (2001)

    Article  MathSciNet  Google Scholar 

  15. Deng, J., Hou, T.Y., Yu, X.: Geometric and nonblowup of 3D incompressible Euler flow. Commun. P.D.E 30, 225–243 (2005)

    Article  MATH  Google Scholar 

  16. He, X.: A sufficient condition for a finite-time L 2 singularity of the 3d Euler Equations. To appear in Math. Proc. Camb. Phil. Soc. (2005)

  17. Kato, T.: Nonstationary flows of viscous and ideal fluids in ℝ3. J. Funct. Anal. 9, 296–305 (1972)

    Article  MATH  Google Scholar 

  18. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)

    MATH  Google Scholar 

  19. Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations. Commun. Math. Phys. 214, 191–200 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Liu, H., Tadmor, E.: Spectral dyanamics of the velocity gradient field in restricted flows. Commun. Math. Phys. 228, 435–466 (2002)

    Article  ADS  MATH  Google Scholar 

  21. Majda, A., Bertozzi, A.: Vorticity and incompressible flow. Cambridge: Cambridge Univ. Press, 2002

  22. Temam, R.: On the Euler equations of incompressible flows. J. Funct. Anal. 20, 32–43 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  23. Temam, R.: Navier-Stokes equations. 2nd ed., Amsterdam: North-Holland, 1986

  24. Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rat. Mech. Anal, 145, 197–214 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  25. Vishik, M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Scient. Éc. Norm. Sup., 4e série, t. 32, 769–812 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, Korea

    Dongho Chae

Authors
  1. Dongho Chae
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongho Chae.

Additional information

Communicated by P. Constantin

Part of this work was done while the author was visiting CSCAMM, University of Maryland, USA. The author would like to thank to Professor E. Tadmor for his hospitality

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chae, D. On the Spectral Dynamics of the Deformation Tensor and New A Priori Estimates for the 3D Euler Equations. Commun. Math. Phys. 263, 789–801 (2006). https://doi.org/10.1007/s00220-005-1465-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1465-8

Keywords

Search

Navigation