Advertisement

Communications in Mathematical Physics

, Volume 275, Issue 3, pp 861–872 | Cite as

The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations

  • Qionglei Chen
  • Changxing MiaoEmail author
  • Zhifei Zhang
Article

Abstract

We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely
$$\sup_{j\in\mathbb{Z}}\int_0^T\|\Delta_j(\nabla\times u)\|_\infty dt,$$
where Δ j is the frequency localization operator in the Littlewood-Paley decomposition.

Keywords

Vorticity Smooth Solution Besov Space Regularity Criterion Logarithmic Sobolev Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beale J.T., Kato T. and Majda A.J. (1984). Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94: 61–66 zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Caflisch R.E., Klapper I. and Steele G. (1997). Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Commun. Math. Phys. 184: 443–455 zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Cannone M., Chen Q. and Miao C. (2007). A losing estimate for the Ideal MHD equations with application to Blow-up criterion. SIAM J. Math. Anal. 38: 1847–1859 CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chemin J.-Y. (1998). Perfect Incompressible Fluids. Oxford University Press, New York zbMATHGoogle Scholar
  5. 5.
    Chemin J.-Y. and Lerner N. (1992). Flot de champs de vecteurs non lipschitziens et equations de Navier-Stokes. J. Diff. Eq. 121: 314–328 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Constantin P. and Fefferman C. (1993). Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42: 775–788 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hasegawa A. (1985). Self-organization processed in continuous media. Adv. in Phys. 34: 1–42 CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    He C. and Xin Z. (2005). On the regularity of weak solutions to the magnetohydrodynamic equations. J. Diff. Eq. 213: 235–254 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kato T. and Ponce G. (1988). Commutator estimates and Euler and Navier-Stokes equations. Comm. Pure. Appl. Math. 41: 891–907 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kozono H. and Taniuchi Y. (2000). Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235: 173–194 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kozono H., Ogawa T. and Taniuchi Y. (2002). The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242: 251–278 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Majda A.J. and Bertozzi A.L. (2002). Vorticity and Incompressible Flow. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  13. 13.
    Meyer Y. (1992). Wavelets and operators. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  14. 14.
    Ogawa T. (2003). Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow. SIAM J. Math. Anal. 34: 1318–1330 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Planchon F. (2003). An extension of the Beale-Kato-Majda criterion for the Euler equations. Commun. Math. Phys. 232: 319–326 zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Politano H., Pouquet A. and Sulem P.L. (1995). Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 2: 2931–2939 CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Sermange M. and Temam R. (1983). Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36: 635–664 zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Stein E.M. (1971). Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ Google Scholar
  19. 19.
    Triebel, H.: Theory of Function Spaces. Monograph in Mathematics, Vol. 78. Basel: Birkhauser Verlag, 1983Google Scholar
  20. 20.
    Wu J. (2000). Analytic results related to magneto-hydrodynamics turbulence. Phys. D 136: 353–372 zbMATHGoogle Scholar
  21. 21.
    Wu J. (2002). Bounds and new approaches for the 3D MHD equations. J. Nonlinear Sci. 12: 395–413 zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Wu J. (2004). Regularity results for weak solutions of the 3D MHD equations. Discrete. Contin. Dynam. Syst. 10: 543–556 zbMATHCrossRefGoogle Scholar
  23. 23.
    Wu, J.: Regularity criteria for the generalized MHD equations. PreprintGoogle Scholar
  24. 24.
    Zhang Z. and Liu X. (2004). On the blow-up criterion of smooth solutions to the 3D Ideal MHD equations. Acta Math. Appl. Sinica E 20: 695–700 zbMATHCrossRefGoogle Scholar
  25. 25.
    Zhou Y. (2005). Remarks on regularities for the 3D MHD equations. Discrete. Contin. Dynam. Syst. 12: 881–886 zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingP. R. China
  2. 2.School of Mathematical SciencePeking UniversityBeijingP. R. China

Personalised recommendations