Acta Mathematicae Applicatae Sinica

, Volume 20, Issue 4, pp 695–700 | Cite as

On the Blow-up Criterion of Smooth Solutions to the 3D Ideal MHD Equations

  • Zhi-fei ZhangEmail author
  • Xiao-feng Liu
Original Papers


In this paper, we consider the blow-up of smooth solutions to the 3D ideal MHD equations. Let (u, b) be a smooth solution in (0, T). It is proved that the solution (u, b) can be extended after t = T if \( {\left( {\nabla \times u,\nabla \times b} \right)} \in L^{1} {\left( {0,T;\ifmmode\expandafter\dot\else\expandafter\.\fi{B}^{0}_{{\infty ,\infty }} } \right)} \). This is an improvement of the result given by Caflisch, Klapper, and Steele [3].


Ideal MHD equations blow-up littlewood-paley decomposition besov space 

2000 MR Subject Classification

76W05 35B65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beale, J.T., Kato, T., Majda, A.J. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys., 94: 61–66 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bergh, J., Löfstrom, J. Interpolation spaces, An Introduction. Springer-Verlag, New York, 1976Google Scholar
  3. 3.
    Caflisch, R.E., I. Klapper, I., Steele, G. Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Comm. Math. Phys., 184: 443–455 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Frazier, M., Torres, R., Weiss, G. The boundedness of Caldern-Zygmund operators on the spaces \( \ifmmode\expandafter\dot\else\expandafter\.\fi{F}^{{\alpha ,q}}_{p} . \) Rev. Mat. Iberoamericana 4: 41–72 (1988)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Kozono, H., Ogawa, T., Taniuchi, Y. The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z., 242: 251–278 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Majda, A.J. Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Sciences, 53, Springer-Verlag, New York, 1984Google Scholar
  7. 7.
    Majda, A.J. Bertozzi, A.L. Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambridge, 2002Google Scholar
  8. 8.
    Stein, E.M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, 1993Google Scholar
  9. 9.
    Tribel, H. Theory of Function Spaces. Monograph in mathematics, Vol.78 , Birkhauser Verlag, Basel, 1983Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems Sciencethe Chinese Academy of SciencesBeijingChina
  2. 2.Department of MathematicsEast China Normal UniversityShanghaiChina

Personalised recommendations