On the Well-posedness of the Ideal MHD Equations in the Triebel–Lizorkin Spaces

Article

Abstract

In this paper, we prove the local well-posedness for the ideal MHD equations in the Triebel–Lizorkin spaces and obtain a blow-up criterion of smooth solutions. Specifically, we fill a gap in a step of the proof of the local well-posedness part for the incompressible Euler equation in Chae (Comm Pure Appl Math 55:654–678 2002).

References

  1. 1.
    Bergh J., Löfstrom J.: Interpolation Spaces, An Introduction. Springer, New York (1976)MATHGoogle Scholar
  2. 2.
    Beale J.T., Kato T., Majda A.J.: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Comm. Math. Phys. 94, 61–66 (1984)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Bony J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246 (1981)MATHMathSciNetGoogle Scholar
  4. 4.
    Caflisch R.E., Klapper I., Steele G.: Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. Comm. Math. Phys. 184, 443–455 (1997)MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Cannone M., Chen Q., Miao C.: A losing estimate for the Ideal MHD equations with application to Blow-up criterion. SIAM J. Math. Anal. 38, 1847–1859 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen Q., Miao C., Zhang Z.: The Beale-Kato-Majda criterion for the 3D Magneto-hydrodynamics equations. Comm. Math. Phys. 275, 861–872 (2007)MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Chae D.: On the well-posedness of the Euler equations in the Triebel–Lizorkin spaces. Comm. Pure Appl. Math. 55, 654–678 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chae D.: On the Euler equations in the critical Triebel–Lizorkin spaces. Arch. Ration. Mech. Anal. 170, 185–210 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chemin J.-Y.: Régularité de la trajectoire des particules d’un fluide parfait incompressible remplissant l’espace. Math. Pures Appl. 71, 407–417 (1992)MATHMathSciNetGoogle Scholar
  10. 10.
    Chemin J.-Y.: Perfect Incompressibe Fluids. Oxford University Press, New York (1998)Google Scholar
  11. 11.
    Fefferman C., Stein E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Frazier M., Torres R., Weiss G.: The boundedness of Calderón-Zygmund operators on the spaces \({{\dot F}^{\alpha, q}_p}\) . Rev. Math. Iber. 4, 41–72 (1988)MATHMathSciNetGoogle Scholar
  13. 13.
    Kato T.: Nonstationary flows of viscous and ideal fluids in R 3. J. Funct. Anal. 9, 296–305 (1972)MATHCrossRefGoogle Scholar
  14. 14.
    Majda, A.J.: Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Sciences, vol.~53. Springer, New York, 1984Google Scholar
  15. 15.
    Meyer Y.: Wavelets and operators. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  16. 16.
    Planchon F.: An extension of the Beale-Kato-Majda criterion for the Euler equations. Comm. Math. Phys. 232, 319–326 (2003)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Stein E.M.: Singular Integrals and Differentiability Propertyies of Functions. Princeton University Press, Princeton (1970)Google Scholar
  18. 18.
    Triebel, H.: Theory of Function Spaces. Monograph in mathematics, vol. 78. Birkhauser, Basel, 1983Google Scholar
  19. 19.
    Wu J.: Generalized MHD equations. J. Differ. Equ. 195, 284–312 (2003)MATHCrossRefGoogle Scholar
  20. 20.
    Wu J.: Bounds and new approaches for the 3D MHD equations. J. Nonlinear Sci. 12, 395–413 (2002)MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Wu J.: Regularity results for weak solutions of the 3D MHD equations. Discrete. Contin. Dynam. Syst. 10, 543–556 (2004)MATHCrossRefGoogle Scholar
  22. 22.
    Wu J.: Regularity criteria for the generalized MHD equations. Comm. PDE. 33, 285–306 (2008)MATHCrossRefGoogle Scholar
  23. 23.
    Zhang Z., Liu X.: On the blow-up criterion of smooth solutions to the 3D Ideal MHD equations. Acta Math. Appl. Sinica, E 20, 695–700 (2004)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.School of Mathematical SciencePeking UniversityBeijingChina

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