Mathematische Annalen

, Volume 347, Issue 1, pp 15–41 | Cite as

On the global well-posedness for the axisymmetric Euler equations

Article
First Online:
Received:
Revised:

Abstract

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in critical Besov spaces \({B_{p,1}^{1+3/p}}\). In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity.

Mathematics Subject Classification (2000)

76D03 35B33 35Q35 76D05 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bergh J., Löfström J.: Interpolation spaces. An introduction. Springer, Berlin (1976)MATHGoogle Scholar
  3. 3.
    Bony J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. de l’École Norm. Sup. 14, 209–246 (1981)MATHMathSciNetGoogle Scholar
  4. 4.
    Chae D.: Local existence and blow-up criterion for the Euler equations in the Besov spaces. Asymptot. Anal. 38(3–4), 339–358 (2004)MATHMathSciNetGoogle Scholar
  5. 5.
    Chemin J.-Y.: Perfect Incompressible Fluids. Clarendon Press, Oxford (1998)MATHGoogle Scholar
  6. 6.
    Constantin P., Fefferman C., Majda A.: Geometric constraints on potentially singular solutions for the 3D Euler equations. Commun. Partial Differ. Equ. 21(3-4), 559–571 (1996)MATHMathSciNetGoogle Scholar
  7. 7.
    Danchin R.: Axisymmetric incompressible flows with bounded vorticity. Russian Math. Surv. 62(3), 73–94 (2007)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hmidi, T., Keraani, S.: Incompressible viscous flows in borderline Besov spaces. ARMA (to appear)Google Scholar
  9. 9.
    Kato T.: Nonstationary flows of viscous and ideal fluids in \({\mathbb{R}^3}\). J. Funct. Anal. 9, 296–305 (1972)MATHCrossRefGoogle Scholar
  10. 10.
    O’Neil R.: Convolution operators and L(p,q) spaces. Duke Math. J. 30, 129–142 (1963)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Pak H.C., Park Y.J.: Existence of solution for the Euler equations in a critical Besov space \({B_{\infty,1}^1(\mathbb{R}^n)}\). Commun. Partial Differ. Equ. 29, 1149–1166 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Peetre, J.: New Thoughts on Besov spaces. Mathematical Series, vol. 1. Duke University, Durham N. C. (1976)Google Scholar
  13. 13.
    Saint Raymond X.: Remarks on axisymmetric solutions of the incompressible Euler system. Commun. Partial Differ. Equ. 19(1–2), 321–334 (1994)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Serfati P.: Solutions C ∞ en temps, n-log Lipschitz bornées en espace et équation d’Euler. C. R. Acad. Sci. Paris Sér. I Math. 320(5), 555–558 (1995)MATHMathSciNetGoogle Scholar
  15. 15.
    Shirota T., Yanagisawa T.: Note on global existence for axially symmetric solutions of the Euler system. Proc. Jpn. Acad. Ser. A Math. Sci. 70(10), 299–304 (1994)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ukhovskii M.R., Yudovich V.I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. Prikl. Mat. Mekh. 32(1), 59–69 (1968)Google Scholar
  17. 17.
    Vishik M.: Hydrodynamics in Besov spaces. Arch. Ration. Mech. Anal. 145, 197–214 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.IRMARUniversité de Rennes 1Rennes CedexFrance

Personalised recommendations