Archive for Rational Mechanics and Analysis

, Volume 224, Issue 3, pp 871–905 | Cite as

On the Critical One Component Regularity for 3-D Navier-Stokes System: General Case



Let us consider initial data \({v_0}\) for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to \({L^{\frac 32}\cap L^2}\). We prove that if the solution associated with \({v_0}\) blows up at a finite time \({T^\star}\), then for any p in \({]4,\infty[}\), and any unit vector e of \({\mathbb{R}^3}\), the L p norm in time with value in \({\dot{H}^{\frac 12+\frac 2 p }}\)of \({(v|e)_{\mathbb{R}^3}}\) blows up at \({T^\star}\).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bahouri, H., Chemin, J. Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations, Grundlehren der mathematischen Wissenschaften 343, Springer, Berlin, 2011Google Scholar
  2. 2.
    Bony J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales de l’École Normale Supérieure 14, 209–246 (1981)MathSciNetMATHGoogle Scholar
  3. 3.
    Chemin J.-Y., Desjardins B., Gallagher I., Grenier E.: Fluids with anisotropic viscosity. Modélisation Mathématique et Analyse Numérique 34, 315–335 (2000)MathSciNetMATHGoogle Scholar
  4. 4.
    Chemin J.-Y., Zhang P.: On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations. Commun. Math. Phys. 272, 529–566 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chemin J.-Y., Zhang P.: On the critical one component regularity for 3-D Navier-Stokes system. Annales de l’École Normale Supérieure 49, 133–169 (2016)MathSciNetMATHGoogle Scholar
  6. 6.
    Escauriaza, L., Seregin, G., Sverák, V.: \({L^{3,\infty}}\)-solutions of Navier-Stokes equations and backward uniqueness, (Russian) Uspekhi Mat. Nauk, 58, no. 2(350), pp. 3–44, 2003. translation in Russian Math. Surveys, 58, 211–250, 2003Google Scholar
  7. 7.
    Fujita H., Kato T.: On the Navier-Stokes initial value problem I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Paicu M.: Équation anisotrope de Navier-Stokes dans des espaces critiques. Revista Matemática Iberoamericana 21, 179–235 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Laboratoire J.-L. Lions, UMR 7598Université Pierre et Marie CurieParis Cedex 05France
  2. 2.Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of MathematicsThe Chinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.School of Mathematical SciencePeking UniversityBeijingPeople’s Republic of China

Personalised recommendations