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

Article

Abstract

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 \({T^\star}\).

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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

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