Archive for Rational Mechanics and Analysis

, Volume 202, Issue 3, pp 919–932

Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor


  • Chongsheng Cao
    • Department of MathematicsFlorida International University
    • Department of Mathematics and Department of Mechanical and Aerospace EngineeringUniversity of California
    • Department of Computer Science and Applied MathematicsWeizmann Institute of Science

DOI: 10.1007/s00205-011-0439-6

Cite this article as:
Cao, C. & Titi, E.S. Arch Rational Mech Anal (2011) 202: 919. doi:10.1007/s00205-011-0439-6


In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, that is, the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier–Stokes equations in the whole space, as well as for the case of periodic boundary conditions.

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© Springer-Verlag 2011