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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Finite time blowup for an averaged three-dimensional Navier-Stokes equation
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by Terence Tao
J. Amer. Math. Soc. 29 (2016), 601-674
Published electronically: June 30, 2015


The Navier-Stokes equation on the Euclidean space $\mathbb {R}^3$ can be expressed in the form $\partial _t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation property $\langle B(u,u), u\rangle =0$ (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modification $\partial _t u = \Delta u + \tilde B(u,u)$ of this equation, where $\tilde B$ is an averaged version of the bilinear operator $B$ (where the average involves rotations, dilations, and Fourier multipliers of order zero), and which also obeys the cancellation condition $\langle \tilde B(u,u), u \rangle = 0$ (so that it obeys the usual energy identity). By analyzing a system of ordinary differential equations related to (but more complicated than) a dyadic Navier-Stokes model of Katz and Pavlovic, we construct an example of a smooth solution to such an averaged Navier-Stokes equation which blows up in finite time. This demonstrates that any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions has to use a finer structure on the nonlinear portion $B(u,u)$ of the equation than is provided by harmonic analysis estimates and the energy identity. We also propose a program for adapting these blowup results to the true Navier-Stokes equations.
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Bibliographic Information
  • Terence Tao
  • Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095-1555
  • MR Author ID: 361755
  • ORCID: 0000-0002-0140-7641
  • Email:
  • Received by editor(s): February 3, 2014
  • Received by editor(s) in revised form: March 31, 2015
  • Published electronically: June 30, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 29 (2016), 601-674
  • MSC (2010): Primary 35Q30
  • DOI:
  • MathSciNet review: 3486169