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Finite time blowup for an averaged three-dimensional Navier-Stokes equation

Author: Terence Tao
Journal: J. Amer. Math. Soc. 29 (2016), 601-674
MSC (2010): Primary 35Q30
Published electronically: June 30, 2015
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Abstract: 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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Terence Tao
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095-1555

Received by editor(s): February 3, 2014
Received by editor(s) in revised form: March 31, 2015
Published electronically: June 30, 2015
Article copyright: © Copyright 2015 American Mathematical Society