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

Tao, Terence

doi:10.1090/jams/838

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

DOI: https://doi.org/10.1090/jams/838

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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