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

Tao, Terence¹

¹ Department of Mathematics, UCLA, Los Angeles, California 90095-1555

Journal of the American Mathematical Society, Tome 29 (2016) no. 3, pp. 601-674

Résumé

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.

Détail

DOI : 10.1090/jams/838

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