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

Abstract

The Navier-Stokes equation on the Euclidean space R 3 \mathbb {R}^3 can be expressed in the form ∂ t u = Δ u + B ( u , u ) \partial _t u = \Delta u + B(u,u) , where B B is a certain bilinear operator on divergence-free vector fields u u obeying the cancellation property ⟨ B ( u , u ) , u ⟩ = 0 \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 ∂ t u = Δ u + B ~ ( u , u ) \partial _t u = \Delta u + \tilde B(u,u) of this equation, where B ~ \tilde B is an averaged version of the bilinear operator B B (where the average involves rotations, dilations, and Fourier multipliers of order zero), and which also obeys the cancellation condition ⟨ B ~ ( u , u ) , u ⟩ = 0 \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 ) 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.