  The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ 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 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 analysing a system of ODE
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 a 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 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.
