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.