Finite time blowup for an averaged Navier-Stokes equation

The Navier-Stokes equation in three dimensions can be expressed in the form $u_t =\Delta u +B(u,u)$ for a certain bilinear operator $B$. It is a notorious open question whether finite time blowup solutions exist for this equation. We do not address this question directly, but instead study an averaged Navier-Stokes equation
$u_t = \Delta u +B’(u,u)$, where $B’$ is a certain average of $B$ (where the average involves rotations and Fourier multipliers of order $0$). This averaged Navier-Stokes equation obeys the same energy identity as the original Navier-Stokes equation, and the nonlinear term $B’(u,u)$ obeys essentially the same function space estimates as the original nonlinearity $B(u,u)$. By using a modification of a dyadic Navier-Stokes model of Katz and Pavlovic, which is “engineered” to generate “self-replicating machine” or “von Neumann machine” type solutions, we can construct an example of an averaged Navier-Stokes equation which exhibits finite time blowup. This demonstrates a “barrier” to establishing global regularity for the true Navier-Stokes equations, in that one cannot hope to prove global regularity by relying purely on function space estimates on the nonlinearity $B$, combined with the energy identity.