“Finite Time Blowup for an Averaged Three-Dimensional Navier-Stokes Equation”, 2014-02-03 (; backlinks):
The Navier-Stokes equation on the Euclidean space 𝐑3 can be expressed in the form ∂tu = Δu + B(u, u), where B is a certain bilinear operator on divergence-free vector fields u obeying the cancellation property 〈B(u, u), u〉 = 0 (which is equivalent to the energy identity for the Navier-Stokes equation).
In this paper, we consider a modification ∂tu = Δu + B̃(u, u) of this equation, where B̃ 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 〈B(u, u), u〉 = 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 & 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 3 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.