The author announces a new paper on arXiv titled 'Finite time blowup for an averaged three-dimensional Navier-Stokes equation', submitted to J. Amer. Math. Soc.. The main purpose is to formalise the 'supercriticality barrier' for the global regularity problem for the Navier-Stokes equation, asserting that global regularity cannot be established by abstract approaches using only upper bound function space estimates on the nonlinear part combined with the energy identity.

This is achieved by constructing a modification of the Navier-Stokes equations with a nonlinearity that obeys essentially all function space estimates the true Navier-Stokes nonlinearity does, follows the energy identity, but admits solutions that blow up in finite time. Previous results by Montgomery-Smith, Gallagher-Paicu, Li-Sinai, Katz-Pavlovic, and Cheskidov established blowup for variants without the energy identity or in higher dimensions, but this is the first blowup result for a three-dimensional Navier-Stokes type equation that also obeys the energy identity.

The method hints at a possible route to establishing blowup for the true Navier-Stokes equations, which the author increasingly believes is the case for a very small set of initial data. The global regularity problem for Navier-Stokes is equivalent to the global regularity problem for the evolution equation involving a bilinear operator with a cancellation law leading to the fundamental energy identity.