Finite time blowup for an averaged three-dimensional Navier-Stokes equation

The research paper titled "Finite time blowup for an averaged three-dimensional Navier-Stokes equation" has been published in the Journal of the American Mathematical Society. The authors aim to address the long-standing problem of global regularity in the Navier-Stokes equations by constructing a modified version of the equations that still satisfies many of the original nonlinearity and energy identity properties, but allows for solutions that become unbounded in a finite amount of time.

This is significant because previous attempts to prove global regularity for the Navier-Stokes equations relied solely on upper bound estimates of the nonlinearity and the energy identity, neither of which are applicable to the modified equations. The authors introduce a new averaged bilinear operator, which is a finite linear combination of local cascade operators, and demonstrate that this modified Navier-Stokes equation admits solutions that blow up in finite time.

This result provides a significant obstacle to any attempt to prove global regularity for the true Navier-Stokes equations using only the energy identity and upper bound estimates for the nonlinearity.