Abstract: It remains an open problem whether or not solutions to the 3D Navier-Stokes equations with smooth data remain smooth for all time.

Recently, Hou gave a compelling numerical candidate singularity formation. Quantities associated with regularity criteria are numerically computed with growth observed.  All previously known regularity criteria are formulated in times of a blow-up time (where the solution loses smoothness), whichmake it practically impossible to use such necessary conditions to test the viability of numerically computed candidates.

Motivated by these issues, we give the first quantitative classification of potentially singular solutions at any given time in the region of potential blow-up times. Moreover, the quantitative bounds in the vicinity of any potential blow-up time are in principle amenable to numerical testing.