Quantitative classification of potential unforced Navier-Stokes singularities beyond the blow-up time

It remains an open problem whether or not solutions to the unforced 3D Navier-Stokes equations with smooth data remain smooth for all time. All previously known regularity criteria are formulated in terms of a blow-up time (where the solution loses smoothness), which make it practically impossible to use such necessary conditions to test the viability of certain 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. I will also discuss sufficient conditions for instantaneous blow-up to occur at a potential blow-up time (blow-up to the right).