is regular on some time-interval $(T^*-\epsilon, T^*)$ and the limit of the maximum of the velocity (or–equivalently–the vorticity) magnitude is infinite as the time variable approaches $T^*$. Such singularity could potentially form in either of the following two cases i) in Eulerian dynamics (zero viscosity), where a formation of ever smaller scales–paired with the formation of ever larger velocity gradients could continue indefinitely or ii) in the viscous case, if the formation of ever smaller scales stays above the threshold of the dissipation scale (shrinking to zero slower than the dissipation scale, as the flow approaches the singularity).

The NS regularity problem is *supercritical*; in other words there is a ‘scaling gap’ between any known regularity criterion and the corresponding *a priori* bound. Here a ‘regularity criterion’ refers to an analytic or geometric property of the solution sufficient to rule out a singularity, while an ‘*a priori* bound’ refers to an analytic or geometric property of the solution that can be rigorously derived from the NSE. Moreover, since the fundamental independent works of Ladyzhenskaya, Prodi and Serrin, as well as Kato and Fujita, in the 1960s [@La67; @P59; @S62; @FK64], no one has improved upon the regularity criteria, with respect 