  In this short survey paper, we focus on some new developments in the study of
the regularity or potential singularity formation for solutions of the 3D
Navier-Stokes equations. Some of the motivating questions are: Are certain
norms accumulating/concentrating on small scales near potential blow-up times?
At what speed do certain scale-invariant norms blow-up? Can one prove explicit
quantitative regularity estimates? Can one break the criticality barrier, even
slightly? We emphasize that these questions are closely linked together. Many
recent advances for the Navier-Stokes equations are directly inspired by
results and methods from the field of nonlinear dispersive equations.

Analysis of PDEs