Blow-up of dynamically restricted critical norms near a potential Navier–Stokes singularity

In this paper we develop new methods to obtain regularity criteria for the three-dimensional Navier–Stokes equations in terms of dynamically restricted endpoint critical norms: the critical Lebesgue norm in general or the critical weak Lebesgue norm in the axisymmetric case. This type of results is inspired in particular by a work of Neustupa (Arch Ration Mech Anal 214(2):525–544, 2014), which handles certain non endpoint critical norms. Our work enables to have a better understanding of the nonlocal effect of the pressure on the regularity of the solutions.