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

TL;DR

This paper introduces new regularity criteria for 3D Navier-Stokes equations based on dynamically restricted critical norms, enhancing understanding of pressure's nonlocal effects near potential singularities.

Contribution

It develops novel methods to establish regularity criteria using endpoint critical norms, extending prior work on non endpoint norms and clarifying pressure's role.

Findings

Regularity criteria in terms of critical Lebesgue norms

Insights into pressure's nonlocal influence on solution regularity

Extension of previous non endpoint norm results

Abstract

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 (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.