Frequency localized regularity criteria for the 3D Navier-Stokes equations

TL;DR

This paper develops frequency-specific regularity criteria for the 3D Navier-Stokes equations, identifying key frequency ranges influencing potential singularities in weak solutions.

Contribution

It introduces new frequency localized criteria that refine existing regularity conditions by focusing on specific Littlewood-Paley frequency bands.

Findings

Identifies critical frequency ranges associated with singularity formation.

Refines Ladyzhenskaya-Prodi-Serrin criteria to finite frequency windows.

Shows the lower bound of these frequency windows diverges near singularities.

Abstract

Two regularity criteria are established to highlight which Littlewood-Paley frequencies play an essential role in possible singularity formation in a Leray-Hopf weak solution to the Navier-Stokes equations in three spatial dimensions. One of these is a frequency localized refinement of known Ladyzhenskaya-Prodi-Serrin-type regularity criteria restricted to a finite window of frequencies the lower bound of which diverges to $+\infty$ as $t$ approaches an initial singular time.