On blowup of nonendpoint borderline Lorentz norms for the Navier-Stokes equations

TL;DR

This paper proves that if the Navier-Stokes equations blow up at a finite time, then the velocity's Lorentz norm in a specific space must become unbounded, providing a new blowup criterion.

Contribution

It establishes a novel blowup criterion involving Lorentz norms for the Navier-Stokes equations near potential singularities.

Findings

Lorentz norm $L^{3,q}$ diverges at blowup time

Provides a new criterion for singularity formation

Enhances understanding of Navier-Stokes blowup behavior

Abstract

Assuming $T$ is a potential blow up time for the Navier-Stokes system in $\mathbb{R}^3$ or $\mathbb{R}^3_+$, we show that the $L^{3,q}$ Lorentz norm, with $q$ finite, of the velocity field goes to infinity as time $t$ approaches $T$.