Necessary conditions of potential blow up for Navier-Stokes equations

TL;DR

This paper investigates the conditions under which solutions to the Navier-Stokes equations blow up, demonstrating that the $H^{1/2}$-norm of the velocity must diverge as the potential blow-up time is approached.

Contribution

It establishes a necessary condition for blow-up in Navier-Stokes solutions, linking blow-up to the divergence of the $H^{1/2}$-norm of velocity.

Findings

The $H^{1/2}$-norm of velocity tends to infinity at potential blow-up time.

Provides a mathematical criterion for identifying blow-up scenarios.

Enhances understanding of singularity formation in fluid dynamics.

Abstract

Assuming that $T$ is a potential blow up time, we show that $H^\frac 12$-norm of the velocity field goes to $\infty$ as time $t$ approaches $T$