A certain necessary condition of potential blow up for Navier-Stokes equations

TL;DR

This paper establishes that for the Navier-Stokes equations, the potential blow-up time must be accompanied by the unbounded growth of the L3 norm of the velocity field.

Contribution

It proves a necessary condition linking blow-up time to the divergence of the L3 norm, advancing understanding of singularity formation in Navier-Stokes equations.

Findings

If blow-up occurs at time T, then the L3 norm of the velocity must tend to infinity as t approaches T.

The result provides a criterion to identify potential singularities in solutions.

It clarifies the relationship between solution norms and finite-time blow-up in fluid dynamics.

Abstract

We show that a necessary condition for $T$ to be a potential blow up time is $\lim\limits_{t\uparrow T}\|v(\cdot,t)\|_{L_3}=\infty$.