A note on necessary conditions for blow-up of energy solutions to the Navier-Stokes equations

TL;DR

This paper investigates the behavior of the $L_3$-norm of velocity fields near blow-up times in Navier-Stokes equations, establishing necessary conditions for blow-up of type I solutions.

Contribution

It proves that for type I blow-ups, the $L_3$-norm's lower limit also tends to infinity, providing new necessary conditions for blow-up.

Findings

The $L_3$-norm's upper limit must be infinite at blow-up.

For type I blow-ups, the lower limit of the $L_3$-norm is also infinite.

The results give necessary conditions for the blow-up behavior of solutions.

Abstract

In the present note, we address the question about behavior of $L_3$-norm of the velocity field as time $t$ approaches blow-up time $T$. It is known that the upper limit of the above norm must be equal to infinity. We show that, for blow-ups of type I, the lower limit of $L_3$-norm equals to infinity as well.