On regularity and singularity for $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ solutions to the Navier-Stokes equations

TL;DR

This paper establishes a new regularity criterion for weak solutions to the Navier-Stokes equations in a weak Lebesgue space, allowing for finite blowup points and encompassing type I singularities without smallness assumptions.

Contribution

It introduces a novel regularity criterion based on $L^ abla(0,T;L^{3,w}( abla^3))$ norms, extending understanding of singularity formation in Navier-Stokes solutions.

Findings

Finite number of blowup points at any singular time.

Regularity criterion holds without smallness assumption.

Includes type I singularities, weaker than Ladyzhenskaya-Prodi-Serrin condition.

Abstract

We study local regularity properties of a weak solution $u$ to the Cauchy problem of the incompressible Navier-Stokes equations. We present a new regularity criterion for the weak solution $u$ satisfying the condition $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ without any smallness assumption on that scale, where $L^{3,w}(\mathbb{R}^3)$ denotes the standard weak Lebesgue space. As an application, we conclude that there are at most a finite number of blowup points at any singular time $t$. The condition that the weak Lebesgue space norm of the veclocity field $u$ is bounded in time is encompassing type I singularity and significantly weaker than the end point case of the so-called Ladyzhenskaya-Prodi-Serrin condition proved by Escauriaza-Sergin-\v{S}ver\'{a}k.