On the possible time singularities for the 3D Navier-Stokes equations

TL;DR

This paper establishes a local regularity criterion for the 3D Navier-Stokes equations, providing bounds on the Hausdorff dimension of potential singular times without requiring weak solutions to be suitable.

Contribution

It introduces a new regularity criterion for Leray-Hopf solutions, bounding the dimension of singular times without assuming solution suitability.

Findings

Hausdorff dimension of singular times is bounded by a specific formula

Regularity criterion applies to solutions in certain Besov spaces

No assumption of solution suitability is needed

Abstract

We prove a local-in-time regularity criterion for the 3D Navier-Stokes equations. In particular, it follows from the criterion that the Hausdorff dimension of possible singular times of Leray-Hopf weak solutions $u\in L^r_t B^\alpha_{s,\infty}$ for some $\alpha>0$, $ s > 3$ and $r> 2 $ is less than $\frac{r}{2}(\frac{3}{s} + \frac{2}{r} -\alpha-1 )$. The main contribution is that we do not assume the suitability of weak solutions.