Remarks on the singular set of suitable weak solutions to the 3D Navier-Stokes equations

TL;DR

This paper refines the upper bounds on the size of the singular set of suitable weak solutions to the 3D Navier-Stokes equations, extending classical regularity results with logarithmic improvements.

Contribution

It improves the known upper box-counting dimension of the singular set and extends the Caffarelli-Kohn-Nirenberg theorem with a logarithmic factor, inspired by a new epsilon-regularity criterion.

Findings

Upper box-counting dimension of the singular set is reduced to approximately 1.286.

Established that a certain measure of the singular set with a logarithmic factor is zero for specific parameters.

Extended classical regularity results with a logarithmic factor in the context of Navier-Stokes solutions.

Abstract

In this paper, let $\mathcal{S}$ denote the possible interior singular set of suitable weak solutions of the 3D Navier-Stokes equations. We improve the known upper box-counting dimension of this set from $360/277(\approx1.300)$ in [24] to $975/758(\approx1.286)$. It is also shown that $\Lambda(\mathcal{S},r(\log(e/r))^{\sigma})=0(0\leq\sigma<27/113)$, which extends the previous corresponding results concerning the improvement of the classical Caffarelli-Kohn-Nirenberg theorem by a logarithmic factor in Choe and Lewis [3, J. Funct. Anal., 175: 348-369, 2000] and in Choe and Yang et al. [4, Comm. Math. Phys, 336: 171-198, 2015]. The proof is inspired by a new $\varepsilon$-regularity criterion proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017].