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

Yanqing Wang, Gang Wu

Published 2017-09-05Version 1

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 $\Lambda(\mathcal{S},r(\log(e/r))^{\sigma})=0(0\leq\sigma<7/23)$, which extends the previous corresponding results concerning the improvement of the classical Caffarelli-Kohn-Nirenberg theorem by a logarithmic factor. The proof rests on a new $\varepsilon$-regularity criterion proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017] and establishing associated decay-type estimates.