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

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 $360\mathrm{/}277(\approx 1.300)$ in [24] to $975\mathrm{/}758(\approx 1.286)$. It is also shown that $\mathrm{\Lambda }(\mathcal{S},r(\log (e\mathrm{/}r){)}^{\sigma })=0(0\le \sigma <27\mathrm{/}113)$, 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 $\epsilon$-regularity criterion proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017].