Fractal dimension of potential singular points set in the Navier-Stokes equations under supercritical regularity

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal{S}$ of suitable weak solution $u$ belonging in $L^q(0,T;L^p({\mathbb{R}}^3))$ for $1\le \frac{2}{q}+\frac{3}{p}\le \frac{3}{2}$ with $2\le q<\mathrm{\infty }$ and $2<p<\mathrm{\infty }$ is at most $\max \{p,q\}(\frac{2}{q}+\frac{3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^2(0,T;{\dot{H}}^{s+1}({\mathbb{R}}^3))$ for $0\le s\le \frac{1}{2}$ is zero, whose proof relies on Caffarelli-Silvestre’s extension. Inspired by Baker-Wang’s recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity under some supercritical regularity.