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The main objective of this paper is to answer the questions posed by Robinsonand Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokesequations. Firstly, we prove that the upper box dimension of the potentialsingular points set $\mathcal{S}$ of suitable weak solution $u$ belonging in $L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for $1\leq\frac{2}{q}+\frac{3}{p}\leq\frac32$ with $2\leq q<\infty$ and $2<p<\infty$ is at most$\max\{p,q\}(\frac{2}{q}+\frac{ 3}{p}-1)$ in this system. Secondly, it is shownthat $1-2 s$ dimension Hausdorff measure of potential singular points set ofsuitable weak solutions satisfying $ u\inL^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for $0\leq s\leq\frac12$ is zero,whose proof relies on Caffarelli-Silvestre's extension. Inspired byBaker-Wang's recent work [1], this further allows us to discuss the Hausdorffdimension of potential singular points set of suitable weak solutions if thegradient of the velocity under some supercritical regularity.

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