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 [22, p. 505, Commun. Math. Phys., 2010] for the Navier-Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set S of suitable weak solution u belonging to L-q(0,T; L-p(R-3)) for 1 2q + 3p 32 with 2 q < 8 and 2 < p < 8 is at most max{p, q}(2q + 3p - 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 ? L2(0,T; ?Hs+1(R3)) for 0 s 21 is zero, whose proof relies on Caffarelli-Silvestre's extension. Inspired by Barker-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 is under some supercritical regularity.