On partial regularity for weak solutions to the Navier–Stokes equations

In this paper, we study the partial regularity of the general weak solution u∈L∞(0,T;L2(Ω))∩L2(0,T;H1(Ω)) to the Navier–Stokes equations, which include the well-known Leray–Hopf weak solutions. It is shown that there is a absolute constant ε such that for the weak solution u, if either the scaled local Lq(1⩽q⩽2) norm of the gradient of the solution, or the scaled local Lq(1⩽q⩽103) norm of u is less than ε, then u is locally bounded. This implies that the one-dimensional Hausdorff measure is zero for the possible singular point set, which extends the corresponding result due to Caffarelli et al. (Comm. Pure Appl. Math. 35 (1982) 717) to more general weak solution.