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 is an element of L-x (0, T; L-2(Omega)) boolean AND L-2 (0, T; H-1(Omega)) to the Navier-Stokes equations, which include the well-known Leray-Hopf weak solutions. It is shown that there is a absolute constant epsilon such that for the weak solution it, if either the scaled local L-q(1 less than or equal to q less than or equal to 2) norm of the gradient of the solution, or the scaled local L-q(1 less than or equal to q less than or equal to 10/3) norm of u is less than epsilon, 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. (C) 2003 Elsevier Inc. All rights reserved.