A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier–Stokes equations

Abstract

The partial regularity of the suitable weak solutions to the Navier–Stokes equations in Rn with n=2,3,4 and the stationary Navier–Stokes equations in Rn for n=2,3,4,5,6 are investigated in this paper. Using some elementary observation of these equations together with De Giorgi iteration method, we present a unified proof on the results of Caffarelli, Kohn and Nirenberg [1], Struwe [17], Dong and Du [5], and Dong and Strain [7]. Particularly, we obtain the partial regularity of the suitable weak solutions to the 4d non-stationary Navier–Stokes equations, which improves the previous result of [5], where Dong and Du studied the partial regularity of smooth solutions of the 4d Navier–Stokes equations at the first blow-up time.