A note on regularity of weak solutions of the Navier-Stokes equations in Rⁿ

Abstract

In this paper we consider the n dimensional Navier-Stokes equations and we prove a new regularity criterion for weak solutions. More precisely, if n=3, 4 we show that the “smallness” of at least n-1 components of the velocity in L^∞ (O, T; Lⁿ_w (Rⁿ)) is sufficient to ensure regularity of the weak solutions.