On regularity and singularity for $$L^{\infty }(0,T;L^{3,w}(\mathbb {R}^3))$$ solutions to the Navier–Stokes equations

Abstract

We study local regularity properties of a weak solution u to the Cauchy problem of the incompressible Navier–Stokes equations. We present a new regularity criterion for the weak solution u satisfying the condition $$L^{\infty }(0,T;L^{3,w} (\mathbb {R}^3))$$ without any smallness assumption on that scale, where $$L^{3,w}(\mathbb {R}^3)$$ denotes the standard weak Lebesgue space. As an application, we conclude that there are at most a finite number of blowup points at any singular time t.