An anisotropic partial regularity criterion for the Navier-Stokes equations

In this paper, we address the partial regularity of suitable weak solutions of the incompressible Navier–Stokes equations. We prove an interior regularity criterion involving only one component of the velocity. Namely, if $(u,p)$ is a suitable weak solution and a certain scale-invariant quantity involving only $u_3$ is small on a space-time cylinder $Q_r(x_0,t_0)$, then $u$ is regular at $(x_0,t_0)$.