Remarks on regularity conditions of the Navier-Stokes equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier-Stokes equations in a space-time domain containing z0 = (x0, t0), and Qz0,r = Bx0,r×(t0−r 2, t0) be a parabolic cylinder in the domain. We show that if v × ω |ω| ∈ L γ,α x,t (Qz0,r) or ω × v |v| ∈ L γ,α x,t (Qz0,r), where L γ,α x,t denotes the Serrin type of class, then z0 is a regular point for v. This refines previous local regularity criteria for the suitable weak solutions.