Regularity criteria for the Navier-Stokes equations based on one component of velocity

We study the regularity criteria for the incompressible Navier-Stokes equations in the whole space R3 based on one velocity component, namely u3, ∇u3 and ∇2u3. We use a generalization of the Troisi inequality and anisotropic Lebesgue spaces and prove, for example, that the condition ∇u3∈Lβ(0,T;Lp), where 2/β+3/p=7/4+1/(2p) and p∈(2,∞], yields the regularity of u on (0,T].