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

Abstract

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].