The regularity criterion for the 3D Navier–Stokes equations involving end-point Prodi–Serrin type conditions

Abstract

We consider the Prodi–Serrin type regularity criterion involving ∂ 3 u h and the third component of velocity (or the gradient of velocity). In particular, if the ∂ 3 u h satisfies the end-point Prodi–Serrin type condition, one can show that Leray’s weak solutions of the three-dimensional Navier–Stokes equations become regular with the help of additional assumption on the third component of velocity (or the gradient of velocity field).