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

We consider the Prodi-Serrin type regularity criterion involving ∂3uh and the third component of velocity (or the gradient of velocity). In particular, if the ∂3uh 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).