A new local regularity criterion for suitable weak solutions of the Navier--Stokes equations in terms of the velocity gradient

We study the partial regularity of suitable weak solutions to the three dimensional incompressible Navier--Stokes equations. There have been several attempts to refine the Caffarelli--Kohn--Nirenberg criterion (1982). We present an improved version of the CKN criterion with a direct method, which also provides the quantitative relation in Seregin's criterion (2007).