Regularity criteria for the three-dimensional Navier-Stokes equations

In this paper we consider the three-dimensional Navier-Stokes equations subject to periodic boundary condi- tions or in the whole space. We provide suYcient conditions, in terms of one component of the velocity field, or alternatively in terms of one component of the pressure gradient, for the regularity of strong solutions to the three-dimensional Navier- Stokes equations.