Regularity Criteria for the Three-dimensional Navier–Stokes Equations C HONGSHENG C AO & E DRISS S. T ITI ABSTRACT. In this paper we consider the three–dimensional Navier–Stokes equations subject to periodic boundary condi- tions or in the whole space. We provide sufficient 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. In honor of Professor Ciprian Foias in his 75th birthday 1. I NTRODUCTION The three-dimensional Navier–Stokes equations (NSE) of viscous incompressible fluid read: ∂u ∂t − ν ∆u + (u ·∇)u +∇p = 0, (1.1) ∇· u = 0, (1.2) u(x 1 ,x 2 ,x 3 , 0) = u 0 (x 1 ,x 2 ,x 3 ), (1.3) where u = (u 1 ,u 2 ,u 3 ), the velocity field, and p, the pressure, are the unknowns, and ν> 0, the viscosity, is given. We set ∇ h = (∂ x 1 ,∂ x 2 ) to be the horizontal gradient operator and ∆ h = ∂ 2 x 1 + ∂ 2 x 2 the horizontal Laplacian, while ∇ and ∆ are the usual gradient and the Laplacian operators, respectively. We equip the system 2643 Indiana University Mathematics Journal c , Vol. 57, No. 6 (2008), Special Issue