Regularity criteria for the Navier-Stokes equations involving the ratio pressure-gradient of velocity

A number of bounds upon the pressure are known to guarantee regularity of the solutions of the Navier-Stokes equations. Since the pressure is the potential whose source is the product of the velocity and its gradient, it is worth to consider bounds depending on the quotient of the pressure and some quantity measuring the size of this source. Estimates involving the ratio pressure-velocity are known. Our result includes the velocity gradient: if the ratio p/1+vertical bar v vertical bar+vertical bar del v vertical bar(r) remains bounded for some r < 1, so does the velocity and therefore it retains its regularity. Copyright (c) 2009 John Wiley & Sons, Ltd.