On regularity criteria in terms of pressure for the Navier-Stokes equations in R-3

In this paper we establish a Serrin-type regularity criterion on the gradient of pressure for the weak solutions to the Navier-Stokes equations in R-3. It is proved that if the gradient of pressure belongs to L-alpha,L-gamma with 2/alpha + 3/gamma <= 3, 1 <= gamma <= infinity, then the weak solution is actually regular. Moreover, we give a much simpler proof of the regularity criterion on the pressure, which was showed recently by Berselli and Galdi.