Remarks on regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure

In this article, we establish a Serrin-type regularity criterion on the gradient of pressure for weak solutions to the NavierStokes equation in R3. It is proved that if the gradient of pressure belongs to , where is the multiplier space (a definition is given in the text) for 0?=?r?=?1, then the weak solution is actually regular. Since this space is wider than , our regularity criterion covers the previous results given by Struwe [M. Struwe, On a Serrin-type regularity criterion for the NavierStokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2007), pp. 235242], Berselli-Galdi [L.C. Berselli and G.P. Galdi, Regularity criteria involving the pressure for the weak solutions to the NavierStokes equations, Proc. Amer. Math. Soc. 130 (2002), pp. 35853595] and Zhou [?Y. Zhou, On regularity criteria in terms of pressure for the NavierStokes equations in R3 , Proc. Am. Math. Soc. 134 (2006), pp. 149156].