Abstract
In this article, we establish a Serrin-type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes 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 Navier–Stokes equations in terms of the pressure, J. Math. Fluid Mech. 9 (2007), pp. 235–242], Berselli-Galdi [L.C. Berselli and G.P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations, Proc. Amer. Math. Soc. 130 (2002), pp. 3585–3595] and Zhou [ Y. Zhou, On regularity criteria in terms of pressure for the Navier–Stokes equations in ℝ3
, Proc. Am. Math. Soc. 134 (2006), pp. 149–156].
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Acknowledgements
I thank Professor Yong Zhou for his interest, helpful comments and constructive suggestions. I also thank the referee for valuable comments.