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Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 1
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Articles

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

Pages 96-103 | Received 30 Jul 2010, Accepted 20 May 2011, Published online: 07 Jul 2011
 

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.

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