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

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].

Keywords:

• Navier–Stokes equations
• regularity criterion
• integrability of pressure

AMS Subject Classifications:

• Primary 35Q35
• Secondary 35B65
• 76D05

Acknowledgements

I thank Professor Yong Zhou for his interest, helpful comments and constructive suggestions. I also thank the referee for valuable comments.