# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 30, Issue 1, 2011, pp. 83–93**

**DOI: 10.4171/ZAA/1425**

Regularity Criteria in Terms of the Pressure for the Navier-Stokes Equations in the Critical Morrey-Campanato Space

Yong Zhou^{[1]}and Sadek Gala

^{[2]}(1) School of Mathematics, Shanghai University of Finance and Economics, 200433, Shanghai, China

(2) Mostaganem University, Department of Mathematics, BP 227, 27000, Mostaganem, Algeria

In this paper, we establish a Serrin-type regularity criterion in terms of the pressure for Leray weak solutions to the Navier–Stokes equation in ℝ^{3}. It is proved that the solution is regular if the associate pressure satifies

*p* ∈ *L*^{2/2-r} ((0, *T*); M_{2,3/r} (ℝ^{3})) or ∇*p* ∈ *L*^{2/3-r} ((0, *T*); M_{2,3/r} (ℝ^{3}))

for 0 < *r* < 1, where M_{2,3/r} (ℝ^{3}) is the critical Morrey–Campanto space. Regularity criteria for the 3D MHD equations are also given.

*Keywords: *Navier–Stokes equations, Morrey–Campanato space, weak solution, regularity criterion

Zhou Yong, Gala Sadek: Regularity Criteria in Terms of the Pressure for the Navier-Stokes Equations in the Critical Morrey-Campanato Space. *Z. Anal. Anwend.* 30 (2011), 83-93. doi: 10.4171/ZAA/1425