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

Abstract

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 ℝ3. It is proved that the solution is regular if the associate pressure satifies
p ∈ L_2/2-r ((0, T); M2,3/r (ℝ3)) or ∇_p ∈ _L_2/3-r ((0, T); M2,3/r (ℝ3))
for 0 < r < 1, where M2,3/r (ℝ3)
is the critical Morrey–Campanto space. Regularity criteria for the 3D MHD equations are also given.