Abstract
In this paper, we obtain the n n th-logarithmically improved regularity criterion of smooth solutions for the incompressible Leray- α \alpha -MHD model in terms of the magnetic field B B . Meanwhile, the new logarithmically improved regularity criterion for the 3D Navier-Stokes equation in terms of the pressure π \pi and gradient of velocity ∇ u \nabla u can also be established. Especially, we explore a new logarithmically improved Serrin’s criterion for the 3D Navier-Stokes equations, which improves the results of Lei and Zhou [Commn. Pure Appl. Anal. 12 (2013), pp. 2715–2719].
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