Regularity criteria for the 3D Navier-Stokes and MHD equations

Abstract

We prove that a solution to the 3D Navier-Stokes or MHD equations does not blow up at $t=T$ provided $\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|\Delta_q(\nabla \times u)\|_\infty \, dt$ is small enough, where $u$ is the velocity, $\Delta_q$ is the Littlewood-Paley projection, and $\mathcal T_q$ is a certain sequence such that $\mathcal T_q \to T$ as $q \to \infty$. This improves many existing regularity criteria.