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 $\underset{q\to \infty }{\mathrm{limsup}}{\int }_{{\mathcal{T}}_q}^T\Vert {\Delta }_q(\nabla \times u){\Vert }_{\infty }dt$ 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.

30 pages

References in corpus (4)

• On the regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations
• The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations
• Determining modes for the 3D Navier-Stokes equations
• Determining modes for the surface Quasi-Geostrophic equation

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• Regularity Criterion for the Three-dimensional Boussinesq Equations
• Regularity criterion for the 3D Hall-magneto-hydrodynamics
• Application of harmonic analysis techniques to regularity problems of dissipative equations