Regularity Criterion to the axially symmetric Navier-Stokes Equations

Abstract

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\Vert r{u}_{\theta }(r,z,t){\Vert }_{L^{\infty }}\le \Vert r{u}_{\theta }(r,z,0){\Vert }_{L^{\infty }}.$ We first prove the global regularity of solutions if $\Vert r{u}_{\theta }(r,z,0){\Vert }_{L^{\infty }}$ or $\Vert r{u}_{\theta }(r,z,t){\Vert }_{L^{\infty }(r\le r_0)}$ is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang \cite{1}. As a corollary, we also prove the global regularity under the assumption that $|r{u}_{\theta }(r,z,t)|\le |\ln r{|}^{-3/2},\forall 0<r\le {\delta }_0\in (0,1/2).$

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References in corpus (4)

• Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II
• Criticality of the Axially Symmetric Navier-Stokes Equations
• Regularity of 3D axisymmetric Navier-Stokes equations
• A priori bound on the velocity in axially symmetric Navier-Stokes equations