Regularity Criterion to the axially symmetric Navier-Stokes Equations

Dongyi Wei

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arXiv · 1508.03318

Regularity Criterion to the axially symmetric Navier-Stokes Equations

Dongyi Wei

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Abstract

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$ We first prove the global regularity of solutions if $\|ru_\theta(r,z,0)\|_{L^\infty}$ or $ \|ru_\theta(r,z,t)\|_{L^\infty(r\leq 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 $|ru_\theta(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\ 0<r\leq\delta_0\in(0,1/2).$

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Dongyi Wei. 2015-08-13. Regularity Criterion to the axially symmetric Navier-Stokes Equations. https://arxiv.org/abs/1508.03318

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