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2017, Volume 37, Issue 4: 1923-1939. Doi: 10.3934/dcds.2017081

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Regularity of 3D axisymmetric Navier-Stokes equations

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

Received: April 2016

Revised: November 2016

Published: May 2017

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Abstract

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{ω^{r}}{r},\frac{ω^{θ}}{r})$, we get several Prodi-Serrin type regularity criteria based on the angular velocity, $u^θ$. Moreover, we obtain the global well-posedness result if the initial angular velocity $u_{0}^{θ}$ is appropriate small in the critical space $L^{3}(\mathbb{R}^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $ω^3$ or $u^3$.

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Mathematics Subject Classification: Primary:35K15, 35K55;Secondary:35Q35, 76A05.

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