Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations

Zijin Li; Xinghong Pan

doi:10.3934/cpaa.2019064

Article Contents

2019, Volume 18, Issue 3: 1333-1350. Doi: 10.3934/cpaa.2019064

This issue Previous Article Solvability of nonlocal systems related to peridynamics Next Article Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up

Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations

Zijin Li^1, , and
Xinghong Pan^2,

1.
Department of Mathematics and IMS, Nanjing University, Nanjing, 210093, China
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

* Corresponding author
* Corresponding author

Received: May 31, 2018

Revised: August 31, 2018

Published: May 2019

The first author is supported by China Scholar Council (NO. 201606190089). The second author is supported by the Research Foundation of Nanjing University of Aeronautics and Astronautics (NO. 1008-YAH17070)

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Abstract

Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^θ$ of the velocity. Second, the precise Green function for the operator $-(Δ-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component $w^θ$ of the vorticity.

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Mathematics Subject Classification: 35Q30, 76N10.

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