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

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    * Corresponding author 

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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  • 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.

    Mathematics Subject Classification: 35Q30, 76N10.


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