Applications of Mathematics, Vol. 67, No. 4, pp. 485-507, 2022
Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations
Zujin Zhang, Chenxuan Tong
Received November 30, 2020. Published online November 22, 2021.
Abstract: We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that
$$
|\omega^r(x,t)|+|\omega^z(r,t)|\leq\frac{C}{r^{10}},\quad0<r\leq\frac12.
$$
By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing $\omega^r$, $\omega^z$ and $\omega^\theta/r$ on different hollow cylinders, we are able to improve it and obtain
$$
|\omega^r(x,t)|+|\omega^z(r,t)|\leq\frac{C|{\rm ln} r|}{r^{17/2}},\quad0<r\leq\frac12.
$$
Keywords: axisymmetric Navier-Stokes equations; weighted a priori bounds
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Affiliations: Zujin Zhang (corresponding author), Chenxuan Tong, School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, Jiangxi, P. R. China, e-mail: zhangzujin361@163.com, 1342745285@qq.com
