Geodesic
Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations
Applications of Mathematics, Tome 67 (2022) no. 4, pp. 485-507.

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We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that $$ |\omega ^r(x,t)|+|\omega ^z(r,t)|\leq \frac {C}{r^{10}},\quad 0\leq \frac {1}{2}. $$ 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 $$ |\omega ^r(x,t)|+|\omega ^z(r,t)|\leq \frac {C|{\rm ln} r|}{r^{17/2}},\quad 0\leq \frac 12. $$
DOI : 10.21136/AM.2021.0344-20
Classification : 35B65, 35Q35, 76D03
Keywords: axisymmetric Navier-Stokes equations; weighted a priori bounds 
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Zhang, Zujin; Tong, Chenxuan. Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations. Applications of Mathematics, Tome 67 (2022) no. 4, pp. 485-507. doi : 10.21136/AM.2021.0344-20. http://geodesic.mathdoc.fr/articles/10.21136/AM.2021.0344-20/

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