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Title: Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations (English)
Author: Zhang, Zujin
Author: Tong, Chenxuan
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 67
Issue: 4
Year: 2022
Pages: 485-507
Summary lang: English
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Category: math
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Summary: We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that $$ |\omega ^r(x,t)|+|\omega ^z(r,t)|\leq \frac {C}{r^{10}},\quad 0<r\leq \frac {1}{2}. $$ 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 $$ |\omega ^r(x,t)|+|\omega ^z(r,t)|\leq \frac {C|{\rm ln} r|}{r^{17/2}},\quad 0<r\leq \frac 12. $$ (English)
Keyword: axisymmetric Navier-Stokes equations
Keyword: weighted a priori bounds
MSC: 35B65
MSC: 35Q35
MSC: 76D03
idZBL: Zbl 07584082
idMR: MR4444789
DOI: 10.21136/AM.2021.0344-20
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Date available: 2022-06-28T13:22:19Z
Last updated: 2024-09-02
Stable URL: http://hdl.handle.net/10338.dmlcz/150439
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