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Keywords:
axisymmetric Navier-Stokes equations; weighted a priori bounds
axisymmetric Navier-Stokes equations; weighted a priori bounds
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 $$ |\omega ^r(x,t)|+|\omega ^z(r,t)|\leq \frac {C|{\rm ln} r|}{r^{17/2}},\quad 0<r\leq \frac 12. $$
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