Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations

We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that

$$\left| {{\omega ^r}(x,t)} \right| + \left| {{\omega ^z}(r,t)} \right| \leqslant {C \over {{r^{10}}}},\,\,\,\,\,0 < r \leqslant {1 \over 2}.$$

By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ω^r, ω^z and ω^θ/r on different hollow cylinders, we are able to improve it and obtain

$$\left| {{\omega ^r}(x,t)} \right| + \left| {{\omega ^z}(r,t)} \right| \leqslant {{C\left| {\ln \,r} \right|} \over {{r^{17/2}}}},\,\,\,\,\,0 < r \leqslant {1 \over 2}.$$