Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations
Tóm tắt
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 ωr, ωz and ωθ/r on different hollow cylinders, we are able to improve it and obtain
$$\left| {{\omega ^r}(x,t)} \right| + \left| {{\omega ^z}(r,t)} \right| \leqslant {{C\left| {\ln \,r} \right|} \over {{r^{17/2}}}},\,\,\,\,\,0 < r \leqslant {1 \over 2}.$$
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