We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that $$|{\omega }^r(x,t)|+|{\omega }^z(r,t)|\le \frac{C}{r^{10}},0<r\le \frac{1}{2}.$$ 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 $$|{\omega }^r(x,t)|+|{\omega }^z(r,t)|\le \frac{C\left|\ln r\right|}{r^{17/2}},0<r\le \frac{1}{2}.$$

We study the Cauchy problem for the MHD system, and provide two regularity conditions involving horizontal components (or their gradients) in Besov spaces. This improves previous results.

This article recalls the results given by A. Dutrifoy, A. Majda and S. Schochet in [1] in which they prove an uniform estimate of the system as well as the convergence to a global solution of the long wave equations as the Froud number tends to zero. Then, we will prove the convergence with weaker hypothesis and show that the life span of the solutions tends to infinity as the Froud number tends to zero.