UC Riverside

Relevant results and theory in the Axially Symmetric Navier Stokes Equations are reviewed. Then we obtain pointwise, a priori bounds for the $r$, $\theta$ and $z$ components of the vorticity of axially symmetric solutions to the three-dimensional Navier-Stokes equations, which improves on an earlier bound in \cite{BZ:1}. Finally, we show that, for any Leray-Hopf solution, $v$, we can use the $\theta$ component of vorticity to bound the velocity and derive

\begin{align*}

|v(x,t)|\leq\frac{C|\ln{r}|^{1/2}}{r^2},\qquad 0 < r \leq 1/2,

\end{align*}

where $r$ is the distance from the $z$ axis.