  A domain in $\mathbb{R}^3$ that touches the $x_3$ axis at one point is found
with the following property. For any initial value in a $C^2$ class, the
axially symmetric Navier Stokes equations with Navier slip boundary condition
has a finite energy solution that stays bounded for any given time, i.e. no
finite time blow up of the fluid velocity occurs. The result seems to be the
first case where the Navier-Stokes regularity problem is solved beyond
dimension 2.
