You certainly know this one, but some readers could ignore it. The fact that NS iw globally well-posed in 2D is due to the so-called Ladyzhenskaia inequality
$$\|f\|_4^2\le c\|f\|_2\|\nabla f\|_2.$$
The fact that this does not hold in 3D is the reason why there is a 1M-dollars problem...

But if the flow is axisymmetric, even with swirl, **and** if the domain is a container between two cylinders ($0<r_0<\sqrt{x^2+y^2}<r_1<\infty$), then LI is still valid, and the solution exists, is unique and smooth whenever the initial energy is finite.

To summarize, the only major difficulty is when the domain reaches the symmetry axis.

By the way, I have proved (see my notes at the Compte Rendus in 1991 and 1999) that in absence of viscosity, that is for the Euler equation, the vorticity of such a fluid (incompressible, axisymmetric, with swirl), generically increases linearly in time. If the flow exists globally, of course.