Vortex Stretching and the Edge of Blow-Up: The engine that concentrates a spin, the brakes that might hold it back, and what is actually proven

Start with a fluid in motion and ask not how fast it is going but how it is spinning — locally, point by point. That local spin is a field: a value at every where and every when. Where the fluid rotates tightly the spin is large; where it slides without turning the spin is near zero. A region where the spin bunches into a coherent rotating column is a vortex tube — a tornado, a bathtub drain, the curl trailing off a wingtip. (In the literature the local spin is called vorticity, written ω.)
The question this note circles is simple to state and, for the full viscous case, has gone unanswered for a century: can a smooth, well-behaved flow spontaneously concentrate its own spin so violently that, in a finite amount of time, the spin at some point runs to infinity? That hypothetical event is a finite-time singularity, or blow-up. It is the heart of the Clay Millennium problem for the three-dimensional Navier–Stokes equations.
The picture below separates the thing that drives concentration — the engine — from the things that might stop it — the brakes. Getting the engine and the brake the right way round is the whole point, because the intuitive story is easy to get backwards.