Navier-Stokes: has an AI really solved the problem?

The Navier-Stokes equations describe how a fluid flows, and they turn up everywhere: weather and climate, the air around an aircraft wing or a car, blood in our arteries, ocean currents, the cooling of a computer or combustion in an engine. They are solved every day in numerical simulations, yet, nearly two centuries after they were written, one basic question is still open: in three dimensions, can their solutions "blow up", that is, become infinite at a point?

In 2000 the Clay Institute made it one of its seven Millennium Problems, with a prize of one million dollars. On 8 September, OpenAI announced an answer: a 166-page proof, produced by thousands of AI agents in 88 hours, showing a blow-up in finite time when the fluid is driven by an external force.

Two months on, enthusiasm has given way to caution. The proof is hard to read, even for specialists, and researchers in London and Cambridge have pointed out that its computer-checked version does not say exactly the same thing as the text. Nothing shows it is wrong, but nothing has officially validated it yet. And even if confirmed, it would leave aside the case of a fluid left to itself.

My view

Physically, there is no such thing as infinite velocity: if the equations "blow up", the model has reached its limits, at scales so small that the fluid no longer behaves as a continuous medium. The real question then becomes whether this can happen in a real flow, with no external force, at the heart of turbulence.