As far as I can tell, the Navier-Stokes regularity problem has now been resolved in the most annoying way possible:
- we have a single counterexample, not a general theorem;
- we have no real idea how it was found, but it seems to have been existing ideas and a lot of elbow grease;
- this makes no difference in practical applications, which use finite-element numerical methods;
- there's no way to apply this to other PDEs, or to improve our understanding of turbulence;
- the possibility of being scooped by an AI company (and possibly one that's spying on its customers) will discourage people from sharing their work-in-progress on big problems.
In other words, the opposite of what the Millennium Prizes were supposed to achieve! Am I missing something?
EDIT: after a bit more reading, I don't think the situation is quite this bad. The "existing ideas" (the construction of Córdoba and Martínez-Zoroa) have also been used to construct blowups in several related PDEs, so there's an obvious follow-on question about how far they can be pushed - and they've been demonstrated to be more important than most mathematicians realised. I know even less about numerical methods than I know about fluid dynamics, but apparently the way the construction builds ever-smaller perturbations might have implications for building better subgrid models. But we've lost any insight that we might have got from the "elbow grease" phase of the construction, and the way the proof came out will cause a lot of mathematicians to be less transparent.