Two mathematicians used an AI-heavy process to build the first examples of finite-time blowup for the 3D incompressible Euler equations, a step toward the unsolved Navier-Stokes problem

Mathematicians Levent Alpöge and Tristan Buckmaster announced the construction of the first examples of finite-time blowup for the 3D incompressible Euler equations, a step toward the unsolved Navier-Stokes problem. The work, inspired by Diego Córdoba and Luis Martínez-Zoroa and aided by extensive LLM iteration, was partly formalized in Lean. Terence Tao noted the method could potentially extend to Navier-Stokes with significant compute and detail-checking.

Interesting developments tonight, as Levent Alpöge and Tristan Buckmaster announce https://mathstodon.xyz/@tristanbuckmaster@mastodon.social/117233413735526010 that after a fair amount of work they have constructed examples of finite-time blowup for a broad class of PDEs including 3-d incompressible Euler, inspired by of Diego Córdoba and Luis Martínez-Zoroa, and using plenty of LLM iteration in order to get the details right. This is, of course, a problem in the neighborhood of Navier-Stokes in the negative direction of finding a counterexample to the conjecture, which I have over the years heard many PDE folks saying was the right way to bet , and Terry Tao says in a Mastodon thread https://mathstodon.xyz/@tao/117233527638291447 that in principle this method doesn’t seem so far from showing blowup for Navier-Stokes too, though a large amount of compute and detail-checking would be involved. At least part of this has already been Lean-formalized, though perhaps eccentrically I find I care a little less about that. What matters is not whether there’s an example but whether the example has something to teach us. An interesting but incorrect example would surely be of more value than an uninteresting but correct one. Well, I suppose the latter would have more financial value https://www.claymath.org/millennium/navier-stokes-equation/ . Though even on that Millennium Prize page, one sees: “Why ask for a proof? Because a proof gives not only certitude, but also understanding.” Very true We mustn’t settle for mere certitude. Certainly the work of Alpöge, Buckmaster, Córdoba, and Martínez-Zoroa seems to offer understanding as well.