With heavy AI help, two mathematicians proved three fluid equations can break down

On 7 September 2026, Terence Tao described new work by Levent Alpöge and Tristan Buckmaster (NYU). They showed that three equations used to model fluids can blow up in finite time: develop a singularity, a point where the solution stops making sense. This happens when a smooth forcing term (a steady outside push) is added. The three are the incompressible porous medium equation, the 2D Boussinesq equation and the 3D incompressible Euler equations. The work builds on “infinite cascade” constructions by Diego Córdoba and Luis Martínez-Zoroa.

Tao says the arguments were heavily AI-assisted and have been formalised in Lean, a language in which a computer checks every step. According to Wikipedia’s timeline, the results were obtained on 15 August 2026 and Lean-verified about a week later. Alpöge is an Anthropic employee.

The work is a stepping stone toward the Navier–Stokes problem, one of the best-known open questions in maths. Tao judged it highly likely that the method extends to Navier–Stokes. He noted that the authors had not yet managed that, even with smooth starting data and forcing, let alone with no force at all. The post came a day before OpenAI’s own Navier–Stokes announcement, and a public priority dispute followed.

What it means

AI tools gave heavy help on research-level maths here, and a computer checked the result, so readers do not have to trust the AI. The results cover equations with an outside push, so the famous question about fluids left to themselves is still unanswered. Which AI tools were used, and how, has not been made public.

Caveats

• These are forced results. Blow-up of the 3D Euler equations with no force and no boundary is not claimed.
• Tao says the preprints were released before they were polished, because of outside events.
• Tao’s post does not name the AI tools used.
• Some outlets date the preprints to 8 September.