Finite-time blowup for the IPM equation with a uniformly space-time smooth force
Córdoba and Martínez-Zoroa proved finite-time singularity formation for the two-dimensional incompressible porous media equation from smooth initial data with a force smooth in space but merely bounded in time, that is in . Their Remark 1 anticipates joint smoothness in space and time but does not prove it. Can the force be taken uniformly smooth in space and time?
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Fluid dynamics; incompressible porous media equation
- Posed by
- Diego Córdoba and Luis Martínez-Zoroa, Remark 1 of arXiv:2410.22920, where joint space-time smoothness is anticipated but not part of the theorem
- Year posed
- 2024
- Years open
- 2y
- Solved
- 2026-09-08
- Model
- Claude, Codex with GPT-5.6 Sol
- Vendor
- Anthropic / OpenAI
- Collaborators
- Levent Alpöge, Tristan Buckmaster, Matei P. Coiculescu
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Yes. Theorem 2.1: there are a smooth odd initial density of zero spatial mean, an odd force , and a solution smooth on for every with in for every , yet and both diverging as . The advance over Córdoba and Martínez-Zoroa is precisely the force class, from to uniformly space-time smooth, on the torus rather than the plane. Not the Clay Millennium problem: that problem is Navier-Stokes with viscosity, and Fefferman's official description states that the Euler equation "is not on the Clay Institute's list of prize problems".
What the AI did
This is the earliest of the three results and the one the authors describe as feeding the others: the Boussinesq AI statement says the Boussinesq work "involved inputting ideas from previous joint work of ours on blowup for the IPM equation following Córdoba-Martínez-Zoroa". Buckmaster's public statement covers the whole project: "For most of the past year progress was slow. We worked through the literature and upgraded various preliminary results, up to obtaining finite time blow up for the Incompressible Porous Media equation (with smooth forcing)", using "Anthropic's Claude, OpenAI's Codex, especially with GPT-5.6 Sol". The IPM paper itself does not break the contribution down per step, and defers a fuller account: "The complete human-readable proofs will be released shortly by the first and second authors, together with an account of the role of artificial intelligence in this work."
Verification
No independent check. Unlike the Boussinesq and Euler results, this one has no Lean formalisation: tristanbuckmaster/fluid_lean contains projects for Boussinesq (twice) and Euler and none for IPM, confirmed by listing the repository tree on 8 September 2026. The paper is a 57-page manuscript on the second author's university page, not on arXiv and not peer reviewed, and no independent expert reading is on record. It is the most conventional of the three write-ups, being the one the authors had time to prepare.