Finite-time blowup for the IPM equation with uniformly space-time smooth forcing
The incompressible porous medium equation on the two-dimensional torus is shown to develop a finite-time singularity under a force that is smooth in space and time together, extending a result previously available only for spatially smooth forcing.
- Lab
- Independent
- Model
- Not named; the arXiv comment field states only that the proof is LLM-assisted
- Field
- Mathematics
- Date
- 2026-09-15
- Human collaborators
- Levent Alpöge, Tristan Buckmaster, Matei P. Coiculescu
What was found
Cordoba and Martinez-Zoroa proved finite-time blowup for the incompressible porous medium equation with a spatially smooth force. This paper adapts their technique to a force that is uniformly smooth in space and time together, which is the stronger requirement. It exhibits a smooth odd initial density, a smooth odd force on the unit time interval crossed with the two-dimensional torus, and a classical solution on the half-open interval whose density gradient and spatial velocity gradient both diverge in the supremum norm as time approaches one, while the density itself still converges in every Holder class below Lipschitz. It is one of three companion forced-blowup results from the same group covering IPM, Boussinesq and three-dimensional Euler; the other two are not yet in this registry.
Novelty check
The paper frames itself in its own abstract as extending the Cordoba and Martinez-Zoroa IPM blowup from spatially smooth to uniformly space-time smooth forcing, so the prior state of the art is named rather than left implicit. Terence Tao's 2026-09-07 post describes the same group's work across IPM, Boussinesq and Euler and notes that the IPM case was already handled by Cordoba and Martinez-Zoroa, with the contribution here being a variant of their method that also extends to the other two equations. This is a different claim from the OpenAI results already in this registry at 2026-09-08-navier-stokes-forced-blowup and 2026-09-08-euler-unforced-blowup, which concern different equations, and the four should not be read as competing claims on one theorem.
Caveats and known objections
The AI disclosure is thin and its location matters: it is not in the abstract but in the arXiv comment field, which reads in full, 57 pages, LLM-assisted and Lean formalized proof. No model or vendor is named anywhere, which is why the model field records the absence rather than a guess, and autonomy is graded ai-assisted as the weakest reading the phrase LLM-assisted will carry. The 2026-09-21 watch run recorded this result under the date 2026-09-08 and without the third author; both were corrected here against the arXiv record on 2026-09-22, which gives a submission date of 15 September 2026 and names Coiculescu alongside Alpoge and Buckmaster. The Lean formalization is asserted in the comment field but no repository is linked from the abstract, so verification is capped at claimed under editorial rule 4 and nothing here is machine-checked for this registry's purposes. Tao's post records that the authors released preliminary preprints before the work was polished, describing their own earlier write-up in severe terms, so a reader should expect the exposition to be in flux.
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Entry history (1 event)
- AddedEntered the registry graded Claimed and AI-assisted.
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Graded claimed for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Finite-time blowup for the IPM equation with uniformly space-time smooth forcing. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-09-15-ipm-forced-blowupBibTeX
@misc{whataifound-independent-2026-blowup,
title = {Finite-time blowup for the IPM equation with uniformly space-time smooth forcing},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Claimed. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2026-09-15-ipm-forced-blowup}
}