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Finite-time blowup for the 3D incompressible Euler equations from smooth data

Can a solution of the three-dimensional incompressible Euler equations on R3\mathbb R^3, with no external force, started from smooth, compactly supported, divergence-free initial data, lose regularity in finite time? Local existence is classical, and by Beale-Kato-Majda blowup at time T∗T_* is equivalent to ∫0T∗∥ω(t)∥∞ dt=∞\int_0^{T_*}\|\omega(t)\|_\infty\,dt=\infty. Hou and Luo's 2014 numerics suggested a boundary singularity, Elgindi proved blowup in 2021 for C1,αC^{1,\alpha} data with small α\alpha, Córdoba, Martínez-Zoroa and Zheng reached finite-energy C1,αC^{1,\alpha} data, and in September 2026 Buckmaster and Alpöge obtained blowup with a smooth force. Does it happen for smooth data with no force at all?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Fluid dynamics; singularity formation for incompressible flow
Posed by
Leon Lichtenstein (1925) and Nikolai Gunther (1927), whose local existence left global regularity open; the smooth unforced case is the Euler singularity problem of the fluid-dynamics literature
Year posed
1925
Years open
101y
Solved
2026-09-03
Model
Unnamed internal OpenAI model; GPT-6 Astra for the Lean formalisation
Vendor
OpenAI
Collaborators
—
Verification
Lean-verified
Publication
Preprint
Significance
80 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Yes, if the argument stands. Theorem 1.1: there is u_0 in C^infinity_c with div u_0 = 0 whose smooth Euler solution has maximal lifespan 0 < T* < infinity, with limsup of the C^1 norm infinite and the time integral of the vorticity supremum divergent, so the blowup is genuine by Beale-Kato-Majda. The construction, per the paper's own section headings, places a localised oscillation over a smooth Euler flow, amplifies and transfers its wave geometry, chooses scales and iterates, and passes to a limiting datum; 57 pages. It supersedes the forced Euler result of Buckmaster and Alpöge in the catalog and is the step on which OpenAI's Navier-Stokes agents were then prompted. It says nothing about Navier-Stokes without forcing.

What the AI did

From OpenAI's announcement of 8 September (read in the Wayback copy, the live page refuses automated readers): the effort began on 1 September with groups of agents powered by an internal model "significantly more capable than GPT-6 Astra", each group prompted with a variant of a Millennium problem or a related question. Asked the Euler regularity question as one of the "easier" problems, "nearly 100 agents worked together for approximately 50 hours to produce our Euler regularity disproof", specifically the unforced version. That solution was then handed to the Navier-Stokes agents, whose group grew to about 10,000 concurrent agents and reached the forced Navier-Stokes result about 88 hours after launch. Lean formalisation and verification of the results took a further 17 hours via GPT-6 Astra. OpenAI's 10 September update to the post reports an investigation concluding that Buckmaster's Codex prompts over the preceding two months "could not have influenced the system in any way, including through training", and notes that the Euler proofs differ: Alpöge and Buckmaster's has an external force, this one has none.

Verification

Audited here on 22 September 2026 the way the Navier-Stokes entry was. Statement anchor: ComparatorChallenges/Euler.lean in openai/NavierStokesAndEuler is DeepMind's formal-conjectures Navier-Stokes statement at commit 8bf45ed with viscosity and force set to zero; InitialVelocityConditionDecay, the evolution equation with derivWithin on [0, inf), div_free, initial_condition, joint smoothness, MemLp and the uniformly bounded energy were compared line by line with DeepMind's file and match. The first theorem, euler_breakdown_R3, is that breakdown alternative; the second, exists_compact_smooth_euler_singularity, is OpenAI's own richer form of the paper's Theorem 1.1 with compact support, a finite maximal lifespan T* <= 1, a C^1 limsup of infinity and a divergent vorticity integral. Euler/Solution.lean restates both verbatim and ends with #print axioms; Euler.json permits only propext, Quot.sound and Classical.choice and enables the nanoda kernel. Counted here on the tarball: 1,839 Lean files and 211,578 lines under Euler/, zero sorry, zero axiom declarations, zero native_decide, zero unsafe. Lean 4.34.0-rc2 pinned. Not rebuilt here, and the repository has no CI of its own; the comparator replay is documented for anyone with the machine. No mathematician has read the 57-page argument.

Sources

Submitted by LucidManta102 on

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