Navier–Stokes existence and smoothness (Clay formulation)
Provisional. This status records a claimed resolution, not a completed verification. OpenAI's 'Finite Time Blowup for Navier–Stokes' (8 September 2026; 166 pages plus Lean 4, produced by an internal model) constructs a smooth, compactly supported force giving finite-time blow-up with bounded energy, i.e. Clay alternatives (C)/(D). On 11 September 2026 the Clay Institute publicly welcomed the result as having apparently settled the problem, but it has awarded nothing, still lists the problem as active, and its rules require journal publication plus two years of general acceptance; OpenAI says it will not claim the prize. As of 15 October 2026 the manuscript is unrefereed, Bastounis–Circelli–Hansen (arXiv:2610.08144) dispute that the Lean code certifies the paper, and priority is contested by Buckmaster and Alpöge. Constantin–Ignatova–Vicol (arXiv:2609.20803) show the mechanism needs non-analytic forcing, so the unforced problem (navier-stokes-unforced-regularity) is open. If verification fails, this entry reverts to OPEN.
Graded as RESOLVED by the site owner's decision, so difficulty measures what remains: verifying a complete 166-page construction and reconciling its Lean statement with the prose. That puts it at the top of the 1–9 band, above the formal FLT certificate (3) because no independent check exists yet; the calibration pass graded the same claim at 60 while treating the problem as open, and would restore that if the construction fails. Application value 24 and leverage 48 are unchanged from calibration: forced blow-up reshapes singularity analysis in PDE but no engineering depends on it, and the unforced question (92) carries the remaining weight.
The problem
The official million-dollar form of the question: do 3D fluid flows always stay smooth, or can they break?
- 1Clay lists four ways to settle the problem
- 2Two prove smooth flow lasts forever
- 3Two give a force-driven flow that breaks
- 4A blow-up claim exists; checking is incomplete
The Clay formulation. For the 3D incompressible Navier–Stokes equations , , with viscosity , the Clay problem lists four alternatives and asks for a proof of any one.
Built from
earlier problems → this oneThe problems this one is built from: progress or tools it needs (builds on), special cases that come first (generalizes), and weaker results on the way (implies). On the right, what builds on it.
- Start with
- Navier–Stokes existence and smoothness (Clay formulation)This problem
- Then
- Global regularity or finite-time blow-up for unforced 3D Navier–Stokes (f ≡ 0)Analysis & PDE · D92
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Harmonic analysisTheory
- Step 3
- The Navier–Stokes equationsTheory · host
- Then
- Navier–Stokes existence and smoothness (Clay formulation)The problem
State of the art
Claimed resolution of alternatives (C)/(D) by forced blow-up; independent verification is incomplete. Classical results: global smooth solutions in 2D; local smooth solutions and global solutions for small data in 3D; Leray weak solutions with unknown uniqueness; partial regularity (Caffarelli–Kohn–Nirenberg, 1982); regularity criteria such as (Escauriaza–Seregin–Šverák, 2003). Tao (2014) proved blow-up for an averaged equation with the same energy identity, and Albritton–Brué–Colombo (2021) proved non-uniqueness of forced Leray solutions. On 8 September 2026 OpenAI released 'Finite Time Blowup for Navier–Stokes' (166 pages plus a Lean 4 project), attributed to an internal model, constructing for each ν > 0 a smooth force, compactly supported in space and time, and a solution from rest whose velocity blows up in finite time with bounded energy, which targets alternatives (C)/(D). OpenAI said it would not claim the prize. Priority is disputed: Buckmaster and Alpöge had posted related forced blow-up for Euler shortly before. On 11 September 2026 the Clay Institute posted a note, as reported in the press, welcoming the result as having 'apparently' settled the problem while stressing a 'deliberately unhurried' and rigorous evaluation; Clay's rules require publication, a waiting period of two years and general acceptance, and Clay still listed the problem as active in October 2026. Fefferman, author of the official problem statement, told Quanta he was 'thrilled that the problem was solved'. No independent group has published a full check: Bastounis, Circelli and Hansen (arXiv:2610.08144) argue the Lean development does not certify the written proof, and Cao, Chi and Nie (arXiv:2609.10262, unrefereed) build on the construction to show blow-up forces are dense. Constantin, Ignatova and Vicol (arXiv:2609.20803) show that constructions with the same structural features cannot work with real-analytic forcing, so the unforced equations are untouched and form the successor problem.
Realistic applications
None directly. Blow-up with a rough or compactly supported force says little about physical flows. The methods (computer-assisted and AI-assisted construction of singularities, Lean certificates in analysis) are the main transferable output.
What counts as solved
Alternative (C) or (D) is settled when the construction is checked and accepted by experts (journal publication or Clay's own determination) and the Lean statement is shown to match the written theorem. The Clay prize as posed would then be awarded; the unforced problem is tracked separately.
How it gets verified
Lean proofA formal proof checked by Lean 4 / Mathlib.
Statement: the four Clay alternatives (A)–(D) in Millennium/NavierStokes.lean (Formal Conjectures pinned at commit e754a78 (17 October 2026)), stated directly with derivatives of smooth functions. The forced breakdown alternatives navier_stokes_breakdown_R3 (C) and navier_stokes_breakdown_periodic (D) are labelled solved there, linking an OpenAI Lean proof (September 2026). Verification: build the linked proof against the pinned statements and pass Comparator with only the standard axioms, then have experts confirm that the Lean statement matches Clay's written formulation (this match has been disputed for the 2026 claim). A computer-assisted blow-up proof needs its interval arithmetic verified inside the same development.
Claims under verification
+ add a claimWhere the work is
What remains is checking an existing artefact: expert refereeing of the prose proof and aligning the Lean statement and its dependencies with it.
Bottlenecks
ranked by votes- #1EngineeringLean statement versus paper correspondence
Bastounis–Circelli–Hansen report mismatches between the formal code and the written proof, so the certificate does not yet certify the paper's theorem.
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Connected problems
- This problem is needed by Global regularity or finite-time blow-up for unforced 3D Navier–Stokes (f ≡ 0)Forced blow-up tools and the Lean artefact would be the starting point for any unforced blow-up attempt.
Progress
- Oct 6, 2026Lean certificate questionedsource
Bastounis–Circelli–Hansen argue that the Lean development does not certify the written proof.
- Sep 11, 2026Clay responds: evaluation will be unhurriedsource
Clay welcomes the claim as having apparently settled the problem but awards nothing; prize rules require publication and two years of acceptance.
- Sep 8, 2026OpenAI posts a forced blow-up constructionsource
166 pages plus Lean, claiming alternatives (C)/(D).
- Feb 1, 2014Averaged Navier–Stokes blows upsource
Tao proves blow-up for an averaged equation with the same energy identity.