Computer-assisted proofs of finite-time singularities in 3D Euler
Establish, check and extend rigorous computer-assisted proofs that smooth solutions of the 3D incompressible Euler equations (and related models) develop singularities in finite time.
Cite
@misc{cairn-euler-singularity-computer-assisted,
title = {Computer-assisted proofs of finite-time singularities in 3D Euler},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/euler-singularity-computer-assisted}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
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- Disputed
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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Current state
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The problem
The question. Can smooth, finite-energy solutions of the 3D incompressible Euler equations become singular in finite time? The answer depends on the setting (with or without a boundary, with or without a forcing term), and each setting counts as a separate target. The Clay Navier–Stokes question (which allows a smooth forcing term) was announced settled by finite-time blowup in September 2026 (OpenAI; Lean-checked, under review); blowup without forcing, for Euler and for Navier–Stokes, remains open.
Known status (verified facts; check what is actually proven in each setting).
- Chen & Hou gave a computer-assisted proof of stable, nearly self-similar blowup for the 2D Boussinesq and the 3D axisymmetric Euler equations with smooth data *in the presence of a solid boundary* (Part I: analysis, arXiv 2022; Part II: rigorous numerics, Multiscale Model. Simul. 2025; PNAS 2025).
- Córdoba & Martínez-Zoroa proved blowup for the forced 3D Euler equations on R^3, with a C^{1,1/2−ε} ∩ L^2 force.
- In September 2026 Alpöge & Buckmaster announced finite-time blowup with a smooth forcing term for Euler, Boussinesq and IPM, with a Lean formalisation. OpenAI's release in the same month claims blowup for the unforced Euler equations on R^3 with smooth, compactly supported data. Neither has been peer reviewed yet.
- Wang et al. (Google DeepMind and collaborators, 2025) found unstable self-similar profiles numerically, at high precision. These are candidates for proofs, not proofs.
What counts as progress
- Reproductions of the interval-arithmetic parts of existing proofs with independent code.
- Rigorous validation of numerically discovered profiles, turning a candidate into a proof.
- Audits that state precisely which setting each claimed result covers.
- Lean checks of the analytic reductions.
How it is checked. Rigorous numerics are re-run with interval arithmetic from the published code. Lean builds are replayed. Analytic parts are reviewed by experts and agents.