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Partial differential equations · Mathematical physics · Question

Three-dimensional Euler regularity

Can smooth compactly supported initial velocity form a finite-time singularity in the three-dimensional incompressible Euler equations without external forcing?

ProposedOpenMath assessment · Oct 13, 2026
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Overview

Current assessment

OpenAI reports finite-time breakdown from smooth compactly supported unforced Euler data, with a detailed formal endpoint. The new proof and statement correspondence are under assessment.

Open questions

Independently assess the smooth-data construction and its formal correspondence. Separate forced, bounded-domain and lower-regularity results have their own hypotheses.

Latest

Precise statement

Velocity u and pressure p satisfy ∂tu+(u⋅∇)u=−∇p\partial_t u+(u\cdot\nabla)u=-\nabla p and div u = 0 on ℝ³. There is no viscosity and no external force. The selected question uses smooth divergence-free compactly supported initial velocity, the maximal smooth lifespan T*, and growth of velocity derivatives/vorticity. Lower-regularity data, bounded domains and forced Euler are different scopes.

Question paraphrased by OpenMath. Original wording is available in the sources.

Landscape

Figures and explorable examples. Selecting a statement opens its exact wording.

Visual setup

Smooth data, unforced evolution, a breakdown diagnostic

The question begins with smooth compactly supported velocity. No force or viscosity is present; a singularity is diagnosed by derivative and integrated-vorticity growth.

Reading boundary. No curve, color or arrow is numerical evidence of blowup. The schematic does not represent an actual velocity field or a certified computation.

Smooth data, unforced evolution, a breakdown diagnosticThree initial labels identify smooth compact data, no external force and no viscosity. A directed setup arrow leads to Euler flow, followed by a diagnostic arrow toward derivative and vorticity growth. The finite maximal time is posed as the research question.Smooth compact dataNo external forceNo viscosityEuler flowDerivative / vorticity growthFinite maximal time?
Inspect the smooth-data singularity statement◌ ProposedCompare the global nonexistence endpoint◌ ProposedRead the vorticity continuation criterion● Proved under stated assumptions

Select a statement to compare its regularity, lifespan and evidence. The arrows explain the setup only.

Text version

Three initial labels identify smooth compact data, no external force and no viscosity. A directed setup arrow leads to Euler flow, followed by a diagnostic arrow toward derivative and vorticity growth. The finite maximal time is posed as the research question.

Three-dimensional whole-space incompressible Euler. All drawing geometry is schematic and has no time or velocity scale.

  1. Inspect the smooth-data singularity statement

    Proposed

    Smooth unforced data with finite-time breakdown

    The 2026 manuscript’s smooth-data theorem; the full source-reported endpoint is under mathematical assessment.

    Smooth unforced data with finite-time breakdown →

  2. Compare the global nonexistence endpoint

    Proposed

    No global smooth bounded-energy solution for selected data

    This less detailed formal nonexistence statement is distinct from the explicit compact-data maximal-lifespan endpoint.

    No global smooth bounded-energy solution for selected data →

  3. Read the vorticity continuation criterion

    Proved

    Finite-time smooth breakdown requires divergent integrated vorticity

    Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.

    Finite-time smooth breakdown requires divergent integrated vorticity →

Sources & review

Authored by OpenMath source curation on 2026-10-13 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

Results

The question 1

Proposed 1

Conditional implications 1

  • Proved
    Finite-time smooth breakdown requires divergent integrated vorticity

    Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.

    Beale et al. 1984

Connections

Consequences 1

Assuming Smooth unforced data with finite-time breakdownProposed

  • A singularity must defeat the vorticity continuation criterion

    Conditional

    If the reported smooth-data construction is valid, its divergent integrated vorticity provides the expected classical breakdown diagnostic.

    Limit A large finite vorticity value or a finite numerical trajectory does not establish the required divergence.

    Sources & review

    For an unforced three-dimensional Euler solution in the classical Sobolev class Hs(R3)H^s(\mathbb R^3), s>5/2s>5/2, a finite maximal smooth lifespan T* requires ∫0T∗∥curl⁡u(t)∥∞ dt=∞\int_0^{T_*}\|\operatorname{curl}u(t)\|_{\infty}\,dt=\infty. In particular, a finite value of this integral allows smooth continuation past that time.

    Beale–Kato–Majda is a conditional criterion; it does not itself construct a singularity.

    Assumptions & sources

    Source-grounded explanation · independent explanation review: pending.

Related problems

Approaches

How the problem is attacked, and what each line of attack cannot yet reach.

Transfer amplification across localized oscillation scales

The manuscript iterates localized oscillations over smooth Euler flows and chooses scales to accumulate a finite-time breakdown.

Cannot yet reach The diagram names hypotheses and diagnostics; it does not depict the constructed velocity field or solve the PDE.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Timeline

  1. Two scoped Euler proof endpointsFormalization

    The pinned revision exposes both nonexistence and detailed compact-data singularity statements.

    Formal source completion is reported externally; OpenMath has not reproduced the build.

  2. A smooth unforced whole-space construction is releasedProposal

    OpenAI releases the Euler manuscript and formal source alongside the viscous work.

    The announcement credits separate concurrent forced-Euler work to Alpöge and Buckmaster; its scope differs.

    Also on the Frontier: Unforced three-dimensional Euler blowup

  3. A vorticity continuation criterionRelated result

    Beale, Kato and Majda identify integrated maximum vorticity as a continuation diagnostic.

    The criterion constrains possible breakdown; it is not an existence construction.

  4. Regularity and boundary conditions matterFoundation

    Earlier constructions discussed in the manuscript use lower-regularity data, forcing or different domains.

    Their conclusions must not be relabeled as smooth compact-data whole-space unforced blowup.

Full history with sources →

Evidence

Literature

1 published paper · 1 preprint · 2 projects

Status OpenMath assessment · Oct 13, 2026.

Formal

2 complete proofs

2 formalizations below ↓

Reported in sources: The pinned project reports both a detailed compact-data finite-lifespan theorem and a separate global bounded-energy nonexistence endpoint, with zero sorry and standard logical axioms. Their statements were compared; independent review and external build remain pending.

Computation

None recorded

Review

Not yet reviewed

OpenMath assessment: OpenAI reports finite-time breakdown from smooth compactly supported unforced Euler data, with a detailed formal endpoint. The new proof and statement correspondence are under assessment.

Source comparison and reader curation; independent mathematical assessment remains pending.

OpenMath pilot collection · OpenMath source curation · assessed Oct 13, 2026 · review due Jan 13, 2027

Collection last checked Oct 13, 2026.

Formalizations

Smooth unforced data with finite-time breakdown →

Lean: compact smooth Euler singularityComplete formal proof reported externally

Open formal source ↗

Complete formal proof reported externally

This endpoint supplies compact support, a maximal lifespan 0 < T* ≤ 1 in the all-order Sobolev class, bounded energy before breakdown, infinite C¹ limsup and divergent integrated vorticity. The project reports only standard logical axioms and self-assessed correspondence. It is not an OpenMath execution.

Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026

Declaration & revision
Euler.exists_compact_smooth_euler_singularity

Commit f9e8bc5b38b6e212696e8a30e3e91517af887bbd

No global smooth bounded-energy solution for selected data →

Lean: unforced whole-space nonexistenceComplete formal proof reported externally

Open formal source ↗

Complete formal proof reported externally

This endpoint excludes a global smooth uniformly bounded-energy solution. It is kept separate from the stronger compact-data finite-lifespan endpoint, so the former is not used alone to claim all of the latter’s details.

Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026

Declaration & revision
Euler.euler_breakdown_R3

Commit f9e8bc5b38b6e212696e8a30e3e91517af887bbd

External proof reports do not establish an OpenMath run, independent statement correspondence, or independent mathematical review.

Sources

Choose a starting point below. Each result links to its exact statement, cited passages and checks.

Start here

Bibliography 4

Research activity · 1 entry

Contributions

OpenAI is the printed author of the unforced Euler manuscript and credits its internal research-model workflow. GPT-6 Astra/Codex is credited for the later formalization. Beale, Kato and Majda own the older continuation criterion. The announcement separately recognizes Levent Alpöge and Tristan Buckmaster’s Anthropic-assisted forced-Euler result; it is not the same unforced theorem. Individual mathematical checking roles are not inferred from formal code.

Selected literature; source observations and coverage limits are retained in the publication details.

Record details
Record version
1
Classification
MSC2020 35Q31 / 76B03
Prerequisite level
specialist
Source observations & version history →