Source · Paper

Finite Time Blowup for the Euler Equation

OpenAI

PreprintPublication date Sep 8, 2026

Reading source record 2026-09-08. View default cited record →

Overview

The manuscript states smooth compactly supported unforced Euler data with finite maximal lifespan.

Source publication and attributed assessments are distinct from independent OpenMath review.

2 cited passages ↓1 problem ↓Record 2026-09-08 ↓

Passages

The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.

Passage index · 2

§1.1, Historical context; references [6], [7], [10], [15]
Theorem 1.1 and Section 1.1

Passage 1 · OpenMath paraphrasePermalink

§1.1, Historical context; references [6], [7], [10], [15]

Elgindi (Annals 2021) proved finite-time blowup on ℝ³ for C^{1,α} axisymmetric velocities without swirl, for small α > 0. Córdoba and Martínez-Zoroa (2023) constructed finite-energy forced Euler solutions on ℝ³ that blow up. For smooth initial data, Chen and Hou (2022; rigorous numerics 2025) proved blowup in an axially periodic cylinder with an impermeable wall.

Read at source ↗7 recorded uses ↓

Passage details

Passage ID: euler-regularity:entry:history:v1

Source item: historical-context

Source record: euler-regularity:source:paper:v1

Passage 2 · OpenMath paraphrasePermalink

Theorem 1.1 and Section 1.1

The manuscript states smooth compactly supported unforced Euler data with finite maximal lifespan.

Read at source ↗12 recorded uses ↓

Passage details

Passage ID: euler-regularity:entry:paper:v1

Source item: paper

Source record: euler-regularity:source:paper:v1

Used in

Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.

Three-dimensional Euler regularity →

Problem publication v9

Smooth-data blowup with a boundary →

Source assessment

Proved

Smooth-data blowup with a boundary (Chen & Hou 2022)

Only the exact statement and passage cited here; not an independent OpenMath assessment.

Observed Oct 21, 2026

Assessment source ↗Statement v1 →

§1.1, Historical context; references [6], [7], [10], [15] ↑

Statement

Statement v1 →

§1.1, Historical context; references [6], [7], [10], [15] ↑

Blowup with external forcing →

Source assessment

Proved

Blowup with external forcing (Córdoba & Martínez-Zoroa 2023)

Only the exact statement and passage cited here; not an independent OpenMath assessment.

Observed Oct 21, 2026

Assessment source ↗Statement v1 →

§1.1, Historical context; references [6], [7], [10], [15] ↑

Statement

Statement v1 →

§1.1, Historical context; references [6], [7], [10], [15] ↑

Blowup from C^{1,α} data in ℝ³ →

Source assessment

Proved

Blowup from C^{1,α} data in ℝ³ (Elgindi 2021)

Only the exact statement and passage cited here; not an independent OpenMath assessment.

Observed Oct 21, 2026

Assessment source ↗Statement v1 →

§1.1, Historical context; references [6], [7], [10], [15] ↑

Statement

Statement v1 →

§1.1, Historical context; references [6], [7], [10], [15] ↑

Smooth unforced data with finite-time breakdown →

Source assessment

Proposed

Smooth unforced Euler singularity reported

Compact smooth data on ℝ³; finite maximal lifespan and derivative/vorticity blowup.

Source date Sep 8, 2026 · Observed Oct 13, 2026

Assessment source ↗Statement v3 →

Theorem 1.1 and Section 1.1 ↑

Scope

Smooth unforced data with finite-time breakdown

Statement v3 →

Theorem 1.1 and Section 1.1 ↑

Statement

Statement v3 →

Theorem 1.1 and Section 1.1 ↑

Finite-time smooth breakdown requires divergent integrated vorticity →

Scope

Finite-time smooth breakdown requires divergent integrated vorticity

Statement v1 →

Theorem 1.1 and Section 1.1 ↑

Statement

Statement v1 →

Theorem 1.1 and Section 1.1 ↑

No global smooth bounded-energy solution for selected data →

Scope

No global smooth bounded-energy solution for selected data

Statement v3 →

Theorem 1.1 and Section 1.1 ↑

Statement

Statement v3 →

Theorem 1.1 and Section 1.1 ↑

Problem context · 6 citations

Question

Three-dimensional Euler regularity

View in problem →

§1.1, Historical context; references [6], [7], [10], [15] ↑
Theorem 1.1 and Section 1.1 ↑

History

A smooth unforced whole-space construction is released

View in problem →

Theorem 1.1 and Section 1.1 ↑

History

A vorticity continuation criterion

View in problem →

Theorem 1.1 and Section 1.1 ↑

History

Regularity and boundary conditions matter

View in problem →

Theorem 1.1 and Section 1.1 ↑

Further reading

Read the mathematical argument; inspect its Claim for assessment and scope.

View in problem →

Theorem 1.1 and Section 1.1 ↑

Credits

Recorded authorship

OpenAI

AI involvement reported by the source

OpenAI attributes the argument to its internal research model. Bibliographic authorship is printed as OpenAI; individual human preparation and checking roles are not inferred.

Versions

OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.

2026-09-08 · selectedObserved Oct 13, 2026 · used in current publications
Finite Time Blowup for the Euler Equation

Record details

Source ID: euler-regularity-paper
Record ID: euler-regularity:source:paper:v1
Recorded title: Finite Time Blowup for the Euler Equation
Revision: 2026-09-08
Observed: Oct 13, 2026
Source updated: Not recorded
Content SHA-256: a0c234518e6c489e16996805023eb2e75c00b7c03455f7a3a5be2c124954bfdd
Update method: Compare the exact source and publish a new immutable curation revision.
Identifier policy: Stable source identity with exact versioned passages.
Check interval: 90 days
Rights: metadata-link
License: Original source license and document rights apply.

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