Source · Paper

Finite Time Blowup for the Euler Equation

OpenAI

PreprintPublication date Sep 8, 2026

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.

OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.

Source terms ↗