Selected research history · publication v1

Three-dimensional Euler regularity

Follow the ideas, results and limitations over time. This is a selected history, not a complete literature survey or a measure of how close the problem is to being solved.

Current state & remaining question →

Viewing the timeline from pinned publication v1.

4 selected milestones · Source coverage checked 2026-10-13 · Historical annotation and mathematical review remain separate.

1984 publication
Related result
A vorticity continuation criterion
Beale, Kato and Majda identify integrated maximum vorticity as a continuation diagnostic.
Scope & review: The criterion constrains possible breakdown; it is not an existence construction.
Statements & sources
- Finite-time smooth breakdown requires divergent integrated vorticity →
- Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations ↗Communications in Mathematical Physics 94 (1984), 61–66; criterion also recalled in the 2026 manuscript Section 1.1
- Finite Time Blowup for the Euler Equation ↗Theorem 1.1 and Section 1.1
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.
Sep 8, 2026 announcement
Proposal
A smooth unforced whole-space construction is released
OpenAI releases the Euler manuscript and formal source alongside the viscous work.
Scope & review: The announcement credits separate concurrent forced-Euler work to Alpöge and Buckmaster; its scope differs.
Statements & sources
- Smooth unforced data with finite-time breakdown →
- Finite Time Blowup for the Euler Equation ↗Theorem 1.1 and Section 1.1
- On the Navier–Stokes Millennium Prize Problem ↗Our process; concurrent work; Euler and Navier–Stokes scope
Historical annotation review: pending. The date refers to the announcement, with the precision supported by the source.
Sep 10, 2026 formalization
Formalization
Two scoped Euler proof endpoints
The pinned revision exposes both nonexistence and detailed compact-data singularity statements.
Scope & review: Formal source completion is reported externally; OpenMath has not reproduced the build.
Statements & sources
- Smooth unforced data with finite-time breakdown →
- No global smooth bounded-energy solution for selected data →
- Navier–Stokes and Euler formal proof sources ↗formalization.yaml; exact main declarations and Comparator instructions
- Navier–Stokes and Euler formal proof sources ↗Euler/Solution.lean: exists_compact_smooth_euler_singularity
- Navier–Stokes and Euler formal proof sources ↗Euler/Solution.lean: euler_breakdown_R3
Historical annotation review: pending. The date refers to the formalization, with the precision supported by the source.
Date unavailable publication
Foundation
Regularity and boundary conditions matter
Earlier constructions discussed in the manuscript use lower-regularity data, forcing or different domains.
Scope & review: Their conclusions must not be relabeled as smooth compact-data whole-space unforced blowup.
Statements & sources
- Finite Time Blowup for the Euler Equation ↗Theorem 1.1 and Section 1.1
Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

← Back to the Problem

A vorticity continuation criterion

A smooth unforced whole-space construction is released

Two scoped Euler proof endpoints

Regularity and boundary conditions matter