[#P29] Three-dimensional Euler regularity

Problem. Every smooth divergence-free initial velocity field $u_0:\mathbb{R}^3\to\mathbb{R}^3$ with suitable decay gives a smooth solution of the incompressible Euler equations for all $t\ge0$.

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

1Context

Unlike Navier-Stokes, the Euler equations contain no viscosity. Local smooth solutions are known, while global smoothness in three dimensions is unresolved.

2Remarks

Remark 1. The incompressible Euler equations describe inviscid flow and impose divergence u = 0.

Remark 2. A finite-time singularity would be a loss of smoothness from initially smooth data.

3What counts as a solution

Prove global smoothness for every initial field satisfying a standard whole-space or periodic formulation, or construct admissible smooth data and rigorously prove finite-time breakdown.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Rigorous blowup exists with a boundary and for whole-space C^{1,alpha} data smooth away from one point. Smooth finite-energy whole-space or periodic data remain unresolved. Exact unresolved remainder: Prove global smoothness for every admissible smooth whole-space or periodic datum, or rigorously prove finite-time breakdown for one.^[1][2][3]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Rigorous blowup exists with a boundary and for whole-space C^{1,alpha} data smooth away from one point. Smooth finite-energy whole-space or periodic data remain unresolved. Exact unresolved remainder: Prove global smoothness for every admissible smooth whole-space or periodic datum, or rigorously prove finite-time breakdown for one.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Rigorous blowup exists with a boundary and for whole-space C^{1,alpha} data smooth away from one point. Smooth finite-energy whole-space or periodic data remain unresolved. Exact unresolved remainder: Prove global smoothness for every admissible smooth whole-space or periodic datum, or rigorously prove finite-time breakdown for one.^[1][2][3] Relevance to this problem For Three-dimensional Euler regularity, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Rigorous blowup exists with a boundary and for whole-space C^{1,alpha} data smooth away from one point. Smooth finite-energy whole-space or periodic data remain unresolved. Exact unresolved remainder: Prove global smoothness for every admissible smooth whole-space or periodic datum, or rigorously prove finite-time breakdown for one. Open full record →
Prove global smoothness for every initial field satisfying a standard whole-space or periodic formulation, or construct admissible smooth data and rigorously prove finite-time breakdown.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove global smoothness for every initial field satisfying a standard whole-space or periodic formulation, or construct admissible smooth data and rigorously prove finite-time breakdown.^[1] Relevance to this problem For Three-dimensional Euler regularity, record three-dimensional-euler-regularity-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove global smoothness for every initial field satisfying a standard whole-space or periodic formulation, or construct admissible smooth data and rigorously prove finite-time breakdown. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove global smoothness for every initial field satisfying a standard whole-space or periodic formulation, or construct admissible smooth data and rigorously prove finite-time breakdown. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
No records match these filters.

Notes and companion material

Original intake status. The cited survey identifies finite-time singularity formation for three-dimensional incompressible Euler flow as open, and current public status was checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Weak and low-regularity solutions can behave differently. A settlement must use a standard classical-solution formulation.

Computational notes

High-resolution simulations can identify candidate blow-up scenarios but cannot alone distinguish a singularity from rapidly growing smooth behavior.

2See also

Simple Laplace spectrum for a generic metric in a Kahler classpartial differential equations
Planar drums whose spectra differ only finitelypartial differential equations
Navier-Stokes existence and smoothnesspartial differential equations

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Three-dimensional Euler regularity.” TheoremDB. P29. Problem statement; statement text SHA-256 82186740e8f81e52d7e73c0e47666d5a3d4809e12ae843bd3cb231e9474680fe. https://theoremdb.org/statement/?ref=P29

BibTeX

@misc{theoremdb-problem-82186740e8f81e52d7e73c0e47666d5a3d4809e12ae843bd3cb231e9474680fe,
  title = {{Three-dimensional Euler regularity}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 82186740e8f81e52d7e73c0e47666d5a3d4809e12ae843bd3cb231e9474680fe},
  url = {https://theoremdb.org/statement/?ref=P29}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Dongho Chae, “Incompressible Euler Equations: the blow-up problem and related results”. arXiv:math/0703405 (2007). Dongho Chae, arXiv:math/0703405, abstract and survey. ↗preprint · primary source · arXiv:math/0703405, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited survey identifies finite-time singularity formation for three-dimensional incompressible Euler flow as open, and current public status was checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract and survey.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked standard problem framing.Source named by the research packet.
Jiajie Chen and Thomas Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis. authors.library.caltech.edu checked 2026-08-01. Abstract. ↗website · primary source · checked 2026-08-01Source use: original summary.Smooth finite-energy axisymmetric Euler blowup with boundary.
Diego Córdoba, Luis Martinez-Zoroa, and Fan Zheng, “Finite Time Singularities to the 3D Incompressible Euler Equations for Solutions in$$\:C^{\infty}(\mathbb{R}^3 \setminus \{0\})\cap C^{1,\alpha}\cap L^2$$”. Annals of PDE 11(2) (2025), 19. DOI 10.1007/s40818-025-00214-2. Abstract and introduction. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Whole-space finite-energy blowup below global smooth regularity.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P29] Three-dimensional Euler regularity

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance