[#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 on this problem in ChatGPT

Definitions and notation

1Context

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

2Problem setup

Definition 1 (The incompressible Euler equations describe inviscid flow and impose divergence u = 0). The incompressible Euler equations describe inviscid flow and impose divergence u = 0.

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

Remark 1. Unlike Navier-Stokes, the Euler equations contain no viscosity. Local smooth solutions are known, while global smoothness in three dimensions is unresolved.

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

What counts as a solution

Current 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	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

2Problem setup

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

2Problem setup

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