# P29: Three-dimensional Euler regularity

- ID: `P29`
- Reference: `three-dimensional-euler-regularity`
- Page: https://theoremdb.org/statements/P29
- Record maturity: Reviewed problem with recorded work

## 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 \(t\ge0\).

### Context

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

### Problem setup

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

### What 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.

## Status

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](#reference-1) [2](#reference-2) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The smooth-data blow-up question and current status were checked on 2026-07-22.
- Weak and low-regularity solutions can behave differently. A settlement must use a standard classical-solution formulation.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `three-dimensional-euler-regularity`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Dongho Chae, “Incompressible Euler Equations: the blow-up problem and related results”. arXiv:math/0703405 (2007). Dongho Chae, arXiv:math/0703405, abstract and survey https://arxiv.org/abs/math/0703405
   - Also cited at Abstract and survey
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:math/0703405, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://authors.library.caltech.edu/records/tnytb-q5n55
   - website; primary source; checked 2026-08-01
   - Source use: original_summary
   - Smooth finite-energy axisymmetric Euler blowup with boundary.
3. <a id="reference-3"></a>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 https://doi.org/10.1007/s40818-025-00214-2
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Whole-space finite-energy blowup below global smooth regularity.