[#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 \(t\ge0\).
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
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]
1Records
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
- Planar drums whose spectra differ only finitelypartial differential equations
- Navier-Stokes existence and smoothnesspartial differential equations
- Simple Laplace spectrum for a generic metric in a Kahler classpartial differential equations
How to cite
TheoremDB contributors, “Three-dimensional Euler regularity,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/three-dimensional-euler-regularityThis page as plain text: three-dimensional-euler-regularity.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.
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