Global regularity or finite-time blow-up for unforced 3D Navier–Stokes (f ≡ 0)
The original supercritical question, untouched by the forced construction, so it is graded with the Millennium problems: 92, below Riemann (96) and Yang–Mills (93), exactly the value the calibration pass gave for the unforced question alone. Application value 23, just below the Clay entry (24): it would validate or limit the continuum model, with no engineering dependence. Leverage 45 is slightly below the Clay entry's 48 because fewer results are conditional on the exact answer.
The problem
Start a 3D fluid smoothly with nobody stirring it. Can it ever form an infinitely sharp spike?
- 1Smooth start, nobody stirring
- 2Whirls can feed smaller whirls
- 3Energy alone cannot stop a runaway
- 4Proof of either outcome is unknown
Statement. For the unforced equations , , with on or the 3-torus, decide whether every smooth, divergence-free initial datum with finite energy (and decay at infinity, or periodicity) yields a smooth solution for all , or whether some such datum develops a singularity in finite time. A proof of regularity must supply control beyond the energy. A proof of blow-up must construct with at a finite time, in a way that survives the smoothing effect of viscosity.
Built from
earlier problems → this oneThe problems this one is built from: progress or tools it needs (builds on), special cases that come first (generalizes), and weaker results on the way (implies). On the right, what builds on it.
- Start with
- Navier–Stokes existence and smoothness (Clay formulation)Analysis & PDE · D9 · solved
- Then
- Global regularity or finite-time blow-up for unforced 3D Navier–Stokes (f ≡ 0)This problem
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Harmonic analysisTheory
- Step 3
- The Navier–Stokes equationsTheory · host
- Then
- Global regularity or finite-time blow-up for unforced 3D Navier–Stokes (f ≡ 0)The problem
State of the art
Open. Smooth solutions are global in 2D and local in 3D, with global existence for small critical-norm data (Fujita–Kato). Leray weak solutions exist globally; uniqueness is unknown. Partial regularity: the singular set has zero 1D parabolic Hausdorff measure (Caffarelli–Kohn–Nirenberg, 1982), and Escauriaza–Seregin–Šverák (2003) showed boundedness in L^∞_t L^3_x forbids blow-up. Albritton–Brué–Colombo (2021) proved non-uniqueness of Leray solutions with a force. Tao (2014) showed blow-up for an averaged Navier–Stokes equation, indicating that any regularity proof must use the exact structure of the nonlinearity beyond energy-type identities. For Euler, Chen–Hou (2022–2023) gave a computer-assisted proof of blow-up from smooth data in a domain with boundary. OpenAI's 8 September 2026 manuscript constructs finite-time blow-up with a smooth, compactly supported force; it is unrefereed and under verification (see navier-stokes-regularity). Constantin–Ignatova–Vicol (arXiv:2609.20803) show that solutions with the construction's structural features (anisotropic Type II bounds, exact axisymmetry in a collapsing core) cannot blow up under real-analytic forcing, in particular f ≡ 0, so the unforced problem is untouched.
Realistic applications
Regularity or singularity formation for the unforced equations underpins the validity of Navier–Stokes as a model of turbulence and informs numerical-analysis guarantees, but no engineering workflow awaits the answer. Methods (a priori estimates in supercritical regimes, computer-assisted proofs for fluids) transfer to other PDEs.
What counts as solved
Either a proof that every smooth finite-energy datum with f ≡ 0 gives a global smooth solution, or an explicit smooth finite-energy datum whose solution blows up in finite time, accepted by experts. Results for the forced equation, averaged models, Euler, or non-smooth data do not count.
How it gets verified
Lean proofA formal proof checked by Lean 4 / Mathlib.
Statement: the unforced Clay alternatives, NavierStokes.navier_stokes_existence_and_smoothness_R3 (A, on ℝ³) and navier_stokes_existence_and_smoothness_periodic (B, on the torus), in Millennium/NavierStokes.lean (Formal Conjectures pinned at commit e754a78 (17 October 2026)), both with force f := 0. Global regularity is a Lean proof of either statement for all admissible data; finite-time blow-up is a proof of its negation, i.e. an explicit smooth initial velocity with no smooth finite-energy solution. The file states the equations directly with derivatives of smooth functions. A proof is accepted when it compiles against the pinned corpus and passes Comparator; expert refereeing of the paper version runs alongside. The forced alternatives (C) and (D) in the same file are labelled solved with a linked OpenAI Lean proof (September 2026); they do not settle this unforced question.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
A resolution needs a new a priori estimate or an unstable singular profile; computation and formal proof support but do not substitute.
Bottlenecks
ranked by votes- #1TheorySupercriticality of the energy
The energy is supercritical with respect to the Navier–Stokes scaling, so no known coercive quantity prevents concentration at small scales.
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Connected problems
- This problem builds on Navier–Stokes existence and smoothness (Clay formulation)Forced blow-up tools and the Lean artefact would be the starting point for any unforced blow-up attempt.
Progress
- Sep 29, 2026Forced mechanism cannot work for analytic forcesource
Constantin–Ignatova–Vicol show why the new construction needs a non-analytic force (arXiv v1 29 September 2026).
- Feb 1, 2014Averaged equation blows upsource
Tao's averaged model shows that energy structure alone cannot prove regularity.
- Jan 1, 1934Leray's weak solutionssource
Global weak solutions in 3D; regularity left open (date approximate to the year).