Navier–Stokes existence and smoothness

Overview

Current assessment

OpenAI reports smooth forced breakdown in the whole space and on the three-torus, with complete formal proofs reported by the project. The new arguments are under assessment; unforced regularity and official evaluation have separate scopes.

Global regularity for small initial dataProvedper Fefferman (official description)
Smooth solutions on a short time intervalProvedper Fefferman (official description)
The singular set is small (Caffarelli–Kohn–Nirenberg)Provedper Fefferman (official description)

Open questions

Unforced alternatives A/B remain separate unresolved targets. The stronger manuscript construction has properties beyond the formal C/D nonexistence statements.

Latest

CMI acknowledges the announcementSep 11, 2026
Pinned formal proof revisionSep 10, 2026

Precise statement

Velocity u and pressure p satisfy $\partial_t u+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f$ and div u = 0, with viscosity ν > 0. The official question distinguishes whole-space ℝ³ from the periodic three-torus and unforced global regularity (A/B) from forced breakdown (C/D). Smooth, rapidly decaying data/force and uniformly bounded kinetic energy are required in the whole-space formulation.

Definitions

Viscosity and forcing: Positive viscosity is the coefficient of the Laplacian term; forcing is an external input. Forced breakdown does not answer every unforced global-regularity question.
Existence and Smoothness of the Navier–Stokes Equation →

Question paraphrased by OpenMath. Original wording is available in the sources.

Landscape

Figures and explorable examples. Selecting a statement opens its exact wording.

Where smoothness is guaranteed

A solution bounded in a norm on or below the scale-invariant line is smooth. Every weak solution is only known to be bounded on the higher line. The band between them is the gap: no general solution is known to have any norm there.

The gap between what is controlled and what is neededBounded here ⇒ smoothEvery weak solution is bounded hereL∞L³ endpoint

Controlling a norm at or below the line 2/p + 3/q = 1 gives smoothness.
The endpoint L∞L³: a bounded L³ norm rules out blowup.
Every Leray–Hopf weak solution controls the norms on 2/p + 3/q = 3/2.
Closing the gap for all smooth data would answer alternative (A).

Dataset navier-stokes-critical-scaling v1 · Exact rational exponent pairs (3/q, 2/p) on 2/p + 3/q = 1 for q ∈ {∞, 12, 6, 4, 10/3, 3} and on 2/p + 3/q = 3/2 for q ∈ {6, 5, 4, 3, 5/2, 2}; layers are the regions those statements describe. Software: python 3.12.13. Exact rationals, stored as floating point. SHA-256 0db7ec759c01… · Download JSON

Data

Layer	Status
Open: the gap between what is controlled and what is needed	open
Proved: bounded here ⇒ smooth (Ladyzhenskaya–Prodi–Serrin)	proved
Proved: every weak solution is bounded here (energy)	proved
Proved: L∞L³ endpoint (Escauriaza–Seregin–Šverák)	proved

Visual setup

The four official alternatives

The Clay problem is settled by proving any one of four statements: global regularity without forcing (A, B) or breakdown with forcing (C, D), in the whole space or on the torus. Select one to read it exactly.

Reading boundary. A proposed answer to C or D does not settle A or B; the forced constructions need external forcing and assert nothing about unforced flows. Statuses are the recorded ones: the C and D constructions are proposed, with source-reported formal proofs pending independent assessment.

The four official alternatives
Domain	Global regularity · no forcing	Breakdown · with forcing
Whole space ℝ³	(A) Global regularity · ℝ³○ Open	(C) Forced breakdown · ℝ³◌ Proposed Singular forced flow from rest◌ Proposed
Periodic torus 𝕋³	(B) Global regularity · 𝕋³○ Open	(D) Forced breakdown · 𝕋³◌ Proposed

◌ Proposed○ Open

Text version

Rows are the domain: the whole space ℝ³ with decaying data, or the periodic torus 𝕋³. Columns are the kind of answer: smooth global solutions for every smooth unforced initial datum (alternatives A and B), or a smooth initial datum and smooth forcing for which no smooth global solution exists (alternatives C and D). A proposed construction is recorded for C, with a stronger version starting from rest, and for D; A and B remain open.

Incompressible Navier–Stokes equations with positive viscosity in three dimensions. The official problem statement fixes the energy and smoothness conditions for each alternative.

(A) Global regularity · ℝ³
Open
Unforced global regularity in the whole space
The official alternative A is not answered by the selected forced C/D constructions.
Unforced global regularity in the whole space →
(B) Global regularity · 𝕋³
Open
Unforced global regularity on the three-torus
The official alternative B; a separate domain from whole-space regularity.
Unforced global regularity on the three-torus →
(C) Forced breakdown · ℝ³
Proposed
Forced whole-space nonexistence
Theorem 1.1 and official alternative C. The manuscript and complete formal theorem are source-reported; mathematical assessment remains pending.
Forced whole-space nonexistence →
Singular forced flow from rest
Proposed
A singular forced flow starting from rest
The manuscript gives this stronger explicit construction. The linked formal nonexistence theorem does not by itself formalize every property in this statement.
A singular forced flow starting from rest →
(D) Forced breakdown · 𝕋³
Proposed
Forced breakdown on the three-torus
Corollary 10.6 and official alternative D; no unforced conclusion is asserted.
Forced breakdown on the three-torus →

Sources & review

Authored by OpenMath curation on 2026-10-21 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

Navier–Stokes and Euler formal proof sourcesformalization.yaml; exact main declarations and Comparator instructionsProject documentationOriginal passage ↗
Existence and Smoothness of the Navier–Stokes EquationEquations (1)–(9) and alternatives (A)–(D)PublishedOriginal passage ↗
Navier–Stokes and Euler formal proof sourcesNavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_periodicProject documentationOriginal passage ↗
Navier–Stokes and Euler formal proof sourcesNavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_R3Project documentationOriginal passage ↗
Finite Time Blowup for Navier–StokesTheorem 1.1; Corollary 10.6; Section 1.1PreprintOriginal passage ↗

Visual setup

What is known in each setting

Rows are the setting, columns the kind of solution. Smoothness for all time is known in two dimensions and for small data; for arbitrary three-dimensional data it is open. Select a result to read it exactly.

Reading boundary. Each cell has its own hypotheses. Two-dimensional regularity gives no hint about three dimensions, small-data and short-time results do not extend to all data and times, and weak solutions are not known to be smooth or unique.

What is known in each setting
Setting	Smooth · short time	Smooth · all time	Weak · all time	Possible singular set
Two dimensions	No result indexed in this selection	Global smooth solutions · 2D● Proved under stated assumptions	No result indexed in this selection	No result indexed in this selection
3D · small initial data	No result indexed in this selection	Global smooth solutions · small data● Proved under stated assumptions	No result indexed in this selection	No result indexed in this selection
3D · any initial data	Smooth on a short time interval● Proved under stated assumptions	(A) Global smoothness · open○ Open	Global weak solutions (Leray)● Proved under stated assumptions	P¹(E) = 0 (Caffarelli–Kohn–Nirenberg)● Proved under stated assumptions

● Proved under stated assumptions○ Open— No result indexed in this selection

Text version

In two dimensions, and in three dimensions with small initial data, smooth solutions exist for all time. For arbitrary three-dimensional data, smooth solutions exist on a short time interval, weak solutions exist for all time (Leray) and their possible singular set is small (Caffarelli–Kohn–Nirenberg), but smoothness for all time is the open alternative (A).

Incompressible Navier–Stokes equations with positive viscosity and no external force, unless a result states otherwise. Results are as summarized in the official problem description.

Global smooth solutions · 2D
Proved
Global regularity in two dimensions
A different dimension. The official description notes it gives no hint about three dimensions, where the main difficulties lie.
Global regularity in two dimensions →
Global smooth solutions · small data
Proved
Global regularity for small initial data
Global smooth solutions exist when the initial velocity is small.
Limit. Large initial data are not covered; the smallness threshold does not grow to all data.
Global regularity for small initial data →
Smooth on a short time interval
Proved
Smooth solutions on a short time interval
Every smooth initial velocity has a smooth solution on a short time interval.
Limit. Whether this interval can always be extended to all time is exactly the problem.
Smooth solutions on a short time interval →
(A) Global smoothness · open
Open
Unforced global regularity in the whole space
The official alternative A is not answered by the selected forced C/D constructions.
Unforced global regularity in the whole space →
Global weak solutions (Leray)
Proved
Global weak solutions (Leray)
Weak solutions need not be smooth, and their uniqueness is not known.
Global weak solutions (Leray) →
P¹(E) = 0 (Caffarelli–Kohn–Nirenberg)
Proved
The singular set is small (Caffarelli–Kohn–Nirenberg)
Possible singularities of suitable weak solutions form a set of zero one-dimensional parabolic measure.
Limit. This does not exclude singular points; it is the best partial regularity theorem known, and global smoothness remains open.
The singular set cannot contain a spacetime curve, but singular points are not excluded.
The singular set is small (Caffarelli–Kohn–Nirenberg) →

Sources & review

Authored by OpenMath curation on 2026-10-21 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

Existence and Smoothness of the Navier–Stokes Equationp. 3, weak solutions (12)–(13)PublishedOriginal passage ↗
Existence and Smoothness of the Navier–Stokes Equationp. 4, Theorem (A), (B)PublishedOriginal passage ↗
Existence and Smoothness of the Navier–Stokes Equationpp. 2–3, blowup timePublishedOriginal passage ↗
Existence and Smoothness of the Navier–Stokes Equationp. 2, partial results: three dimensionsPublishedOriginal passage ↗
Existence and Smoothness of the Navier–Stokes Equationp. 2, partial results: two dimensionsPublishedOriginal passage ↗
Existence and Smoothness of the Navier–Stokes EquationEquations (1)–(9) and alternatives (A)–(D)PublishedOriginal passage ↗
Finite Time Blowup for Navier–StokesTheorem 1.1; Corollary 10.6; Section 1.1PreprintOriginal passage ↗

Results

Conditional results 2

Proposed 3

Conditional implications 1

Proved
Unbounded compactly supported velocity prevents smooth continuation
A conditional consequence of continuity on a compact space-time set; it does not certify existence of a singular Navier–Stokes solution.
OpenAI 2026
Theorem · Conditional implication
ProvedOpenAI 2026
Unbounded compactly supported velocity prevents smooth continuation
If a smooth velocity field on ℝ³ × [0,T) is supported in one fixed compact spatial set and its spatial supremum norm has infinite limsup as t ↑ T < ∞, then it has no smooth continuation to ℝ³ × [0,∞) that agrees with it before T.
Assumptions
- The same compact set contains the velocity support for every time before T.
- Continuation must agree with the original velocity before the finite time T.
Domain
One fixed compact spatial support
Force
Not specified by this continuity implication
Viscosity
Not an assumption of this implication
Data / conclusion
Unbounded velocity before T obstructs a smooth agreeing continuation; no existence construction is certified
In this problem
A conditional consequence of continuity on a compact space-time set; it does not certify existence of a singular Navier–Stokes solution.
Sources
- Finite Time Blowup for Navier–Stokes →
Open statement page →

Supporting results 2

Variants and analogues 2

Compare statements · 6

Each row is a separate assertion. Its assumptions determine which question it answers.

Statement scopes and hypotheses
Statement	Domain	Force	Viscosity	Data / conclusion
Unforced global regularity in the whole space → Open Exact statement & sources	Whole space ℝ³	None	Every ν > 0	Rapidly decreasing smooth data; global smooth solution with uniformly bounded energy
Unforced global regularity on the three-torus → Open Exact statement & sources	Unit three-torus	None	Every ν > 0	Smooth periodic data; global smooth periodic solution
Forced whole-space nonexistence → Proposed Exact statement & sources	Whole space ℝ³	Smooth, rapidly decreasing in space-time	Every ν > 0	Rapidly decreasing smooth data; no global smooth solution with uniformly bounded energy
A singular forced flow starting from rest → Proposed Exact statement & sources	Whole space ℝ³	Smooth and compactly supported	Every ν > 0	Starts from rest; velocity and pressure supported in one fixed compact spatial set before time one
Forced breakdown on the three-torus → Proposed Exact statement & sources	Unit three-torus	Smooth periodic forcing	Every ν > 0	Smooth periodic data; forced breakdown
Unbounded compactly supported velocity prevents smooth continuation → Proved Exact statement & sources	One fixed compact spatial support	Not specified by this continuity implication	Not an assumption of this implication	Unbounded velocity before T obstructs a smooth agreeing continuation; no existence construction is certified

Unforced global regularity in the whole space →

Unforced global regularity on the three-torus →

Forced whole-space nonexistence →

A singular forced flow starting from rest →

Finite Time Blowup for Navier–Stokes →

Forced breakdown on the three-torus →

Unbounded compactly supported velocity prevents smooth continuation →

Finite Time Blowup for Navier–Stokes →

Connections

Consequences 1

Assuming A singular forced flow starting from restProposed

Finite energy alone does not ensure a smooth continuation
Conditional
If the reported compactly supported flow is valid, bounded kinetic energy can coexist with a velocity singularity that prevents a smooth continuation.
Limit This consequence is conditional on the reported singular flow and supplies no unforced example.
- Unbounded compactly supported velocity prevents smooth continuationProved
Sources & review
If a smooth velocity field on ℝ³ × [0,T) is supported in one fixed compact spatial set and its spatial supremum norm has infinite limsup as t ↑ T < ∞, then it has no smooth continuation to ℝ³ × [0,∞) that agrees with it before T.
A conditional consequence of continuity on a compact space-time set; it does not certify existence of a singular Navier–Stokes solution.
Assumptions & sources
- Finite Time Blowup for Navier–Stokes ↗Theorem 1.1; Corollary 10.6; Section 1.1
Source-grounded explanation · independent explanation review: pending.

Approaches

How the problem is attacked, and what each line of attack cannot yet reach.

Amplify localized oscillations while controlling the force

The manuscript builds a leading flow, high-order corrections and localized oscillations, then constructs smooth compact forcing and a whole-space breakdown example.

Cannot yet reach The PDE-term schematic explains the setup, not the analytic estimates of the proof.

Forced whole-space nonexistence (Proposed)
Forced breakdown on the three-torus (Proposed)

Sources & review

Supporting statements

Finite Time Blowup for Navier–Stokes ↗Theorem 1.1; Corollary 10.6; Section 1.1

Source-grounded explanation · independent explanation review: pending.

Timeline

Sep 11, 2026CMI acknowledges the announcementAssessment
The Institute states that evaluation will follow its deliberately unhurried process.
The announcement records no prize award or completed evaluation.
- Forced whole-space nonexistenceProposed
- Forced breakdown on the three-torusProposed
- Clay Mathematics Institute: Navier–Stokes announcement ↗Announcement dated 11 September 2026; evaluation process
Sep 10, 2026Pinned formal proof revisionFormalization
The formal project exposes the exact C/D statements and self-assessed metadata.
This is a repository revision date, not a new discovery or an OpenMath execution.
- Forced whole-space nonexistenceProposed
- Forced breakdown on the three-torusProposed
- Navier–Stokes and Euler formal proof sources ↗formalization.yaml; exact main declarations and Comparator instructions
- Navier–Stokes and Euler formal proof sources ↗NavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_R3
- Navier–Stokes and Euler formal proof sources ↗NavierStokes/ComparatorSolution.lean: navier_stokes_breakdown_periodic
Sep 8, 2026Manuscript and formal source releasedProposal
The public report gives a whole-space theorem and periodic corollary for every positive viscosity.
It uses external forcing and does not establish unforced viscous blowup.
- Forced whole-space nonexistenceProposed
- Forced breakdown on the three-torusProposed
- Finite Time Blowup for Navier–Stokes ↗Theorem 1.1; Corollary 10.6; Section 1.1
- On the Navier–Stokes Millennium Prize Problem ↗Our process; concurrent work; Euler and Navier–Stokes scope
Sep 5, 2026A forced breakdown argument is producedProposal
OpenAI’s later announcement dates its model-produced Navier–Stokes argument to 5 September.
This is a first-party account of discovery; public release and formal checking are later events.
- Forced whole-space nonexistenceProposed
- Forced breakdown on the three-torusProposed
- On the Navier–Stokes Millennium Prize Problem ↗Our process; concurrent work; Euler and Navier–Stokes scope
Also on the Frontier: Forced three-dimensional Navier–Stokes singularity
2000Four official existence/breakdown alternativesFoundation
Fefferman specifies decay, periodicity, forcing and energy hypotheses for the Millennium question.
The four alternatives must not be collapsed into one unqualified regularity assertion.
- Existence and Smoothness of the Navier–Stokes Equation ↗Equations (1)–(9) and alternatives (A)–(D)

Full history with sources →

Evidence

Evidence by kind
Literature	2 published papers · 1 preprint · 2 projects Status OpenMath assessment · Oct 13, 2026. Proposed: Finite Time Blowup for Navier–Stokes · Forced whole-space breakdown reported ↗ · source dated Sep 8, 2026 · checked Oct 13, 2026Every positive viscosity; compact smooth force, zero initial velocity, bounded kinetic energy. Proposed: Navier–Stokes and Euler formal proof sources · Forced periodic breakdown reported ↗ · source dated Sep 8, 2026 · checked Oct 13, 2026Every positive viscosity; smooth periodic data and forcing.
Formal	2 complete proofs 2 formalizations below ↓ Reported in sources: Pinned Lean C/D nonexistence theorems are reported complete, with zero sorry and standard logical axioms. They are distinguished from the manuscript’s stronger detailed singular-flow construction; OpenMath has not executed the project.
Computation	None recorded
Review	Not yet reviewed OpenMath assessment: OpenAI reports smooth forced breakdown in the whole space and on the three-torus, with complete formal proofs reported by the project. The new arguments are under assessment; unforced regularity and official evaluation have separate scopes. Source comparison and reader curation; independent mathematical assessment remains pending. OpenMath pilot collection · OpenMath source curation · assessed Oct 13, 2026 · review due Jan 13, 2027 Collection last checked Oct 13, 2026.

Formalizations

Forced whole-space nonexistence →

Lean: forced whole-space breakdownComplete formal proof reported externally

Open formal source ↗

Complete formal proof reported externally

The formal theorem matches official alternative C with explicit decay/force and energy assumptions. The stronger from-rest compact singular-flow statement is separate and has no complete formalization asserted by this link. Metadata reports zero sorry and standard logical axioms only.

Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026

Declaration & revision

NavierStokes.Comparator.navier_stokes_breakdown_R3

Commit f9e8bc5b38b6e212696e8a30e3e91517af887bbd

Forced breakdown on the three-torus →

Lean: forced periodic breakdownComplete formal proof reported externally

Open formal source ↗

Complete formal proof reported externally

Exact periodic nonexistence theorem for every positive viscosity. The reference definitions and independent execution are separate checks; no unforced theorem is linked to this declaration.

Statement match: Linked by OpenMath · independent review separate · checked Oct 13, 2026

Declaration & revision

NavierStokes.Comparator.navier_stokes_breakdown_periodic

Commit f9e8bc5b38b6e212696e8a30e3e91517af887bbd

External proof reports do not establish an OpenMath run, independent statement correspondence, or independent mathematical review.

Sources

Choose a starting point below. Each result links to its exact statement, cited passages and checks.

Start here

Original question
Existence and Smoothness of the Navier–Stokes Equation →
Consult the original question and its hypotheses.
Proof source
Finite Time Blowup for Navier–Stokes →
Read the mathematical argument; inspect its statement for assessment and scope.
Formal source
Navier–Stokes and Euler formal proof sources →
Inspect the formal statement, assumptions and reported checks.

Bibliography 7

Existence and Smoothness of the Navier–Stokes Equation →
Charles L. Fefferman · Published · 2000
Equations (1)–(9) and alternatives (A)–(D)
Finite Time Blowup for Navier–Stokes →
OpenAI · Preprint · Sep 8, 2026
Theorem 1.1; Corollary 10.6; Section 1.1
Clay Mathematics Institute: Navier–Stokes announcement →
Clay Mathematics Institute · Project documentation · Sep 11, 2026
Announcement dated 11 September 2026; evaluation process
On the Navier–Stokes Millennium Prize Problem →
OpenAI · Project documentation · Sep 8, 2026
Our process; concurrent work; Euler and Navier–Stokes scope
Navier–Stokes and Euler formal proof sources →
OpenAI · Project documentation · Sep 8, 2026
formalization.yaml; exact main declarations and Comparator instructions
Kang–Protas · Ladyzhenskaya–Prodi–Serrin conditions →
Di Kang · Bartosz Protas · Published · Sep 6, 2022
Tao · quantitative bounds for critically bounded solutions →
Terence Tao · Published · Oct 14, 2021

Research activity · 1 entry

Formalization

Compare manuscript, comparator statement and proof dependencies

Check the official decay/periodic predicates, force regularity, energy condition and imported proof adapters at the pinned theorems.

The inspected metadata is self-assessed; a local build and independent correspondence check are not attached.

Related statements · context, not additional proved progress

Existence and Smoothness of the Navier–Stokes Equation · sources & artifacts →

Navier–Stokes and Euler formal proof sources · sources & artifacts →

Selected literature; source observations and coverage limits are retained in the publication details.

Contributors

OpenAI is the printed manuscript author and credits an internal multi-agent research-model workflow for the mathematical argument. The announcement and formal metadata credit GPT-6 Astra/Codex for formalization. Fefferman and CMI own the official statement and evaluation process; Lean, Mathlib and Formal Conjectures/Comparator contributors support the proof infrastructure. Alpöge and Buckmaster’s separately credited forced-Euler work is not this Navier–Stokes theorem.

Authors & roles

Existence and Smoothness of the Navier–Stokes Equation →

Charles L. Fefferman

No specific AI contribution is attributed by this source record.

Finite Time Blowup for Navier–Stokes →

OpenAI

OpenAI attributes the argument to its internal research model. Bibliographic authorship is printed as OpenAI; individual human preparation and checking roles are not inferred.

Clay Mathematics Institute: Navier–Stokes announcement →

Clay Mathematics Institute

No specific AI contribution is attributed by this source record.

On the Navier–Stokes Millennium Prize Problem →

OpenAI

OpenAI reports an internal multi-agent research-model workflow and later GPT-6 Astra formalization. It recognizes Alpöge and Buckmaster’s concurrent forced-Euler work, which is distinct from its unforced Euler result.

Navier–Stokes and Euler formal proof sources →

OpenAI

The project metadata names GPT-6 Astra with Codex for formalization and marks alignment review as self-assessed. Lean, Mathlib and independent statement/Comparator infrastructure have separate supporting roles.

These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.

Record details

Record version: 7
Classification: MSC2020 35Q30 / 76D03
Prerequisite level: specialist

Assess the forced C/D proofs and exact statement correspondence. CMI’s official evaluation is a separate process.

Source observations & version history →

Overview

Current assessment

Open questions

Latest

Definitions

Landscape

Where smoothness is guaranteed

The four official alternatives

Unforced global regularity in the whole space

Unforced global regularity on the three-torus

Forced whole-space nonexistence

A singular forced flow starting from rest

Forced breakdown on the three-torus

What is known in each setting

Global regularity in two dimensions

Global regularity for small initial data

Smooth solutions on a short time interval

Unforced global regularity in the whole space

Global weak solutions (Leray)

The singular set is small (Caffarelli–Kohn–Nirenberg)

Results

The question 1

Assumptions

In this problem

Sources

Special cases 2

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Bounds 1

Assumptions

In this problem

Sources

Conditional results 2

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Proposed 3

Assumptions

In this problem

Formal proofs

Sources

Assumptions

In this problem

Sources

Assumptions

In this problem

Formal proofs

Sources

Conditional implications 1

Assumptions

In this problem

Sources

Supporting results 2

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Variants and analogues 2

Assumptions

In this problem

Sources

Assumptions

In this problem

Sources

Connections

Consequences 1

Related problems

Approaches

Amplify localized oscillations while controlling the force

Timeline

Evidence