Partial differential equations · fluid dynamics · harmonic analysis

Navier–Stokes Existence and Smoothness

Collaboration beta

For smooth, rapidly decaying, divergence-free initial flow in three dimensions, must the incompressible Navier–Stokes equations produce a smooth solution for all future time?

\partial_{t} u + (u \cdot \nabla) u + \nabla p = ν Δ u, \nabla \cdot u = 0 \Rightarrow u remains smooth for all t \geq 0

Clay Millennium Prize Problems

Known results and sources

A smooth incompressible flow enters an unknown future: does viscosity prevent every finite-time singularity?

Research problem

Exact mathematical statement

For every viscosity $\nu>0$ and every smooth, divergence-free, rapidly decreasing finite-energy initial field $u_0: \mathbb R^3\to\mathbb R^3$ , the solution of

\begin{aligned} \partial_{t}u+(u\cdot\nabla)u+\nabla p&=\nu\Delta u,\\ \nabla\cdot u&=0,\\ u(\cdot,0)&=u_0 \end{aligned}

remains smooth for all $t\ge 0$ .

Problem infographic

Problem at a glance

A problem-first scientific infographic introduces a smooth divergence-free velocity field on three-dimensional space, the incompressible Navier–Stokes evolution with positive viscosity, finite kinetic energy and rapid decay, and the unresolved fork between global smoothness and finite-time breakdown. — The conjecture asks whether every smooth, rapidly decaying, finite-energy incompressible flow in three dimensions stays smooth forever; the retained source develops constraints on any hypothetical breakdown but reports no resolution.

Current mathematical picture

Where work on Navier–Stokes Existence and Smoothness stands

Open problem

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

A moment-defined characteristic frequency by itself does not supply the Fourier-support hypothesis needed for wake decoupling. The retained architecture would need a production-bearing component with enough compactness and inherited growth to enter a narrow rigidity theorem, together with a reduction of collective multiplicity and a separate treatment of the zero-viscosity corridor.

Route status · Narrowed route

Main reductionCorridor-generated active increment

The source labels the corridor-generated increment theorem as derived in Revision 6.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeSelect a production-bearing component from the corridor-generated response.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Navier–Stokes Existence and Smoothness in numbers

4.8kretained lines of mathematical investigation4,790 in the current working snapshot

Argument development: 4,175 · 87%
Explored or eliminated routes: 80 · 2%
Open obligations: 222 · 5%
Definitions and setup: 313 · 7%

9selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byClay Mathematics InstituteProposed2000Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Select a production-bearing component from the corridor-generated response.

Suggested move: Carry out Work Order NS-V6-B on the generated increment: split its energy into the bounded, subcritical, and critical branches, then prove joint time–space–frequency concentration or a quantified temporal multiplicity alternative.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Select a production-bearing component from the corridor-generated response.Suggested move: Carry out Work Order NS-V6-B on the generated increment: split its energy into the bounded, subcritical, and critical branches, then prove joint time–space–frequency concentration or a quantified temporal multiplicity alternative.

Ready to work on

Unify spatial, frequency, and temporal production multiplicity.Suggested move: Construct a positive localized spacetime production measure, select bounded derivative-mass cylinders with a fixed measure share, and either obtain a finite-energy bubble or contradict the global energy and packet budgets.

Ready to work on

Rule out the narrow viscous or Euler profiles produced by the reduction.Suggested move: Prove rigidity only for the source-generated profile classes—including the required inherited equality or finite-interval enstrophy change—while testing steady fields, traveling fields, separated packets, derivative leakage, and coordinate normalization.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen problem

Clay Mathematics Institute continues to label the problem Unsolved. Global Leray weak solutions, two-dimensional regularity, partial regularity, conditional continuation criteria, numerical scenarios, averaged-model blowup, and weak-solution nonuniqueness do not establish any of Fefferman's smooth-data global-regularity or breakdown alternatives.

[1][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025–2026
PreprintHou, Wang, and Yang claim a computer-assisted proof of nonuniqueness for unforced Leray–Hopf solutions. Their compactly supported L^2 initial datum is singular at the origin, and the checked source remains a…[10]
2023
Computational resultHou published high-resolution numerical evidence of potentially singular behavior in an axisymmetric scenario, explicitly as potential numerical evidence rather than a proof of singularity.[9]
2022
Peer reviewedAlbritton, Brué, and Colombo proved nonuniqueness of Leray solutions with the same forcing and zero initial data; forced weak nonuniqueness is distinct from Clay's smooth-solution alternatives.[8]
2016
Peer reviewedTao proved finite-time blowup for an averaged three-dimensional Navier–Stokes equation retaining energy cancellation, demonstrating a barrier to arguments based only on coarse energy structure rather than…[7]

Mathematical neighborhood

Related results and reusable starting points

Current focusExistence and Smoothness of the Navier–Stokes Equation

Weaker or relaxed formGlobal Leray weak existence

Global Leray weak solutions exist in three dimensions, but weak existence permits insufficient regularity and does not supply the uniqueness or smoothness demanded by the target.

[4]

Solved special caseTwo-dimensional incompressible Navier–Stokes

The corresponding two-dimensional incompressible regularity problem is classically solved, but the proof does not control three-dimensional vortex stretching.

[3]

Dependency or reductionPartial regularity and critical continuation criteria

Partial-regularity and critical-norm criteria sharply constrain any singularity; proving the required critical control for all smooth initial data would rule out blowup, but that bound remains unavailable.

[5][6]

Related problemAveraged Navier–Stokes blowup model

Blowup in an averaged model shows that energy cancellation alone cannot prove regularity, but the averaged equation is not the original Navier–Stokes equation.

[7]

Related problemForced weak-solution nonuniqueness

Forced Leray-solution nonuniqueness concerns weak solutions and nonzero forcing, so it is a neighboring phenomenon rather than a smooth-data blowup or global-regularity result.

[8]

Related problemUnforced Leray–Hopf weak-solution nonuniqueness

The recent unforced Leray–Hopf nonuniqueness claim uses singular L^2 initial data and remains a computer-assisted preprint; it does not address the official smooth-data alternatives.

[10]

Related problemNumerical singularity scenarios

High-resolution numerical potentially singular scenarios can guide analysis, but finite numerical resolution neither proves blowup nor certifies global regularity.

[9]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

computation · not independently reproducedPotentially singular axisymmetric Navier–Stokes computation
Published high-resolution computation supports a potentially singular scenario but is not a mathematical certificate and was not independently rerun for this collection.
[9]
software · not independently reproducedHou–Wang–Yang weak-nonuniqueness code
Public code accompanies the computer-assisted preprint on unforced Leray–Hopf weak nonuniqueness. ProofAtlas did not run it, and its singular-data weak-solution target is not the Clay statement.
[10][11]
dataset · source linked; not reproduced by ProofAtlasJohns Hopkins Turbulence Databases
Large direct-numerical-simulation datasets and query software support empirical turbulence study, but finite-resolution data do not certify global regularity or blowup.
[12]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA statement-aligned formalization of Fefferman's exact R^3 and periodic alternatives, including the official smoothness, decay, energy, pressure, and forcing hypotheses.
Formalization targetDistributions, weak derivatives, Bochner and Sobolev spaces, divergence-free vector fields, and nonlinear PDE solution concepts on R^3 and T^3.
Formalization targetLocal and global regularity, pressure reconstruction, energy inequalities, and checked continuation or blowup criteria.
Formalization targetA verified bridge distinguishing smooth classical solutions, suitable weak solutions, and Leray–Hopf solutions so neighboring nonuniqueness results cannot be mistaken for a Clay solution.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction4 of 94
lemma4 of 94

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementGlobal smoothness for 3D incompressible Navier–Stokes
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementQuasi-record production selectionintermediate
retained route statementSupported wake decouplingintermediate
retained route statementBranch-II logarithmic invariantintermediate
retained route statementPositive-viscosity reservoirintermediate
retained route statementFinite-energy response promotionintermediate
retained route statementCorridor-generated active incrementintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetSelect a production-bearing component from the corridor-generated response.open
Research targetUnify spatial, frequency, and temporal production multiplicity.open
Research targetRule out the narrow viscous or Euler profiles produced by the reduction.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.2 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
Narrowed routeSource-reported limitationA moment-defined characteristic frequency by itself does not supply the Fourier-support hypothesis needed for wake decoupling. The retained architecture would need a production-bearing component with enough compactness and inherited growth to enter a narrow rigidity theorem, together with a reduction of collective multiplicity and a separate treatment of the zero-viscosity corridor.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeSelect a production-bearing component from the corridor-generated response.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
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Prepared starting pointSelect a production-bearing component from the corridor-generated response.

Navier–Stokes Existence and Smoothness · ready to start

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Research contextPrepared context for any AI agent

For smooth, rapidly decaying, divergence-free initial flow in three dimensions, must the incompressible Navier–Stokes equations produce a smooth solution for all future time?

Exact question and boundaries
Current routes and known obstacles
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Sources and references12 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Navier-Stokes Equationmaintained problem list · Clay Mathematics Institute · Clay Mathematics Institute · accessed Aug 7, 2026
2
The Millennium Prize Problemsmaintained problem list · Clay Mathematics Institute · Clay Mathematics Institute · accessed Aug 7, 2026
3
Existence and Smoothness of the Navier–Stokes Equationoriginal source · Charles L. Fefferman · Clay Mathematics Institute · 2006 · accessed Aug 7, 2026
4
Sur le mouvement d’un liquide visqueux emplissant l’espacepeer reviewed result · Jean Leray · Acta Mathematica · 1934 · DOI 10.1007/BF02547354 · accessed Aug 7, 2026
5
Partial regularity of suitable weak solutions of the Navier-Stokes equationspeer reviewed result · Luis Caffarelli, Robert Kohn, Louis Nirenberg · Communications on Pure and Applied Mathematics · 1982 · DOI 10.1002/cpa.3160350604 · accessed Aug 7, 2026
6
L3,∞-solutions of Navier–Stokes equations and backward uniquenesspeer reviewed result · Luis Escauriaza, Gregory Seregin, Vladimír Šverák · Russian Mathematical Surveys · 2003 · DOI 10.1070/RM2003v058n02ABEH000609 · accessed Aug 7, 2026
7
Finite time blowup for an averaged three-dimensional Navier–Stokes equationpeer reviewed result · Terence Tao · Journal of the American Mathematical Society · 2016 · DOI 10.1090/jams/838 · accessed Aug 7, 2026
8
Non-uniqueness of Leray solutions of the forced Navier-Stokes equationspeer reviewed result · Dallas Albritton, Elia Brué, Maria Colombo · Annals of Mathematics · 2022 · DOI 10.4007/annals.2022.196.1.3 · accessed Aug 7, 2026
9
Potentially Singular Behavior of the 3D Navier–Stokes Equationspeer reviewed result · Thomas Y. Hou · Foundations of Computational Mathematics · 2023 · DOI 10.1007/s10208-022-09578-4 · accessed Aug 7, 2026
10
Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equationpreprint · Thomas Hou, Yixuan Wang, Changhe Yang · arXiv · 2025; revised 2026 · ARXIV 2509.25116 · accessed Aug 7, 2026
11
Code for 3D Navier–Stokes nonuniquenesssoftware or dataset · Hou Group · GitHub · accessed Aug 7, 2026
12
Johns Hopkins Turbulence Databasessoftware or dataset · Johns Hopkins University · Johns Hopkins University · accessed Aug 7, 2026

Important qualifications

The Navier–Stokes literature is enormous; this record selects statement-shaping milestones rather than attempting completeness.
The target is Fefferman's Clay formulation for three-dimensional incompressible flow on R^3 or the three-torus, not every Navier–Stokes model or solution concept.
Recent manuscripts claiming full resolution outside authoritative review were not treated as status evidence.
The Hou–Wang–Yang weak-nonuniqueness result remains at preprint posture and was not independently reproduced.
Negative formalization findings reflect a scoped search and are not proofs of nonexistence.
No packet source or submitted mathematical claim was read or used as authority.

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