Analysis and PDE · fluid dynamics · geometric analysis · singularity formation

Three-Dimensional Euler Regularity and Blow-Up

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Can every smooth finite-energy three-dimensional incompressible Euler flow remain regular for all time, or can vorticity become singular in finite time?

\partial_{t} u + (u \cdot \nabla) u = - \nabla p, \nabla \cdot u = 0, ω = \nabla \times u

Known results and sources

Vortex stretching can amplify vorticity in three-dimensional incompressible flow; whether smooth finite-energy evolution can become singular remains open.

Research problem

Exact mathematical statement

For smooth divergence-free initial velocity $u_0$ of finite kinetic energy on $\mathbb R^3$ , consider a classical solution of the incompressible Euler equations

\partial_t u+(u\cdot\nabla)u=-\nabla p,\qquad \nabla\cdot u=0.

Must every such solution remain smooth for all forward time, or can a smooth finite-energy solution develop a singularity at some finite time $T$ ? Equivalently, the open problem asks whether the vorticity $\omega=\nabla\times u$ can undergo the nonintegrable growth required for classical breakdown. The workspace contains neither a global-regularity proof nor a smooth boundaryless blow-up construction.

Problem infographic

Problem at a glance

Scientific explainer of the open three-dimensional Euler problem, showing the incompressible equations, vortex stretching, the unresolved global-smoothness versus finite-time-singularity fork, the established BKM continuation criterion, what that criterion leaves open, and the known status landscape. — The incompressible Euler equations allow vorticity stretching, while BKM identifies the vorticity growth required for breakdown; neither global smoothness nor smooth boundaryless blow-up is known.

Current mathematical picture

Where work on Three-Dimensional Euler Regularity and Blow-Up stands

Open problem

The v8 packet retains the material-vorticity oscillator and logarithmic pressure-Hessian threshold two as the strongest unconditional post-v6 advance, along with finite-energy pressure estimates. Its square-root endpoint remains conditional and is corrected to a Q4-anchored mixed-origin skeleton. The audit withdraws local pressure-child, pointwise-threshold, and unsupported ancestry readings. A bundle-level near-saturation, energy-overlap, or summation theorem for repeated coefficient-two histories remains the main frontier.

Strongest supported footholdCritical-coherence continuation candidate assembled

A full source-reported Hölder/Dini continuation chain has survived two internal audits, while external validation and literature review remain explicitly open.

Evidence posture · Reported result

Leading routeCritical-coherence continuation audit

A complete source-reported proof chain and two internal audits exist, but the route remains an externally unverified candidate and must be checked line by line before theorem-level presentation.

Route status · Active route

Useful failureReused-channel pressure-Hessian rigidity

Use the exact Cauchy–Green identities to distinguish pressure-Hessian forcing from strain-square and frame-rotation effects along a repeatedly reused material channel.

Route status · Narrowed route

Main reductionHigh-order packets and material capacity connected

The source combines near-quadratic coherent packet weights with same-time Hilbert–Piola capacity and common-label multiplicity.

Evidence posture · Reported reduction

Evidence footholdArbitrary-amplitude weighted charge theorem

For packet lower weights $a_j=c\varepsilon_j^{p_m}$ , total charge $W_J$ , weighted net dose $\mathcal A_J^{(w)}$ , and multiplicity $K_J$ , the source derives $K_J\ge W_J\exp(-2\mathcal A_J^{(w)}/W_J)$ , equivalently $\mathcal A_J^{(w)}\ge (W_J/2)\log_+(W_J/K_J)$ . This covers irregular amplitude distributions without assuming a square-root lower envelope.

Evidence posture · Manually checked · provisional

Priority open bridgeClose the current load-bearing frontier

Prove a bundle-level near-saturation, energy-overlap, or summation theorem for repeated coefficient-two pressure histories, together with ancestry and coherence-to-Q4 selection.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. 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

Three-Dimensional Euler Regularity and Blow-Up in numbers

3.7kretained lines of mathematical investigation3,682 in the current working snapshot

Argument development: 3,277 · 88%
Explored or eliminated routes: 41 · 1%
Computational analysis: 2 · 0%
Open obligations: 144 · 4%
Definitions and setup: 269 · 7%

14selected mapped statements8routes investigated6reported milestones7open questions6contribution-ready tasks

Evidence attached to the current work7 manually checked claims

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.

Status checkedAug 6, 2026

Source recordedAug 7, 2026

Recommended next task

Close the current load-bearing frontier

Prove a bundle-level near-saturation, energy-overlap, or summation theorem for repeated coefficient-two pressure histories, together with ancestry and coherence-to-Q4 selection.

Suggested move: Start from the exact material oscillator and formulate a finite-block theorem whose quantifiers do not assume nesting, common spatial localization, or pointwise saturation.

What would count as progress

Control repeated coefficient-two histories at the bundle level rather than one endpoint at a time.
Supply valid ancestry or replacement selection into the Q4-anchored skeleton.
Derive a finite-energy contradiction or summable alternative without pointwise reinterpretation.

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

27 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.

Active routeCritical-coherence continuation audit

A complete source-reported proof chain and two internal audits exist, but the route remains an externally unverified candidate and must be checked line by line before theorem-level presentation.

Route status · Active route

Active routeSummable-tree ancestry

Finite-energy capacity permits summable packet charge; the route seeks a dynamical parent–child charge through circulation, flux, pressure work, Cauchy amplification, or channel reuse.

Route status · Active route

Active routeRecord-hierarchy bridge and selection

The coherence and Sobolev record theories are separately useful but cannot be combined until time, scale, center, and irregular-amplitude selection are proved.

Route status · Active route

Active routeCoefficient-two history summation

Aggregate the exact material-oscillator lower bounds across a rigorously selected Q4-anchored packet skeleton and close the finite-energy overlap budget.

Route status · Active route

Explored alternatives

Other routes

4 recorded

Narrowed routeReused-channel pressure-Hessian rigidity

Use the exact Cauchy–Green identities to distinguish pressure-Hessian forcing from strain-square and frame-rotation effects along a repeatedly reused material channel.

Route status · Narrowed route

Narrowed routePower-sublinear packet dose or secondary profile

The exact linear endpoint is understood; a closing route now needs a genuine power saving, a sharp constant mismatch, or an explicit secondary profile from the defect.

Route status · Narrowed route

Narrowed routeComplete-profile inheritance

A complete finite-energy Euler profile is not contradictory by itself; this route seeks additional inherited record, localization, circulation, helicity, packet, or ancestry structure.

Route status · Narrowed route

Browse 1 more explored route

Narrowed routeArbitrary-amplitude charge classification

Revision v6 replaces an overstrong finite list of mechanisms by the exact summable-versus-nonsummable charge split, weighted water filling, and dyadic amplitude ledgers.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementCritical Hölder/Dini continuation proof candidate

Let $u$ be a smooth finite-energy Euler solution on $[0,T)$ . If for some $0<\alpha<1$ , $\sup_{t<T}(T-t)^{1+2\alpha/5}[\omega(t)]_{C^\alpha}<\infty$ , then the current work's proof candidate concludes that $u$ continues through $T$ . It also proposes a common-Dini-modulus extension. The source marks this [PC-A]: internally audited but not externally checked or peer reviewed.

Manually checked · provisional

Route statementArbitrary-amplitude weighted charge theorem

Manually checked · provisional

Route statementCoherence and Sobolev records remain unbridged

The coherence-frequency and high-order Sobolev-record hierarchies are separately retained. No result in the current work places a high-order packet inside a chosen long coherence octave; a temporal, scale, and center-selection bridge remains open.

Manually checked · provisional

Route statementMaterial-vorticity oscillator

Along a material trajectory, the vorticity evolution yields a second-order pressure-Hessian oscillator and a logarithmic continuation estimate with sharp coefficient two in its stated mean form.

Source-reported route statement · dependencies incomplete

Route statementFinite-energy pressure derivative cascade

The current work retains real-variable pressure estimates derived from finite energy, including the pressure derivative cascade and a local energy inverse mechanism.

Source-reported route statement · dependencies incomplete

Route statementQ4-anchored mixed-origin skeleton

Under explicit finite-block hypotheses, the square-root endpoint reduces to a mixed collection of order-four coherent cores and original high-order packets, all controlled relative to Q4.

Source-reported route statement · dependencies incomplete

Route statementLocal pressure-child inference withdrawn

The pressure inverse theorem can identify a local kinetic-energy core or a large global third pressure derivative, but it does not localize that derivative or create a parent-child pressure tree.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks

Close the current load-bearing frontier

Prove a bundle-level near-saturation, energy-overlap, or summation theorem for repeated coefficient-two pressure histories, together with ancestry and coherence-to-Q4 selection.

Suggested move: Start from the exact material oscillator and formulate a finite-block theorem whose quantifiers do not assume nesting, common spatial localization, or pointwise saturation.

Ready to work on

Bridge coherence and Sobolev record hierarchies

Construct a common-subsequence theorem that aligns a sufficiently long coherence-frequency octave with a high-order Sobolev-record event, or quantifies the local-energy and pressure-work smallness of the remaining component.

Suggested move: State and prove a precise temporal, spatial-center, and scale-selection lemma rather than comparing dimensional frequencies informally.

Ready to work on

Independently audit the critical-coherence criterion

Verify every analytic, geometric, packet-separation, and exponential-moment step of the [PC-A] continuation chain, and perform a current literature/status comparison before treating it as an established theorem.

Suggested move: Prepare a standalone line-by-line manuscript and return an explicit pass/fail finding for each item in Sections 7 and Appendix C.

Ready to work on

Prove a dynamic ancestry theorem for summable-energy trees

Charge each genuinely fresh high-frequency child branch to inherited circulation, energy flux, pressure work, or reuse of an energy-bearing material channel.

Suggested move: Formulate a parent–child lemma using Kelvin circulation, Cauchy transport, local energy flux, the componentwise Carleson law, or terminal endpoints.

Ready to work on

Prove reused-channel pressure-Hessian rigidity

Convert large material condition-number growth, after accounting for strain-square and singular-frame rotation terms, into a forbidden pressure-Hessian burden or a nonzero secondary energy profile.

Suggested move: Derive a scalar or packet-weighted inequality from the exact Cauchy–Green identities along a reused material channel.

Ready to work on

Obtain power-sublinear net dose or a secondary profile

At the order-adaptive square-root endpoint, prove a power-sublinear upper bound for total net denominator dose except for an explicitly extracted nonzero secondary profile.

Suggested move: Test intrinsic-frequency orthogonality, zero-mean selector-curl cancellation, packet-weighted square functions, and profile-defect estimates against the exact linear endpoint.

Ready to work on

Strengthen complete-profile inheritance

When a nonzero complete profile is extracted, retain additional record, endpoint, packet, circulation, helicity, recurrence, or ancestry data capable of supporting a rigidity theorem.

Suggested move: Audit which localized and material quantities survive profile extraction on both negative and positive record-time directions.

Prerequisites still open

Sourced mathematical context

The known mathematical landscape

Context collected Aug 6, 2026

Current statusOpen problem

The smooth finite-energy three-dimensional incompressible Euler regularity/blow-up problem remains open on R^3 and in the standard periodic setting: no accepted theorem proves global smoothness for all smooth data or constructs smooth boundaryless finite-time blow-up. Major neighboring advances include C^{1,alpha} blow-up on R^3 and smooth finite-energy blow-up in an axisymmetric cylinder with boundary. Those results establish genuine singularity mechanisms but do not settle the canonical smooth whole-space or periodic problem.

[6][10][11]

External progress

What the literature has established

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

2025-01-06
Peer reviewedChen and Hou completed a computer-assisted proof of stable nearly self-similar finite-time blow-up for smooth finite-energy axisymmetric Euler data in a cylinder with boundary. This rigorously resolves the Hou–Luo boundary scenario, not the boundaryless whole-space or periodic problem.[9][12]
2025
Peer reviewedChen, Elgindi, Ghoul, and Masmoudi constructed finite-energy blow-up solutions on R^3 that are smooth away from the origin but only C^{1,alpha} globally. Their result reduces the nonsmooth set to one point while leaving smooth finite-energy blow-up open.[11]
2022
Computational resultHou presented numerical evidence for a distinct, nearly self-similar potential singularity in the interior of an axisymmetric domain from smooth finite-energy data. It remains a computational scenario rather than a singularity theorem.[8]
2021
Peer reviewedElgindi proved finite-time singularity formation on R^3 for axisymmetric no-swirl solutions with C^{1,alpha} velocity for sufficiently small positive alpha. This landmark theorem lies below the smooth-data threshold of the canonical problem.[7]

Mathematical neighborhood

Related results and reusable starting points

Current focusThree-dimensional incompressible Euler regularity and blow-up problem

Solved special caseSmooth axisymmetric Euler blow-up with boundary

Smooth finite-energy blow-up is proved for axisymmetric Euler flow in a periodic cylinder with a solid boundary. The boundary suppresses normal advection at the singular point and is part of the mechanism, so this is not the free-space or torus result.

[9][10]

Weaker or relaxed formC^{1,alpha} finite-energy Euler singularity formation on R^3

Finite-time blow-up is known for finite-energy data with globally C^{1,alpha} velocity, including constructions smooth away from a single point. Weakening smoothness changes the problem and does not settle the smooth-data alternative.

[7][11]

Solved special caseSmooth axisymmetric no-swirl Euler flow

For smooth axisymmetric flow without swirl and without boundary, classical global regularity is known. Blow-up constructions in this symmetry class therefore exploit lower initial regularity, a boundary, or additional structure.

[11]

Dependency or reductionBeale–Kato–Majda vorticity continuation criterion

The Beale–Kato–Majda criterion reduces finite-time breakdown to a nonintegrable growth of maximum vorticity. It is a necessary-and-sufficient continuation diagnostic within the classical solution framework, not a decision of whether that growth can occur.

[2]

Dependency or reductionVorticity-direction coherence criteria

Geometric depletion criteria constrain how vorticity directions may remain coherent near intense vortex stretching. They exclude classes of singularity scenarios without proving global regularity for arbitrary smooth data.

[3]

Related problemNavier–Stokes existence and smoothness

The Clay Navier–Stokes existence and smoothness problem concerns viscous flow and is a distinct official Millennium Prize problem. Euler is the inviscid neighboring equation; neither problem should be labeled as the other.

[13]

Formal and computational footholds

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

formal library support · partial resource linkedScoped Lean ecosystem support
Current public Lean resources contain substantial real analysis, measure theory, geometry, and differential-equation infrastructure, but the bounded search located no exact reviewed Lean statement or proof of the smooth finite-energy 3D Euler regularity/blow-up problem.
[15][16]
computation · not independently reproducedLuo–Hou boundary singularity computation
Adaptive axisymmetric simulations reported more than 3×10^8 vorticity amplification and effective mesh resolution above (3×10^12)^2 near a candidate boundary singularity. ProofAtlas has not rerun the computation, and the 2014 result is numerical evidence.
[4][5]
computation · not independently reproducedHou interior-domain potential-singularity computation
The computation reports a stable one-scale, nearly self-similar potential singularity in the interior of an axisymmetric domain from smooth finite-energy data. ProofAtlas has not reproduced it, and it does not constitute a blow-up theorem.
[8]
software · source linked; not reproduced by ProofAtlasChen–Hou computer-assisted boundary blow-up code
The authors provide MATLAB code supporting rigorous numerical bounds used in their smooth-data boundary blow-up proof. Availability does not mean ProofAtlas independently reproduced the calculations in this metadata lane.
[9][12]

Formalization opportunities

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

Formalization targetA reviewed exact Lean statement fixing the domain (R^3 or T^3), smoothness and decay class, finite-energy hypothesis, pressure normalization, maximal classical solution, and the regularity-versus-finite-time-breakdown alternative.
Formalization targetA formal local well-posedness and continuation framework for the three-dimensional incompressible Euler equations in an appropriate Sobolev or Hölder class.
Formalization targetFormal vector-calculus and singular-integral infrastructure sufficient for vorticity evolution, Biot–Savart recovery, and the Beale–Kato–Majda logarithmic estimate.
Formalization targetA formal statement-alignment boundary separating smooth whole-space/periodic data from C^{1,alpha} data and from smooth boundary-domain theorems.
Formalization targetA checked proof of either global continuation for every admitted smooth datum or a smooth finite-energy boundaryless blow-up construction; neither was located.

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Material-oscillator frontier clarifiedThe v8 audit preserves the coefficient-two estimate and pressure cascade while narrowing the conditional skeleton and retracting unsupported ancestry claims.

Changed the research frontierLater mathematical revision

Research stage 6

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

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 updated the highlighted open task or route. 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.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

4 mapped milestonesretained argument map

Browse all 4 mapped stages

stage 1Open Euler problem and proof-candidate boundary fixed
stage 2Critical Hölder/Dini continuation chain retained for audit
stage 3Exact packet-capacity and arbitrary-charge program retained
stage 4Four decisive Euler interfaces isolated

Open Euler problem and proof-candidate boundary fixedThe baseline separates the unresolved global regularity/blow-up question from the internally audited but externally unverified critical-coherence continuation candidate.

Mapped research milestoneInitial research sequence

Research stage 1

Critical Hölder/Dini continuation chain retained for auditA complete internally audited continuation chain is recorded as a candidate with one explicit independent-review obligation.

Mapped research milestoneInitial research sequence

Research stage 2

Exact packet-capacity and arbitrary-charge program retainedHigh-order packet weights, common-time material capacity, and arbitrary-amplitude charge now constrain both nonsummable and summable packet families.

Mapped research milestoneInitial research sequence

Research stage 3

Four decisive Euler interfaces isolatedCandidate validation, reused-channel rigidity, packet-dose or ancestry control, and record-hierarchy selection are the four current closing interfaces.

Mapped research milestoneInitial research sequence

Research stage 4

Detailed research inventory

Claims, milestones, and routes in the current map

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

11 standing statements3 proposed statements6 mathematical milestones7 open questions4 narrowed routes2 conditional results

Statements by mathematical role14 selected mapped statements

theorem candidate3 of 143
reduction2 of 142
lemma6 of 146
negative result3 of 143

Checks attached to these statementsPositive checks available

Manually checked7

Selected mathematical clusters3 mathematical clusters

Euler regularity and continuation boundaryThe unresolved full problem, known BKM diagnostic, provisional coherence criterion, and conditional Type-II consequence are kept in distinct trust and dependency states.9 displayed rows · 1 route included

retained route statementThree-dimensional Euler regularity or blow-up
retained route statementBeale–Kato–Majda continuation diagnostic
retained route statementCritical Hölder/Dini continuation proof candidateintermediate
retained route statementConditional noncompact Type-II necessityconditional
DerivationThe current work's proposed proof passes from a common critical modulus to normalized vorticity/strain control, a trapping step, coherent energy cores, a terminal energy measure and atom tubes, exterior enstrophy damping, and infinitely many same-time disjoint Kelvin–Piola packets contradicting finite energy.active reported
DerivationIf the common-modulus continuation criterion is correct, a surviving singularity cannot retain any fixed critical Hölder/Dini coherence and the normalized coherence frequency must escape the Type-I scale.proposed
ChallengeThe source reports two internal audits but no external mathematical review, current literature/status search, or independently reproduced line-by-line verification. The continuation chain must remain a proof candidate until each listed analytic and geometric interface is externally checked.unsupported step · open
Research targetIndependently audit the critical-coherence criterionopen
Active routeCritical-coherence continuation auditA complete source-reported proof chain and two internal audits exist, but the route remains an externally unverified candidate and must be checked line by line before theorem-level presentation.

Exact packet, capacity, and charge programSource-designated exact high-order packet extraction, common-time capacity, arbitrary-amplitude charge, and rollback boundaries form a reusable program independent of the continuation candidate.11 displayed rows · 3 routes included

retained route statementHigh-order coherent packet extractionintermediate
retained route statementHilbert–Piola material-capacity inequalityintermediate
retained route statementArbitrary-amplitude weighted charge theoremintermediate
retained route statementExact packet program survives candidate failureintermediate
DerivationInsert the current work-energy lower weights into the common-time capacity inequality and apply Jensen's inequality with probability weights a_j/W_J to obtain the exact weighted dose/reuse lower bound.active reported
Useful failureAdd energy lower bounds obtained at different emission times.reported failure
Research targetObtain power-sublinear net dose or a secondary profileopen
Research targetProve a dynamic ancestry theorem for summable-energy treesopen
Narrowed routePower-sublinear packet dose or secondary profileThe exact linear endpoint is understood; a closing route now needs a genuine power saving, a sharp constant mismatch, or an explicit secondary profile from the defect.
Active routeSummable-tree ancestryFinite-energy capacity permits summable packet charge; the route seeks a dynamical parent–child charge through circulation, flux, pressure work, Cauchy amplification, or channel reuse.
Narrowed routeArbitrary-amplitude charge classificationRevision v6 replaces an overstrong finite list of mechanisms by the exact summable-versus-nonsummable charge split, weighted water filling, and dyadic amplitude ledgers.

Current multiscale closing interfacesPressure-Hessian rigidity, dose or ancestry control, record-hierarchy selection, and inherited profile structure are the retained full-problem routes.10 displayed rows · 3 routes included

retained route statementExact Cauchy–Green and pressure-Hessian identitiesintermediate
retained route statementCoherence and Sobolev records remain unbridgedintermediate
Useful failureTreat large material condition-number growth as an Euler contradiction by itself.reported failure
Useful failurePlace high-order Sobolev packets inside long coherence-frequency octaves by comparing dimensional scales alone.reported failure
Research targetProve reused-channel pressure-Hessian rigidityopen
Research targetBridge coherence and Sobolev record hierarchiesopen
Research targetStrengthen complete-profile inheritanceopen
Narrowed routeReused-channel pressure-Hessian rigidityUse the exact Cauchy–Green identities to distinguish pressure-Hessian forcing from strain-square and frame-rotation effects along a repeatedly reused material channel.
Active routeRecord-hierarchy bridge and selectionThe coherence and Sobolev record theories are separately useful but cannot be combined until time, scale, center, and irregular-amplitude selection are proved.
Narrowed routeComplete-profile inheritanceA complete finite-energy Euler profile is not contradictory by itself; this route seeks additional inherited record, localization, circulation, helicity, packet, or ancestry structure.

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 bridgeProve a bundle-level near-saturation, energy-overlap, or summation theorem for repeated coefficient-two pressure histories, together with ancestry and coherence-to-Q4 selection.

4 approaches have 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.

Control repeated coefficient-two histories at the bundle level rather than one endpoint at a time.
Supply valid ancestry or replacement selection into the Q4-anchored skeleton.
Derive a finite-energy contradiction or summable alternative without pointwise reinterpretation.

Continue the mathematics

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Prepared starting pointClose the current load-bearing frontier

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Research contextPrepared context for any AI agent

Can every smooth finite-energy three-dimensional incompressible Euler flow remain regular for all time, or can vorticity become singular in finite time?

Exact question and boundaries
Current routes and known obstacles
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Sources and references16 cited works · next context review by Nov 6, 2026

The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.

1
Principes généraux du mouvement des fluidesoriginal source · Leonhard Euler · Mémoires de l'Académie des Sciences de Berlin; Euler Archive mirror · 1757 · accessed Aug 6, 2026
2
Remarks on the breakdown of smooth solutions for the 3-D Euler equationspeer reviewed result · J. Thomas Beale, Tosio Kato, Andrew J. Majda · Communications in Mathematical Physics · 1984 · DOI 10.1007/BF01212349 · MR 763762 · accessed Aug 6, 2026
3
Geometric constraints on potentially singular solutions for the 3-D Euler equationspeer reviewed result · Peter Constantin, Charles Fefferman, Andrew J. Majda · Communications in Partial Differential Equations · 1996 · DOI 10.1080/03605309608821197 · accessed Aug 6, 2026
4
Potentially singular solutions of the 3D axisymmetric Euler equationspeer reviewed result · Guo Luo, Thomas Y. Hou · Proceedings of the National Academy of Sciences · 2014-08-25 · ARXIV 1310.0497 · DOI 10.1073/pnas.1405238111 · accessed Aug 6, 2026
5
Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigationpeer reviewed result · Guo Luo, Thomas Y. Hou · Multiscale Modeling & Simulation · 2014 · DOI 10.1137/140966411 · accessed Aug 6, 2026
6
Fefferman on Fluid Dynamicssurvey or monograph · Diego Córdoba · Notices of the American Mathematical Society · 2017-11 · accessed Aug 6, 2026
7
Finite-time singularity formation for C^{1,alpha} solutions to the incompressible Euler equations on R^3peer reviewed result · Tarek M. Elgindi · Annals of Mathematics · 2021-11-02 · DOI 10.4007/annals.2021.194.3.2 · MR 4334974 · accessed Aug 6, 2026
8
Potential Singularity of the 3D Euler Equations in the Interior Domainpeer reviewed result · Thomas Y. Hou · Foundations of Computational Mathematics · 2022 · ARXIV 2107.05870 · DOI 10.1007/s10208-022-09585-5 · accessed Aug 6, 2026
9
Stable Nearly Self-Similar Blowup of the 2D Boussinesq and 3D Euler Equations with Smooth Data II: Rigorous Numericspeer reviewed result · Jiajie Chen, Thomas Y. Hou · Multiscale Modeling & Simulation · 2025-01-06 · ARXIV 2305.05660 · DOI 10.1137/23M1580395 · accessed Aug 6, 2026
10
Singularity formation in 3D Euler equations with smooth initial data and boundarypeer reviewed result · Jiajie Chen, Thomas Y. Hou · Proceedings of the National Academy of Sciences · 2025-06-27 · DOI 10.1073/pnas.2500940122 · accessed Aug 6, 2026
11
Finite Time Singularities to the 3D Incompressible Euler Equations for Solutions in C-infinity(R^3 minus {0}) intersect C^{1,alpha} intersect L^2peer reviewed result · Jiajie Chen, Tarek M. Elgindi, Tej-Eddine Ghoul, Nader Masmoudi · Annals of PDE · 2025 · DOI 10.1007/s40818-025-00214-2 · accessed Aug 6, 2026
12
MATLAB codes for computer-assisted proofs in Stable Nearly Self-Similar Blowupsoftware or dataset · Jiajie Chen, Thomas Y. Hou · Jiajie Chen and Thomas Y. Hou · accessed Aug 6, 2026
13
Navier–Stokes Equationmaintained problem list · Clay Mathematics Institute · accessed Aug 6, 2026
14
Euler equations (fluid dynamics)encyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
15
Mathlib documentation indexformalization · The Mathlib Community · Lean mathematical library · accessed Aug 6, 2026
16
Formal Conjectures repositoryformalization · The Formal Conjectures Authors · Google DeepMind · accessed Aug 6, 2026

Important qualifications

The canonical scope is the smooth finite-energy Cauchy problem on R^3, with the periodic T^3 formulation as a standard neighboring version. Boundary-domain theorems and lower-regularity blow-up results are not silently promoted to this scope.
The current work's critical Hölder/Dini continuation criterion is an internally audited proof candidate. This administrative review did not validate its proof or establish novelty.
Luo–Hou and Hou interior-domain calculations are numerical evidence and were not independently reproduced by ProofAtlas in this lane.
Chen–Hou's smooth finite-energy blow-up theorem is for an axisymmetric cylinder with a solid boundary. The boundary is part of the proved mechanism and the theorem does not resolve the boundaryless R^3 or periodic T^3 problem.
Elgindi and Chen–Elgindi–Ghoul–Masmoudi prove blow-up below the globally smooth threshold; their initial velocities are only C^{1,alpha} at one or more points.
The scoped formalization search covered current public mathlib documentation, Google DeepMind's Formal Conjectures repository, and bounded public Lean searches. It found no exact reviewed Lean statement or formal proof of this smooth Euler problem, but does not establish global nonexistence.
No exact membership was verified in Epoch AI's public FrontierMath Open Problems collection. This bounded negative search does not establish permanent non-membership.
The Euler problem is related to but distinct from the Clay Millennium Navier–Stokes problem; it must not receive a Clay badge.
Recent manuscripts, packet claims, and numerical scenarios do not change the accepted open status without authoritative mathematical review.

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