Partial differential equations · incompressible flow · harmonic analysis

Navier–Stokes Existence and Smoothness

Collaboration beta

Does viscosity keep every permitted smooth three-dimensional incompressible flow smooth for all future time? The current source studies constraints on a hypothetical breakdown through conditional exact-core and neutral Euler profiles. It reports useful carrier, variance and value-current restrictions, while keeping their assumptions separate. A missing PDE estimate and an exhaustive transfer from arbitrary initial data to all relevant alternatives still prevent a proof of the full problem.

\forall ν > 0, \forall u_{0} \in S (ℝ^{3}; ℝ^{3}), \nabla \cdot u_{0} = 0 : \exists (u, p) smooth on ℝ^{3} \times [0, \infty), \partial_{t} u + (u \cdot \nabla) u + \nabla p = ν Δ u, \nabla \cdot u = 0, u (0) = u_{0}, \sup_{t \geq 0} | u (t) |_{2}^{2} < \infty

Clay Millennium Prize Problems

Known results and sources

Illustration: Does positive viscosity preserve smoothness for every permitted three-dimensional initial flow?

Research problem

Exact mathematical statement

For every viscosity $\nu>0$ and every smooth, divergence-free initial velocity $u_0: \mathbb R^3\to\mathbb R^3$ that is rapidly decreasing with all derivatives, find smooth velocity and pressure on $\mathbb R^3\times[0,\infty)$ satisfying the unforced incompressible Navier–Stokes equations, the prescribed initial data and a uniform kinetic-energy bound:

\partial_tu+(u\cdot\nabla)u+\nabla p=\nu\Delta u,\qquad\nabla\cdot u=0,\qquad u(0)=u_0,\qquad\sup_{t\ge0}\int_{\mathbb R^3}|u(x,t)|^2\,dx<\infty.

This is the whole-space existence and smoothness alternative for every permitted datum. Periodic, forced, Euler, weak-solution or restricted-data statements do not replace it. The current source reports no resolution.

Problem infographic

Problem at a glance

Illustration: Smooth three-dimensional flow, incompressibility, positive viscosity and finite energy lead to an open question: smooth flow for all future time, or finite-time breakdown. The ribbons are schematic. — Illustration: For every positive viscosity and every smooth, divergence-free initial velocity on ℝ³ that decreases rapidly with all derivatives, the unforced problem asks for global smooth velocity and pressure with the prescribed initial data and uniformly bounded kinetic energy. The source establishes neither an all-data proof nor a counterexample.

Current mathematical picture

Where work on Navier–Stokes Existence and Smoothness stands

Open problem

recorded v56 frontier for the unforced all-data whole-space existence target: retain unresolved upstream extraction, baseline-EC current/carrier routes and additional MP/positive-defect hypotheses. Preserve v6 factual scope, the inactive historical computation and all source corrections without promoting reported audits or unopened proof owners.

Leading routeDirect exact-core angular and carrier routes

This route keeps the direct angular obstruction at EC-WEAK scope and the centered local carrier identity at baseline-EC scope. Neither requires MP. Angular rank two remains possible. The centered carrier has baseline windows T=L a^(−4/3), improved under EC-WEAK to T=L/a, with fixed-L endpoint error O(1/L) and no implicit interchange of limits. A vanishing complete signed interface mean on the appropriate clock would close that core, but the required depletion is open; absolute-value replacements lose the cancellation.

Route status · Active route

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 bridgeV49-I

Adjudicate actual dissipative profiles after the exact MP provider, not an arbitrary Euler profile.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. 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

6kretained lines of mathematical investigation1,196 in the current working snapshot

Argument development: 5,158 · 86%
Explored or eliminated routes: 149 · 2%
Computational analysis: 11 · 0%
Open obligations: 276 · 5%
Definitions and setup: 392 · 7%

23selected mapped statements5routes investigated20open questions20contribution-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 recordedSep 7, 2026

Recommended next task

V49-I

Adjudicate actual dissipative profiles after the exact MP provider, not an arbitrary Euler profile.

Suggested move: Adjudicate actual dissipative profiles after the exact MP provider, not an arbitrary Euler profile.

What would count as progress

Supply the missing mathematical implication with all stated source hypotheses and retained boundary cases.
Retain exact evidence for an independent source-fidelity and mathematical review; a model, calculation or branch-only result does not complete the full target.

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

28 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 routeDirect exact-core angular and carrier routes

Route status · Active route

Active routeActual MP-derived neutral profiles

This route first needs the conditional MP construction of an actual canonical Euler profile with positive local defect. The source reports targeted MP checks rather than a full reconstruction. Compensated neutrality and rough rank deficiency do not automatically supply the regularity or conservation needed for rigidity, and a smooth special-case theorem does not close the rough profile route.

Route status · Active route

Active routeTemporal simplification and pressure order zero

The temporal route concerns the fixed core U's affine-L² variance: its lower bound uses the actual MP diagonal to contradict conservation of stationary retained profiles. The fixed-rank quantifiers remain essential; allowing rank to grow with T does not supply the requested simplification. The pressure route separately needs the additional pressure-time order-zero premise in the actual retained canonical frame and normalization. The logarithmic ceiling and fixed-rank estimates do not provide that premise or a simultaneous arbitrary-frame/rank theorem; no completed EC exclusion follows.

Route status · Active route

Active routeJoint value-current incompatibility at baseline EC scope

The joint value-current route starts from baseline EC without MP. Its prospective contradiction requires a PDE-specific annular estimate from the common origin of velocity, vorticity and pressure. Finite action, the no-atom result and the critical tail admit the recorded measure model, so those abstract estimates alone do not supply the missing bound.

Route status · Active route

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementConditional exact-core and profile chain

The source's conditional chain is EC-WEAK to the exact MP provider, then V49-T10 and an actual canonical neutral Euler profile with positive defect. The provider keeps the same retained sequence, frames, fields, pressure trace, carrier decomposition and limit order. Upstream extraction and closure of all remaining singularity branches are still separate requirements.