[#P17] Navier-Stokes existence and smoothness

Problem. Let \(u_0:\mathbb{R}^3\to\mathbb{R}^3\) be smooth, divergence-free, and rapidly decaying. For every viscosity \(\nu>0\), the three-dimensional incompressible Navier-Stokes equations with initial data \(u_0\) have a globally defined smooth solution satisfying the standard energy bounds.

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1Context

Smooth solutions are known locally in time under standard hypotheses. The unresolved issue is whether singular behavior can occur in three dimensions.

2Problem setup

Definition 1 (The incompressible Navier-Stokes equations are partial differential equations for a velocity field u and pressure p, with div(u) = 0 and positive viscosity). The incompressible Navier-Stokes equations are partial differential equations for a velocity field u and pressure p, with div(u) = 0 and positive viscosity.

Definition 2 (A global smooth solution exists for every nonnegative time and has the regularity and energy bounds required by the standard problem formulation). A global smooth solution exists for every nonnegative time and has the regularity and energy bounds required by the standard problem formulation.

Remark 1. Smooth solutions are known locally in time under standard hypotheses. The unresolved issue is whether singular behavior can occur in three dimensions.

3What counts as a solution

Prove global existence and smoothness under one of the official three-dimensional formulations, or construct initial data and a solution behavior satisfying an official breakdown alternative, with every analytic condition verified.

1Status

What counts as a solution

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Global Leray-Hopf weak solutions exist in three dimensions. Smooth solutions are global in two dimensions and local, or global for suitable small data, in three dimensions. Exact unresolved remainder: For every admissible smooth finite-energy three-dimensional datum, prove global smooth existence and uniqueness, or construct admissible data producing finite-time breakdown.^[1][2]

1Packet records

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Global Leray-Hopf weak solutions exist in three dimensions. Smooth solutions are global in two dimensions and local, or global for suitable small data, in three dimensions. Exact unresolved remainder: For every admissible smooth finite-energy three-dimensional datum, prove global smooth existence and uniqueness, or construct admissible data producing finite-time breakdown.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Global Leray-Hopf weak solutions exist in three dimensions. Smooth solutions are global in two dimensions and local, or global for suitable small data, in three dimensions. Exact unresolved remainder: For every admissible smooth finite-energy three-dimensional datum, prove global smooth existence and uniqueness, or construct admissible data producing finite-time breakdown.^[1][2] Relevance to this problem For Navier-Stokes existence and smoothness, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Global Leray-Hopf weak solutions exist in three dimensions. Smooth solutions are global in two dimensions and local, or global for suitable small data, in three dimensions. Exact unresolved remainder: For every admissible smooth finite-energy three-dimensional datum, prove global smooth existence and uniqueness, or construct admissible data producing finite-time breakdown. Open full record →
Prove global existence and smoothness under one of the official three-dimensional formulations, or construct initial data and a solution behavior satisfying an official breakdown alternative, with every analytic condition verified.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove global existence and smoothness under one of the official three-dimensional formulations, or construct initial data and a solution behavior satisfying an official breakdown alternative, with every analytic condition verified.^[1] Relevance to this problem For Navier-Stokes existence and smoothness, record navier-stokes-existence-and-smoothness-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove global existence and smoothness under one of the official three-dimensional formulations, or construct initial data and a solution behavior satisfying an official breakdown alternative, with every analytic condition verified. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove global existence and smoothness under one of the official three-dimensional formulations, or construct initial data and a solution behavior satisfying an official breakdown alternative, with every analytic condition verified. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited authoritative source listed this problem as unsolved when checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

The official problem allows specified Euclidean and periodic formulations. Consult it for exact decay, forcing, energy, and breakdown conditions.

Computational notes

Numerical simulations can test proposed mechanisms and bounds, but finite-resolution calculations cannot alone rule out or establish a singularity.

2See also

Simple Laplace spectrum for a generic metric in a Kahler classpartial differential equations
Planar drums whose spectra differ only finitelypartial differential equations
Three-dimensional Euler regularitypartial differential equations

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Plain text

“Navier-Stokes existence and smoothness.” TheoremDB. P17. Problem statement; statement text SHA-256 d10504df1fae84481a090cd2cd1804be14f0e12f84af4af90e4c903bf6b07139. https://theoremdb.org/statement/?ref=P17

BibTeX

@misc{theoremdb-problem-d10504df1fae84481a090cd2cd1804be14f0e12f84af4af90e4c903bf6b07139,
  title = {{Navier-Stokes existence and smoothness}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 d10504df1fae84481a090cd2cd1804be14f0e12f84af4af90e4c903bf6b07139},
  url = {https://theoremdb.org/statement/?ref=P17}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from claymath.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Clay Mathematics Institute, Navier-Stokes Equation, official Millennium Prize Problem page, checked 2026-08-01. Official Problem Description by Charles L. Fefferman; listed under Unsolved Millennium Prize Problems. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Unsolved label and official overview.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides the authoritative current status and links the official problem description.Source named by the research packet.
Charles L. Fefferman, Existence and Smoothness of the Navier-Stokes Equation, official Clay Mathematics Institute problem description. claymath.org checked 2026-08-01. Official problem description. ↗website · primary source · checked 2026-08-01Source use: original summary.Defines the exact alternatives and summarizes established theory.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P17] Navier-Stokes existence and smoothness

1Context

2Problem setup

3What counts as a solution

1Status

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Computational notes

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