OP-6262OpenPartial differential equations

Navier-Stokes existence and smoothness

TheoremDB P17 · partial differential equations

Inferred from the community field; finer classification has not been reviewed.

Source catalogue assessmentAs of 2026-09-08Source & evidence ↓

Read original source ↗ 0 Submit progress Lean 4 ↓Discussion (0)

Problem statement

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.

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Progress & source assessment

Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.

Open. Not Lean-verified

References

[1]TheoremDB P17 link
[2]Official Problem Description by Charles L. Fefferman; listed under Unsolved Millennium Prize Problems link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.