Navier–Stokes

 https://openai.com/index/navier-stokes-solution/?utm_source=copilot.com

See my thinking

Short answer: no complete classical “for all data, all time” solution is known yet—but something big just happened.

Where things stand (September 2026)

Aspect	Status
Clay Millennium problem (existence & smoothness)	Claimed partially resolved (forced blow‑up)
OpenAI result	Finite‑time singularity with smooth forcing in 3D
Unforced global regularity	Still open
Clay prize	Not (yet) awarded; problem still listed “active”

What the Navier–Stokes problem asks

For 3D incompressible flow on ${\mathbb{R}}^3$, with viscosity $\nu >0$:

$${\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u=\nu \mathrm{\Delta }u-\mathrm{\nabla }p+f,\mathrm{\nabla }\cdot u=0,$$

given smooth, divergence‑free initial data $u_0$, the Clay problem asks whether:

• Either every such smooth solution with $f\equiv 0$ stays smooth for all time (global regularity),

• Or there exists smooth data (possibly with smooth forcing) that lead to finite‑time blow‑up (a singularity).

Clay formulated four alternatives (A–D); proving any one of them rigorously is enough for a “solution” in their sense.

The recent AI claim

OpenAI has released:

• An analytical proof + Lean formalization showing a 3D incompressible Navier–Stokes flow, starting from rest, with smooth compactly supported forcing, develops a singularity in finite time while kinetic energy stays bounded.

• This corresponds to Clay’s forced blow‑up alternatives (C/D), not the unforced global regularity case.

Clay has said the problem is “apparently settled” in that forced sense, but:

• The proof is still under community scrutiny.

• The unforced global regularity question remains open.

• The $1M prize has not been formally awarded.

So, “full solution”?

• We now (apparently) have a rigorous counterexample with smooth forcing: smooth data → finite‑time singularity.

• We still do not have a complete answer for the unforced 3D Navier–Stokes global regularity problem.

• Mathematically, the landscape has changed, but the classical “does every smooth unforced flow stay smooth forever?” is still an open frontier.

If you want, I can walk through the structure of the claimed blow‑up construction (vorticity, scaling, vortex filament picture) rather than just the headlines.