For the three-dimensional incompressible Navier–Stokes equations partial_t u+(u·nabla)u=-nabla p+nu Delta u+f, nabla· u=0, with smooth divergence-free initial data u(0,x)=u_0(x) of finite energy and smooth forcing f, determine whether the solution remains smooth for every finite time, or whether there exist such data and forcing for which a smooth solution develops a singularity at some finite time T<∞.

Navier–Stokes existence and smoothness problem

The problem asks whether smooth three-dimensional incompressible fluid flow can develop a singularity in finite time, or whether smoothness persists forever. No verified resolution is established in the retrieved material.

September 2026 claimed resolution and profile construction

On September 8, 2026, OpenAI announced that an internal system produced an analytical proof and Lean formalization claiming finite-time singularity formation from smooth data with smooth forcing and finite energy, allegedly resolving statements $C$ and $D$ . The claim is unconfirmed; the released account does not supply independent mathematical verification. Separately, Lei and Ren’s Part I constructs smooth axisymmetric profiles with a structured residual, but explicitly leaves a companion correction and therefore does not establish blowup.

Community submission (unverified)

A September 2026 submission reports a Lagrangian consistency check of the announced construction, but says the decisive issue is whether the required first-derivative structure survives the complete correction and localization architecture; it does not verify the full claim.

Current status (as of September 2026): OpenAI’s claimed counterexample is unverified, while the profile construction and its required correction do not yet settle the problem.

Navier–Stokes existence and smoothness problem

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