Navier–Stokes existence and smoothness and Euler blow-up problems

For the three-dimensional incompressible Navier–Stokes equations $\partial_t u+(u\cdot\nabla)u=-\nabla p+\nu\Delta u$ , $\nabla\cdot u=0$ , with $\nu>0$ and smooth divergence-free initial data $u(0,x)=u_0(x)$ , determine whether every smooth solution remains smooth for all $t\ge 0$ , or whether there exist smooth initial data for which a finite-time singularity occurs. For the three-dimensional incompressible Euler equations $\partial_t u+(u\cdot\nabla)u=-\nabla p$ , $\nabla\cdot u=0$ , determine whether there exist smooth divergence-free initial data and a finite time $T<\infty$ such that the solution becomes singular as $t\uparrow T$ ; equivalently, whether every smooth Euler solution remains smooth for all $t\ge 0$ .

References

Primary source

Zenodo (CERN European Organization for Nuclear Research)

Additional references

The OpenAI Navier-Stokes and Euler Blow-Up Proofs: A Physical Reading, Not a Physical Refutation (v5.8.0) — Zenodo (CERN European Organization for Nuclear Research) — Xavier Callens

fluid-dynamics mathematical-physics partial-differential-equations

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

OpenAI announced claimed breakthroughs on both fluid-flow problems, but neither has independent mathematical validation and both remain unconfirmed.

These problems ask whether three-dimensional fluid equations stay smooth forever, and whether the corresponding inviscid flow can develop a singularity in finite time.

September 19, 2026 repository deposit and September claims

On September 8, 2026, OpenAI announced claimed constructions for finite-time singularity in forced three-dimensional Navier–Stokes and for blow-up in unforced three-dimensional Euler, with Lean formalization involving GPT-6 Astra. The accompanying manuscripts state the claimed theorems, but independent validation is absent. A September 17 preprint imposes regularity constraints on features attributed to the Navier–Stokes construction, without refuting it. On September 19, 2026, a repository version again claimed proofs for both problems; its title and physical framing do not establish acceptance.

Current status (as of September 2026): OpenAI’s Navier–Stokes and unforced Euler resolutions remain claimed but unverified; no accepted proof or independent validation has been found.

Sources

No solutions have been posted yet.