Navier-Stokes Problem
Do flows of water and air stay smooth forever, or can they blow up in finite time? In September 2026 OpenAI claimed a blow-up proof, but only with an external force and not yet independently verified.
Open in the interactive tree →The Navier-Stokes equations (1822-1845) describe nearly every flow of viscous fluids, from coffee to hurricanes. Unknown is whether in three dimensions smooth starting data can ever produce infinite velocities. The Clay Institute lists four variants (A to D): two smoothness proofs with no external force and two blow-up proofs with a smooth force; proving any one earns one million US dollars.
As of October 2026
On 8 September 2026 OpenAI released a 166-page proof, with a Lean formalization, that three-dimensional Navier-Stokes flow under a smooth external force develops infinite velocity in finite time (Clay statement C; the periodic case D follows); about 10,000 AI agents worked roughly 88 hours, plus 17 more for the Lean check. Tristan Buckmaster and Levent Alpöge had posted related blow-up results (including forced Euler flow) shortly before, and a priority dispute followed. On 11 September the Clay Institute said the problem had 'apparently been settled' while stressing a deliberately unhurried review; as of 18 September the proof was not independently verified, the problem is still listed as active, and OpenAI said it will not claim the prize. The problem as experts meant it (no external force, variants A and B) remains open.
What is missing
- A proof (or counterexample) for the case without external force: viscosity damps the blow-up mechanisms found so far for Euler flow, so they do not carry over (Vicol and coauthors, February 2026)
- Expert review of the 166 pages and of whether the Lean statement faithfully encodes the Clay problem: this is still under way
- Mechanisms humans can follow instead of a computing swarm (Tao: the proof is only a proxy for understanding)
- A link to physics: whether real fluids made of molecules can reach such singularities at all
Becomes possible once solved
- Flow computation with mathematical guarantees (weather, aircraft, blood flow)
- An understanding of turbulence, one of the last unexplained areas of classical physics
- New proof methods for other nonlinear equations
Open steps
- Blow-up or smoothness without force Medium AI leverageSettle the Clay variants A and B with no external force: prove smooth solutions persist, or construct blow-up from smooth starting data.
- Unstable singularities, precisely High AI leverageCompute self-similar blow-up profiles of Euler, Boussinesq and related equations to near machine precision, as seeds for computer-assisted proofs.
- Does the Lean statement match Clay? Medium AI leverageIndependently confirm that the formalized hypotheses and conclusion encode Clay's variant C faithfully, and review the 166-page argument.
- An understandable blow-up mechanism Medium AI leverageDistil the machine proof into a mechanism people can follow and reuse, so it explains why and how the flow blows up.
- Can real fluids reach singularities? Low AI leverageDetermine whether molecular fluids, where continuum equations fail at small scales, can approach such singularities, using simulation and experiment.
Where AI could help
Medium AI leverage. AI found unstable singularities and checked long proofs; the no-force Clay case, a clear mechanism and expert review remain open.
- Find blow-up solutions numerically at near machine precision with neural networks
- Write and Lean-check very long proofs, including computer-assisted estimates
- Test whether blow-up mechanisms from Euler flow survive viscosity
- Rewrite machine proofs into mechanisms that humans can follow
Shown so far
- In September 2025 DeepMind and academic partners used physics-informed neural networks to find new families of unstable singularities in related fluid equations (Boussinesq, porous media). source
- On 8 September 2026 OpenAI claimed an AI-generated, Lean-formalized proof of finite-time blow-up of 3D Navier-Stokes with a smooth external force (about 10,000 agents, about 88 hours); expert review is still under way. source
- In early September 2026 Buckmaster and Alpoge posted Lean-verified blow-up results with smooth forcing for related equations, including 3D Euler, found with LLM-generated proofs. source
Prerequisites
- Calculus (Analysis)1684
- Newtonian Mechanics1687
- Partial Differential Equations1747The Navier-Stokes equations are PDEs whose smooth solutions are still unproven
- Fluid Equations from Atoms2024-2025Derives fluid equations from particles, but does not show their solutions stay smooth
- AI Proves Open Problems2026