Navier-Stokes Equations – Instagram

OpenAl claims its Al agents have solved the Navier-Stokes equations, asserting that fluid speeds can become unbounded in finite time using smooth external forces. This breakthrough has sparked a credit dispute with mathematicians Tristan Buckmaster and Levent Alpöge, who accuse OpenAI of potentially misusing their unpublished research. While OpenAI provided computer-checked proofs, the mathematical community and the Clay Mathematics Institute have yet to formally validate these findings.

The Navier—Stokes equations use Newton’s second law of motion (“F=ma”) to describe how fluids move. Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules. These equations are used for aircraft design, weather forecasting, and the study of blood flow. A fundamental open question for these dynamical equations has been whether the continuum approximation of the fluid can break down. Specifically, can the Navier—Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time. The development of a singularity would have to happen despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, this would mark a breakdown in how the equations model the fluid. To continue modeling the system, one would then need to track the behavior of each particle individually. [1]

The OpenAI system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier—Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound. [2]

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Equations

For an incompressible Newtonian fluid with constant density and viscosity,

ρ(∂𝐮∂t+(𝐮⋅∇)𝐮)=−∇p+μ∇2𝐮+ρ𝐠\rho\left( \frac{\partial \mathbf{u}}{\partial t} +(\mathbf{u}\cdot\nabla)\mathbf{u} \right) = -\nabla p+\mu\nabla^2\mathbf{u}+\rho\mathbf{g}

 

together with conservation of mass,

∇⋅𝐮=0\nabla\cdot\mathbf{u}=0

 

Their relative importance is characterized by the Reynolds number:

Re=ρULμ=ULν\mathrm{Re}=\frac{\rho UL}{\mu}=\frac{UL}{\nu}

 

Equations

The Navier–Stokes equations describe how fluid velocity changes under pressure, viscosity, and body forces:

ρ(∂𝐮∂t+(𝐮⋅∇)𝐮)=−∇p+μ∇2𝐮+ρ𝐠\rho\left(\frac{\partial \mathbf{u}}{\partial t} +(\mathbf{u}\cdot\nabla)\mathbf{u}\right) =-\nabla p+\mu\nabla^2\mathbf{u}+\rho\mathbf{g}

 

For incompressible flow:

∇⋅𝐮=0\nabla\cdot\mathbf{u}=0

 

In essence, they apply Newton’s second law to a continuously moving fluid.

baultro. “NAVIER-STOKES BLOW-UP.” Instagram. Accessed September 12, 2026. https://www.instagram.com/reels/DdFFuQmRh4g/.

OpenAI’s 166-page paper builds an explicit smooth force that drives the 3D incompressible Navier–Stokes equations from perfect rest to infinite velocity in finite time, settling alternatives (C) and (D) of Fefferman’s Clay Millennium problem statement. This reel animates the mechanism: a self-similar vortex on a vertical axis whose inward spiral conserves angular momentum per unit mass r·uθ and so spins up without limit, while incompressibility fires two axial jets that keep the core from clogging, and the core narrows as ℓ_r ≍ τ^(1/2) with its kinetic energy ≍ τ^(1/2−3h) falling to zero even as the speed diverges. Roughly 10,000 agents ran for 88 hours to produce it and the argument was formalized in Lean, but Clay has not certified the result and the unforced problem is still open.

A snapshot of local incompressible motion. Orange marks faster angular rotation; teal marks slower rotation.
Circulating speed also depends on radius. The trajectories show inward spiraling and axial stretching.

math.visualizations. “Navier-Stokes Equations.” Instagram. Accessed September 12, 2026. https://www.instagram.com/reels/DdK6gMVhV5F/.

In September 2026 OpenAI announced a computer-assisted proof of finite-time blowup for the 3D Navier-Stokes equations under a smooth, compactly supported force, produced by an internal model it described only as more capable than GPT-6 Astra.

Vortex stretching is the lengthening of a vortex tube in a moving fluid, and the rise in spin that follows from it. It is the mechanism at the centre of the Navier-Stokes regularity problem.

References

[1] Leung, Barry. “I Don’t ******* Care That AI or ChatGPT Solved The Navier Stokes Equations.” Math Games, September 12, 2026. https://medium.com/math-games/i-dont-fucking-care-that-ai-or-chatgpt-solved-the-navier-stokes-equations-63751d812937.

[2] “On the Navier–Stokes Millennium Prize Problem.” OpenAI, September 8, 2026. https://openai.com/index/navier-stokes-solution/.

Additional Reading

∑ “Navier-Stokes Equations Explained.” Mathematical Mysteries, September 21, 2026. https://mathematicalmysteries.org/navier-stokes-equations-explained/.

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