Navier–Stokes
The Navier–Stokes research made me want to see what the math is describing.
How to read this & what to try
These equations describe how fluids move—how a whirlpool spins, stretches, and spreads out.
Imagine dropping tiny specks into a whirlpool and tracing where they go. That’s what the moving lines show. The 3D vortex pulls specks inward and sends them out through the top and bottom. Viscosity describes how much a fluid resists layers sliding past each other—think honey versus water.
Try this: Drag to look around the 3D vortex. Increase axial stretch to pull the flow inward faster, then increase viscosity to widen the core. The 2D playground lets you stir dye yourself.
Here, a mathematical “blowup” means predicted speed growing without limit in a finite time. The 3D view uses a classical Burgers vortex, which does not do that. The separate Paper explorer illustrates selected research ideas; neither animation reproduces or verifies the whole proof.
Moving tracers spiral inward, rotate around the core, and leave along the axis. Drag to orbit. Arrow keys rotate the camera.
The equation, and what you’re seeing
∂u/∂t + (u · ∇)u = −∇p + ν∇²u + f · ∇ · u = 0
Velocity carries itself through the fluid. Pressure keeps the flow incompressible, viscosity smooths it, and external forces push it around. The dye is a passive tracer that makes the motion visible.
The playground computes two-dimensional velocity on a 96 × 64 grid, with dye transported separately on a 768 × 768 GPU texture with reflecting walls, semi-Lagrangian advection, implicit diffusion, and a pressure projection. Units are dimensionless. Numerical diffusion remains even with viscosity set to zero.
The Millennium problem concerns three-dimensional equations. This small two-dimensional simulation cannot demonstrate finite-time blowup. The paper explorer evaluates selected scaling laws and shows schematic construction layers.
The research behind the idea
On September 8, 2026, OpenAI published a proposed proof of finite-time singularity formation for forced, three-dimensional incompressible Navier–Stokes flow. Its announcement also acknowledges Levent Alpöge and Tristan Buckmaster’s prior, separate work on forced Euler equations.
As of September 12, 2026, Clay’s Navier–Stokes page lists the problem as active. OpenAI says it does not intend to claim the Millennium Prize. This app is inspired by the questions raised by that research.
OpenAI: announcement & proof ↗Clay: the Millennium problem ↗