The governing equations
Where viscosity enters
A simplified form of the incompressible Navier–Stokes system is
∂u/∂t + (u · ∇)u = −∇p + ν∇²u + f
∇ · u = 0
The velocity u is transported by the flow itself; pressure p helps enforce incompressibility, and viscosity ν smooths velocity gradients. Dragging the cursor applies an external force f and injects passive dye. The computation uses semi-Lagrangian advection, numerical diffusion, and a Fourier-based pressure projection, broadly following the real-time Stable Fluids method of Jos Stam.
The Millennium Problem
What this lab cannot prove
The celebrated Navier–Stokes existence and smoothness question concerns sufficiently general three-dimensional solutions. In 2026, an AI-generated blow-up construction using specially chosen external forcing attracted major attention, but its ultimate mathematical assessment and relation to unforced flows require care.
This is a 2D, finite-resolution educational approximation, not a test of a 3D singularity or a proof of the Millennium Problem. Two-dimensional flows have fundamentally different mathematical regularity properties. Numerical grids cannot establish whether exact continuum equations develop singularities.
The Taylor–Green preset is a standard exact-solution benchmark: its unforced 2D velocity amplitude decays proportionally to exp(−8π²νt) on the unit periodic square. Real-time semi-Lagrangian simulations exhibit additional numerical dissipation.
Clay Mathematics Institute ↗ · September 2026 announcement ↗