Navier–Stokes Equations
Definition
The master equations of fluid flow — Newton's second law written for every drop of fluid. Fully solving them is a million-dollar problem.
Technical definition
ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f: momentum conservation for viscous fluids. Smooth global 3D solutions are a Clay Millennium Prize problem.
Example
Weather models solve simplified Navier–Stokes equations on supercomputers — every forecast is an approximate solution of these equations.
Common confusion
Engineers use Navier–Stokes daily — the million-dollar prize is for PROVING smooth solutions always exist, not for using the equations. Applied and pure math ask different questions.