Navier-Stokes equations
The PDEs governing viscous fluid flow. Nonlinear, exact, and generally unsolved — smoothness in 3D is a Millennium Problem.
Definition
The Navier-Stokes equations describe the motion of a viscous Newtonian fluid, combining Newton's second law (mass × acceleration = sum of forces) with the viscous stress tensor implied by the velocity gradient. In vector form for an incompressible fluid: ρ(∂v/∂t + v·∇v) = −∇p + η∇²v + f.
The first term on the left is the convective acceleration (the nonlinearity that makes turbulence hard); on the right we have pressure gradient, viscous diffusion, and external body forces. Combined with the continuity equation ∇·v = 0, they are the equations of all fluid mechanics at continuum scales.
Proving that smooth initial conditions always produce smooth solutions in three dimensions is one of the seven Millennium Prize Problems posted by the Clay Institute in 2000. A positive answer would be a major contribution to turbulence theory; a negative answer (a blow-up in finite time) would be physically extraordinary.