Navier–Stokes existence and smoothness

The statement. The Navier–Stokes equations describe how the velocity u(x, t) of a fluid changes: ∂u/∂t + (u·∇)u = −∇p + ν∇²u, with ∇·u = 0 (incompressible). Given smooth initial velocity in three dimensions, do smooth solutions exist for all time? Or can the velocity become infinite somewhere in finite time?

In plain words. We use these equations for weather, aircraft and blood flow every day, and nobody knows whether they are even mathematically well-behaved in 3D. In two dimensions the answer is yes (Ladyzhenskaya). In three, energy could in principle cascade into ever-smaller whirls until something breaks — and we cannot rule it out.

What you need first. Stage 6 (derivatives, the chain rule, integrals), Stage 7 (vectors and matrices, because ∇ acts on vector fields), Stage 11 (differential equations — Navier–Stokes is a system of nonlinear partial ones), Stage 12 (limits, convergence, what “smooth” means precisely).

The parts. (1) Reading each term: the time change of velocity, the fluid carrying its own velocity (the nonlinear (u·∇)u term that causes all the trouble), pressure pushing, viscosity smoothing. (2) Energy: the total kinetic energy can only decrease — the one thing we can prove. (3) Scaling: the equations look the same at every scale, so a solution could concentrate into a smaller and smaller region without violating the energy bound. (4) Weak solutions (Leray, 1934) exist forever but may not be smooth or unique; the question is whether the smooth ones survive.

What is known. Global smooth solutions exist for small initial data and for short times in general; 2D is fully solved; partial regularity says singularities, if any, are “small” (Caffarelli–Kohn–Nirenberg). A Millennium Prize problem; both a proof of regularity and a blow-up example would win it.