Navier–Stokes Existence and Smoothness
$1,000,000 USD
1822
Claude-Louis Navier · George Gabriel Stokes
Open
TL;DR
The Navier–Stokes equations describe how fluids move — water in a pipe, air over a wing, blood through arteries. They're derived from Newton's second law applied to a fluid, plus the assumption that stress is proportional to strain (a Newtonian fluid). We've used them for nearly 200 years to design ships, airplanes, and weather models.
The problem: we don't actually know if they always work.
Given smooth initial conditions in three dimensions, does a smooth solution always exist for all time? Or can the equations produce a "singularity" — a finite point in space where the velocity becomes infinite, or the solution simply ceases to exist? Physically, this would mean the mathematical model of fluid motion breaks down for some perfectly ordinary starting configuration.
In two dimensions, existence and smoothness are known. In three dimensions — the world we actually live in — nobody knows. The Clay Prize asks for a proof of global existence and smoothness, or a specific counterexample. Turbulence, one of the deepest unsolved problems in physics, is intimately tied to this question.
Formal statement
For any smooth, divergence-free initial velocity field on with rapidly decaying tails, prove that a smooth solution to the incompressible Navier–Stokes equations exists for all , with bounded energy — or provide a counterexample.
Why it matters
Almost all of computational fluid dynamics assumes the answer is yes. If it's no, our theoretical foundations for turbulence, weather prediction, and aerospace engineering rest on a mathematically unverified assumption. Either resolution transforms our understanding of continuum physics.
History
Claude-Louis Navier derived the equations in 1822, working from molecular arguments. George Gabriel Stokes gave them their modern form in 1845 using continuum mechanics. Together they've described fluid motion for almost two centuries with extraordinary empirical accuracy.
Jean Leray proved in 1934 that "weak solutions" always exist globally in three dimensions — but weak solutions may not be unique, and may develop singularities that render them non-smooth. Olga Ladyzhenskaya proved global existence and smoothness for the two-dimensional case in 1959.
Vladimir Scheffer (1976) and Luis Caffarelli–Robert Kohn–Louis Nirenberg (1982) proved that the set of possible singularities in 3D weak solutions has parabolic Hausdorff dimension ≤ 1 — meaning if singularities exist, they're constrained. Still, nobody has ruled them out.
Current status of research
Recent developments have gone in a surprising direction. Terence Tao proposed a "supercriticality barrier" in 2007: standard PDE techniques appear intrinsically unable to solve the problem because the equation is supercritical at the natural energy scale. In 2016, Tao proved global regularity fails for a modified Navier–Stokes system, suggesting singularities in the true system may be possible in principle.
On the constructive side, Terence Tao and others continue to develop machinery for potential singularity formation. On the negative side, no convincing candidate for a singularity has been produced. The community is roughly split on which way the answer will go.
Notable attempts
Mukhtarbay Otelbaev announced a proof in 2014; it was found flawed within a year. Numerous other attempts have followed the same pattern. The problem is considered one of the most difficult in analysis.
What a resolution would mean
A proof of global smoothness would justify centuries of engineering practice and open new tools for understanding turbulence. A proof of finite-time singularity formation would mean our continuum model of fluids is fundamentally incomplete — a physical singularity in a mathematical model would demand explanation, and probably lead to a new physical theory of fluid behaviour at the small scale.