What Clay is actually asking, and what counts as a solution
The prize
In 2000, the Clay Mathematics Institute picked seven of the hardest unsolved problems in mathematics and put $1 million on each one. The Navier-Stokes existence and smoothness problem made the list.
The question, stripped down: do the equations of fluid motion always produce smooth, well-behaved solutions, or can they blow up?
Nobody has claimed the prize. Not even close. There's been real progress in understanding what a solution (or a breakdown) would look like, but the problem itself remains wide open.
The precise statement
Here's what the problem actually asks, in plain language:
Setup: Take any initial fluid velocity that's perfectly smooth (no sharp edges, no discontinuities) and dies off at infinity. Far from the action, the fluid sits still.
Question: Does the velocity remain smooth and finite for all future time? Or can it blow up?
Two answers. Only two.- Yes, always smooth. Prove that no matter what smooth initial state you pick, the solution stays smooth forever. Every initial condition, every time.- No, blowup happens. Find one specific smooth starting configuration, possibly together with a smooth external force, where the solution breaks down. Just one is enough.
What makes it a Millennium Problem?
Three things put Navier-Stokes on that shortlist:- Practical importance. These equations run most of fluid dynamics: aircraft design, climate models, blood flow, ocean currents. Even without a complete proof, engineers use these equations successfully in many regimes; the open problem is about whether the 3D equations can always be justified mathematically.- Mathematical depth. It draws on analysis, geometry, topology, and physics simultaneously.- Sheer stubbornness (explore why). Over 180 years of effort by some of the greatest mathematicians who ever lived, and we still don't know the answer.A bright undergrad can state the question in five minutes. No one has found an answer. That gap between a simple statement and an unreachable proof is what defines a Millennium Problem.
History of progress
The essential milestones:- 1822: Navier derives the equations from molecular considerations.- 1845: Stokes gives the modern derivation from continuum mechanics.- 1934: Leray proves that "weak" solutions always exist. A massive result, but these solutions might not be smooth.- 1982: Caffarelli, Kohn, and Nirenberg prove that singularities (more on partial regularity), if they exist, are extremely small: in the parabolic geometry natural to these equations, the singular set has zero one-dimensional Hausdorff measure.- 1984: Beale, Kato, and Majda prove (originally for Euler, with Navier-Stokes analogues) that blowup can only happen if the vorticity becomes infinite.- 2000: Clay names it a Millennium Problem.- Today: Still open. Active work on critical-space approaches, Type-I/II blowup classification, and computer-assisted proof.
Continue exploring
This article is part of The Problem.
If you came here wondering whether someone already solved it, start with Is the Navier-Stokes Problem Solved?.
Then explore why it's so hard, or see how mathematicians have broken it into subproblems. For the structural reasons the 2D problem is tractable while 3D remains open, see Why 2D Is Easier Than 3D.
Originally published on navier-stokes.org. The site covers the Navier-Stokes existence and smoothness problem with toggleable simple/rigorous explanations.
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