One of the central unsolved questions in fluid dynamics, and one of the seven Millennium Prize Problems
The question
No, it's not solved.
The Navier-Stokes problem asks a deceptively simple question: if you start a 3D fluid flowing smoothly, does it stay smooth forever? Or can the motion become so wild that the equations break down, with smoothness breaking down in finite time?
Nobody knows.
This is the Navier-Stokes existence and smoothness problem, one of the deepest open questions in all of mathematics, and it has resisted every attempt at a proof since the equations took shape in the 19th century. People have claimed solutions. None survived. For the full status, see Is It Solved?
What we know
Unsolved doesn't mean untouched. Nearly a century of deep mathematical work has mapped the terrain and revealed exactly where the difficulty lies and why it won't yield to the tools we have:- Weak solutions exist globally (Leray, 1934). Relax the notion of "solution" to allow rough, averaged-out behavior and solutions exist for all time. Smooth? Nobody can prove it. More on approaches →- 2D is solved. Smooth solutions always exist globally in two dimensions, but three dimensions is an entirely different beast. Why 3D is harder →- Singularities, if they exist, are rare (CKN, 1982). Caffarelli, Kohn, and Nirenberg proved that the set of possible singularities has zero one-dimensional measure, meaning they can't fill even a single curve in spacetime. Subproblems and partial results →- Smooth solutions exist briefly. Start with smooth data and you get a unique smooth solution for some time interval, but whether that interval can always be extended to infinity is exactly what's unknown.- The precise formulation was set out by Charles Fefferman for the Clay Mathematics Institute. Read the Millennium Problem statement →
Why it resists proof
Here's the core difficulty. A fluid's own motion can push activity to smaller and smaller scales faster than current estimates can control. In three dimensions, the math doesn't give us enough control to rule this out. It doesn't let us prove it happens, either.
This isn't about cleverness. It isn't about computing power. The known mathematical tools are fundamentally insufficient, and that tension between concentration and dissipation is exactly why solving the problem would require genuinely new mathematics.
Supercriticality, the scaling gap, why 3D turbulence is fundamentally different: for the full story, see Why the Navier-Stokes Problem Is So Hard.
The Clay Millennium Prize
In 2000, the Clay Mathematics Institute named Navier-Stokes existence and smoothness one of seven Millennium Prize Problems, offering $1,000,000 for a correct proof or disproof. Twenty-six years later, the prize is unclaimed.
Read about the Millennium Problem →
Dive deeper
This page is a map. The territory runs deep. Pick a thread:- Is It Solved? No. Here's the current status, major published claims and the technical reasons they failed under expert scrutiny.- The Millennium Problem Demands. Precise ones.- Why It's Hard Supercriticality, turbulence, and the scaling gap that blocks every known approach from getting anywhere near a proof.
What comes next
Mathematicians haven't just stared at the problem. They've developed powerful tools, partial results, and entirely new fields of analysis trying to crack it. The work continues.
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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