The Millennium Prize asks for smooth solutions. All we can prove exist globally for arbitrary data are weak solutions. That gap is the entire problem.
What is a weak solution?
Here's the situation. The Millennium Prize offers a million dollars for resolving whether 3D Navier-Stokes always has smooth solutions that last forever, or whether things can blow up. Smooth means the velocity field is perfectly well-behaved: no sudden jumps, no infinite speeds, no points where the math breaks down. But the best existence result anyone has ever proven, in nearly a century of trying, only guarantees something weaker. These are called weak solutions.
So what's a weak solution? It's not an approximation. It's not "almost right." It's an exact solution to the equations, but one that plays by relaxed rules. A normal ("classical") solution requires the velocity to be smooth enough that you can compute its rate of change at every single point. A weak solution skips that requirement. Instead of checking the equations point by point, you check them "on average" across regions of space.
Here's an analogy. A classical solution is a student who solves every exam problem by showing all their work, step by step. A weak solution is a student who can't show you the intermediate steps, but whose final answers are provably correct for every possible question you could ask. You can't watch them work, but the answers always check out.
Why would you accept that? Because sometimes the equations are too wild for classical solutions. The fluid might develop regions where the velocity changes so sharply that you simply can't compute a rate of change there. The math breaks. Weak solutions let you keep going where classical solutions give up. They're the safety net that keeps the equations alive when things get rough.
The catch: weak solutions might not be unique. You could get multiple weak solutions starting from the exact same flow, and nobody can tell you which one is "the real answer." That's a problem, because physics says the fluid should do one specific thing, not several. And weak solutions might not be smooth. Smoothness is what the Millennium Prize demands, and it's what nobody can prove.
Leray and the first existence proof (1934)
In 1934, Jean Leray did something that still defines the field. In a single 73-page paper, he proved that weak solutions to the 3D Navier-Stokes equations exist for all time, starting from any reasonable initial flow. Any. As long as the starting velocity isn't infinitely energetic or physically nonsensical, Leray guarantees you'll get a solution that lasts forever. This was the first time anyone proved a global existence result for the 3D equations, and over ninety years later, it's still the strongest unconditional existence theorem we have.
His strategy was clever. The actual equations are too nasty to solve directly because of how the fluid's velocity feeds back into itself (that's the nonlinearity). So Leray blurred the equations slightly, like adding a tiny Gaussian filter to an image. The blurred equations are tame enough to solve. Then he dialed the blur down toward zero and showed that the solutions don't fly apart. They settle into something that satisfies the original, unblurred equations in the weak sense.
But here's what Leray did NOT prove. Uniqueness. His method produces at least one weak solution, but there might be others starting from the same flow. He couldn't rule that out. He also didn't prove smoothness. His solutions have finite energy and satisfy an energy inequality: friction can drain energy away, but energy can't spontaneously appear from nowhere. That's it. Nothing more.
Leray himself suspected that singularities might form. He sketched what one might look like: the fluid collapsing toward a point, faster and faster, concentrating all its energy into a tinier and tinier region, like a whirlpool shrinking to a point at infinite speed. In 1996, Nečas, Růžička, and Šverák proved that this exact self-similar collapse can't happen. Leray's guess about the shape of potential blowup was wrong. Whether blowup happens at all, in any form? Nobody knows.
In 1951, Eberhard Hopf extended Leray's construction to fluids in bounded containers (not just all of infinite space), and the resulting class became known as Leray-Hopf weak solutions: weak solutions that satisfy the energy inequality. This is the standard notion. When researchers say "weak solutions" without further qualification, they almost always mean this.
One more thing. Even within Leray-Hopf weak solutions, there's a pickier subclass called suitable weak solutions. These don't just satisfy the energy inequality globally (total energy doesn't grow). They satisfy it locally too: energy can't secretly pile up in one corner of the fluid while draining from another. Caffarelli, Kohn, and Nirenberg (CKN) proved their famous partial regularity result in 1982 specifically for this smaller class. Don't confuse the two: CKN applies to suitable weak solutions, not to all Leray-Hopf solutions.
Strong solutions and regularity
Weak solutions exist globally. But they might not be unique, and they might not be smooth. Can we do better?
Yes, but only temporarily. Strong solutions are the upgrade: solutions where the equations hold exactly at every point, not just "on average." For smooth initial data in 3D, strong solutions exist for a short time. How short? That depends on how wild the starting flow is. Calm, gentle flows get longer guarantees. Violent, turbulent starting conditions? Microseconds.
And nobody can prove that these strong solutions don't eventually blow up.
In 1962, James Serrin proved something like a promotion rule. It goes like this: if a weak solution happens to stay well-behaved enough (not too large, not concentrating its energy into smaller and smaller regions), then it was secretly smooth the whole time. You can promote it. And by a principle called weak-strong uniqueness, it's also the only weak solution with those starting conditions. One solution, smooth and unique, case closed. But if you can't verify that the solution stays tame? Nothing. You're stuck.
This is a conditional result. IF the solution isn't too wild, THEN it's perfectly well-behaved. The entire difficulty is proving the IF.
In two dimensions, the energy estimates are strong enough that every weak solution automatically passes Serrin's test. Done. That's why 2D is solved. In 3D, the estimates fall just barely short of what you'd need, and closing that gap is the whole game.
Researchers have found other conditional tests too, each one a different angle of attack: "Prove this one specific thing about the solution, and I'll give you smoothness for free." Proving any single one of them unconditionally would solve the Millennium Problem. Nobody has managed it. For a survey of the different proof strategies people have tried, there's a whole page on that.
Smooth solutions and the Millennium Problem
Smooth solutions are the gold standard. The velocity field is perfectly well-behaved everywhere, for all time. No sudden jumps. No infinite speeds. Zoom in as far as you want, and the solution just keeps being nice.
The Clay Millennium Prize Problem, formulated by Charles Fefferman in 2000, asks a question that fits on an index card. Start with any smooth, physically reasonable velocity field filling three-dimensional space. Does the Navier-Stokes equation always produce a smooth solution that lasts forever, or can you find a starting flow where the solution eventually blows up?
Either answer is worth a million dollars.
Here's where we stand. Nobody has proven that smooth solutions always exist globally in 3D, and nobody has constructed a blowup either. We've been stuck in between since Leray's 1934 paper, over ninety years of one of the hardest open questions in all of mathematics, and we still don't know which side the answer falls on.
Short-term? Fine. For smooth starting data, the equations do produce a smooth solution for some stretch of time. The fluid starts moving, the math works, everything is clean. But what happens later? Does the solution stay smooth forever, or does it hit a point where the velocity rockets off to infinity?
If it does stay smooth, something nice happens. That smooth solution automatically satisfies the relaxed rules for weak solutions too, so it's a weak solution. And by weak-strong uniqueness, no other weak solution with those starting conditions can exist. So if someone proved global smoothness, the entire hierarchy would collapse: weak, strong, and smooth would all turn out to be the same thing, a single unique solution that's perfectly well-behaved for all time. That's what makes this problem so appealing and so hard. The gap between what we can prove exists (weak solutions) and what we want (smooth solutions) is exactly the content of the million-dollar question.
Why the distinction matters
If weak solutions exist and describe the fluid, why should anyone care about smoothness?
Three reasons.
First, uniqueness. Physics demands one answer. Give me the initial state of a fluid, and I should be able to tell you exactly what it does next. Not "here are several possibilities, pick whichever you like." But weak solutions don't guarantee that. Multiple weak solutions might emerge from the same starting flow with no way to tell which one the real fluid follows. The equations would become a menu instead of a recipe. That's not physics.
Second, numerical reliability. Many important fluid simulations are based on Navier-Stokes or closely related models: weather forecasts, aerodynamics, blood flow through arteries, and more. Make the grid finer and the simulation should converge toward the true answer. Without a smoothness-and-uniqueness guarantee? No theorem says that actually happens in every 3D scenario. The simulations work. We can't fully explain why.
Third, extreme physics. If singularities can form, that's nature sending us a message. The Navier-Stokes equations, our best model of fluid motion, would have a built-in expiration date: at some extreme scale the model itself stops working, and the equations are telling us, "You need new physics."
This isn't a technicality. It's the fault line running through everything. Existence on one side (Leray, 1934, done). Smoothness on the other (open, one million dollars). Why is crossing so hard? The energy estimates in 3D fall just barely short of what's needed, and ninety years of effort, hundreds of papers, entire careers spent trying, nobody has closed that gap.
Every proof strategy being pursued right now is an attempt to bridge this divide. Prove weak solutions are smooth. Or prove they aren't. Two words or three.
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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