Surfers and AIs chase Navier-Stokes blowup breakthrough

Navier-Stokes blowup – A $1-million Millennium Prize question—whether the Navier-Stokes equations can “blow up”—is drawing renewed attention. New computer-assisted work hints at infinities for frictionless fluids, while other results warn those hints won’t survive once viscosity is

The stakes are famously simple: solve one of the seven Millennium Problems and you take home a $1-million prize. Since the Clay Mathematics Institute published the list in 2000, it has happened only once.

For decades. the unanswered question that keeps pulling mathematicians back like gravity isn’t about abstract number games or string theory. It is about fluid motion—how water. air. and everything that flows can be predicted by equations powerful enough to govern everything from a crashing wave to the weather.

Now the race has gathered new momentum, with some prominent researchers arguing that victory may be close. Others are less convinced—pointing to how quickly artificial intelligence can appear to solve something, then fail when the last, physical details are restored.

At the center is a single test: whether the Navier-Stokes equations can “blow up.” That phrase refers to hypothetical solutions where the fluid’s speed becomes infinite. In the language of the problem. it would mean that. starting from simple laws of physics. it might be possible to construct strange fluid behaviors that “tear reality apart.”.

The problem asks whether the Navier-Stokes equations can “blow up” in this way—whether rare, nightmare-like solutions lurk inside the “limitless menagerie” of possible flows that the equations permit.

To understand why this is so hard, start with the familiar physics setup. If you want to capture a river’s flow mathematically. you’d need a perfect snapshot at one point in time—every droplet’s position and velocity. Then conservation laws of energy and momentum are supposed to determine what happens next.

For an “incompressible” fluid—water is the example that matters most for this story—those laws take the form of four coupled equations known as the three-dimensional Navier-Stokes equations. Each possible way the fluid can swirl corresponds to its own distinct solution.

The challenge is not just finding a solution. It is proving that the equations always make sense, never failing to describe reality. If they can be pushed into blowup behavior, the equations would no longer be trustworthy for that class of motion.

The usual route—friction first—targets a simpler set of equations. Viscosity. which is part of what makes fluids able to flow and exchange momentum with their surroundings. is central to the difficulty. Viscosity is the feedback between the motion of the fluid and the medium. It also makes the math especially hard to tame.

So mathematicians look to remove friction entirely, first. In that frictionless world, fluids are governed by the Euler equations. “The natural path to Navier-Stokes would be to go through Euler,” Javier Gómez-Serrano, a mathematician at Brown University, said.

That approach has already scored a milestone: in 2022, it worked for a variant of the Millennium Problem dealing with compressible fluids such as air. The feat earned mathematician Frank Merle this year’s Breakthrough Prize in Mathematics.

But incompressible fluids bring extra complications that make transfer from Euler to Navier-Stokes far from automatic. A simple example illustrates why. A person diving into one end of a swimming pool can raise the level of a floating buoy at the other. Everything affects everything else, which makes the equations much harder to control.

Gómez-Serrano’s group and others have increasingly turned to machines to search for blowup behavior. In recent months, those efforts have created a feeling of being on the edge of something—along with an urgent question: how close is “hint” to “proof,” and how much does viscosity change the outcome?

Last September. Gómez-Serrano and a group of mathematicians including him reported a preprint describing “glimpses of infinities” seen on computer screens. They were simulating a frictionless fluid trapped within a cylinder—like coffee swirling in a cup—and using AI built in collaboration with Google’s DeepMind team. The system located a point near the cup’s edge where the fluid’s speed appeared infinite.

Gómez-Serrano had framed the goal in plain terms. During a colloquium at Columbia University in March, he said, “I want to discover a blowup. I don’t care whether it’s with or without AI.” He added, “This is a tool that allowed me to go farther, so I used it.”

It could take years for the team to mathematically prove that the purported blowup truly satisfies the Euler equations. The Clay problem. after all. requires a blowup in an infinite fluid—not a cup of coffee. but something closer to a boundless sea. Still, the reported result suggested that computer-assisted searching might eventually crack the Euler equations and perhaps, in time, reach Navier-Stokes.

The optimism doesn’t come only from mathematics. Some experts have cited the work as an omen that computers could claim Navier-Stokes first—and then move on to other open problems. That prospect, however, has met skepticism.

In February, three mathematicians argued in another preprint that the AI revolution may be nowhere near providing Navier-Stokes blowups. Their target was a key simplifying assumption baked into the simulations: axial symmetry. where the fluid’s motion spins around a central axis. Many mathematicians agree that blowup solutions of the Euler equations can have this kind of symmetry—and that it helps make the simulations tractable.

But the new proof showed that practically any Euler blowup with this symmetry won’t carry over to Navier-Stokes once friction is restored. Adding viscosity would turn an infinity into a finite value. In other words, the coffee’s viscosity would keep a “tiny tornado” from ever erupting.

“It doesn’t look promising,” said Vlad Vicol, a mathematician at New York University, who co-authored the preprint. He called the situation one that would “really require a miracle” for axial symmetry to rescue the idea for Navier-Stokes. Merle agreed on the failure of the method “as it stands. ” saying. “The paper shows that the method. as it stands. does not work.”.

Even so. Vicol emphasized that if the DeepMind team truly found a blowup of the Euler equations for an infinite fluid. it would be “an incredible achievement.” He also argued that it might still be within reach for the program he and others are watching closely. “And I think that maybe this is actually within the reach of this program,” Vicol said. “Our paper basically says that just because you understood the Euler equations. you don’t get the Navier-Stokes equations for free.”.

If blowup is possible for Navier-Stokes, the February result implies it could depend on the intricate feedback between viscosity and flow. “It has to be some kind of interplay,” Vicol said. “That’s what we’re seeing.”

The trouble is that to get at that interplay. mathematicians may have to avoid the kinds of simplifying tricks computers rely on. Finding blowup from the depths of the equations might demand what machines can’t supply well: an intuition for how fluids “come together. ” and an understanding of the squishiness that doesn’t translate cleanly into today’s broad numerical search.

For Steve Shkoller, that intuition is not theoretical—it is something he has pursued in the ocean for as long as he can remember.

Shkoller has been developing what he describes as a feel for motion since he was five years old in San Diego. He is now a mathematics professor at the University of California. Davis. but he still spends at least two hours a day on the ocean near his home in Marin County. using the water for thinking. “When you surf from when you’re young. the ocean gives you this feel for motion that the equations alone do not. ” he said. He described sensing “timing. geometry. position. ” saying. “you kind of feel like the wave is a living thing. ” and “You just get these ideas.”.

That relationship between water and math hit a brutal interruption last fall when a muscle tear immobilized him for the better part of a year. Surfing and mathematics both suddenly felt out of reach.

In October, convalescing on his couch, he closed his eyes and tried to imagine himself back in the water. “I was trying to feel the energy. And then, simultaneously, I was thinking about math,” he said.

As he “mind-surfed” on what he described as a gargantuan wave, he began imagining it as a feature with infinite velocity—a shifting, life-sized picture of blowup. He also came to believe that the computer-driven approach and others like it were missing something essential.

Many recent computer-driven Euler breakthroughs have pictured blowup as a somewhat static feature called a “self-similar” shape—an image where zooming in forever on a wave’s crest always reveals the same form of curling tip. There is a reason mathematicians suspected this might be the right structure. Despite fluids’ chaotic behavior, near-identical whorls and gyres tend to emerge at almost every scale. Time and again, mathematicians have found features with astonishing symmetry and used them to make the math manageable.

Shkoller’s argument was that the assumption of self-similarity lets go of what fluid behavior really is: change.

He compared it to what water does in reality. An ocean wave can look like one cohesive form from afar, but no drop of water actually ventures far from its starting point. The wave’s contents turn over every moment.

Maybe, he mused, that ongoing turnover wasn’t an obstacle to be ignored. Maybe it was the ingredient that produces the blowup itself.

He grabbed an iPad and turned that suspicion into math.

Over the next week, he laid on his back for 12 hours a day with the tablet held aloft, scribbling equations and sketches to build a simplified picture of a blowup from his intuition. “The first three days, I was so excited, I couldn’t even sleep,” he said.

His “wave”—the shape of the infinite-velocity fluid feature he derived—was not self-similar or static. “You’re making a movie rather than one frame of the film,” Shkoller said. He described deriving his blowup from constant turnover: “Imagine every frame of the movie. you bring in an entirely new cast.” Then he proved that a blowup of the true Euler equations could precisely follow his sketch.

In March, Shkoller posted a proof to the preprint server arXiv.org. He described it as more than 100 pages long. filled with dense mathematics. and said it will likely take the community many months to verify it. Initial reactions have been promising. “No one thought it would be possible to really prove this,” said Scott Armstrong, a mathematician at New York University. “Steve appears to have done it.”.

The proof does not rely on a boundary, unlike the DeepMind work. It uses other shortcuts that the Millennium Problem won’t allow. And even with those differences. the same warning from Vicol still applies: adding friction is likely to kill Shkoller’s blowup as well. so it probably won’t carry over to Navier-Stokes.

Still, Shkoller believes his central insight will transfer—because it taps into something he says he has felt throughout his life. “You’re in an environment that’s constantly changing. Every wave is different,” he said. “They just kind of hit you—like, why didn’t I think of this before?”

The picture that emerges from these developments is not a tidy march toward the finish line. It is a push and pull between what computers can hunt and what viscous physics may refuse to surrender. On one front. simulations reported “glimpses” of infinities in the frictionless Euler world. while a February preprint warned that axial symmetry may not help once viscosity is put back in. On another front. Shkoller’s surf-driven intuition produced a proof for Euler blowups that challenges the idea that the key structure has to be self-similar.

For Navier-Stokes, the question remains brutally specific: can viscosity be navigated without turning infinity into something finite?. As the community digests proofs and critiques. the feeling among mathematicians is still the same as Shkoller’s lifelong relationship with waves—moves that look inevitable from close range. and surprises that only arrive when you stop watching for the wrong shape.

Navier-Stokes Euler equations Millennium Problem Clay Mathematics Institute blowup viscosity DeepMind artificial intelligence arXiv fluid dynamics mathematics breakthroughs Breakthrough Prize in Mathematics Steve Shkoller