Why the Navier–Stokes Equations Push Mathematics and Physics to the Edge

These are the Navier-Stokes equations. They're some of the most important and the most challenging equations in physics. Their purpose is simple, to describe the motion of fluids in their countless forms. But hidden in this elegant formulation is a profound uncertainty. It's one of the cleanest, simplest models of a supercritical equation that we don't understand. That's because within this term lies the seed of turbulence, one of the deepest and most elusive mysteries in physics. Turbulence is so complex that we can't find exact solutions to the Navier-Stokes equations. Instead, we use them as a lighthouse to guide our calculations of fluid motion using numerical simulations. But even after centuries of success, mathematicians still couldn't prove something far more basic, that solutions to the equations always behave well. That challenge became one of seven Millennium Prize problems, each worth one million dollars. And it's the only one grounded in classical physics. And now an AI-powered proof has finally answered the question. So why are these still among the hardest equations in physics? And how does the new proof change our understanding of fluid motion? Seventy years after Newton changed the world with his laws of motion, Swiss mathematician Leonard Euler built upon these laws to describe how fluids move. Euler saw that fluids, like solids, should obey Newton's second law. But fluids are not solid objects. They move and flow and deform. They require a new way of thinking. Euler imagined breaking the fluid into a grid of infinitesimally tiny volumes. He then tracked the velocity of the fluid flow passing through each volume in this grid. Taken together, these velocities build up a vector field, a snapshot of the motion of the entire fluid. By evolving this vector field over time, Euler could forecast how the fluid would flow. He expressed his insights in two equations. The first comes from the conservation of mass. Mass flowing into a given volume must equal the mass flowing out. So the total flux of the fluid velocity across all directions must add up to zero. Then, building on Newton's second law, Euler derived a similar equation for conservation of momentum. He turned mass times acceleration into density times the rate of change of the fluid's velocity. This term is called the material derivative, and we'll dig into it more soon. On the other side, Euler accounts for forces acting on the fluid. This is the force on the fluid due to pressure. This triangle, called nabla, indicates the direction of the steepest pressure change. This part accounts for any external forces acting on the fluid. Often this is gravity, which is density times gravitational acceleration. With these equations, Euler founded the field of fluid mechanics, the study of how liquids, gases, and plasmas move. To this day, Euler's equations remain remarkably accurate for describing streamlined flows, where layers of fluid glide smoothly past one another in predictable patterns. But they fail to capture the vast majority of real-world fluid motion. Because they leave out something essential, something at the very heart of what defines a fluid, and makes its behavior so difficult to predict. For nearly a century, the world's greatest mathematical minds searched for this missing link. In 1822, French civil engineer Claude-Louis Navier proposed a bold idea. What if fluids experienced internal friction? He was on to something, but his formulation wasn't complete. In 1845, the Irish mathematical physicist George Stokes took up the problem. He refined the equations, grounding them in a more rigorous framework. The collective work produced the missing term, viscosity, the measure of a fluid's internal stress. This viscosity arises because when layers of a fluid slide past each other, they create friction. This coefficient is a measure of a fluid's internal resistance to flow, how thick or sticky it is. It's high for honey and low for air. This term, called the Laplacian, captures how much the velocity is changing locally between neighboring layers. Where velocity varies rapidly between neighboring fluid layers, the fluid experiences more internal friction due to viscosity. With the addition of this term, we've arrived at the modern Navier-Stokes equations. This is the simplified form of the equations for fluids that are incompressible, meaning their densities don't change. Scientists and engineers have used these equations to successfully capture the flow of countless real-world fluids with exceptional accuracy. But if we've been using the Navier-Stokes equations for over 100 years, why have they posed such a profound mathematical challenge? The answer has to do with turbulence, the most unpredictable motion in physics. Consider a calm river. The entire fluid moves in the same direction at the same speed. But if the river fractures into an eddy, part of the flow starts moving in a different direction at a different velocity. That eddy will fracture into another and another, launching an unpredictable cascade of directions, all intermixing at various scales. This is turbulent flow. To understand why turbulence pushes math to its edge, we must unpack this term, the material derivative. It measures how a fluid accelerates. This part tells us how a fluid's velocity changes over time at one fixed point in space. That approach works fine for solids. If the velocity changes here, the whole object moves too. But to understand the movement of a fluid, you have to look beyond a single spot and track how the surrounding flows reshape what you observe. That's where the convective term comes in. It accounts for how a fluid's velocity changes as it moves. To calculate this term, we first measure how the flow's velocity varies across space. These rates of change are spatial derivatives. Then we combine those derivatives according to the speed and direction of the flow. This gives us the convective term. Calculating this convection term is like trying to measure the wind while inside a hot air balloon being blown by the wind. You can't get an accurate reading of the isolated wind velocity because you're trying to measure it while being affected by it. This feedback loop turns the equations non-linear. Small changes amplify unpredictably into chaotic motion. Turbulence is a problem. Famously, weather prediction becomes unreliable after one or two weeks. And it is because the equations are very chaotic. This chaos is why we can't find general exact solutions to the Navier-Stokes equations. Instead, scientists rely on simulations. They break the fluid into many discrete chunks and compute how each piece interacts with and affects its neighbors. This process demands immense computing power and still only gives us approximations. But even beyond the limits of computing, there's a deeper puzzle, one that demands a mathematical proof. In turbulent flow, energy dissipates from large eddies into smaller and smaller ones until eventually they diffuse into heat. But in theory, the opposite could happen. Instead of dissipating, energy might focus inward, funneling into a vanishing point and causing the fluid there to reach infinite velocity. In this scenario, the equations don't just fail to predict the flow. They blow up to infinity and stop making sense at all. In the real world, we know that nature has limits and that fluids don't spontaneously accelerate to infinite speeds. But until recently, no one had proved that mathematically the equations won't break under extreme conditions. This was the question at the heart of the Millennium Prize problem. Prove that solutions to the Navier-Stokes equations always exist and remain smooth, meaning that the flow stays continuous, free of infinite spikes, and with well-defined velocities at every point, no matter how close you look. Or find a counterexample, a situation where the equations blow up and the math fails to hold. In 1934, Jean Larrey made a breakthrough. He proved that weak solutions, which are lower-resolution, averaged versions of the velocity field, always exist. From there, a new approach emerged. Start with Larrey's weak solutions and try to smooth them out, gradually extending them to sharper, more detailed versions of the equations. Over the following decades, mathematicians learned when the equations behaved well and which tools weren't powerful enough to settle the full three-dimensional case. In 2016, mathematician Terence Tao offered a twist. Instead of trying to rule out a singularity, he created one. He constructed a modified version of the Navier-Stokes equations and showed, for the first time, that they can blow up. What I showed is that if you cheated, if you turned off some of the nonlinear effects, but you just kept some of them, the nonlinear is still strong enough to create blow-up. But I rigged the game. You know, it's like I had this casino where you just sort of shut off all the parts of the world that would lead to you losing and only lead to you winning. In 2021, Diego Córdova and Luis Martínez Zorro took an unconventional approach to the problem. Working with the Euler equations, the mathematicians developed an analytic strategy to study singularities without relying on computer simulations. By 2023, they had constructed an infinite sequence of fluid flows. Each layer remained well-behaved on its own, driven by a smooth external force, a technique mathematicians call forcing. But when the layers were combined, they produced an infinite cascade, a vortex that concentrated motion into an ever-smaller region until its velocity blew up, forming a singularity. But there was a problem. The forcing function they got when they combined their layers wasn't smooth. That meant their construction could produce a singularity in Euler, but it didn't meet the smooth forcing conditions needed to satisfy the criteria of the Millennium Prize problem. In September 2026, mathematicians Tristan Buckmaster and Levent Alpuguet, working with AI models, found a way forward. They were able to smooth out the external force, causing the Euler equations to blow up. Then, in a stunning development, OpenAI claimed it had achieved the same result for Navier-Stokes. The company deployed roughly 10,000 AI agents working in parallel, which arrived at the proposed solution in just 88 hours. The solution appears to show that, mathematically, a fluid can form a vortex that spirals inward, moving faster and faster as it shrinks, until its velocity blows up to infinity. The proof has been formalized in Lean, a mathematical language that verifies each logical step. If the result withstands further scrutiny, it answers the million-dollar Millennium Prize question. Under some conditions, the Navier-Stokes equations have solutions that break down mathematically. Navier-Stokes isn't just about solving a Millennium Prize problem. It's about understanding the limits of our ability to describe the physical world around us with equations. Solving the prize problem doesn't mean we've solved turbulence. We still don't fully understand how turbulent flows evolve across scales, or how to predict them and all of their complexity. At the same time, the blow-up proof reveals something strange about the equations themselves. They can describe real fluids with remarkable accuracy, while still allowing behavior that could never actually happen in the physical world. Even if mathematically you can create a singularity, it doesn't mean that physically the singularity actually happens. It just means that you have to use different laws of physics. In this way, Navier-Stokes still stands as both a beacon and a warning of the power and the limits of mathematical description.