A Million Dollar Solution: The Navier-Stokes Equations

For a mechanical engineering student such as myself, the Navier-Stokes equations are seen as the epitome of fluid mechanics. These equations describe how any liquid or gas flows and are notoriously difficult to deal with, so much so that there is a whole field dedicated to computationally implementing them.

So what does it mean hearing that the company behind ChatGPT recently “solved” Navier-Stokes? What about the heat from mathematicians claiming the solution was stolen? How will this change the field of fluid dynamics and the world?

But first, a bit of Fluids 101. A fluid is a material that moves continuously, has no fixed shape, and flows under an applied force.¹ All liquids and gases are fluids under this definition. A solid object would not be considered a fluid because it has a fixed shape and can resist an applied shearing force. For example, a solid brick will hold its shape even if you push the top of it. If you ran your hand along the top of a pool of water, it would flow and change shape.

Even a still glass of water is composed of septillions of molecules (that is a 1 followed by 24 zeros!), all of which are moving, colliding, and interacting. Tracking and modeling this many water molecules is nearly impossible, so to study fluid behaviour a better approach is to treat it as a continuous substance and take the average of properties like density, temperature, velocity, and pressure.

By treating a fluid as a continuous substance, mathematicians can apply known laws. Newton’s second law relates force as a product of mass and acceleration, which is where the Navier-Stokes equation originates from.² In simple terms, it relates how the pressure, gravity, and viscosity of a fluid affect its acceleration. They might also be referred to as a set of equations, each in a different x-y-z dimension.

Considering how much of the world is liquid or gas, it becomes evident why the Navier-Stokes equations are so vital. Although they can be incredibly useful, these equations are also very difficult to exactly solve. This is because they are nonlinear partial differential equations, meaning that a variable is multiplied by its own derivative.

In the context of Navier-Stokes, the velocity term is multiplied by its own derivative, which helps model how the fluid pushes and interacts with itself. When fluids move quickly, they become turbulent and seemingly more unpredictable. Even a slight change in the initial conditions can lead to extreme changes in the overall movement. This trait is captured by the nonlinear velocity term.

Challenges also arise when Navier-Stokes is applied in real world situations, where fluids are interacting with structures such as the walls of a pipe or rocks in a river.³ Oftentimes multiple types of fluids exist in the same system, like bubbles of air in water, which also complicates things.

Mathematicians have been able to find solutions for extremely simplified systems, such as the case for liquid flow through a straight, circular pipe.⁴ Complex cases require approximating a solution.

Instead of solving the equation over the entire system, the space can be broken down into smaller chunks. These are called “cells” that make it easier to analyze the fluid over. Similarly, time is also broken down into smaller steps. This allows mathematicians to replace the complicated calculus with simpler algebraic operations. As the size of the cells and time intervals gets smaller, the approximation becomes more and more accurate, but also takes more time to solve.

This is the field of computational fluid dynamics, and it has shaped the way scientists analyze the world. Using approximations allows nearly impossible calculus to be approximated with simple algebra.

Computational fluid dynamics has been used to predict weather, to model airplanes and rockets, and to figure out how pollution disperses in the atmosphere. The applications extend beyond aerospace and atmospheric sciences. The blood in your veins is also a fluid, and biologists can use Navier-Stokes to study how it flows throughout the human body. Artists use them to model how water moves in animations. Geologists can model the fluid-like behaviour of plate tectonics and discover how continents were formed.⁵

Scientists have put a lot of faith into these equations and they have led to the advancements of many different fields. The equations work in many applications, however there is still a gap in our understanding of the pure mathematics behind them.

The big unanswered question is whether the Navier-Stokes equations will always produce smooth, realistic solutions or if there are cases where the mathematics can break down and not reflect true reality. Smoothness refers to quantities, like velocity or pressure, remaining finite and continuous.

In other words, the challenge is to determine the existence and smoothness of the equations. Is there always a solution? And if so, is the solution always smooth?

This question is so important that the prize for solving it is a million dollars. It is one of seven millennium problems from the Clay Institute of Mathematics.⁶ These problems cover unanswered mathematical questions in a wide range of fields, including topology, theoretical physics, and number theory.

The millennium problems are known to be incredibly difficult to solve, and their solutions could very well shape their respective fields. For example, a solution to the P vs NP problem would break nearly all cryptosystems that keep information encoded.⁷

Previously, only one millennium had been solved in 2003, of which the prize was turned down. Many mathematicians have dedicated their lives to solving these problems, whether it is for the money, the prestige, or the chance to make a discovery at the edge of our understanding of mathematics.

After roughly 90 years since the question of Navier-Stokes existence and smoothness was formally posed, OpenAI announced that they found a solution.⁸ On September 5th, 2026, the company behind ChatGPT had generated a solution using advanced AI models.

They had been testing and experimenting with an internal model beyond the latest GPT-6 Astra, and found that it performed well for mathematics problems. The millennium problems had recently gained traction following rumors that groups were close to a solution. Their team successfully solved a simpler version of the full Navier-Stokes problem, which encouraged focus on the millennium prize. Following the announcement, OpenAI Researcher Sébastien Bubeck reflected on their decision to tackle the problem.

“So we thought to ourselves: ‘We have such a strong model. Why don’t we try to solve also a Millennium Prize problem?”⁹

OpenAI reported using 10,000 AI agents which could read from the internet and run code. These agents had the ability to send messages to each other, which allowed them to work in parallel and build from the group’s breakthroughs. Codex monitored the agents and was able to guide ideas from promising subgroups to other agents. Groups also served to peer-review each other's work, emulating the guiding principles of scientific advancements.

Over the course of 88 hours, they exchanged 2.7 million messages, consumed 130 billion output tokens, and reached a conclusion.

On September 8th, 2026, OpenAI announced an analytical proof that an initially smooth fluid can develop solutions that do not follow the expected laws of the universe. They determined that by applying the Navier-Stokes equations, a fluid can reach a “singularity” where the velocity becomes infinite.

The conditions in their proof examine a fluid initially at rest that is subject to a force which twists the fluid into a tightly spinning vortex, which somewhat looks like spaghetti wrapped around a fork. The force continues to be applied, stretching the “spaghetti” out and making it thinner and thinner.

Eventually, the vortex collapses and the velocity at the center of the vortex becomes infinite. Infinities are a concern because they do not reflect our understanding of nature. It is physically impossible for anything to move at an infinite speed, including the molecules in air or water. This implies that there is a gap between the math and laws of physics.

This finding does not mean that Navier-Stokes should be discarded or considered as completely incorrect. The reason that this problem took many years to solve is because for the vast majority of cases, the equations provide possible and accurate solutions. The conditions that caused the equations to break down are extreme and unnatural. For current applications of Navier-Stokes, such as modeling lift of an airplane, the equations still function and predict fluid motion precisely.

However, the fact that existence and smoothness have been disproven means that the Navier-Stokes equations are mathematically incomplete. This is one of the most exciting times of science, when an unexpected result challenges the current body of knowledge.

It is unclear at the moment where this uncertainty will lead science. Perhaps the solution will help develop more accurate computational fluid dynamics solvers, or force scientists to rewrite how models behave at extreme limits.

This is a breakthrough for the field of fluid dynamics, and for the role of AI in academia. A model that can answer a 90 year mystery in under one week holds incredible opportunities for the future of science.

OpenAI’s solution has sparked controversy, particularly from mathematicians Tristan Buckmaster and Levent Alpöge. The two claim to have been working on the Navier-Stokes millennium problem using OpenAI’s Codex.⁸ They claim to have solved a simpler version of the problem and were in the process of publishing a proof.

Buckmaster reached out to OpenAI after hearing that they were working on a similar solution. He was supposedly offered sole ownership of the solution so long as he removed Alpöge from the author list and credited OpenAI’s Codex. Buckmaster did not accept this offer.

After the solution was released, questions were raised about the AI agents potentially using Buckmaster and Alpöge’s work that was stored in Codex. In response, OpenAI claimed that their work “could not have influenced the system in any way”.

Even if OpenAI’s proof is independent of Buckmaster and Alpöge’s work, the proof was built on work from hundreds of prior mathematicians. Some scientists express concern that AI generated proofs fail to teach human scientists and can erode true understanding. Could there reach a point where AI proofs are beyond the understanding of any human?

Award-winning mathematician Terence Tao mentions that previous human-directed effort towards intense mathematics problems has helped push many fields forward. Tao argues that overuse of AI in mathematics could hinder this effect.

“Prematurely solving the problem by purely AI-powered methods - particularly without full transparency into the solution process - can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.”¹⁰

It is difficult to determine how the field of mathematics, and more broadly science, will be helped or harmed by AI. We have access to tools that could diminish human critical thinking, as well as solve problems we thought insurmountable, and so we must tread forward with both caution and hope.