What is Navier-Stokes, the 90-year-old maths problem OpenAI says it solved?

The Navier-Stokes equation is one of seven Millennium Prize Problems identified by the Clay Mathematics Institute in 2000

OpenAI on Tuesday announced that it had solved a nearly 90-year-old mathematics problem called the “Navier-Stokes existence and smoothness problem”, which asks whether the equations used to describe the movement of fluids such as water and air can break down under extreme conditions.

 

The problem, commonly referred to as Navier-Stokes, is named after the 19th-century mathematicians Claude-Louis Navier and George Gabriel Stokes. It is one of seven Millennium Prize Problems identified by the Clay Mathematics Institute in 2000. The institute offered a $1 million prize for solving each problem. Before OpenAI’s announcement, only one of the seven had been solved.

 

OpenAI said its internal AI system had shown that smooth fluid flow can develop a singularity — a point where the fluid’s velocity grows without bound in a finite amount of time. 

 

 

But what exactly is the problem, and why has it remained one of mathematics’ biggest unanswered questions for decades?

 

What is the Navier-Stokes problem?

 

The Navier-Stokes equations are a set of mathematical equations used to describe how fluids move. This includes liquids such as water and gases such as air.

 

The equations use Newton’s second law of motion — which states that force is equal to an object’s mass multiplied by its acceleration — and treat a fluid as a continuous substance rather than tracking individual molecules. They are used to understand and predict the movement of fluids in the real world, from weather and ocean currents to the flow of air around an aircraft.

 

But the problem is not simply about solving these equations. There is a basic question that mathematicians have not been able to answer for decades: Can these equations break down under extreme conditions?

 

In other words, if a fluid starts moving smoothly, could its motion become so extreme at some point that the equations can no longer describe what is happening?

 

Where does the difficulty lie?

 

The difficulty lies in understanding whether fluid motion can become infinitely fast in a finite amount of time — a situation mathematicians call a singularity.

 

This is where viscosity comes in. Viscosity is the resistance a fluid has to flowing. Honey, for example, is more viscous than water. It generally helps smooth out fluid motion.

 

The question is whether this smoothing effect is always enough to prevent a singularity, or whether a fluid can still reach a point where its speed becomes unbounded.

 

If such a singularity can form, it would mean the Navier-Stokes equations can no longer fully describe the fluid’s behaviour under those conditions.

 

What OpenAI claims to have solved

 

OpenAI said its internal AI system produced an analytical proof showing that an initially smooth fluid flow can develop a singularity in a finite amount of time. In other words, it showed that there is a scenario in which the fluid’s velocity can grow without bound, causing the equations to break down.

 

The solution involves a vortex — a spinning swirl of fluid — that spirals inward and becomes increasingly elongated. As the central region shrinks, the fluid speeds up while its energy remains finite, OpenAI said.

 

The mathematical challenge was to show that this breakdown could emerge from the fluid’s own motion, rather than by simply applying an infinitely large force. OpenAI said the different terms in the equations, including acceleration, pressure, momentum transfer and viscosity, become very large but cancel each other out in a precise way. This allows the external force to remain smooth even as the fluid’s velocity grows without bound.

 

OpenAI said the result corresponds to statements “C” and “D” in the Clay Mathematics Institute’s formulation of the Navier-Stokes problem. It also produced a formalisation of the proof in Lean, a computer-based system used to verify mathematical proofs.

 

The company, however, said it does not intend to claim the $1 million Millennium Prize for the result.