Navier-Stokes problem

Water in a pipe, smoke above a candle, air around a wing: all of these are flows. For such flows, a set of computational rules has existed since the 19th century. They are named after their discoverers Claude Navier and George Stokes and describe how velocity and pressure in a fluid change from moment to moment. These rules work superbly in practice, and engineers use them daily. Nevertheless, mathematics still lacks a proof that they always behave sensibly. This very gap in proof is the Navier-Stokes problem.

A million dollars for a proof

In the year 2000, the Clay Mathematics Institute in the USA selected seven particularly hard open questions. They are called Millennium Problems, and a prize of one million US dollars has been offered for each one. The Navier-Stokes problem is one of them. To this day, only a single one of these seven problems has been solved, and Navier-Stokes is not among them.

The question is not merely academic. If no one can prove that the equations always have a clean solution, one also cannot know for certain when computer calculations based on them can be trusted. Weather models, climate models, and simulations of aircraft wings all rely on these equations. They deliver useful results, but without a mathematical guarantee.

Behind this also lies a physical question: turbulence. This refers to the wild, chaotic disorder within a flow, for instance behind a bridge pier in a river. Turbulence is considered one of the last great unexplained phenomena of classical physics. Many researchers hope that a proof would also bring new insights here.

Why the proof falls short

The equations do not directly tell you what a flow looks like. They only tell you how it changes in the next second. One specifies a starting state and calculates step by step from there. The statement being sought is: for every harmless starting state, there exists a solution that remains smooth for all time. Here, smooth means that velocity and pressure never jump to infinity anywhere.

The problem is a feedback effect within the equations. A fast flow transports itself and can, in doing so, compress into an ever smaller space. Purely computationally, the velocity at a point could thereby become infinite within finite time. This is called a singularity. No one has ever found such a singularity, but no one has been able to rule it out either.

For comparison: in two dimensions, that is, for an imagined flat flow, the question has long been settled, and the answer is yes. The three-dimensional case, that is, the real world, has resisted for over 90 years. There are partial successes: it is known that solutions exist at least for a short span of time. For all time, it remains open.

From weather models to AI research

In practice, you encounter Navier-Stokes constantly without the name ever being mentioned. Every weather forecast is an approximate calculation of these equations. The same is true for the flow simulation of a car in a virtual wind tunnel, for blood flow models in medicine, and for the water animations in films and video games.

In AI news, the term comes up for two reasons. First, research groups are trying to replace the costly flow calculations with trained models. Such models deliver an estimate in seconds where a classical computer would take hours. Second, the Millennium Problems serve as a benchmark for how far AI has come in proving mathematics. When a provider claims that its model can perform genuine research, people like to ask whether it comes anywhere close to something like this.

A common misconception is that the Navier-Stokes problem is a computational problem that faster computers could solve. This is not the case. What is sought is a logical proof covering all possible starting states, of which there are infinitely many. Simulations can at most provide hints as to where one should search for a singularity.