The Navier-Stokes Equations

A million-dollar science scandal

On 8 September, 2026, OpenAI announced to the world that it had conquered one of the most formidable problems to exist: the Navier-Stokes equations. Since the emergence of AI technology, its power has been continuously measured relative to its mathematical feats. Of course, the impact and potential ramifications were immediately felt by mathematicians everywhere. However, to those outside the field, this stream of news may merely be bewildering and be dismissed for lack of significance.

It will come as a relief that comprehensive knowledge of fluid flows and vector calculus is unnecessary to appreciate the scale of this problem. Instead, we can look back to the turn of the century, when the Clay Mathematics Institute sought to “herald a new millennium of mathematics” by announcing the seven Millennium Problems… each with a prize of one million US dollars. This effort sought to highlight the greatest mathematical challenges, one of which was the Navier-Stokes existence and smoothness problem. Yet only one has ever been definitively solved: the Poincaré Conjecture by Grigori Perelem. To take the Navier-Stokes equations, whose origins extend back nearly two centuries, their irresolution has endured through the lives of Gauss, Riemann, Hilbert, and almost every great technological advancement. But now modernity has prevailed, and the conquest of a lifetime has been claimed by an intelligence which is not human. 

Or has it? The fate of Navier-Stokes is liable for verification and credibility. Verifying one of the hardest problems to exist is an understandably daunting task. OpenAI has already claimed complete verification of the proof, but in light of these considerations, it is evident that formal validation through the peer review process has an inestimable timeline. Furthermore, the entire solution is subject to the issue of rightful credibility. The methodology of AI is to utilise the work that exists, and hence the conclusion reached draws powerfully on the works of many mathematicians. Two names in particular have heralded controversy, Tristan Buckmaster and Levent Alpöge. Working in collaboration, Buckmaster, prior to the OpenAI statement, made the central claim that “information about our progress had been passed to OpenAI". Whilst the trail of data remains obscure, it can only be hoped that in unpacking the methodology and proof, time will reveal the truth of the matter. 

If OpenAI has in actuality solved the Navier-Stokes equations, it raises the question of the future of fluid dynamics. The equations are fundamental to the field, providing a description of fluid movement through the relation of velocity, pressure, temperature, and density. The issue lies in their extended governance of fluid behaviour; in particular, will a fluid that begins smoothly always remain mathematically well-behaved? The proof seems to establish that they do not; instead, from perfectly smooth initial conditions, a fluid in finite time can develop a singularity, the breakdown of a physically meaningful mathematical description. But fluid dynamics has never depended upon finding exact solutions; Computational Fluid Dynamics (CFD) already uses numerical approximations to model flows without exact solution via numerical approximation. Furthermore, predicting the chaotic behaviour of turbulent flow remains unresolved, leaving arguably the field's greatest challenge intact. 

The significance of Navier-Stokes resolution is derived more from what it represents than the transformative effects it will have in applied practice. If verified, an unsolvable problem defining the history of human mathematics will have been defeated in 88 hours by an intelligence we have created, but that is ultimately not our own. This demonstration of AI power extends beyond the scope of mathematicians, forcing us all to consider our role in a future where we are no longer the pioneers of our own knowledge.

Image from Wikimedia Commons