Sept 11, 2026

The (almost) two-century-old prestigious Navier–Stokes mathematics problem has reportedly been solved using AI.

The Navier–Stokes equations were devised in the early 19th century (1820s–1840s) to describe the motion of fluids. These equations quintessentially demonstrate the power of mathematical language. Just a handful of equations (mathematical statements) model the intricacies of the physical world: atmospheric weather patterns, turbulence around airplanes, and the flow of blood through the human body, among other things.

As an applied mathematics problem, it asks whether a fluid modeled by the equations can start in a smooth (well-behaved) state, but develop infinite velocity in a finite time. Developing infinite velocity in a finite time is referred to as a blowup or singularity. Two mathematical possibilities exist: a proof that no smooth starting state for the fluid will ever cause a blowup, or a counterexample of a particular, albeit pathological, smooth starting state that causes a blowup.

In addition to its practical relevance, the problem has been of immense theoretical importance in mathematics. Over the centuries, it has driven advances in partial differential equations and other areas of mathematics.

The current news is that OpenAI claims to have found a counterexample: On the Navier–Stokes Millennium Prize Problem

A blowup does not happen in practice, because a liquid does not spontaneously explode. So, what does discovering a counterexample mean? It implies that while the Navier–Stokes equations model most practical situations in fluid mechanics, they do not place enough constraints to disallow some unrealistic corner cases from creeping in.