What Is The Navier-Stokes Problem, & Why Is Everyone Suddenly Talking About It?

The Navier-Stokes equations explain how fluids, including water, air, blood, and weather systems, move. These equations, used daily in aircraft design, ocean modelling, and meteorology, have one unresolved question for nearly a century: can a smooth flow, described by these equations, suddenly develop an infinite, uncontrollable spike - known as a "blow-up" - in a finite amount of time?

No one has been able to prove whether this can or cannot happen for the three-dimensional version of the equations, which describes the real world. Recently, OpenAI claimed it has solved the equation, triggering concern among mathematicians. The question was formalized in the early 2000s by the Clay Mathematics Institute as one of seven "Millennium Prize Problems," with a $1 million reward for a solution.

Only one of the seven problems has been solved so far. Mathematicians find the Navier-Stokes problem difficult to solve partly because fluid behavior is chaotic and non-linear - small changes can have unpredictable, compounding effects, making it extremely challenging to prove whether a solution remains smooth forever or to create an example where it doesn't.

OpenAI claims an internal AI model helped produce a proof that a version of the Navier-Stokes equations can blow up in finite time. The proof was created using around 10,000 AI agents working in parallel, costing millions of dollars, and formally checked using the verification language Lean. However, the proof applies to a "forced" version of the equations, which include an external forcing term, rather than the actual "unforced" version specified by the Clay Institute's prize.

It remains unclear whether this satisfies the Millennium Prize criteria. Beyond pure mathematics, if AI systems can meaningfully contribute to such a complex problem, it suggests a shift in how mathematical research is conducted. This could potentially compress years of human effort into weeks, which has significant implications for fields reliant on fluid dynamics, such as climate modelling and aerospace engineering.

Better theoretical guarantees could lead to more reliable predictions and designs. However, some mathematicians are uneasy about the episode, accusing OpenAI of building on their unpublished ideas and warning that treating landmark open problems as benchmarks for AI capability might discourage mathematicians from sharing early-stage ideas.

These ideas, though often unpublished, have historically played a crucial role in shaping major breakthroughs.