Major breakthrough made on famous Millennium maths problem

The Navier-Stokes equations describe how fluids flow

WEIQUN ZHANG/STAN WOOSLEY/SCIENCE PHOTO LIBRARY

Two human mathematicians have made, with a “great deal of help” from AI, three important steps towards solving one of the world’s most famous outstanding mathematical problems. Other researchers say the approach could lead to a full solution in time and that it may already be enough to land the pair a $1m Millennium Prize.

The Navier-Stokes equations have been used to model the flow of fluids for two centuries – whether that is used to design more efficient and stable aircraft wings, modelling blood flow through arteries or building space rockets – despite their habit of occasionally breaking and outputting nonsense in certain scenarios.

Solving this problem – whether the equations completely tally with the real world, or if their modelled smoothness and turbulence can deviate from it – is one of the six remaining Millennium Problems, the thorny mathematical puzzles published by the Clay Mathematics Institute that will net the solver a $1 million prize.

Now, Tristan Buckmaster at New York University has announced groundbreaking results which edge us towards that larger solution via a document uploaded to his website, developed with Levent Alpöge at Harvard University and AI company Anthropic.

The pair claim three results: two of which were published along with Lean formalisation – a process of converting mathematical theories into computer code that allows it to be rigorously checked for logical flaws and errors – and one which is not yet published as the pair await a finished formalisation. The findings relate to close cousins of Navier-Stokes, the Boussinesq approximation and the Euler equations, but are not yet generalised to the wider Navier-Stokes problem.

In the papers, they describe how they have taken previous work by Diego Córdoba and Luis Martínez-Zoroa to a conclusion with a “great deal of help from LLMs”, including models from Anthropic and OpenAI.

Buckmaster outlines in his announcement that he sees the results as less important than the fact that AI is rapidly becoming a powerful amplifier of human mathematical effort and leading to faster progress. “This is a a Deep Blue-Kasparov moment,” he said, alluding to the famous chess match in 1996 where an IBM supercomputer beat chess world champion Garry Kasparov. Neither Buckmaster or Alpöge immediately responded to New Scientist‘s request for comment.

The results can be thought of as “stepping stones” to a full solution of Navier-Stokes, says David Silvester at the University of Manchester.

Silvester says that one of the papers, which focuses on the Euler equations, shows that spontaneous “blow-ups” or turbulence can appear.

“It’s a really hard problem because when it was stated it wasn’t clear whether the result was true: that is that there are smooth solutions and it stays forever stable, or in fact there is some blow up. So it’s not like you’re trying to prove something. You don’t know whether you’re trying to prove it or trying to find a counterexample,” says Silvester.

Silvester says that expanding this result to Navier-Stokes will not be trivial, but that the current work alone may be enough for the pair to claim the Millennium Prize.

Terence Tao at the University of California, Los Angeles wrote in a social media post that he believes the work takes us very close to a full Navier-Stokes solution.

“There does not seem to be anything in principle preventing the methods from extending all the way to Navier-Stokes,” says Tao. “At this point, I would not be surprised if one could batter out such an extension by pouring an enormous amount of compute and AI assistance at such a task.”

Camilla Nobili at the University of Surrey says that they key question now is whether a similar example can be found in Navier-Stokes, which builds upon Euler equations by adding more detail like friction and dissipation – elements which have a natural tendency to smooth out simulations and keep them more regular. She believes that running the same technique on Navier-Stokes will not be enough, and that there will be more to the solution than that, but is optimistic. “I’m confident, as Terence [Tao is], that this is a good way to go,” says Nobili.

Despite the fame and longstanding insolubility of Navier-Stokes, there is unlikely to be any practical effect in a working result, says Silvester. Computer models on fluid dynamics are already so good that they have in large part rendered wind tunnels obsolete.

“Nothing will change in the applications where [Navier-Stokes] is used because of this result,” says Silvester. “It’s a mathematical nicety, honestly.”