The Clay Mathematics Institute has described the Navier-Stokes equation as one of seven exceptionally difficult mathematical problems that have resisted solution for many years, with fundamental questions about the behaviour of fluids still awaiting proof.
CMI medala
The Cambridge, Massachusetts-based institute established the seven Millennium Prize Problems to mark the new millennium and to highlight the continuing existence of major unsolved questions in mathematics.
A check by our correspondent to the CMI website on Tuesday noted that according to the institute, the prizes were created “to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium” and “to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems.”
PUNCH Online earlier on Tuesday reported that the question had returned to the spotlight after OpenAI said on Tuesday that an internal AI system had produced what it described as a solution to the Navier-Stokes problem in about 88 hours, using roughly 10,000 AI agents.
The institute said the problems were also intended “to emphasize the importance of working towards a solution of the deepest, most difficult problems” and “to recognise achievement in mathematics of historical magnitude.”
The seven problems were formally announced at a meeting in Paris on May 24, 2000, at the Collège de France.
The institute said its founding Scientific Advisory Board selected the problems after consulting leading experts worldwide, with its focus on “important classic questions that have resisted solution for many years.”
A total prize fund of $7 million was designated for solutions, with $1 million allocated to each problem.
The Navier-Stokes equation is one of the six problems listed by the institute as still unsolved.
“This is the equation which governs the flow of fluids such as water and air,” the institute said.
But despite the equation’s importance, the institute said some of its most basic mathematical questions remain unanswered.
“However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique?”
The institute also explained why mathematical proof is essential in addressing such questions.
“Why ask for a proof? Because a proof gives not only certitude, but also understanding,” it said.
The Navier-Stokes problem is one of seven questions that the institute identified as Millennium Prize Problems. The others are the Birch and Swinnerton-Dyer Conjecture, Hodge Conjecture, P vs NP, Riemann Hypothesis, Yang-Mills and the Mass Gap, and the Poincaré Conjecture.
The institute lists the Poincaré Conjecture as the only problem already solved.
It said French mathematician Henri Poincaré posed the question in 1904, asking whether the three-dimensional sphere is characterised as the unique simply connected three-manifold.
The institute said mathematician Grigori Perelman’s proof showed that “every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.”
The remaining six problems, including Navier-Stokes, remain on the institute’s list of unsolved Millennium Prize Problems.
The institute also noted that one of the seven problems, the Riemann Hypothesis, has an even longer mathematical history.
It said the hypothesis was formulated in 1859 and also appeared among the 23 problems discussed by German mathematician David Hilbert in his famous address in Paris on August 9, 1900.
The Clay Mathematics Institute said the rules governing the prizes had the endorsement of its Scientific Advisory Board and approval of its Directors, who have responsibility for preserving “the nature, the integrity, and the spirit of this prize.”