Mathematical Challenge Navier-Stokes Problem: Possibility of Resolution

The Clay Mathematics Institute announced that the problem of 'existence and smoothness of the Navier-Stokes equation' has been 'clearly resolved.' A formal review process is currently underway, with the institute working to verify the proof.

This problem is one of the 'Millennium Prize Problems' selected by the Clay Mathematics Institute in 2000. The institute selected seven major unsolved mathematical problems and offered a prize of one million dollars (approximately 150 million yen) for each. Only one has been solved so far, and the Navier-Stokes problem has remained unsolved for over 20 years.

The Navier-Stokes equation is a mathematical description of how fluids (liquids and gases) move. While it is applied in many real-world scenarios such as weather forecasting, aircraft design, and ocean current prediction, whether its solutions always exist 'smoothly'——that is, whether they can be defined mathematically without contradiction——has never been proven. Proving this point is at the core of the prize problem.

It is significant that the Clay Mathematics Institute used the phrase 'clearly resolved.' However, it should be noted that formal review has not yet been completed, and at this point we remain at the stage of 'high likelihood of resolution.' In the mathematical world, submitted proofs are sometimes later found to contain errors, and a solution is only formally recognized upon completion of review.

Of the Millennium Prize Problems, only the 'Poincaré conjecture' has been solved so far, with Russian mathematician Grigory Perelman publishing his proof in 2003. Perelman is also known for declining the prize money. If the Navier-Stokes problem is formally solved, it would be the second such achievement in a quarter-century.

Interest in unsolved mathematical problems has recently intersected with the field of AI. Research in mathematical reasoning using machine learning is advancing, and there are growing instances of AI assisting in the discovery and verification of complex proofs. While it is not yet clear what process led to this proof, as an event in an era where the intersection of mathematics and AI is expanding, attention is also focusing on its background.

When the results of the formal review are published, it could lead to various developments, including the award of the one million dollar prize and impacts on the entire mathematical community. Going forward, the focus will be on when the Clay Mathematics Institute completes its review and how the proof is evaluated.