Why Navier-Stokes remains one of mathematics’ toughest problems — Institute

The Clay Mathematics Institute has recognized the Navier-Stokes equation as one of the seven Millennium Prize Problems, a collection of seven notoriously difficult mathematical questions that have remained unsolved for many years. The institute, based in Cambridge, Massachusetts, established these prize problems in 2000 to highlight the ongoing presence of significant unsolved questions in mathematics and to draw public attention to the fact that the field still has frontier challenges.

A recent AI-generated claim that solved the Navier-Stokes problem came under scrutiny from the institute. The Navier-Stokes equation, which governs the flow of fluids like water and air, remains shrouded in mystery despite its critical importance. The institute's founding Scientific Advisory Board chose the equation as one of the problems after consulting experts worldwide, focusing on classic questions that have resisted solution for many years.

Solving these problems would carry a total prize fund of $7 million, with each problem receiving $1 million. The Navier-Stokes problem asks whether solutions exist and if they are unique, fundamental questions that have yet to be proven. The institute emphasizes that a proof provides not only certainty but also understanding. Among the other six Millennium Prize Problems, the Poincaré Conjecture has been solved, but the remaining five problems, including Navier-Stokes, still await resolution.