Stephen Montgomery-Smith Labors to Find Elusive Proof

Professor Stephen Montgomery-Smith
Kristi Galloway
News Source: 
College of Arts & Science

In mathematics, a proof is a deductive argument for a mathematical conclusion. Proofs typically involve a sequence of statements that show the complete thought process used to reach the verified conclusion. Professor Stephen Montgomery-Smith has spent his career working on complex proofs. However, no problem has captured his attention like the Navier–Stokes equations, which describe the motion of fluids.

The equations may be used to model weather forecasts, ocean currents, airflow around the wing of an airplane, and more. But even with all these practical uses, it has not yet been proven that solutions exist to the Navier–Stokes equations in three dimensions. In 2001, the Clay Mathematics Institute called the Navier–Stokes existence and smoothness problem one of the seven most-important open problems in mathematics and offered a $1 million prize for a solution or counter-example.

Montgomery-Smith first became interested in the Navier–Stokes equations in 1995, long before there was a prize. He overheard some colleagues discussing it in the hallway and was immediately gripped by the intricacies and beauty of the equations. Later that year, Montgomery-Smith had the opportunity to attend a conference in England that focused on the Navier–Stokes equations. “After I wrapped my head around the problem, I got every book and resource I could from the library and started reading all about the equations,” he says. “I stopped doing research in areas that had previously been successful and began putting most of my energy into solving the Navier–Stokes equations; I simply couldn’t think about anything else.”

Many scholars have been working on the Navier–Stokes equations for years, such as Charles Fefferman of the Princeton Mathematics Institute and Terence Tao of the University of California, Los Angeles, so the race is on to find a solution. On numerous occasions, Montgomery-Smith has believed he solved the problem, only to find an error in his work. He is not alone in that experience.

Earlier this year, Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan, proposed a solution to the equations but the international mathematics community is skeptical of all proposed solutions. Montgomery-Smith reviewed Otelbaev’s work and shared it on an online forum. Terence Tao found a counter-example that disproved Otelbaev’s solution and sent him back to the drawing board. Read more about it in an article recently published in Nature here.

Montgomery-Smith now believes that the Navier–Stokes equations are unsolvable, and he is working on a counter-example to prove his theory. Because of Montgomery-Smith’s work on the equations, he knows much more about complex fluids, which has led to additional research opportunities and related projects on products such as injected molded plastics. “The Navier–Stokes equations are still my calling,” says Montgomery-Smith. “I don’t know if I will ever find a counter-example or solution, but I know I’m supposed to work on it.”

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