The Navier-Stokes equations hold some of the biggest questions in mathematics, which is why one funder offers $1 million to anyone who can solve them.
The work of mathematics professor Luis Caffarelli is commonly considered to have laid the foundations for solving the problem, and his lecture appears on the Clay Mathematics Institute website in an overview of the $1 million problem.
The Navier-Stokes equations are a family of equations that fundamentally describe how a fluid flows through its environment. Biomedical researchers use the equations to model how blood flows through the body, while petroleum engineers use them to reveal how oil is expected to flow through a well or pipeline. Animators even apply Navier-Stokes to render realistic looking waves in movies.
Despite Navier-Stokes' importance and wide use across fields, parts of it remain a mathematical mystery. Specifically, it's unknown if all points within a flow have a unique, quantifiable speed, versus whether points could in theory break from that pattern and reach an infinite speed.
Plenty of research is left to be done—and prize money to be won—on the nature of these unknowns, but the work is better defined thanks to Caffarelli.
"In principle, it's interesting to understand phenomena using mathematics," Caffarelli said. "But what has given a lot of value to the field is the ability to numerically simulate problems."
Caffarelli's mathematical models focus on diffusion, meaning particle movement from areas of high concentration to low concentration. He is widely recognized as the world's leading specialist in free-boundary problems for nonlinear partial differential equations, which describe the behavior of matter in "meeting places," such as where an iceberg meets the ocean, or the edge of a forest fire meets a forest.
Diffusion factors into the mysterious Navier-Stokes equations, as well. Because of the complexity of the interactions throughout the fluid space—and the turbulence it creates—understanding fluid speed throughout the flow remains a question. However, collaborating with mathematicians Robert Kohn and Louis Nirenberg, Caffarelli was able to explain how a particle in the flow would behave if its speed became infinite.
"Basically, we showed that if the flow in someplace becomes infinity the points where it is infinite cannot curve in space and time, so you will never see it persist for an interval of time," Caffarelli said. "You can see the speed get bigger, bigger, bigger and at an instant in time --pop!--it reaches infinity and then immediately after it's finite again."
Because the point of flow immediately reverts back to a finite value, it's unlikely to significantly affect other flow regions.
"A singularity appears and disappears, so if they exist they have a minimal effect because you never see them," Caffarelli said.
Caffarelli and his collaborators published their findings in a paper in 1982. Since then, their work has served as a guiding force for other researchers seeking to understand Navier-Stokes equations.
In 2012, Cafferelli received Israel's prestigious Wolf Prize for his achievements. Caffarelli, Kohn and Nirenberg together were awarded the 2014 Steele Prize for Seminal Contribution to research by the American Mathematical Society (AMS).
"The paper has been a kind of textbook for a whole generation of Navier-Stokes researchers, motivating many of the later developments and simplifications," wrote the AMS in its prize announcement. "[It] was and remains a landmark in the understanding of the behavior of the solution to the Navier-Stokes equations and has been a great source of inspiration for a generation of mathematicians."
Although Caffarelli's research has moved away from Navier-Stokes, his continued focus on fundamental mathematics keeps his research focused on the big problems facing multiple fields.
Video by Jeff Mertz. Text adapted from a post by the Institute for Computational Engineering and Sciences. Cafferelli holds the Sid W. Richardson Foundation Regents Chair in Mathematics #1 in the College of Natural Sciences at the University of Texas at Austin.
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