Understanding the unknowns of Navier-Stokes equations is one of the biggest questions in mathematics, which explains why The Clay Mathematics Institute continues offering $1 million to anyone who can solve them.
The work of ICES researcher Luis Caffarelli, a mathematics professor, is commonly considered to have laid the foundations for solving the problem. In a new video Caffarelli briefly describes this work.
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.
But 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 remain “smooth”—that is, have a unique, quantifiable speed—or if some could theoretically break from smoothness and reach an infinite speed. That’s the $1 million question.
While most applications of Navier-Stokes only require a “weak solution”-- a sort of average overview instead of an all-points perspective—having a “strong solutions” that take into account currently unknown information could lead to deeper understanding on the nature of flow, as well as increase solution precision and consistency.
Plenty of research is left to be done—and prize money to be won—on the nature of these unknowns, but the work is now better defined thanks to Caffarelli.
“I work a lot in models that come from very different areas of research, but, they are all diffusion processes” Caffarelli says.
Currently, these include problems such as non-local diffusion, where the movement of a particle is influenced by factors outside of its immediate environment, and free-boundary problems, 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.
But when working in the 1980s, Caffarelli’s diffusion research was targeted on understanding the complexities of Navier-Stokes.
Diffusion, or the movement of particles from an area of high concentration to low concentration, is a governing principle of Navier-Stokes. But unlike simpler diffusion problems, such as the propagation of heat in an iron bar, the variables that affect fluid flow are more complex: viscosity and acceleration have opposing effects on particle diffusion, while the effects of fluid pressure—a side effect of fluid being incompressible—oppose diffusion as a whole.
“The heat equation only has the diffusion, so everything is nice mathematically, but Navier-Stokes has diffusion versus incompressibility,” Caffarelli explained. “The fluid cannot be squeezed because of this property of incompressibility so it fights the diffusion property.”
These sparring properties affect flow at fine scales, with their effects eventually resulting in the flow we see at the macro level. But 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 at large.
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—one of the possible behavior scenarios.
“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 titled “Partial Regularity for Suitable Weak Solutions of the Navier-Stokes Equation” in the journal Communications on Pure and Applied Mathematics in 1982. Since then, their work has served as a guiding force for other researchers seeking to understand Navier-Stokes at fine levels.
The paper’s significance was recently recognized, with Caffarelli, Kohn and Nirenberg being awarded the 2014 Leroy P. 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.
And with the rise of high performance computing enabling the simulation of real-world phenomena, an understanding of the underlying mathematics is more important than ever to ensure that our models—whether it’s the flow of blood, gush of oil, or the ripple of waves—represent reality.
“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.”