Pushing the limits of fluid equations

The movement of fluids in nature, including the flow of water in our oceans, the formation of tornadoes in our atmosphere, and the flow of air around airplanes, have long been described and simulated by what are known as Navier-Stokes equations.

However, mathematicians do not have a complete understanding of these equations. Although they are a useful tool for predicting fluid flow, we still don’t know if they accurately describe fluids within Everyone possible scenarios. This prompted the Clay Mathematics Institute of New Hampshire to designate the Navier-Stokes equations as one of its Seven Millennium Problems: the seven most pressing unsolved problems in all of mathematics.

The Millennium Problem of the Navier-Stokes equation challenges mathematicians to prove whether there are always “smooth” solutions to the Navier-Stokes equations. Simply put, smoothness refers to whether equations of this type behave in a predictable way that makes sense. Consider a simulation where a foot presses the gas pedal on a car and the car accelerates to 10 miles per hour (mph), then to 20 mph, then to 30 mph, and then to 40 mph. However, if the foot presses the gas pedal and the car accelerates to 80 km/h, then to 60 km/h and then immediately to an infinite number of miles per hour, you would say that something is wrong with the simulation.

This is what mathematicians want to find out for the Navier-Stokes equations. Do they always simulate liquids in a meaningful way or are there situations in which they break?

You can find a detailed explanation on this topic in the blog post “Why global regularity is difficult for Navier-Stokes‘ by the Australian mathematician Terence Tao.

In a paper published Oct. 19 on preprint site arXiv, Caltech’s Thomas Hou, the Charles Lee Powell Professor of Applied and Computational Mathematics, and Jiajie Chen (PhD ’22) of New York University’s Courant Institute provide a proof that is a long-standing solution to an open problem for the so-called 3D Euler singularity. The 3D Euler equation is a simplification of the Navier-Stokes equations, and a singularity is the point at which an equation collapses or “explodes,” meaning it can suddenly become chaotic without warning (like the simulated car, which accelerates to infinity). Kilometers per hour). The evidence is based on a scenario first proposed by Hou and his former postdoc Guo Luo in 2014.

Hou’s calculation with Luo in 2014 discovered a new scenario that showed the first convincing numerical evidence for a 3D Euler explosion, while previous attempts to detect a 3D Euler explosion were either inconclusive or unreproducible.

In the latest publication, Hou and Chen present the definitive and irrefutable proof of Hou and Luo’s work involving the 3D Euler equation explosion. “It starts out as something that behaves well, but then kind of evolves to become catastrophic,” says Hou.

A portrait of Thomas Hou.  He is wearing a suit and glasses and is smiling at the camera.

Thomas Hou

Credit: Vicki Chiu/Caltech

“For the first ten years of my work, I believed there was no Euler explosion,” says Hou. After more than a decade of research since then, Hou has not only proved his former self wrong, he has also solved a centuries-old math puzzle.

“This breakthrough is a testament to Dr. Hou’s persistence in solving the Euler problem and the intellectual environment that Caltech nurtures,” said Harry A. Atwater, Otis Booth leadership chair of the Division of Engineering and Applied Science, Howard Hughes Professor of Applied Physics and Materials Science and director of Liquid Sunlight Alliance. “Caltech empowers researchers to continue to creatively work on complex problems – even across decades – to achieve extraordinary results.”

A portrait of Jiajie Chen.  He is wearing a polo shirt and glasses and is smiling at the camera.

Jiajie Chen

Credit: Briana Ticehurst/Caltech

The joint effort by Hou and his colleagues to prove the existence of explosions using the 3D Euler equation is a major breakthrough in itself, but also represents a major step forward in solving the Navier-Stokes Millennium Problem. If If the Navier-Stokes equations could also explode, it would mean that something is wrong with one of the most fundamental equations used to describe nature.

“The overall framework that we have put in place for this analysis would be tremendously helpful for Navier-Stokes,” says Hou. “I recently identified a promising Navier-Stokes explosion candidate. We just need to find the right formulation to prove the Navier-Stokes explosion.”

The paper detailing the proof is entitled “Stable Nearly Self-Similar Blowup of the 2D Boussinesq and 3D Euler Equations with Smooth Data”.

Funding for the research was provided by the National Science Foundation and the Choi Family Postdoctoral Fund, Choi Family Gift Fund, and Choi Family Graduate Fellowship Fund.