Mathematicians solve a long-standing open problem for the so-called 3D Euler singularity

Mathematicians solve a long-standing open problem for the so-called 3D Euler singularity

X. Credit: arXiv (2022). DOI: 10.48550/arxiv.2210.07191″ width=”800″ height=”309″/>
Approximate steady state in the near field. Left: profile ω̅; right: θ̅ X. Credit: arXiv (2022). DOI: 10.48550/arxiv.2210.07191

The movement of fluids in nature, including the flow of water in our oceans, the formation of tornadoes in our atmosphere, and the airflow around airplanes, has long been described and simulated by what we calls the Navier-Stokes equations.

Yet 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 in all possible scenarios. This led the Clay Mathematics Institute in New Hampshire to label 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 asks mathematicians to prove whether there are always “smooth” solutions for the Navier–Stokes equations.

Simply put, regularity refers to whether equations of this type behave in a predictable and logical way. Imagine a simulation in which a foot presses the accelerator pedal of a car, and the car accelerates to 10 miles per hour (mph), then 20 mph, then 30 mph, then 40 mph. However, if the foot presses the accelerator pedal and the car accelerates to 50 mph, then 60 mph, then instantly to an infinite number of miles per hour, you would say that there is something wrong. with simulation.

This is what mathematicians hope to determine for the Navier-Stokes equations. Do they always simulate fluids in a way that makes sense, or are there situations where they break down?

In an article published on the preprint server arXivThomas Hou of Caltech, Charles Lee Powell Professor of Applied and Computational Mathematics, and Jiajie Chen (Ph.D. ’22) of New York University’s Courant Institute provide proof that solves a long-standing open problem for the so-called 3D Euler singularity.

Euler’s 3D equation is a simplification of the Navier-Stokes equations, and a singularity is the point where an equation begins to break down or “explode”, meaning it can suddenly become chaotic without warning ( like the simulated car accelerating to infinite speed). number of miles per hour). The proof is based on a scenario first proposed by Hou and his former postdoc, Guo Luo, in 2014.

Hou’s calculation with Luo in 2014 uncovered a new scenario that showed the first convincing numerical evidence of a 3D Euler explosion, where previous attempts to uncover a 3D Euler explosion were either inconclusive or not. reproducible.

In the final paper, Hou and Chen show definitive and irrefutable proof of Hou and Luo’s work involving the explosion of the 3D Euler equation. “It starts out as something that does well, but somehow evolves in a way where it becomes catastrophic,” Hou said.

“For the first ten years of my work, I thought there was no Euler explosion,” says Hou. After more than a decade of research since, Hou has not only proven him wrong, he’s solved a centuries-old mathematical mystery.

“This breakthrough is a testament to Dr. Hou’s tenacity in solving the Euler problem and the intellectual environment that Caltech nurtures,” says 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 the Liquid Sunlight Alliance. “Caltech enables researchers to apply sustained creative effort on complex problems, even over decades, to achieve extraordinary results.”

The combined effort of Hou and his colleagues to prove the existence of an explosion with the 3D Euler equation is a major breakthrough in itself, but also represents a huge leap forward in solving the millennium problem of Navier-Stokes. 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 whole framework we’ve put in place for this analysis would be extremely helpful for Navier-Stokes,” Hou said. “I recently identified a promising candidate for the Navier-Stokes explosion. We just need to find the right formulation to prove the Navier-Stokes explosion.”

More information:
Jiajie Chen et al, Almost self-similar stable bursting of 2D Boussinesq and 3D Euler equations with smooth data, arXiv (2022). DOI: 10.48550/arxiv.2210.07191

Journal information:

Provided by California Institute of Technology

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