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

x. Attestation: 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 motion 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 equations of Navier-Stokes.

However, mathematicians don’t have a complete understanding of these equations. While they’re a useful tool for predicting fluid flow, we don’t yet know if they accurately describe fluids under all possible scenarios. This led New Hampshire’s Clay Mathematics Institute to label the Navier-Stokes equations one of its Seven Problems of the Millennium—the seven most pressing unsolved problems in all of mathematics.

The Navier-Stokes equation millennium problem asks mathematicians to prove whether there are always “smooth” solutions to the Navier-Stokes equations.

Simply put, uniformity refers to whether equations of this type behave in a predictable and sensible way. Imagine a simulation where one foot presses a car’s accelerator pedal and the car accelerates to 10 miles per hour (mph), then 20 mph, then 30 mph, and finally 40 mph. However, if your foot hits the gas pedal and the car accelerates to 50 mph, then 60 mph, then instantly to an infinite number of miles per hour, you’d say something was wrong with the simulation.

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

In a document published on the prepress 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.

The 3D Euler equation is a simplification of the Navier-Stokes equations, and a singularity is the point at which an equation starts to break down or “explode”, meaning it can suddenly become chaotic without warning (such as the car simulated accelerating to an infinite number of miles 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 uncovered a new scenario that showed the first convincing numerical evidence for a 3D Euler magnification, whereas previous attempts to discover a 3D Euler magnification were inconclusive or unreproducible.

In the latest article, Hou and Chen show definitive and irrefutable proof of Hou and Luo’s work involving the magnification of the 3D Euler equation. “It starts with something that performs well, but then it kind of evolves in a way where it becomes catastrophic,” Hou says.

“For the first ten years of my work, I believed there was no Euler explosion,” Hou says. After more than a decade of research, Hou not only proved himself wrong, but solved an age-old mathematical mystery.

“This breakthrough is a testament to Dr. Hou’s persistence in addressing the Euler problem and the intellectual milieu that Caltech nurtures,” said Harry A. Atwater, Otis Booth Leadership Chair of the Division of Engineering and Applied Sciences, Howard Hughes Professor of Applied Physics and Materials Science and director of the Liquid Sunlight Alliance. “Caltech enables researchers to apply sustained creative effort to complex problems, even spanning decades, to achieve extraordinary results.”

The combined effort of Hou and colleagues to prove the existence of magnification with the 3D Euler equation is in itself a major step forward, but it also represents a huge leap forward in tackling the Navier-Stokes Millennium Problem. 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 created for this analysis would be extremely useful for Navier-Stokes,” says Hou. “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, Nearly self-similar stable magnification of 2D and 3D Euler Boussinesq equations with uniform data, arXiv (2022). DOI: 10.48550/arxiv.2210.07191

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