HomeSciencePhysicsMathematicians resolve a longstanding open problem for the so-called 3D Euler singularity

Mathematicians resolve a longstanding open problem for the so-called 3D Euler singularity

X. Credit: arXiv (2022). DOI: 10.48550/arxiv.2210.07191″ width=”800″ height=”309″/>
Estimated stable 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 flow of air around airplanes, have long been described and simulated by what are known as the Navier-Stokes equations.

Yet mathematicians do not have a complete understanding of these equations. While they are a useful tool for predicting the flow of fluids, we still don’t know if they accurately describe fluids in all possible scenarios. This led the Clay Mathematics Institute of New Hampshire to designate the Navier-Stokes equations as one of the 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 “smooth” solutions always exist for the Navier-Stokes equations.

Simply put, smoothness refers to whether equations of this type behave in a predictable way that makes sense. Imagine a simulation in which a foot presses the gas pedal of a car and the car accelerates to 10 miles per hour (mph), then to 20 mph, then to 30 mph, then to 40 mph. However, if the foot presses the accelerator and the car accelerates to 80 km/h, then to 100 km/h, and then immediately to an infinite number of miles per hour, you would say that there is something wrong with the 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 certain situations where they break down?

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In a paper published on the preprint server arXivCaltech’s Thomas Hou, the Charles Lee Powell Professor of Applied and Computational Mathematics, and Jiajie Chen (Ph.D. ’22) of New York University’s Courant Institute, provide evidence 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 where an equation begins to break down or “blow up,” meaning it can suddenly become chaotic without warning (such as the simulated car accelerating to an infinite number of 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 burst, while previous attempts to discover a 3D Euler burst were either inconclusive or not reproducible.

In the latest paper, Hou and Chen show the definitive and irrefutable proof of Hou and Luo’s work on blowing up the 3D Euler equation. “It starts with something behaving well, but then somehow evolves in a way where it becomes catastrophic,” says Hou.

“For the first 10 years of my work, I believed there was no Euler burst,” says Hou. With more than a decade of research since then, Hou has not only proven him wrong, he has also solved an age-old math mystery.

“This breakthrough is a testament to Dr. Hou’s tenacity in addressing 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 the Liquid Sunlight Alliance. “Caltech enables researchers to apply sustained creative effort on complex problems – even over decades – to achieve extraordinary results.”

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Hou and colleagues’ joint effort to prove the existence of blasts with the 3D Euler comparison is a major breakthrough in its own right, but also represents a huge step forward in tackling the Navier-Stokes millennium problem. If the Navier-Stokes equations could also explode, it would mean 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 to Navier-Stokes,” says Hou. “I recently identified a promising Navier-Stokes explosion candidate. We just need to find the right wording to prove the Navier-Stokes explosion.”

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

Magazine information:

Quote: Mathematicians Solve a Longstanding Open Problem for the So-Called 3D Euler Singularity (2022, Nov. 23) Retrieved Nov. 23, 2022 from -called- 3d.html

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