Mathematicians solve a long-standing open problem of the so-called three-dimensional Euler singularities

يحل علماء الرياضيات مشكلة مفتوحة طويلة الأمد لما يسمى بتفرد أويلر ثلاثي الأبعاد

Approximate steady state in the near field. left: profile ω̅; Right: θ̅ x. attributed to him: 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 hurricanes in our atmosphere, and the airflow surrounding aircraft, has long been described and simulated by what are known as the Navier-Stokes equations.

However, mathematicians do not have a complete understanding of these equations. While it is a useful fluid flow forecasting tool, we still don’t know if it accurately describes fluids in all possible scenarios. This led the Clay Institute of Mathematics in New Hampshire to name the Navier-Stokes equations as one of the Seven Millennium Problems: The Seven Most Burning 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 indicates whether equations of this type behave in a predictable and logical manner. Imagine a simulation where the foot hits the accelerator of a car, and the car accelerates to 10 mph, then to 20 mph, then to 30 mph, then to 40 mph. However, if the foot hits the accelerator and the car accelerates to 50 mph, then to 60 mph, and then immediately to infinite miles per hour, you can say that something is 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 some situations where they fall apart?

In a paper published on a prepress server arXivThomas Hu of Caltech, Charles Lee Powell Professor of Applied and Computational Mathematics, and Jiajie Chen (PhD ’22) of NYU’s Courant Institute present evidence that solves a long-standing open problem of the so-called 3D Euler singularity.

Euler’s 3D equation is a simplification of the Navier-Stokes equations, and unity is the point at which the equation begins to collapse or “explode”, meaning it can suddenly become chaotic without warning (like a simulated car accelerating to infinity miles per hour) . The evidence is based on a scenario first proposed by Hu and his former postdoctoral researcher, Guo Luo, in 2014.

Hou’s account with Luo in 2014 discovered a new scenario that showed the first convincing numerical evidence for a 3D Euler detonation, while previous attempts to detect a 3D Euler detonation were either inconclusive or not reproducible.

In their latest paper, he and Chen show irrefutable evidence of Hu and Lu’s work involving detonating Euler’s equation in 3D. “It starts from something that’s benign, but somehow it develops into a catastrophe,” Ho says.

“For my first 10 years, I thought there was no Euler detonation,” Hu says. After more than a decade of searching since then, Huo has not only proven him wrong previously, but also settled a centuries-old mathematical puzzle.

says Harry Attwater, Otis Booth Principal Chair in the Department of Engineering and Applied Sciences, Howard Hughes Professor of Applied Physics and Materials Science, and Director of the Liquid Sunlight Alliance. “California Institute of Technology enables researchers to apply sustained creative effort to complex problems—even over decades—to achieve extraordinary results.”

He and his colleagues teamed up to prove the existence of an explosion using a 3D Euler apparatus Equation That’s a huge accomplishment in itself, but it’s also a huge leap forward in addressing Navier-Stokes’ millennial problem. If the Navier-Stokes equations can also go awry, then something is awry with one of the most basic equations used to describe nature.

“The whole framework we’ve created for this analysis will be very useful to Navier-Stokes,” Hu says. “I recently identified a promising explosive candidate for Navier-Stokes. We just need to find the right formula to prove the Navier-Stokes explosive.”

more information:
Jiajie Chen et al, Semi-autonomous stable explosion of Boussinesq and 3D Euler equations with smoothed data, arXiv (2022). doi: 10.48550/arxiv.2210.07191

Journal information:
arXiv


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