The motion of fluids in nature, such as the flow of ocean water, the formation of tornadoes in the atmosphere, and the flow of air surrounding an airplane, has long been described and simulated by what are known as the Navier-Stokes equations.
However, mathematicians do not fully understand these equations. They are useful tools for predicting fluid flow, but it is not yet known whether they accurately describe fluids in all possible scenarios. Now labels the Stokes equation as one of the seven Millennium Problems. Here are his seven most pressing open problems in all of mathematics.
The Navier-Stokes Equation Millennium Problem challenges mathematicians to prove whether there always exists a “smooth” solution to the Navier-Stokes equations.
Simply put, smoothness refers to whether this type of equation behaves in a meaningful and predictable way. Imagine a simulation where you put your foot on the gas pedal of a car and the car goes 10 miles per hour (mph), then he accelerates to 20 mph, 30 mph, 40 mph. However, if you put your foot on the gas pedal and the car accelerates to 50 miles per hour, then 60 miles per hour, and then instantly to infinity, then something is wrong with your simulation.
This is what mathematicians want to determine about the Navier-Stokes equations. Do they always simulate fluids in a sensible way or are there situations where they fail?
In a paper published on the preprint server arXiv, Thomas Hou of Caltech, Charles Lee Powell Professor of Applied and Computational Mathematics, and Jiajie Chen (Ph.D. ’22) of the Courant Institute of New York University solve the long-standing open problem of the so-called 3D Euler provide proof. Singularity.
The 3D Euler equation is a simplification of the Navier-Stokes equation, and the singularity is the point at which the equation begins to collapse or “explode”. miles per hour). This proof builds on a scenario originally proposed by Hou and his former postdoc Guo Luo in 2014.
Hou’s calculations with Luo in 2014 found a new scenario that provides the first compelling numerical evidence for 3D Euler blowup, but previous attempts to discover 3D Euler blowup were inconclusive. or it was not reproducible.
In their latest paper, Hou and Chen present conclusive and irrefutable evidence for Hou and Luo’s work involving the explosion of the 3D Euler equation. “It starts out working well, but then somehow evolves in ways that become devastating,” Hou says.
“For the first decade of my research, I believed there was no Euler explosion,” says Hou. More than a decade of research since then, Hou has not only proven his former self wrong, but has also solved a centuries-old mathematical riddle.
“This breakthrough is a testament to Dr. Hou’s tenacity in addressing the Euler problem and the intellectual environment that Caltech fosters, and materials science, and director of the Liquid Sunlight Alliance.” allows us to apply sustained creative effort to complex problems and achieve extraordinary results over decades.”
Hou and Colleagues Collaborate to Prove Existence of Blowup in 3D Euler equation is not only a major breakthrough in itself, but also a giant leap in tackling the Navier-Stokes Millennium Problem. If the Navier-Stokes equations can also blow up, it means something is wrong with one of the most basic equations used to describe nature.
“The whole framework we have set up for this analysis will be very helpful for Navier–Stokes,” says Hou. “I recently identified a promising detonation candidate for Navier-Stokes. We just need to find a suitable formulation to prove the detonation of Navier-Stokes.”
For more information:
Jiajie Chen et al, Stable nearly self-similar blowup and smooth data for 2D Boussinesq and 3D Euler equations, arXiv (2022). DOI: 10.48550/arxiv.2210.07191
California Institute of Technology
Quote: Mathematicians solve the long-standing open problem of the so-called 3D Euler singularity (23 Nov 2022). Retrieved 24 November 2022 from https://phys.org/news/2022-11-mathematicians-longstanding-problem-so-called-. 3d.html
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