The Navier-Stokes Clay problem: external force?

This is a follow up on my last two posts, here and here. 

Another nuance of the OpenAI Clay problem solution is the existence of an external force in the Navier-Stokes equations that can cause the blow-up.  A version of the problem in the absence of any external force remains unsolved, according to this post on Scientific American by Joseph Howlett.

 

More on the Navier-Stokes AI breakthrough

As the dust settles on last week's OpenAI announcement regarding the discovery of a blow-up solution for the incompressible Navier-Stokes equations, Nature has provided additional coverage of note to DTLR readers.  One is another article by Davide Castelvecchi regarding the wider implications on the assignment of credit for mathematical discovery, prompted by the controversy between OpenAI and Buckmaster and Alpoge.  This is certainly an important issue for mathematicians and scientists.

Addressing the implications for the science of fluid mechanics, the other article, by Nicola Jones, points out that the blow-up solution occurs in a situation in which the incompressible Navier-Stokes equations are invalid.  Thus the implications for lower-level modeling, such as using the kinetic theory of gases or molecular dynamics simulations, are unclear (and quite possibly irrelevant).  The article also discusses other uses of AI to study fluid mechanics problems such as turbulence and atmospheric dynamics.

I'm sure there is much additional coverage that I'm missing out on, but I will endeavor to bring the most notable discussions I've come across to readers of DTLR.

A tumultuous week

The past week has been tumultuous.  I never expected to see the Navier-Stokes equations being mentioned in blockbuster coverage in the New York Times and Washington Post.  I speak of course about the claim made by OpenAI to have found a blow-up solution to the Navier-Stokes equations (NSE), announced on Sept. 8, and the ongoing dispute between them and mathematicians Tristan Buckmaster and Levent Alpoge.

These events have garnered wide attention and discussion across multiple platforms.  I haven't been following in detail, but the most helpful articles for laypersons like me that I've come across have been the following:

• Konstantin Kakaes' piece in Quanta Magazine;
• Celina Zhao and Adrian Cho's article in Science magazine; and
• Davide Castelvechhi's coverage for Nature.

Unfortunately the Castelvecchi article is paywalled.  However I want to credit him for calling attention to the work of a third group, led by Anima Anandkumar, who also released a paper on the 7th regarding the Euler problem, but which used a different AI approach (physics informed neural networks).

I have little to add except that, if the OpenAI result is validated and accepted by the mathematical community, it will have shown that the Navier-Stokes equations are in some sense "broken", a result that has been anticipated due to recent developments in the field.  It is important to bear in mind that the impact on physics, engineering, and the geosciences is quite limited.  The Navier-Stokes equations are derived from the principles of mass and momentum balance, plus some simple assumptions about the fluid material itself.  The new blow-up solution would appear to violate the conservation of momentum and of energy, making it unphysical on its face.  The physical principles of mass, momentum, and energy conservation are more "fundamental" than the Navier-Stokes equations, and these underlying principles remain unthreatened by this result.  The NSE can be derived as a model of fluid motion from various perspectives, whether from the kinetic theory of gases, or as an effective field theory, as well as from its original continuum formulation from the 19th century.

On a related note, I want to give a shout out to Quanta Magazine, which has consistently provided some of the best nontechnical exposition on "hot topics" in fluid mechanics research (especially mathematical fluid mechanics, which can be particularly difficult to cover in this manner) that I've seen in recent years.