In the same way that eddies in a stream alter downstream currents, Elgindi’s work itself prompted a new round of mathematical discovery. In October 2019, Hou and Jiajie Chen adapted some of Elgindi’s methods to create a rigorous mathematical proof of a scenario closely related to the one in the 2013 experiment. They proved that in this slightly modified scenario, the singularity they’d observed forming in the Euler equations really does occur.
“They took Elgindi’s ideas and applied them to the scenario from 2013”. The circle was complete.
There’s still more work to be done, of course. Hou’s new proof has some technical qualifications that prevent it from establishing the existence of the singularity in the exact situation he modeled in 2013. But after a remarkable 6-year run and with renewed momentum, Hou believes he’ll soon surmount those challenges, too. “I think we’re very close”.
Now another group has joined the hunt. They’ve found an approximation of their own — one that closely resembles Hou and Luo’s result — using a completely different approach. They’re currently using it to write their own computer-assisted proof. The team’s answer looked a lot like the solution that Hou and Luo had arrived at in 2013. But the mathematicians hope that their approximation paints a more detailed picture of what’s happening, since it marks the first direct calculation of a self-similar solution for this problem. “The new result specifies more precisely how the singularity is formed. You’re really extracting the essence of the singularity,. It was very difficult to show this without neural networks. It’s clear as night and day that it’s a much easier approach than traditional methods.”
Gregor J. Rothfuss 