FYI: The problem at hand is an existence problem. No real construction of any real formula for the solution is provided, only a singular perturbation expansion.

In short: The problem is about whether a solution (of the NS Equations with external driving force) can be found that blows up in finite time. i.e. exhibits infinite velocity at a point even for a viscous flow.

The solution: Take a circular curl ansatz which shrinks in xy-direction and elongates in z-direction and see whether you can find linearized waves so that these waves show a blow up when propagated on the curl. Then prove that the higher orders of the perturbation are regular before the T0 singularity time and you have solved the problem. The external force is just the remainder of the NS-Equation right hand side.

It is a lot of tedious formula juggling of all the higher orders and some singular perturbation expansions. Perfectly suited for algebra systems. OpenAI was using probably python sympy for the formula work and the researchers had to guide the LLM what to do in higher mathematical language.

Here is the paper:

https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8...

when you upload it to chatgpt astra can explain what they do and why it works, have fun.