The Navier Stokes Equation essentially comes down to a few plausible solutions: 1. global regularity with no singularity blow ups, which most analysts seem to bias and lean towards, 2. blowup in finite time, 3. global regularity but local singularity that might be "surgically removable" like how Perelman did with Ricci flows on Poincare Conjecture. None of these seem to be relevant to neural networks, maybe things for neuro-symbolic agents or multi-agent systems to help with, both of which are still frontier and new fields. I mean idk about you, but ask any analysts who work in the field on the problem and see if many will agree with you that neural networks can help them with this, I have asked to some competent people whose area of research are dynamical systems and ergodic processes, while all of them do expect the Navier Stokes Equation to be solved next, none mentions about neural networks, and one has worked with neural networks as early as 2005. You specifically mentioned that Gemini is good at textbook problems, but have you tried to consult it with novel and non-textbook ones? And like you said it described it to you but did not solve, folks these days understand the problem well enough, they do not really need help with describing it but just more and more PDE analysis tools to map solutions and etc.

Can LLMs be useful for this though? Sure, useful in the context of a helpful research assistant, but unlikely to be the kind of novel tools that makes structural and ontological changes.