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Comment by tnecniv | Hacker News Reader

Re: Navier-Stokes

The PDE that most analyses start with does not need to be proven. This has been rigorously derived since forever. The thing that actually needs to be proven is whether or not smooth solutions exist to the PDE for all cases of parameters. In practice, people tend to stick to working with simple cases that can be solved analytically, or use computer simulations to compute approximate solutions (to very high precision).

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What does it even mean "to prove a PDE"? The million dollars is for proving certain properties of a PDE, in this case, whether smooth solutions exist.

I was just using the language of the parent. By proven I meant derived from Newton's laws, and that was my interpretation of the parent's comment.

The NS momentum equation is just Newton’s second law written for all the forces acting on a fluid. We just don’t know if it always works.

> The PDE that most analyses start with does not need to be proven. This has been rigorously derived since forever.

Are you sure?

https://www.quantamagazine.org/famous-fluid-equations-are-in...

https://www.quantamagazine.org/mathematicians-find-wrinkle-i...

To add on that, for all cases really means a lot more than bends in pipe or air off a wing or smoke which we have difficulty with today. You'll need to hit everything, plasma on the Sun, exotic matter, it leads to quasiparticles for the many-body problem.

My interest in this subject is I hope new developments lead to cardiac therapies. I send some buzzing acoustic signal with a pacemaker, I can treat blockages in the brain or tumors in hard to reach organs. A lot better to me than building faster missiles to blow more people up.

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