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"Does there exist a pathological combination of smooth body forces and initial conditions for this set of PDEs, where singularities appear, which by the way is completely impossible to actually create in the real world unless you are a literal God?" is a question of math, not physics. This is a hill I will die on.

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AFAICT the unforced problem is still open. I don't think we've established that you need to be a literal God to create a finite-time blowup.

If you think about what it actually means to have a time-varying smooth body force defined in all of 3-space, you fairly quickly come to that kind of conclusion.

Even if someone comes up with a construction that does not require any forcing, it is going to be some extremely weird initial conditions that you will never be able to even approximate in reality unless you can move all the individual molecules of a fluid around and set their initial velocities from a far distance.

The unforced problem is still open.

You realize that hill is a mathematical argument, not a physical hill.

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