If Navier-Stokes allows simulation of a Turing machine, then Gödel may make existence of a solution undecidable

Let us consider the Navier-Stokes equation of perfect fluid, with no atoms or other type of a cutoff at very short distances.

Solutions of the Navier-Stokes equation easily develop turbulence. Turbulence is a fractal-like phenomenon.

Can we harness turbulence to do complex digital calculations, like on a Turing machine?

If yes, then we might get analogues of the Gödel incompleteness theorem for solutions of the Navier-Stokes equation.

Suppose that a proof of a contradiction from the Peano axioms is equivalent to proving that a certain solution of the Navier-Stokes equation develops a singularity. Then it would be an undecidable problem if a singularity appears.

https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

Terence Tao in his 2007 blog post refers to a possible connection of problems of complexity theory (e.g., P = NP) to solutions of the Navier-Stokes equation. Turbulence develops complex, pseudorandom structures. (See his note "6. Understanding pseudorandomness".)

If we can build a digital computer from turbulence, then complexity theory will pop up.

Real fluid has a cutoff at the atomic scale. We do not expect a turbulence-based Turing machine to have any relevance in the real physical world.

Quantum fields probably have a cutoff at the scale of the Planck length, because mini black holes may turn up. It is unlikely that we can harness microscopic quantum fields to make a Turing machine, but this deserves further thought.