The Navier-Stokes millennium problem and the existence of solutions for the Einstein equations

In this blog we have brought up the question whether the Einstein field equations have a solution for any realistic physical system.

The Einstein equations are nonlinear. Therefore we cannot build complex solutions by summing simple solutions.

The Einstein equations are very strict: they imply results like the Birkhoff theorem, which dictates energy conservation for all matter fields.

https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness

The Navier-Stokes equation is nonlinear, too.

https://arxiv.org/abs/1402.0290

Terence Tao suspects in his 2015 paper that at least some smooth initial problems "blow up", that is, develop singularities.

The problem in proving the existence of solutions for Navier-Stokes is in turbulence. We currently lack tools to understand turbulent behavior mathematically. Turbulence might create a singularity very quickly for almost all initial conditions.

The (assumed) generation of singularities is another common feature of Navier-Stokes and Einstein.

We do not know if solutions of the Einstein equations might contain turbulent behavior. The problem in finding solutions for Einstein seems to lie in the strictness of the equations. Solving the equations requires that "ends meet" when we assemble a puzzle on the FLRW model finite space. How do we know there exists any configuration of pieces such that the ends meet?

Terence Tao remarks that turbulence may be analogous to problems of complexity theory, for instance, to the P = NP millennium problem. We believe that P = NP will never be solved, because the collection of all polynomial-time algorithms is too complicated a system to be tamed and analyzed.