Quantization solves the existence problem of the solutions for the Einstein equations?

https://en.wikipedia.org/wiki/Exact_solutions_in_general_relativity#Existence_of_solutions

In 1993, Demetrios Christodoulou and Sergiu Klainerman were able to prove the stability of the Minkowski vacuum under small perturbations.

But the existence of solutions for the Einstein equations remains unproven for essentially all practical cases - that is, if we have a non-symmetric, non-uniform mass distribution.

The Navier-Stokes equations

It is notoriously hard to prove the existence of smooth solutions for non-linear differential equations. The most famous example is the Clay Millennium Problem about the smooth solutions of the Navier-Stokes equations.

Let us think about a real physical fluid, say, water. A milliliter of water contains some 3 * 10^22 water molecules H2O. The Navier-Stokes equations approximate a viscous flow of a very large number of water molecules. The equations are an idealized effective theory of a macroscopic amount of water.

A priori, there is no reason why the equations would make sense, or have smooth solutions, if we extend them to the case where a water molecule is infinitesimally small. The Clay Millennium problem may have little physical relevance. 

A water molecule size gives a natural cutoff scale for the Navier-Stokes equations. Approximate solutions of the equations are physically relevant provided that features whose size is of the order of a molecule do not affect the solution.

In the case of water, the quantum of water, a single molecule, saves us from the problem of the existence of smooth solutions.

Maxwell's equations

The electromagnetic field is another example of quantization. Maxwell's equations describe the behavior of a macroscopic classical field. We assume that a photon carries an energy hf, where h is the Planck constant and f is the frequency of classical (circularly polarized) macroscopic wave.

A very large number of coherent photons form a classical macroscopic wave. But Maxwell's equations do not describe the absorption of a single photon correctly. The equations are not aware of the quanta.

Maxwell's equations are an effective theory. Does it make sense to study the smoothness of the solutions for features whose size is much less than the photon wavelength? Probably not - a very short wavelength would involve a photon of a high energy. How could such a photon be produced? From where would the energy come from?

The Einstein equations

What about the existence and smoothness of solutions for the equations of General Relativity?

We do not know if the field is quantized. If it is, then we might get a cutoff scale which saves us from considering features smaller than a certain length.

We need to check if the problem in proving the stability of the Einstein equations involves very small features. If the problem is the appearance of singularities, then a cutoff scale would help.

In our earlier post today about a perpetuum mobile we again pointed at the possibility that the Einstein equations, combined to a lagrangian of an infinitely strong vessel, may have no sensible solutions at all. This potential problem is separate from the stability and smoothness of pure Einstein equations.

Conclusions

The smoothness and stability problems in various non-linear physical equations probably are not relevant, once we take into account the quantization of the field.