I am more interested in answers to this question then I can help, but as far as I know  there is no mathematical proof as yet that Navier-Stokes equations actually have solutions for other than trivial cases. There is one million dollars prize for the proof. So I am not sure if your question can be answered at all.

I lived in constant fear that the numerical approaches we use for turbulence and closure were really rubbish and that the use of Fourier modes was a convenient mathematical tool lacking conceptual justification.  Nevertheless, these are the tools to generate publications - and we must have publications!  If it all turns out to be wrong the most prolific publishers will be the journal judges so I don't know if it will get to be known.  In the words of Colonel Kurtz: The horror... the horror...

I agree with Andrew that proofs represent a problem and with Clifford about critics of numerical approaches.

There exist some theorems. Russian mathematician Olga Ladyzhenskaya provided the first rigorous proofs of the convergence of a finite difference method for the Navier-Stokes equations; see https://en.wikipedia.org/wiki/Olga_Ladyzhenskaya .

Navier-Stokes equations are not conservation equations, but rather a dissipative system and thus it is not possible to apply some math tools derived to conservative systems. They are applicable if we want to study the dynamics of the systems with viscosity that can also generate turbulence. Applications include modelling weather, ocean currents, water flow in a pipe and others.

If your problem allows to neglect viscosity, it may be better to use simpler Euler equations for inviscid flow, where there are more analytical results and less problems with convergence of numerical schemes. You can also introduce small parameter and use asymptotic methods to derive an equation that is both asymptotically correct and is more friendly for numerical methods.

Thank you for bringing Ladyzhenskaya to my attention.  In physics the N-S equations are often introduced as momentum conservation equations but, with thermal effects this is no longer true as you say.  There is a big push for "extended thermodynamics" whereby everything become hyperbolic to enforce causality.  I am no a fan of this approach as they usually introduce unphysical degrees of freedom.  I think the latest method by Karlin and Gorban may not have this problem.  

The Euler equations are problematic because they are a singular limit of N-S.  As the viscosity goes to zero the flow should become wildly turbulent so there is no nice way to get to Euler unless on varies time scales very carefully.  Almost all real experiments exhibit viscous boundary layer flow so I don't know how often such results are useful but I see a lot of them in books.  

I've made many attempts at this problem and have been doubting perturbative improvements in N-S for reasons other than Karlin's.  

As a numerical method is concerned, we can only support a solution with the Lax theorem

https://en.wikipedia.org/wiki/Lax_equivalence_theorem.

Furthermore, the Lax and Wendroff theorem state that if a numerical solution of conservative form converges, it does towards a weak solution.

https://en.wikipedia.org/wiki/Lax%E2%80%93Wendroff_theorem