Will a proof of the existence and smoothness of the Navier-Stokes equations contribute to a GUT? Note: throughout the course of the question, the word "describe" will often be used to suggest that a mathematical equation can "describe" what it happening in a physical context.
One of the long term goals for physicists at the moment is the development of a Grand Unified Theorem (GUT), which describes a collection of mathematical descriptions that can be applied to physics on any scale (for the sake of clarity, I will refer to equations that describe quantum mechanics as quantum equations and equations that describe relativistic mechanics simply as relativistic equations).
The need for this stems from the current model of mathematics we utilize in physics being accused of being "outdated", since it still entails quantum equations that cannot be applied to a relativistic context (put simply: equations that we use within the scale of femtometers will not "work" when applied to a scale of kilometers, and obviously everything on a greater scale), and the inverse also applies.
In light of this, various problems have arised in mathematics as a challenge to "catch up" with the physics, of which the two most prominent ones have been awarded a position on the list of the millennium prize problems list, namely the Yang-Mills' Mass Gap Hypothesis and the Existence and Smoothness of the Navier-Stokes equations. The former is one of the problems in the vanguard of the GUT, but my question lies with the latter.
Given that each of the Navier-Stokes equations is a partial differential equation, there is no doubt that the role of differential calculus is pivotal, and with this in mind, if the equations themselves originate from infinitesimals, does this imply that they are applicable to a quantum context? $(1)$
Furthermore, in a lay book called "The Millennium Prize Problems" written by Keith Delvin in $2002$, he mentions that:

"...the [Navier-Stokes] equations themselves have various applications; on the face of it, they can be applied when investigating the the flow of fluids around the hull of a boat. More obvious applications include the flow of fluids around the wing of a plane, in addition to more cryptic applications, ..."

and if this is true, then this must mean that they are applicable to a (more) relativistic context. $(2)$
Ultimately, then: is it true that the equations are applicable to a quantum context, as suggested in $(1)$? And is what has been said about the use of the Navier-Stokes equations in $(2)$ true? If so, does this mean that a proof of the existence and smoothness of the Navier-Stokes equations would be a step closer to a GUT?

Question in short
Are the Navier-Stokes equations applicable to a relativistic and quantum context? If so, are they therefore a step in the direction of a GUT?

 A: Fluids in General Relativity
The Navier Stokes equations are applicable to fluids in general relativity, but only if two conditions are met: 


*

*the gravitational fields are weak

*the speeds involved are considerably smaller than the speed of light


In particular one can start with the stress energy tensor for a perfect fluid:
$$ T^{\alpha\beta} = (\rho+p/c^2)u^\alpha u^\beta + p g^{\alpha \beta} $$
where $\rho$ is the density and $p$ is the pressure of the fluid. Inserting this into Einstein's equation and assumming the metric is in the weak-field form:
$$ ds^2=-e^{2\phi/c^2}dt^2 + dx^2 + dy^2 + dz^2$$
one obtains a certain equation, which in the Newtonian limit (speeds much less than $c$) reduces to the Navier-Stokes equations.
However, in general the equations obeyed by a general relativistic fluid are more complicated and Navier Stokes are not valid in general.
Fluids in Quantum Mechanics
An example of fluid motion which is affected by quantum mechanics is that of a superfluid. You can have a look at this paper, where quantum mechanical analogs of Navier Stokes are discussed. 
Is the Navier Stokes Millenium Prize Problem relevant to a GUT?
It's important to notice that Navier Stokes not only does not hold in general in quantum mechanics or general relativity; it doesn't even hold in general in classical mechanics. The general classical continuum equation is the Cauchy equation. One then can obtain the Navier Stokes equation by assuming that the fluid is incompressible and newtonian.
So now to your question: is the solution of the millenium prize problem on Navier Stokes important towards a Grand Unified Theory? It depends on what you mean by GUT. In the most common use of the term (see Wikipedia) a GUT is a theory which unifies the three non-gravitational interactions that we know of. If one includes gravity, then one gets a Theory of Everything. Notice that this unification is at the moment mostly a mathematical undertaking, since up to now the Standard Model is in principle capable of describing everything that we can do experiments on (see for instance this post by Sean Carroll). The energy scales at which a GUT might become relevant are far beyond our current technological capabilities. So in this sense the answer to your question would be no: they're different things.
Notice, however, that the Standard Model is only in principle capable of explaining all the experiments that we know of. In this sense, if we're making a fluid experiment we're not going to use the Standard Model machinery, but probably Navier Stokes (or the other fluid equation appropriate in the context). However, we do believe that under appropriate assumptions the Standard Model reduces to Navier Stokes (or the right equation for the context).
So if your question is more generally: will it improve our understanding of physics? Then the answer is yes: it is probably a step towards a better understanding of turbulent phenomena, which can be described by Navier Stokes and can have a lot of practical applications.
