Question: At the [109th UCLA Faculty Research lecture](http://youtu.be/i6RutDwO3XU), Seth Putterman gave a talk on Sonoluminescence. During the lecture he emphasized that "The Navier Stokes equations cannot be derived from first principles [of physics]".

In physics there are lots of first principles, and so the first question is what set of first principles would one expect to derive the Navier Stokes equations?

And the second, and main question is why does a derivation fail? Are we missing some yet to be discovered set of first principles in this area of physics?

Answer: 
None of the interesting equations in physics can be derived from simpler principles, because if they could they wouldn't give any new information. That is, those simpler principles would already fully describe the system. Any new equation, whether it's the Navier-Stokes equations, Einstein's equations, the Schrodinger equation, or whatever, must be consistent with the known simpler principles but it has also to incorporate something new.

In this case you appear to have the impression that an attempt to derive the Navier-Stokes equations runs into some impassable hurdle and therefore fails, but this isn't the case. If you [search for derivations of the Navier-Stokes equations](https://www.google.co.uk/search?q=navier%20stokes%20derivation) you will find dozens of such articles, including (as usual) [one on Wikipedia](https://en.wikipedia.org/wiki/Derivation_of_the_Navier%E2%80%93Stokes_equations). But these are not derivations in the sense that mathematicians will derive theorems from some initial axioms because they require some extra assumptions, for example that the stress tensor is a linear function of the strain rates. I assume this is what Putterman means.

**Later:**

[Phil H takes me to task in a comment](https://physics.stackexchange.com/questions/191786/why-cant-the-navier-stokes-equations-be-derived-from-first-principle-physics/191792#comment407840_191792), and he's right to do so. My first paragraph considerably overstates the case as the number of equations that introduce a fundamentally new principle are very small.

My answer was aimed at explaining why Putterman says the Navier-Stokes equations can't be derived but actually they can be, as can most equations. Physics is based on [reductionism](https://en.wikipedia.org/wiki/Reductionism), and while I hesitate to venture into deep philosophical waters physicists basically mean by this that everything can be explained from a small number of basic principles. This is the reason we (some of us) believe that a [theory of everything](https://en.wikipedia.org/wiki/Theory_of_everything) exists. If such a theory does exist then the Navier-Stokes equations could in principle, though not in practice, be derived from it.

Actually the Navier-Stokes equations could in principle be derived from a statistical mechanics treatment of fluids. They don't require any new principles (e.g. relativity or quantum mechanics) that aren't already included in a the theoretical treatment of ideal fluids. In practice they are not derivable because those derivations based on a continuum approach rather than a truly fundamental treatment.
They are derivable from classical mechanics using either the continuum or molecular points of view.

Starting with a continuum view, one applies conservation of mass, momentum, and energy to a control volume and the result is the Navier Stokes equations. The Navier Stokes equations, in the usual form, apply to Newtonian fluids, that is fluids whose stress and rate-of-strain are linearly related. One might regard this as an assumption but it can also be viewed as the first term in a power law expansion.

Starting with a microscopic point of view, one can derive the Navier-Stokes equations from taking moments of the Boltzmann equation. In this approach, the linear relation between stress and rate-of-strain appears naturally as the first term in the Chapman-Enskog expansion.

Many undergraduate fluids textbooks include a derivation from the continuum point of view. The derivation from a molecular point of view is done in first-year graduate textbooks such as *Introduction to Physical Gas Dynamics* by Vincenti and Kruger.