Let me rebuke just one part of the post: the example with Newton's laws. Of course everyone knows that Navier-Stokes is just a very nice approximation. That doesn't mean that mathematical results about Navier-Stokes can't improve or understanding of the physics side. For instance, if we find a solution which blows up, that may lead us to a correction to the model or (less likely) it may be actually physically realizable.