From the perspective of trying to get to the actual physical behavior of such systems, the exercise is purely theoretical since we know there are other mechanisms at work to prevent the “ blow up” ...
(I'm not a fluid flow person and this is probably not a very good comment.)
Yeah, that seems likely. But sometimes it's difficult to draw the line between issues that are "purely theoretical" and issues that make a practical difference. Conceivably at some scale the Navier-Stokes equations match up with molecular dynamics simulations in a way that is illuminating. Then in that situation maybe the problems with the N-S equations become nice indicators of what's going on.
Yeah, that seems likely. But sometimes it's difficult to draw the line between issues that are "purely theoretical" and issues that make a practical difference. Conceivably at some scale the Navier-Stokes equations match up with molecular dynamics simulations in a way that is illuminating. Then in that situation maybe the problems with the N-S equations become nice indicators of what's going on.