Let me make my question a bit more precise. The situation i had in mind is a fluid flow whose mean flow, say, has a low Mach number. As far as i know experiment says then that the turbulent fluctuations will have a low Mach number, too. Now, might the model (in particular the viscous term) need to be adjusted (due to large velocity gradients for example) already in the low Mach number regime?

So, talking about compressible NSE, the first assumption to be violated in an infinite-velocity blow-up is the continuum hypothesis? Shouldn't the viscous term be adjusted before that happens? On the other hand: I guess a cavitation can be observed in experiment. Are there corresponding singular solutions of compressible NSE known to exist?