This is a comment about Michael's answer and the comment by Daniel about compressible NSE.
1- There are several models for viscous fluids, among which we may select incompressible/compressible NSE. It is widely believed that the compressible case behaves wildlier than the incompressible one. Therefore, saying that a blow up singularity in the incompressible NSE means that incompressibility is no longer a valid assumption is not the end of the story. Because then we have to use the compressible model, which likely displays the same trouble.
2- In the compressible case, the issue of cavitation is subtle and so far remains an open problem. Except in one space dimension (ha! ha!), we don't know whether vaccum may occur at positive time if the initial density is strictly positive. But the one-D case is interesting: D. Hoff and myself have a paper in which we show that a constant viscosity coefficient yields an unphysical behaviour at very low densities ; something like an ill-posedness result for the Cauchy problem. This suggests to take in account the dependence of the viscosities upon the state (density, temperature). Actually, the derivation of NSE from Boltzmann yields a dependence upon the temperature alone. In isentropic flows, this amounts to saying that the viscosities depend upon the density ; in which case the viscosity vanishes with the density. This is what is needed to avoid the ill-posedness mentionned above.