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Let us suppose that there exists a (initially smooth) solution of NSE that blows up in finite time. Then, in particular, the corresponding velocity field becomes unbounded as time progresses. Which assumptions in the common modelling of fluid flow (leading to NSE) will be violated and in which way will NSE need to be adjusted?

I read that the viscous term is some kind of expansion and higher derivatives of the velocity field u, like \Delta^2 u, should, in fact, be retained. (It´s well-known that NSE with hyperdissipation admits global smooth solutions.) Unfortunately, i couldn't find any physical details on this expansion. Does someone know about it? Thanks a lot in advance! :-)

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    $\begingroup$ Many non-specialists would be interested in an expert answer to this question, not only for the mathematics-per-se of Navier-Stokes, but for the insinuations about actual ... weather, say? :) $\endgroup$
    – paul garrett
    Commented Aug 22, 2012 at 0:24
  • $\begingroup$ Terry Tao wrote a blog post terrytao.wordpress.com/2007/03/18/… related to this a couple years ago. Specifically related to adjustements see also the (first couple) comments. $\endgroup$
    – user9072
    Commented Aug 22, 2012 at 13:01
  • $\begingroup$ 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? $\endgroup$
    – Daniel Lengeler
    Commented Aug 22, 2012 at 22:27

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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.

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If the velocity becomes infinite, it exceeds the speed of sound, so incompressibility is no longer a valid assumption. On the other hand, we should consider the fact that the pressure also reaches minus infinity in a hypothetical blow-up solution. The practical consequence of this is cavitation.

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  • $\begingroup$ Why stop at the speed of sound, you'll go faster than the speed of light! $\endgroup$
    – user7807
    Commented Aug 22, 2012 at 18:52
  • $\begingroup$ 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? $\endgroup$
    – Daniel Lengeler
    Commented Aug 22, 2012 at 21:43

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