## Implications of a hypothetical blow-up of Navier-Stokes for the mathematical model

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. (ItÂ´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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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? :) – paul garrett Aug 22 at 0:24
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. – quid Aug 22 at 13:01
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? – Daniel Lengeler Aug 22 at 22:27

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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 Why stop at the speed of sound, you'll go faster than the speed of light! – Dan Blazevski Aug 22 at 18:52 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? – Daniel Lengeler Aug 22 at 21:43