most recent 30 from http://mathoverflow.net 2013-05-23T10:41:33Z http://mathoverflow.net/feeds/question/105193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105193/implications-of-a-hypothetical-blow-up-of-navier-stokes-for-the-mathematical-mode Daniel Lengeler 2012-08-21T21:44:13Z 2012-09-05T15:42:39Z

<p>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? </p> <p>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! :-)</p>

http://mathoverflow.net/questions/105193/implications-of-a-hypothetical-blow-up-of-navier-stokes-for-the-mathematical-mode/105242#105242 Michael Renardy 2012-08-22T14:43:54Z 2012-08-22T14:43:54Z

<p>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.</p>

http://mathoverflow.net/questions/105193/implications-of-a-hypothetical-blow-up-of-navier-stokes-for-the-mathematical-mode/106445#106445 Denis Serre 2012-09-05T15:42:39Z 2012-09-05T15:42:39Z

<p>This is a comment about Michael's answer and the comment by Daniel about compressible NSE.</p> <p>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 <em>incompressibility is no longer a valid assumption</em> is not the end of the story. Because then we have to use the compressible model, which likely displays the same trouble.</p> <p>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.</p>