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Given a solution $S$ of the Navier-Stokes equations, is there a way to make mathematically precise a statement like: "$S$ is turbulent in the spacetime region $U$"?

And if such a definition exists, are there any known exact solutions of Navier-Stokes exhibiting turbulence?

Reading Wikipedia makes me also want to ask the following related questions:

  1. Does a single vortex count as turbulence?
  2. Is the appearance of vortices a necessary feature of turbulence?
  3. What's the difference between a vortex and an eddy?

Pointers to (mathematically rigorous) literature are much appreciated.

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  • $\begingroup$ I think one idea would be that in $U$ the wavenumbers (up to a cutoff determined by $U$ itself) follow the 5/3 law. $\endgroup$
    – Steve Huntsman
    Commented Jun 11, 2010 at 12:50
  • $\begingroup$ @Steve: could you make that more precise or add a reference? $\endgroup$
    – Michael Bächtold
    Commented Jun 11, 2010 at 14:33
  • $\begingroup$ See the wiki page on turbulence or micromath.wordpress.com/2008/04/04/kolmogorovs-53-law $\endgroup$
    – Steve Huntsman
    Commented Jun 11, 2010 at 16:33
  • $\begingroup$ books.google.com/… $\endgroup$
    – Steve Huntsman
    Commented Jun 12, 2010 at 0:33
  • $\begingroup$ Thanks everyone for all the references. I could have accepted any of the answers, but chose Andreys since he states more explicitly that the problem seems to be neither solved nor easy. From your answers I also get the impression that the question is basically the same as what physicist call "the problem with turbulence", though I'm not quite sure what they mean by that. $\endgroup$
    – Michael Bächtold
    Commented Jun 23, 2010 at 10:25

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There is probably no universally accepted mathematical definition of turbulence. (By the way, is there a physical one?) Moreover, the prevailing definitions seem to be highly volatile and time-dependent themselves.

A few notable examples.

  • In the Ptolemaic Landau–Hopf theory turbulence is understood as a cascade of bifurcations from unstable equilibriums via periodic solutions (the Hopf bifurcation) to quasiperiodic solutions with arbitrarily large frequency basis.

  • According to Arnold and Khesin, in the 1960's most specialists in PDEs regarded the lack of global existence and uniqueness theorems for solutions of the 3D Navier–Stokes equation as the explanation of turbulence.

  • Kolmogorov suggested to study minimal attractors of the Navier-Stokes equations and formulated several conjectures as plausible explanations of turbulence. The weakest one says that the maximum of the dimensions of minimal attractors of the Navier–Stokes equations grows along with the Reynolds number Re.

  • In 1970 Ruelle and Takens formulated the conjecture that turbulence is the appearance of global attractors with sensitive dependence of motion on the initial conditions in the phase space of the Navier–Stokes equations (link). In spite of the vast popularity of their paper, even the existence of such attractors is still unknown.

Edit 1. Concerning explicit solutions to the Navier-Stokes equations, I don't think any of them really exhibit turbulence features. The thing is that the nonlinear term $v\cdot \nabla v$ is equal to $0$ for most known classical explicit solutions. In other words, these solutions actually solve the linear Stokes equation and don't "see" the nonlinearity of the full Navier-Stokes system. This is probably not what one would expect from a truly turbulent flow.

Edit 2. As for the quick reference, you may find helpful the short survey on turbulence theories by Ricardo Rosa. (Wayback Machine) It appears as an article in the Encyclopedia of Mathematical Physics.

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    $\begingroup$ I interpre what Arnold/Khesin wrote as expressing skepticism that existence/uniqueness theorems can resolve what is turbulence: "In the sixties most specialists in partial differential equations (with the notable exception of V.I. Yudovich) regarded the lack of global existence and uniqueness theorems for solutions of the Navier-Stokes equation as the explanation of turbulence. This point of view was never popular among physicists." Implicitly the physicists' opinion about fluids is more valued than that of pde specialists. I read it as roughly: here's a view with which we don't much agree ... $\endgroup$
    – Dan Fox
    Commented Dec 13, 2011 at 8:52
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Leray himself gave such a definition in his paper in the 1930's. If you want a translation of that, there is one on my web page at https://www.bterrell.net/

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An excellent book on the subject is "Navier-Stokes equations and turbulence", by, among others, Roger Temam, who is an authority on that subject.

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No, there is no universal definition on turbulence, mathematically or physically. Some believe N-S equation is the one, and only one, that governs the fluid motion and induces turbulence, while some stand for the statistical based view of the phenomenon. Among others, Lesieur's book "turbulence in fluids" gives some details on the topic. If there is no agreement among physicists on the definition of turbulence, any claim toward universal mathematical theory on the topic is risky.

The existence of vortices is a must for turbulence, but single vortex definitely does not suffice, the same as digit 24 is not number theory. Eddy is nothing but the name of vortex by tradition. Among turbulence MODELING theory, it usually refers vortex motion outside of mean flow scale.

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I understand turbulence to be defined as a chaotic patch of vorticity. This definition comes from http://books.google.co.uk/books?id=rkOmKzujZB4C&lpg=PP1&dq=davidson%20turbulence&pg=PP1#v=onepage&q=davidson%20turbulence&f=false

This definition implies that vorticity is a necessary feature of turbulence. I suppose the vortex patch would tend towards a stable structure such as a vortex.

A vortex often refers to a mathematically definable point vortex. An eddy is a less rigorous term for an overturning motion that often implies a vortex.

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  • $\begingroup$ Thanks for the reference, looks like a pleasant introduction. Does he explain in a mathematically precise way what "chaotic" means in this context? $\endgroup$
    – Michael Bächtold
    Commented Feb 13, 2012 at 20:27
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    $\begingroup$ I have heard him lecture about this. He says that fluid instabilities can be shown to be chaotic. (break down of horse shoe vortices, comparison with the logistic equation etc.) I don't know if that is in his book, or if it is adequately mathematical or precise for you. $\endgroup$
    – Charlie Hogg
    Commented Jul 25, 2013 at 1:20

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