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Why is it that turbulence is considered to be an unsolved problem of classical mechanics? What is meant by "unsolved"? Don't the Navier-Stokes equations apply to turbulent flows? It's difficult to grasp these concepts because there doesn't seem to be a mathematically precise definition of turbulence.

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I think your question is too broad. To answer it properly, one needs to write a book (or, at least, a survey article). Anyway, see a related MO question and the references provided in the answers… – Andrey Rekalo Dec 3 '10 at 6:16
Short answer: Even if you have a system of equations that seem to model a system well, that does not mean you have a good understanding of the behavior of that system. The Navier-Stokes equations are one of the best examples of this discrepancy. – S. Carnahan Dec 3 '10 at 6:46
The Clay Mathematics problem has an explicit, 4-page description by C. Fefferman of what sort of answer might win the $1M.… – Allen Knutson Dec 3 '10 at 15:35
+1 for Scott's comment. Some mathematicians think that a system of equations in itself is the only meaningfull object and that solutions are just dirty ashes which try desperately to tie with it. I had, two decades ago, a useless discussion with a Field medalist, who did not believe in shock waves, because hydrodynamic-type systems of PDEs have too a nice structure. He just refused to face the real world. My credo: let us study the structure (algebraic, geometric, ...) of differential systems, not just for themselves, but because of their consequences on the Cauchy problem. – Denis Serre Dec 5 '10 at 13:48

Let's take a provocative definition of turbulence:

Turbulence is the part of fluid dynamics that is beyond our capacity of calculation/prediction/explanation.

Of course, this is a moving definition as time varies. But it ensures that turbulence will remain an unsolved problem, forever.

Post scriptum. Never say in a talk "I dont't think we shall understand turbulence within my lifetime". Perhaps someone in the audience will stand up, show a knife and say "I am not so patient to wait that long".

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Wir müssen wissen — wir werden wissen! – Andrey Rekalo Dec 3 '10 at 10:31
Ya. Aber wir werden wissen only the questions that can be posed in mathematical words. I'm not sure this is the case for turbulence. Sorry, but my German does not go far beyond "Das Wandern ist der Müllers Lust". – Denis Serre Dec 3 '10 at 10:44
+1: for the PS! – Suvrit Dec 3 '10 at 13:57

Indeed, from a mathematical point of view, turbulence is what happens to a solution of Navier-Stokes when it develops a singularity. And we do not know that the solution exists to begin with. So turbulence is the regime where something which we are not sure to exist ceases to exist. Quite shady, isn't it :)

More seriously, there have been some attempts to model turbulence. I think one basic problem is that it is not even clear what exactly one should model. Does it make sense to describe the fluid by a field $u(t,x)$, surely a very singular one, governed by some PDE? A classical discussion of this problem is in Chapter III of the sixth volume of Landau-Lifshitz' treatise, and there are some more recent books entirely devoted to this question.

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@Piero. Many people disaggree. Turbulence and development of singularities are two different topics. Turbulence is somehow the fact that in some circumstances, the solution is so sensitive to variation of the initial data that any prediction becomes impossible. Singularities do not necessarily imply this conclusion; think to gas dynamics with heat diffusion, where a flow can be piecewise smooth, with quite a good predictibility despite having shocks. Conversely, a smooth flow may be turbulent: turbulence occurs in systems with finitely many degree of freedom, although solutions are smooth. – Denis Serre Dec 4 '10 at 8:57
@Denis: sure, turbulence is something we can not describe, as you write. You will agree that one possible scenario is a smooth flow breaking into turbulence; of course there are others, and conversely a solution developing a singularity does not necessarily describe turbulence. I was not trying to define turbulence but just pointing at one way to look at it, and half jokingly (I thought that was clear :) – Piero D'Ancona Dec 4 '10 at 17:56

I think we don't need a mathematically precise definition of turbulence to make sense of "turbulence as an unsolved problem of classical mechanics".

The question of singularities in Navier-Stokes solutions is a challenging math problem but almost unrelated to "turbulence as an unsolved problem";

which in my opinion amounts to: what, if anything, is universal in the statistics of actual fluid flows at (extremely) high Reynolds numbers?

No answer yet, so unsolved seems right... (Well, no full answer: Kolmogorov's 4/5th law is a universal feature, but there should be more, for instance a probability measure on a space of velocity fields having suitable invariance properties with respect to Navier-Stokes or Euler dynamics).

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