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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.

flag	asked Dec 3 2010 at 5:46 unknown (google) 21●1

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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 mathoverflow.net/questions/27805/… – Andrey Rekalo Dec 3 2010 at 6:16

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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 2010 at 6:46

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The Clay Mathematics problem has an explicit, 4-page description by C. Fefferman of what sort of answer might win the $1M. claymath.org/millennium/Navier-Stokes_Equations/… – Allen Knutson Dec 3 2010 at 15:35

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+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 2010 at 13:48

Denis Serre · Answer 1 · 2010-12-03 10:15:43Z UTC

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".

Piero D'Ancona · Answer 2 · 2010-12-03 12:58:03Z UTC

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.

turbulence as an unsolved problem of classical mechanics

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