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I know the problem is especially of interest to mathematicians, but I was wondering if a solution to the problem would have any practical consequences.

Upon request: this is the official problem description of the aforementioned problem.

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A description of the challenge or links to the same would help here. I'm familiar with the Navier-Stokes equation, but not with this particular challenge. Others may be in the same boat. –  dmckee Dec 3 '11 at 17:48
The Clay description is awfully loose and allows many different formulations of the problem. Terry Tao gave a pretty good description of the various ways of formulating the Navier-Stokes Millennium Problem and their mutual implications. –  Willie Wong Dec 4 '11 at 2:08
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This is difficult to answer, because the answer depends on whether the answer is positive or negative. Although most people, including myself, expect a positive answer, a negative answer is actually possible, unlike say, for the question of P!=NP, or the well-definedness/mass-gap of gauge theory where we are 100% sure that we know the answer already (in the scientific, not in the mathematical sense).

If the answer is positive, if all Navier Stokes flows are smooth, then the proof will probably be of little practical consequence. It will need to provide a new regularity technique in differential equations, so it will probably be of great use in mathematics, but it will not be surprising to physics, who already expect smoothness from the known approximate statistical falloff of turbulence.

At short distances, the powerlaw falloff in turbulence turns into an exponential falloff, in the dissipative regime, so that the high k modes are suppressed by exponentials in k, and this implies smoothness. So smoothness is the expected behavior. This doesn't provide a proof, because the k-space analysis is too gross-- it is over the entire system. To prove the smoothness locally, my opinion is that you need a wavelet version of the argument.

On the other hand, if there are blow ups in Navier Stokes in finite time, these are going to be strange configurations with very little dissipation which reproduce themselves in finite time at smaller scales. Such a solution might be useful for something conceivably, because it might allow you to find short distance hot-spots in a turbulent fluid, where there are near-atomic-scale velocity gradients, and this could concievably be useful for something (although I can't imagine what). Also, such solution techniques would probably be useful for other differential equations to find scaling type solitons.

So it is hard to say without knowing from which direction the answer will come.

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Thank you for this. –  Max Muller Dec 3 '11 at 21:17
Two additional notes: (a) one possibly useful implication of the instability of Navier Stokes flow would be the following "really bad case scenario": Navier Stokes admits blow-up in finite time for a dense set of initial data, but also admits smooth global solutions for a dense set of initial data. This would basically tell you that any hope of numerically modelling nature using the Navier-Stokes equation is lost. (b) In regards to Ron's second to last paragraph, I would say in fact that blow up will suggest a physical regime where the NS equations are not applicable, and hence points to –  Willie Wong Dec 4 '11 at 2:13
where one should look for new physics. Perhaps even giving a clue on how to partially resolve Hilbert's sixth problem (at least the version that studies the derivation from Newtonian mechanics of the fluid equations). –  Willie Wong Dec 4 '11 at 2:15
@Willie: I don't think that blowups, even if dense, could ever be a real problem, because the NS equations have a scale-separation property due to Galilean invariance. Even if there are little hot-spots where the fluid gets singular, it would just be advected by the average flow. The mean velocity profile in k-space will definitely be powerlaw in the inertial regime, and exponential in the viscous regime, so it will be mostly smooth, in the sense that any singularities will be isolated and tiny and will not affect numerical simulations. This is born out by both experiment and simulation. –  Ron Maimon Dec 4 '11 at 7:33
As far as deriving the fluid equations, they are just momentum conservation, the only thing you need to derive is the constitutive viscosity relation, and the derivation is good-enough for physicists today. It might not be good enough for mathematicians, but I am not sure what the issues are exactly. When you have a fluid with bulk velocity field v, the bulk stress tensor must be $\rho v^iv^j + P \delta^{ij}$, and the pressure maintains $\nabla \cdot p$, and then incompressible Euler equations follow. The NS viscosity term is just the leading correction. What else is left to show? –  Ron Maimon Dec 4 '11 at 7:38
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