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.

  • $\begingroup$ 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. $\endgroup$ Commented Dec 3, 2011 at 17:48
  • 2
    $\begingroup$ 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. $\endgroup$ Commented Dec 4, 2011 at 2:08

3 Answers 3


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.

  • $\begingroup$ 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 $\endgroup$ Commented Dec 4, 2011 at 2:13
  • $\begingroup$ 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). $\endgroup$ Commented Dec 4, 2011 at 2:15
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    $\begingroup$ @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. $\endgroup$
    – Ron Maimon
    Commented Dec 4, 2011 at 7:33
  • $\begingroup$ 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? $\endgroup$
    – Ron Maimon
    Commented Dec 4, 2011 at 7:38
  • $\begingroup$ @Willie: Also, a minor technical point: you say "dense", but dense is not a problem if it is measure zero. The rationals are dense, but if blowups require some Reynolds number to be exactly rational, nobody would think those are real observable blowups. Perhaps you meant that both smooth and blow-uppy solutions are of finite measure, but that's exceedingly unlikely. $\endgroup$
    – Ron Maimon
    Commented Dec 4, 2011 at 8:08

YES! it will have drastic consequences in practice. N-soliton solutions (general solutions) to the Navier-Stokes d.e. has been recently found. Please see the following publication cited as

R. Meulens , "A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation", AIP Advances 12, 015308 (2022) https://doi.org/10.1063/5.0074083

The consequences for physics are then obviously (except of those already listed)

  • in aeronautics (and car manufacturing industries): -better and faster designs (Will save millions in CAD designs)
  • will make expensive wind tunnels up till certain degrees obsolete
  • no need for approximation of lift and description of aeronautical processes with statistical coefficients like the lift coefficient, drag coefficient etc. anymore
  • better understanding of lift. lift occurrence is now postulated by several theories. None of them suffice exactly. "The causal relationship between lift and acceleration will suffice and will complement all 3 postulates"
  • in Astro-physics : -better understanding of gravity and galaxy occurrences which are spirals. These are stream-line solutions of the Navier Stokes d.e.
  • replacement theory for the general relativity solving the cosmological constant problem. -possibly explanation and "bridge-over" to a mechanics for everything (one theory for classic and quantum mechanics and general and special relativity)
  • $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Commented Nov 23, 2022 at 21:40

The Answer to this Question is "NO" There is no positive solution. The reason why there can't be a positive solution, is the surface in fluid; It means, no continuity over surface. No viscous forces but, collision and friction.

This helps to understand the cause of the Turbulence; If you avoid to "damage" the fluid to aparts" you avoid turbulence. This "damage" can be caused by cutting edges, but also from pressure shocks. Velocity is only the way to these pressure shocks.

The practical consequences is therefore that ie.

  1. Reynolds Number becomes meaningless outside the range where it's fully tested.
  2. The Turbines are designed with more attention to continuity laws, ie. Tests of a turbine using this principle. According to the Euler's Turbine equations we are having a efficiency of over 100%. This difference comes from the measured angle and measured power. Cause in this machine the Continuity rules are hold, the fluid doesn't damaged and viscous expansion is able to transfer the deceleration of fluid to an usable pressure; bit like in piston pump. The former counter rotating turbines suffered from a sudden pressure drop, as the first runner accelerated the flow and took energy away. This caused heavy turbulence to second runner -> low efficiency.
  3. Also the open channel flow can be Finally calculated without magical "Experince-factor". This saves the building costs, as the erodible Turbulence can be avoided. An example from these structures is here; You can really see how the flow in the middle remains Laminar-alike. There is more videos about the issue in my channel.
  4. The aerodynamics of any objects can be made better. Ie. the cars frontal edges might have similar components as the Bulpous bow in ships; reducing drag and wake and thus increasing speed, range, fuel efficiency, and stability. The purpose of this bulp, is obviously to make a clean cut on water.
  5. Controlled vortex bursting can reduce the drag, like decribed in this question to be able to do that optimally, the tubulence must be first understood correctly.
  6. Please edit and add your own,,,,, Generally it leads to great improvement on efficiency in most everyday products.

The existince of these surfaces can be proven by optics.

  • $\begingroup$ The question is "would have any practical consequences?" And though I shortly name the thinking path to these consequences, and then even name true results with video or foto, the answer is voted down. I read the up voted answer, but there wasn't any "practical consequences" named. Not a single one. $\endgroup$
    – Jokela
    Commented Jul 3, 2015 at 5:22

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