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.
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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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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.
- Reynolds Number becomes meaningless outside the range where it's fully tested.
- 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.
- 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.
- 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.
- 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.
- Please edit and add your own,,,,, Generally it leads to great improvement on efficiency in most everyday products.
