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December 21st, 2017, 11:29 AM   #1
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Progress on Navier-Stokes

Just the other day I read an outline of the millennium problems, in particular I noticed Navier-Stokes. Now progress is made: https://www.quantamagazine.org/mathe...ions-20171221/

A really good article, if a mite simplistic.
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December 21st, 2017, 01:24 PM   #2
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This seems to open up a huge can of worms.

Suppose for example you have a quantum process whose evolution is described by some differential equation. By Heisenberg we can only have the so called weak solutions and thus mathematically these solutions may not be unique.

Fluid flow with NS is treated classically but there are elements, such as friction, that might have quantum aspects to them at a fine enough scale.

So you are relying on statistics at best to ensure that the classical solutions are unique.

I'm in over my head but perhaps you see what I'm getting at. Are we sure that fluid flow solutions are in fact physically unique?
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December 22nd, 2017, 04:39 AM   #3
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Quote:
I'm in over my head but perhaps you see what I'm getting at. Are we sure that fluid flow solutions are in fact physically unique?
Even classically the 'solutions' to fluid flow equations may not be unique, for instance the in the hydraulic jump it has two solutions at the jump line.
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December 22nd, 2017, 10:57 AM   #4
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Even classically the 'solutions' to fluid flow equations may not be unique, for instance the in the hydraulic jump it has two solutions at the jump line.
Never had to bother much with fluid dynamics. Is there experimental evidence that both these solutions physically occur or is it just an artifact of the mathematics?
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December 22nd, 2017, 09:24 PM   #5
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There seems to be some confusion in the original article as well as the discussion here about the relationship between uniqueness of solutions and existence of blowup solutions.

Namely, there is no relationship. While it isn't known whether or not weak solutions of NS equations are unique, nonuniqueness does not in any solve the issue one way or another. This is because weak solutions need not be smooth and the NS existence and smoothness problem asks about smooth solutions. Also, smooth solutions are necessarily unique.

The reason people have looked at weak solutions is this is a common technique in PDE theory. You want to know if a PDE has a global smooth solution so first prove existence of a weak solution or a viscosity solution. Next you try to obtain bounds on regularity of the weak/viscosity solution and if you are lucky, you may be able to prove smoothness.

Since smooth solutions are unique and smooth solutions are necessarily both weak solutions as well as viscosity solutions, you have succeeded in proving existence and uniqueness of a smooth solution.


Keep in mind that none of this discussion has anything to do with blowup and there is no apparent relationship between smoothness and blowup as even simple examples of analytic vector fields are known to blow up in finite time. Obviously, finite-time blowup implies a non-smooth solution but the converse is far from true and there is no reason to assume that a negative answer to NS existence/smoothness question requires a blowup solution.
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December 23rd, 2017, 02:08 PM   #6
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Never had to bother much with fluid dynamics. Is there experimental evidence that both these solutions physically occur or is it just an artifact of the mathematics?
The flow equations lead to an equation for the free surface of flowing water (or other liquids) flowing in an open channel.

If the gradient is too steep, a situation can arise where the specific energy exceeds a critical value so there has to be a sudden change in the surface height, (and therefore the depth) because the water can't flow fast enough to accommodate the gravitational energy input as velocity head.

According to the equations this rise occurs as a discontinuity step change and the natural rise is nearly also a step change. There are no reliable equations over the short length of very turbulent flow.

Yes this is a real world effect that is used to reduce kinetic energy (and thus scouring) at the bottom of dams.

If you like I can dig out the equations.
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