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Mathematicians Find Wrinkle in Famed Fluid Equations

quantamagazine.org

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Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#41

Post by mlvljr »

floatingatoll wrote: post #24
snakeboy wrote: post #22

Earlier quoted context omitted.

The Uncertainty Principle is a very well established result which is quite generalized mathematically beyond Heisenberg's result. This has nothing to do with that principle.

If you weaken the vector density until a point that you can generate multiple possible outputs from a given weakened input, then what level of weakening guarantees no unique outputs from the equations? These equations were designed for a system in which every 'atom' (vector) has a perfectly knowable 'spin', and they begin to produce unexpected results as the uncertainty is dialed up through weakening. It's just a sha…

"you have no idea how Sciance works" :)

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#42

Post by emileokada »

wiz21c wrote: post #40
CamperBob2 wrote: post #10

Using this approach, Buckmaster and Vicol prove that these very weak solutions to the Navier-Stokes equations are nonunique. They demonstrate, for example, that if you start with a completely calm fluid, like a glass of water sitting still by your bedside, two scenarios are possible. The first scenario is the obvious one: The water starts still and remains still forever. The second is fantastical but mathematically p…

Also FTA : >>> Nonunique Leray solutions would mean that, according to the rules of Navier-Stokes, the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense and implies that the equations aren’t really describing what they’re supposed to describe. This basically say that from one given starting conditions one only expect one (and only one)…

I'm not sure what you're getting at. Classically we expect that given the same initial conditions, the outcome of a fluids experiment should always be the same. If an equation meant to describe fluid flows doesn't have this property, it is probably a bad model.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#43

Post by tanderson92 »

m_mueller wrote: post #38
labster wrote: post #28

Earlier quoted context omitted.

Immediate? Certainly not. Weather models do have incomplete information, but the equations are approximated by Taylor series (1st to 3rd order, depending on the model, last time I checked). It's possible that this can explain some of the differences between models or ensemble runs... but you have to realize that most of the error comes from incomplete data in the initial and boundary conditions. Looking for weather m…

Almost. Most models use some variations of Runge-Kutta approximation, to third order usually. Edit: typos

Convergence, or at least consistency and order, of Runge-Kutta methods is shown via Taylor series, so I don't see the problem or why the GP is only "Almost" correct.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#44

Post by taneq »

pencilhappen wrote: post #30
ianai wrote: post #9

Earlier quoted context omitted.

Aye either they have upturned a theory as august as general relativity or they’ve thrown the baby out with the bath water somewhere.

What's funny is the article says this about the Navier-Stokes equations: "The equations work. They describe fluid flows as reliably as Newton’s equations predict the future positions of the planets" Newton's equations do not in fact reliably predict Mercury's orbit, and it took GR to do it. Lazy journalist!

Sounds like a perfectly accurate statement to me, in both directions.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#45

Post by eru »

Gibbon1 wrote: post #33
eru wrote: post #27

Earlier quoted context omitted.

Navier-Stokes is only incidentally related to real life. It's pure math.

Navier Stokes is probably Turing Complete.

Probably, yes. But the gymnastics required to make it Turing complete are unlikely to be in the regime that describes real fluids.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#46

Post by thethirdone »

tanderson92 wrote: post #43
m_mueller wrote: post #38

Earlier quoted context omitted.

Almost. Most models use some variations of Runge-Kutta approximation, to third order usually. Edit: typos

Convergence, or at least consistency and order, of Runge-Kutta methods is shown via Taylor series, so I don't see the problem or why the GP is only "Almost" correct.

The Runge-Kutta methods are not a Taylor series though. So "almost" is apt.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#48

Post by kindfellow92 »

eru wrote: post #27
snakeboy wrote: post #21

Earlier quoted context omitted.

You missed the point of their comment. The responders' point was that Quantum Mechanics is a different framework for modeling physical phenomena which takes a probabilistic framework, and so if fluid were to be modeled in a similar framework, you could work more naturally with these "vanishingly unlikely" events.

Navier-Stokes is only incidentally related to real life. It's pure math.

Right, smooth vector fields aren’t actually physically realizable.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#49

Post by alephnil »

pencilhappen wrote: post #30
ianai wrote: post #9

Earlier quoted context omitted.

Aye either they have upturned a theory as august as general relativity or they’ve thrown the baby out with the bath water somewhere.

What's funny is the article says this about the Navier-Stokes equations: "The equations work. They describe fluid flows as reliably as Newton’s equations predict the future positions of the planets" Newton's equations do not in fact reliably predict Mercury's orbit, and it took GR to do it. Lazy journalist!

The Navier-Stokes equations assume that the medium is continuous even at infinitely small scales, which obviously not the case for natural fluids, that are made of discrete atoms. Thus the equations are only correct at sufficiently large scales. They work fine for describing the airflow around an aeroplane, but not the airflow around the head of a hard drive, which is small enough that the finite size of atoms must be taken into account. On a more visible scale, you have Brownian motion of small particles, which can be seen even in a low magnification microscope. The Navier-Stokes equations predict that these effect does not exist. The equations are still useful approximation in a lot of cases.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#50

Post by claudius »

emileokada wrote: post #42
wiz21c wrote: post #40

Earlier quoted context omitted.

Also FTA : >>> Nonunique Leray solutions would mean that, according to the rules of Navier-Stokes, the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense and implies that the equations aren’t really describing what they’re supposed to describe. This basically say that from one given starting conditions one only expect one (and only one)…

I'm not sure what you're getting at. Classically we expect that given the same initial conditions, the outcome of a fluids experiment should always be the same. If an equation meant to describe fluid flows doesn't have this property, it is probably a bad model.

Not really. We know that the idea of a "fluid" has to break down eventually (at least due to finite particle sizes, possibly due to discretisations of space or whatever). An equation which describes "fluids" at sufficiently low resolution and breaks down once a resolution is reached where the idea of a "fluid" breaks down as well is then perfectly sufficient and, combined with the idea of a fluid, gives us as good model for reality.

Contrary to what the article claims here:

> the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense

it makes quite a lot of "physical sense" to get two possible outcomes out of one initial state, though we would not expect the Navier-Stokes equations to describe that situation.

The article is simply wrong in arguing that because we expect classical fluids to behave classically and because Navier-Stokes may break down in certain limits, NS may be a bad model. I actually struggle to put together a coherent sentence which comes close to what the article tries to say regarding the relation between "physical sense" and our expectations for the results of Navier-Stokes.

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