Btw this is work relating to one of the famed Millennial Problems - http://www.claymath.org/millennium-problems/navier%E2%80%93s...
As the article pointed out.
Mathematicians Find Wrinkle in Famed Fluid Equations
61–70 of 124 posts
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#62Based off the abstract, there's one class of weak solutions, the Leray solutions, which are known to exist. And now they've shown for a different class of weak solutions to NS that the solutions are not unique. Is that right?
Yeah, they relaxed one of the criterion’s for Leary solutions (the energy inequality) they’ve proved that these solutions are energetically unstable, and that doesn’t strike me as very profound (I need to go read the paper though, I haven’t had time yet).
Just mentioning at the bottom of the article “yeah now we are going to see if the same thing applies to proper Leary solutions, we think it does” means close to nothing, honestly.
And even if it does, the article’s author is right when he remarks that this can be seen entirely as a warning against using approximations that are too broad or coarse.
I’ll add that I find it funny that nowhere in the article (that I can see, but I am reading on mobile Safari, so maybe...) is the Navier-Stokes partial differential equation even displayed, and the relationships it defines are not explained (other than some waffle about ‘derivatives’).
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#63Earlier quoted context omitted.
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 b…
How are Navier Stokes the other way, on the macro scale? e.g. in the context of meteorology. Asking because I had a discussion with somebody recently where they claimed the fundamental flaw to the science underlying Climate Change science is over-reliance on NS at macro scale as a way of predicting climate behaviour, or something. I took it to be Baloney, but I'm wondering if there is some strands of truth to it ...
As you intuit, there is no reason to presume that Navier-Stokes would be unreliable at macro scales relevant to meteorology, simply because it is so thoroughly tested in experimental settings and to such sensitivities that it is known that all relevant factors are accounted for.
(Of course, why would one presume that if it is inaccurate at planetary scales, it biases observations towards the climate change narrative? It's just the typical “God of the gaps” kind argument.)
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#64Earlier 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.
Well general relativity breaks down at very small scales, and imparts the need for quantum mechanics ...
If you are really interested, somebody has bothered to make a special-relativity & quantum-indeterminacy compatible formulation of Navier-Stokes suitable for calculating shockwaves in extremely dense mediums such as the neutronium neutron stars are thought to be made of (where the speed of sound is comparable to the speed of light in a vacuum, hence the relativity, and the matter is degenerate and in states of superposition, hence the quantum indeterminacy). I don’t have the reference right with me at the moment (mainly because I just looked at it and though “oh horror!” and averted my eyes) but I know where and how to dig it up for you if you want.
Of course... for obvious reasons it hasn't been experimentally verified...
For the record: I also saw it coded in FORTRAN. Yeah. It's like catching grandma in starkers.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#65Earlier quoted context omitted.
Financial forecasting and risk-elimination in a system like a blockchain would be my guess — and in that respect it’s probably the most important problem out there IMO.
What??
I think we have just been exposed to somebody's Markov Chain Natural Language Generation experiment.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#66This makes senses to me. If you run a simulation at too low a resolution you run into ambiguity. By refining the grid you can resolve that ambiguity. Turbulent flow is chaotic, so this makes perfect sense to me. What this may lead to is a method of determining criteria for adaptive grid refinement. But I thought that already existed, so maybe just an improvement over what's out there.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#67I wonder what the interplay of the material's EOS has with constraining these solutions. Note I only had time to skim the article.
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#68Earlier quoted context omitted.
How are Navier Stokes the other way, on the macro scale? e.g. in the context of meteorology. Asking because I had a discussion with somebody recently where they claimed the fundamental flaw to the science underlying Climate Change science is over-reliance on NS at macro scale as a way of predicting climate behaviour, or something. I took it to be Baloney, but I'm wondering if there is some strands of truth to it ...
As you intuit, there is no reason to presume that Navier-Stokes would be unreliable at macro scales relevant to meteorology, simply because it is so thoroughly tested in experimental settings and to such sensitivities that it is known that all relevant factors are accounted for. (Of course, why would one presume that if it is inaccurate at planetary scales, it biases observations towards the climate change narrative?…
“God of the gaps” yeah that’s pretty much what I thought!
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#69Earlier quoted context omitted.
What??
I think we have just been exposed to somebody's Markov Chain Natural Language Generation experiment.
Ha... I guess I can be a bit terse. Just think of the “turbulence” in navier-stokes as valuation fluctuations. Without the realworld dampening effects of regulation, slow tranactions, and managed markets — volatility along the lines of the infinite incongruities posed in the article are possible. ( note: Im not referring to a pendactic ‘infinite’ wrt a blockchain).
Re: Mathematicians Find Wrinkle in Famed Fluid Equations
#70Earlier quoted context omitted.
I think we have just been exposed to somebody's Markov Chain Natural Language Generation experiment.
Ha... I guess I can be a bit terse. Just think of the “turbulence” in navier-stokes as valuation fluctuations. Without the realworld dampening effects of regulation, slow tranactions, and managed markets — volatility along the lines of the infinite incongruities posed in the article are possible. ( note: Im not referring to a pendactic ‘infinite’ wrt a blockchain).
Ah OK I get it now.
I’m an applied mathematician and a macroeconomist that studied turbulence in financial markets and crashes thereof. I can assure you that the dynamics are pretty distinct. In economics wealth is not a conserved quantity whereas in physics energy and momentum are.