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Mathematicians Find Wrinkle in Famed Fluid Equations
- aperrien 9y agoDoes anyone have a link to this particular paper?
- deleted 9y ago[deleted]
- foota 9y agoBased 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?
- boxfire 9y agoThat's correct, while taking away the entropy inequality as well. The thing is that very inequality is one of these laws of thermodynamics we so love, so I kind of see this as hot air.
- ianai 9y agoAye either they have upturned a theory as august as general relativity or they’ve thrown the baby out with the bath water somewhere.
- im3w1l 9y agoBased on my fairly layman understanding it could be as simple as the "weak solutions" not being as useful as they were thought to be.
- contravariant 9y agoThat seems to be one of the explanations provided in the article.
- meri_dian 9y agoThe laws of thermodynamics are quite a bit more august than GR
- ianai 9y agoI’ll leave such distinctions to more seasoned minds.
- pencilhappen 9y agoWhat'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!
- yorwba 9y agoThe Navier-Stokes equations do not in fact reliably predict the flow of all real-world fluids. The comparison to Newton's equations is perfectly apt. Good enough for most applications, but may be imprecise in some cases.
- taneq 9y agoSounds like a perfectly accurate statement to me, in both directions.
- alephnil 9y ago
- foota 9y agoIsn't that possible though if the fluid isn't a closed system? (which it probably isn't)
- qubex 9y agoAll of the above arguments obtain if the fluid is a closed system and is only acting as a result of the forces it itself is exerting upon itself, subject to the various boundary conditions around it. Of course if you open the system various outcomes are possible depending on what the external influences do to it (for example, the two solutions mentioned for the still water become entirely plausible if there's somebody roaming around who might put a lighter under the glass and cause the water in it to boil, but that isn't the point of the exercise).
- mar77i 9y agoWho would have guessed that this story about the behavior of fluids seems to kind of leak.
- qubex 9y agoYeah, 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’).
- BucketSort 9y agoBtw this is work relating to one of the famed Millennial Problems - http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation http://www.claymath.org/millennium-problems/navier%E2%80%93s...
- FabHK 9y agoAs the article pointed out.
- westurner 9y agoNavier-Stokes equations: https://en.wikipedia.org/wiki/Navier–Stokes_equations https://en.wikipedia.org/wiki/Navier–Stokes_equations
- CamperBob2 9y agoUsing 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 permissible: The water starts still, erupts in the middle of the night, then returns to stillness. Also permissible, and also vanishingly unlikely, under quantum theory. It's a stretch, but I wonder if it's reasonable to think of data points in the vector field describing fluid motion as probabilities rather than definite measurements. That might allow their behavior to be modeled with different mathematical tools.
- cinquemb 9y agoYeah, I wondered about this too since the typical NS BVP, velocity vector is assumed to be confined to the domain of real numbers[0]. [0] https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/girolami/podes/examples/example03/ns_details.pdf https://warwick.ac.uk/fac/sci/statistics/staff/academic-rese...
- eru 9y ago> Also permissible, and also vanishingly unlikely, under quantum theory. Navier-Stokes has nothing to do with Quantum Mechanics..
- CamperBob2 9y agoNo one said it does. But given that statistics based on complex probabilities are effective in one domain, maybe they're worth investigating in another.
- snakeboy 9y agoYou 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.
- ericbowman 9y agoThat is fascinating. Is there any sort of immediate real-world impact (like to weather forecasting)?
- labster 9y agoImmediate? 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 model effects is like looking for relativistic effects in automobiles.
- m_mueller 9y agoAlmost. Most models use some variations of Runge-Kutta approximation, to third order usually. Edit: typos
- tanderson92 9y agoConvergence, 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.
- thethirdone 9y agoThe Runge-Kutta methods are not a Taylor series though. So "almost" is apt.
- labster 9y agoGeez guys, I chose a term that I thought most people would be more likely to understand. I realize that some of us have worked a lot more on numerical methods, but other hackers never made it past Calculus 2.
- uptownfunk 9y agoSo will they get the million?
- emileokada 9y agoNo. They've shown non-uniqueness of solutions of a strictly weaker problem. They have yet to say anything about the actual Navier-Stokes problem.
- floatingatoll 9y agoIt's disappointed to see that their reaction to non-unique results is "must be broken", rather than "we've rediscovered the quantum physics uncertainty principle in fluid mechanics".
- snakeboy 9y agoThe 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.
- floatingatoll 9y agoIf 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 shame that the considerations stop at "therefore we broke the equations" as opposed to "gee, that looks familiar". What's the Navier-Stokes equivalent of the diffraction experiment? What does the interference pattern of two vector fields even look like? Why aren't they trying to study the interference patterns of the one input, two outputs scenario? I get that this is all "obviously pointless" to others, but no one I've asked can actually explain why these comparisons are unacceptable. You completely dismiss it without any explanation other than "science is well-established", as if somehow that's meaningful. So, yeah, disappointment.
- yorwba 9y agoYou seems to think that there is some kind of analogy between a weakened version of the Navier-Stokes equations having multiple solutions and the double-slit experiment generating an interference pattern. I don't think there is such an analogy. The equivalent of a spreading wave in quantum physics is the vector field describing the flow according to the Navier-Stokes equations. The equivalent of an interference pattern is the pattern of vortices in the fluid. When you do a double-slit experiment, you'll always see the same interference pattern, and it can be predicted exactly. There is only one solution to the equations. Having two different vector fields satisfy the Navier-Stokes equations with the same boundary conditions would be like seeing different interference patterns for no reason at all. There is no place for the Uncertainty Principle in this result, because that is a statement about the standard deviations of two complementary quantities, and there are no such quantities involved here.
- throwwit 9y agoIsn’t Navier-Stokes smoothness just a reframing of the three-body problem? Karl Sundman solved that one in 1909, and n-body was generalized in 1990 by Qiudong Wang. Edit- to answer my own question: http://www.scholarpedia.org/article/N-body_simulations_(gravitational)#Mean_Field_Approach:_analogies_and_differences_with_fluid_dynamics http://www.scholarpedia.org/article/N-body_simulations_(grav...
- qubex 9y agoI'm sorry if it seems I am persecuting you and putting all your ideas down, but I really don't see how this can be true either. The 3- or N-body problem is about point-particles interacting gravitationally at nonzero distance according to an inverse square law. Navier-Stokes is, at root and in the limit, about elastic collisions about infinitesimal corpuscles that transfer momentum between each other. Again, I can see the analogy “lots of things interacting”, but they have quite little in common beyond that.
- throwwit 9y agoJust the inverse n-vectors applied (regardless of space and field strength or possibly just instantaneous field strength). Always good to be called out tho.
- qubex 9y agoI don't understand what the first sentence of your reply is supposed to mean, but the second sentence is a very mature response and belies your wisdom: yes, science and rational thinking is all about putting ideas out there and rejoicing when somebody helps you etch away at those that are not compatible with reality, so that only plausible ones remain.
- throwwit 9y agoWell what I meant was shrinking the n body system to a point then extending that to a field. But it’s beyond me how the math works to invert those same force vectors to an impulse.
- epx 9y agoLuce Irigaray knew it already!
- phkahler 9y agoThis 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.
- reikonomusha 9y agoThis doesn’t have to do with numerical error introduced by discretization. This has to do with uniqueness of solutions.
- Glauc 9y agoI wonder what the interplay of the material's EOS has with constraining these solutions. Note I only had time to skim the article.
- pge 9y agoI'm struggling to understand the significance of this, at least as the N-S equations are used in the real world. Many years ago, I interned with the Navy writing fortran code for fluid dynamics simulations on submarine hulls, and IIRC there were flow dynamics we observed consistently in the real world (e.g. oscillating vortices) that were fundamentally inconsistent with the results coming from our N-S calculations (which would say there could not be an oscillation because it was a steady state flow). There was always a hand-waving of "N-S is actually right; our computer models are just not fine-grained enough." But at the same time, given the computational limits of our grids(particularly at the time -25 years ago), it was understood and accepted that N-S would yield only an approximation. That's only one data point, but it certainly seemed to me that no one was relying on N-S as an accurate predictor of motion (as you would a newtonian model of a ball rolling or something like that), but rather just as a first order approximation. If that impression is accurate, a result that says N-S isn't always accurate is kind of a statement of the obvious. What am I missing?
- eadmund 9y agoWhat I got from the article is that there's a possibility of a chaotic result: it may not be possible to get computed results which are arbitrarily close to the real-world results. I was reminded a bit of the famous Lorenz system (originally for weather prediction, IIRC), which turns out to be chaotic under some conditions.
- incidentnormal 9y agoI think it's not a chaotic result as such (as the Lorenz weather prediction models you're referencing are) - which would mean large differences in outcome from very small differences in input, but a case where multiple different outcomes can arise from identical inputs.
- monochromatic 9y agoRight. It’s already known that NS solutions can be chaotic. It would be a big deal if they turned out to be non-unique.
- justaman 9y agoI'm not going to pretend I understand anything about fluid dynamics. It brings to mind how people say "Bumble bee's defy the laws of physics to fly".
- mannykannot 9y ago"When I meet God, I am going to ask him two questions: Why relativity ? And why turbulence ? I really believe he will have an answer for the first." - Werner Heisenberg.
- FabHK 9y ago"I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic." - Horace Lamb
- tulasinandan 9y agoThe author explaining the paper: https://www.youtube.com/watch?v=F71SRP3MZcw https://www.youtube.com/watch?v=F71SRP3MZcw
- davidw 9y agoTotally unfamiliar with the math and physics, but I recall the name from one of the earlier parts of Cryptonimicon.
- qubex 9y agoNavier & Stokes is the name of the scientists who indecently formulated the equation that bears their name that describes the motion of most fluids. In Cryptonomicon the main character is given a simple question about a boat on a river during his admission to the army, deploys heavy-duty fluid dynamics to give a non-obvious answer, and consequentially gets classified as a moron and relegated to menial duties (which suit him just fine, as I recall).
- davidw 9y agoYeah it was a pretty funny scene - it's supposed to be this simple math question, but the ... "Asperger"... he might be called these days - character, Lawrence Waterhouse, really digs into the problem. He ends up discovering something that he submits to a math journal for publication, but the army folks think he's an idiot.
- bitL 9y agoIs there any online course I can take that would teach me Navier-Stokes?
- derjames 9y agoI would recommend this text book: Transport Phenomena. Bird, Steward and Lightfoot. The first part is related to the motion of fluids/transfer of momentum ( The one you are interested ). The second part is the transfer of energy and the third part is related to the transfer of mass.