Timeline for A basic understanding of the Navier-Stokes, or Terry Tao's "exploding water" problem
Current License: CC BY-SA 3.0
15 events
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| Jun 9, 2017 at 11:24 | audit | Reopen votes | |||
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| May 18, 2017 at 1:30 | comment | added | uhoh | @MichałMiśkiewicz I knew he had a blog but didn't know my way around there. You are right, I find the post is much more accessible than the linked ArXiv preprint. Thank you for your help! | |
| May 17, 2017 at 20:18 | comment | added | Michał Miśkiewicz | Have you read Tao's blog post on finite time blowup for an averaged version of NSE? There's a more mathematical and still accessible explanation (especially at the end). | |
| May 17, 2017 at 1:37 | vote | accept | uhoh | ||
| May 17, 2017 at 1:29 | answer | added | Ian | timeline score: 7 | |
| May 17, 2017 at 0:04 | comment | added | uhoh | @Ian Thanks, that's exactly the kind of answer I was hoping for, pls consider posting as such. Also it reminded me of something in the original article. I've added it in an edit, but you (or anyone) could transplant it instead into an answer if appropriate. | |
| May 17, 2017 at 0:02 | history | edited | uhoh | CC BY-SA 3.0 |
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| May 16, 2017 at 23:31 | comment | added | Ian | As I recall they are carefully set up to vorticial motion to repeatedly move energy from long length scales to successively shorter length scales. | |
| May 16, 2017 at 22:54 | comment | added | uhoh | @Ian Is there any way to approximately describe said initial conditions beyond just "improbable/impossible". I'm guessing it's more than just sticking ones finger in and swirling it around a bit. Or, if there are papers or preprints where one might look, even if advanced, a link or two would be appreciated. Thanks! | |
| May 16, 2017 at 14:45 | comment | added | Winther | As for why water does not explode; even if the equations have the possibility of a blow-up it is know that this would require very special initial conditions that would probably be impossible to generate in nature. Another things is that in nature these equations are only an approximation in certain regimes so a mathematical blow-up does not necessarily imply a physical blow-up. | |
| May 16, 2017 at 14:25 | comment | added | Ian | This is all about the millennium problem, yes. It is just a question of starting with "nice" initial data, do you get "bad" data later? And the problem is that none of the techniques we have to show that "niceness" is preserved over time (which are applicable in many evolution PDE) apply to Navier-Stokes in particular. Moreover, Tao recently showed that Navier-Stokes is at best "right on the cusp" of developing singularities, since a relatively modest modification of it does develop singularities. | |
| May 16, 2017 at 14:23 | comment | added | Malkoun | I did not read all the details in the paper, but roughly speaking, Terry Tao considers a modifed version of Navier-Stokes, and proves that for these equations, there exist initial data which leads to a finite time blow-up, i.e. to the formation of a singularity in finite time. This is related to the Navier-Stokes Millenium problem indeed (albeit it is a modifed version of Navier-Stokes). That being said, his approach in the paper is highly original! | |
| May 16, 2017 at 14:18 | comment | added | tp1 | There was nice development recently where $\int_{a}^{b} f = (b-a)av_f(a,b) $ would provide category theory version of integration via simple average-function and mulltiplication. This would easily extend to navier-stokes, but dunno if it produces anything better than numerical solution -- and the millennium prize problem description says that numerical solution exhibits blowup.... | |
| May 16, 2017 at 14:07 | history | edited | florence |
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| May 16, 2017 at 13:50 | history | asked | uhoh | CC BY-SA 3.0 |