Timeline for How were the Navier-Stokes equations found in the first place if we can't solve them?
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| Oct 19, 2017 at 7:59 | comment | added | Anton Tykhyy | I hear that modifications of the WKB method can be used in non-linear systems, perhaps after suitable transformations which convert a non-linear equation into a more complicated linear equation - AFAIR Cortevega-de Vries (sp?) type equations are handled by this approach. However, I'm no specialist on this. | |
| Oct 18, 2017 at 10:33 | comment | added | Selene Routley | @AntonTykhyy Sorry, yes I should have said "physics" rather than "numerical physics". I only know it from QM and also from modal waveguide theory, and I'm really only familiar with its application to linear systems, hence my question as to whether WKB finds much use in nonlinear problems such as the NS equation - I would find such an assertion surprising | |
| Oct 18, 2017 at 7:16 | comment | added | Anton Tykhyy | In general. WKB originally came from quantum mechanics, and it's an analytical not a numerical method. | |
| Oct 16, 2017 at 22:00 | comment | added | Selene Routley | @AntonTykhyy Are you speaking of WKB in fluid mechanics (which fries my brain a bit too much) in particular or about numerical physics in general? (I'm interested since I'm trying to get a better, albeit surface, understanding of what the fluid people in a group I am linked with do) | |
| Oct 16, 2017 at 17:17 | comment | added | Anton Tykhyy | @Ooker that's one tool, yes. Asymptotic methods like WKB provide other kinds of information. | |
| Oct 16, 2017 at 1:44 | comment | added | Selene Routley | @Ooker ...... is about furthering that knowledge. Note that weak field gravity solutions, post Newtonian theory and the like are all linear approximations to General Relativity that have big uses; Einstein's big 1915 paper made use of a linear approximation to recover Newtonian physics as the limiting form of GTR and to compute the perturbation to Mercury's apsidal precession. I'm much less comfortable with fluid mechanics, but the theory of inompressible, irrotational, linearlized flows are also often useful approximations to the Navier Stokes equations. | |
| Oct 16, 2017 at 1:44 | comment | added | Selene Routley | @Ooker Of course. For example, in linear systems, we have a huge body of theory - the majority of functional analysis: operator theory, spectral theory, classes of problems known to have fully discrete spectums thus solutions are resolvable into generalized Fourier series, distribution theory ....... the list is endless and the knowledge about the behavior of solutions this brings is immense. Often similar idealized problems have full, analytic solutions. The theory behind the behaviors that arise in nonlinear systems - albeit impressive - is much more piecemeal and the Clay prize .... | |
| Oct 15, 2017 at 20:26 | comment | added | Ooker | @AntonTykhyy ah, you mean something like the characteristic equations? | |
| Oct 15, 2017 at 16:40 | comment | added | Anton Tykhyy | Yes, sometimes, although it's difficult and often we can say less than we'd like. E.g. asymptotic methods can show the behavior of solutions in the vicinity of special points, such as infinity. Stephen Hawking got his Nobel prize for work of just this sort -- figuring out the general behavior of certain classes of solutions of gravity equations. | |
| Oct 15, 2017 at 7:50 | comment | added | Ooker | Given a system's equation without computer, can we say anything about its behaviors? Does knowing the equation without solution help us know it any better? | |
| Oct 15, 2017 at 3:54 | history | edited | Selene Routley | CC BY-SA 3.0 |
added 24 characters in body
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| Oct 15, 2017 at 3:14 | history | answered | Selene Routley | CC BY-SA 3.0 |