Timeline for Domain of dependence for Hyperbolic system of PDES
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2016 at 22:36 | comment | added | Igor Khavkine | @WillieWong, ah, then one modern reference that I know is Volevich & Gindikin, Mixed problem for partial differential equations with quasihomogeneous principal part (AMS, 1996) MR1357662. But it's so technical, that I never managed to really sink my teeth into it. | |
| Sep 3, 2016 at 22:12 | comment | added | Willie Wong | @IgorKhavkine: no, I actually mean 3rd, 4th, arbitrary $n$th order PDE systems. I'm curious whether anyone since Leray had thought of treating these systems systematically. | |
| Sep 3, 2016 at 17:45 | comment | added | Igor Khavkine | @WillieWong, if by higher order you just mean wave-like (aka normally hyperbolic, that is with a principal symbol given by the metric tensored with a diagonal matrix) systems, then the books with "wave" in the title do, as well as Courant-Hilbert, Christodoulou and a few other ones. Though, I'm not sure if any of them has a thorough discussion of systems of higher order equations (except maybe Leray). Such a discussion is actually hard to find and is always very technical. | |
| Sep 3, 2016 at 14:55 | comment | added | Willie Wong | @IgorKhavkine: do you know which of the items in your list deal with the case of higher order hyperbolic systems? (Leray is the only one I checked since it was the only one I had in my office; I know also that several on your list only treat first/second order equations.) | |
| Sep 3, 2016 at 7:50 | comment | added | Amir Sagiv | @WillieWong, thanks, here's a full link to Rauch's paper - projecteuclid.org/download/pdf_1/euclid.maa/1175797445 | |
| Sep 3, 2016 at 7:15 | comment | added | Igor Khavkine | As Willie mentioned, the domain of dependence theorem is discussed in many textbooks and monographs (see the answers to MO181630 for a sample of references). Of course, there are many (slightly distinct) notions of hyperbolicity and the proof needs to be adapted to each case. | |
| Sep 2, 2016 at 15:25 | comment | added | Willie Wong | For the sake of giving a reasonably complete reference: the case of strictly hyperbolic systems with variable coefficients is treated by Jean Leray in his notes on Hyperbolic Differential Equations; for first order hyperbolic systems, you can see this article by Rauch and his references. | |
| Sep 2, 2016 at 14:53 | comment | added | Willie Wong | (1) Leveque's use of the word "domain of dependence" is non-standard, usually one refers to the entire space-time domain and not just its intersection with the initial data hypersurface. (2) This result is old and standard (going back to at least Leray), for strictly or symmetric hyperbolic systems; look for example in Volume 2 of Courant-Hilbert where they discuss hyperbolic PDEs. (3) The property you are referring to is also called the "finite speed of propagation" property for hyperbolic PDEs, and is proven in most textbooks in hyperbolic PDEs. | |
| Sep 2, 2016 at 13:52 | history | asked | Amir Sagiv | CC BY-SA 3.0 |