Timeline for Method of characteristics for higher order PDEs in more than two variables
Current License: CC BY-SA 4.0
6 events
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| Mar 22 at 5:30 | comment | added | Phil Harmsworth | @Puzzled The (rare) exceptional cases where characteristics exist for second order PDEs in three independent variables are described in D. H. Parsons, The Extension of Darboux's Method, Mémorial des Sciences Mathématiques, fasc 142 (1960) (numdam.org/issue/MSM_1960__142__1_0.pdf). | |
| Oct 31, 2023 at 11:52 | comment | added | Willie Wong | ... and a theorem (which you can find in Hormander) equating hyperbolicity of a linear PDO with solvability of its corresponding inhomogeneous problem. The keyword here is "hyperbolicity of a differential operator". | |
| Oct 31, 2023 at 11:49 | comment | added | Willie Wong | Yes! There's a lot you can say about the characteristics of the associated differential operator. Some key ideas: (a) in microlocal analysis there is the notion of "propagation of singularities", which states roughly that singularities (including finite but not infinite differentiability) propagate along characteristics for solutions to a PDE. A key related concept is the "wave front set". This underlies a lot of regularity results. (b) Understanding the characteristics structure is key to understanding local solvability of a nonlinear PDE. Key results including the Cauchy-Kovalevskaya Theorem | |
| Oct 31, 2023 at 8:02 | comment | added | Puzzled | Thank you very much for your answer. At this stage I guess the quetion is the following: If in general we can not find the solutions of a PDE via characteristics is there anything we can say about the PDE by knowing the characteristics of the associated differential operator? | |
| Oct 31, 2023 at 3:19 | history | edited | Willie Wong | CC BY-SA 4.0 |
fix minor typos
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| Oct 31, 2023 at 3:12 | history | answered | Willie Wong | CC BY-SA 4.0 |