Timeline for Cauchy problem for an overdetermined system of PDE
Current License: CC BY-SA 3.0
14 events
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| Jan 14, 2015 at 13:03 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Jan 14, 2015 at 13:00 | comment | added | Deane Yang | Sorry but if $E_1$ and $E_2$ are linearly independent, you need to solve both equations. In particular, if $p + q = 1$, then $E_1 + pE_3$ and $E_1 + qE_3$ are linearly dependent and therefore do not imply solutions to $E_1$ and $E_2$. So you have to assume that $p + q \ne 1$ in order to get two linearly independent equations. | |
| Jan 14, 2015 at 11:17 | comment | added | guido giuliani | @DeaneYang And Btw I'm having indeed quite a hard time reading your paper, but it is really worth every single minute spent. Really cannot say how much I appreciate your patience. Regards | |
| Jan 14, 2015 at 11:15 | comment | added | guido giuliani | @DeaneYang as your addendum points at.. Let's say that it is sufficient to solve my system using $E_1$ and $E_2+E_3$, that is $p=0,\;q=1$ using your notations. If I do this I come up again with an overdetermined system, indeed $E_1$ and $E_2+E_3$ are linearly dependent. But then modulo compatibility relations it is sufficient to check hyperbolicity just by inspecting the first equation. Did I understood correctly? | |
| Jan 13, 2015 at 16:06 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Jan 13, 2015 at 15:46 | comment | added | Deane Yang | Yes, this case is handled in section 1.7 of "Involutive Hyperbolic Differential Systems" (Memoirs of the AMS, #370). In the generic case, the so-called reduced Cartan characters are, I believe, $3, 2, 2$. Theorem (1.26) gives an invariant way to identify whether the procedure yields a hyperbolic system of PDE's or not. Alas, as proud as I am of this paper, it's rather difficult to read. My advice is to play around with the calculations described above. I've appended some additional information. | |
| Jan 13, 2015 at 3:19 | comment | added | Deane Yang | I may have misspoken. I'm not sure an invariant way has been worked out. If it has, it would be in the reference cited by Ben McKay above (yes, I wrote it but I don't have a copy handy and I don't remember whether I dealt with this case). I'll look at it when I get a chance. | |
| Jan 12, 2015 at 20:34 | vote | accept | guido giuliani | ||
| Jan 12, 2015 at 20:34 | comment | added | guido giuliani | @DeaneYang Could you please give the references you were hinting at about invariant ways to detect hyperbolic differential systems? Thank you very much in advance | |
| Jan 7, 2015 at 23:50 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Jan 7, 2015 at 23:38 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Jan 7, 2015 at 22:28 | comment | added | Deane Yang | Ach. Thanks for pointing that out. This is what happens when I try to do math in my head. | |
| Jan 7, 2015 at 21:36 | comment | added | Igor Khavkine | In case you hadn't noticed, the system is indeed overdetermined. The coefficient matrices share a common cokernel. Namely, the sum of each column is zero. So your approach won't work directly. The kernel/cokernel of the principle symbol is more obvious from the formulas I gave in my answer to the OP's previous question. | |
| Jan 7, 2015 at 20:12 | history | answered | Deane Yang | CC BY-SA 3.0 |