Timeline for Existence and uniqueness of solutions for a system of first order PDEs [closed]
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| May 9, 2016 at 14:25 | vote | accept | Fernando | ||
| May 8, 2016 at 20:55 | history | closed |
Stefan Kohl♦ Michael Renardy Wolfgang András Bátkai Christian Remling |
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| May 8, 2016 at 3:45 | answer | added | Willie Wong | timeline score: 0 | |
| May 6, 2016 at 23:43 | comment | added | Fernando | @Willie Wong, I have a request. Actually the bibliography I have got for system pde have the Cauchy-Kovalevswkaia theorem, but the curves for the initial data are in $t=0$, as you mentioned in your second comment. Is there a method to change the initial curve for any noncharacteristic curve in the domain of where I'm looking the solutions to apply the CK theorem? or can you mention me a bibliography for the versión of the theorem that I need? Thanks in advance. | |
| May 5, 2016 at 13:32 | comment | added | Fernando | Thanks a lot for all your comments, I finally understand. | |
| May 5, 2016 at 13:06 | comment | added | Willie Wong | It is of course possible that there will be singularities forming in finite time so that the solution is not global, and there's nothing you can do about that. The classical theorems only assert local well-posedness anyway. | |
| May 5, 2016 at 13:05 | comment | added | Willie Wong | @Fernando: for the local problem, as long as $b$ is bounded initially, you can always find a change of variables $\tau = t + \lambda x$ and $\xi = t - \lambda x$ (or something else appropriately chosen) so that the matrix is uniformly invertible along the initial data curve. Then by continuity it will remain uniformly invertible for some small time for the solution to the initial value problem. | |
| May 5, 2016 at 13:01 | comment | added | Fernando | @Igor Khavkine, I see now the hyperbolic system, thanks very much to you and Willie Wong. But I have just a question about your comment (please, be patient with me): the coefficient matrix of $\partial /\partial\tau$ have the form $\pmatrix{1 & 0& 0\\ 0& 1& 0\\ 0&0&1+b(\tau,\varepsilon)}$. So, for that the coefficients to be invertible what I need isn't that $b(x,t)$ don't be too close of $1$? | |
| May 5, 2016 at 11:33 | comment | added | Igor Khavkine | As Willie suggests, change coordinates to get a hyperbolic system in a form that might be more familiar to you. Namely, try $\tau = x+y$ and $\xi = x-y$. As long as $|b(t,x)|$ is not too large, the coefficient matrix of $\partial/\partial\tau$ will be invertible and $\tau = \text{const}$ will give you non-characteristic curves. | |
| May 4, 2016 at 23:57 | history | edited | Fernando |
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| May 4, 2016 at 21:51 | comment | added | Fernando | @Willie Wong, thank you for your comments, but it can be considered a system of hyperbolic equations even when the time derivative in two equations is mising? And if so, could you point out bibliography with the theorem?, because I don't have anything useful for that. | |
| May 4, 2016 at 21:39 | comment | added | Willie Wong | Re edit: If you are prescribing data on non-characteristic curves, then you have a standard nonlinear hyperbolic system for which local existence and uniqueness follow directly. Most sources want $A$ invertible because they want the make $t = 0$ a non-characteristic curve for the initial data. | |
| May 4, 2016 at 21:35 | comment | added | Fernando | you are right, this new edit take in account your comment? | |
| May 4, 2016 at 21:34 | history | edited | Fernando | CC BY-SA 3.0 |
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| May 4, 2016 at 21:16 | comment | added | Willie Wong | Do I read correctly that you are giving initial data at a point? Then chances are you will not get any uniqueness results. Given what you wrote as your system, you can always find a coordinate system $\tau, \xi$ for which you have a hyperbolic first order PDE. and from which with the prescribed initial data at one point you conclude both existence and non-uniqueness. | |
| May 4, 2016 at 20:38 | review | Close votes | |||
| May 8, 2016 at 20:55 | |||||
| May 4, 2016 at 20:35 | history | edited | Fernando | CC BY-SA 3.0 |
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| May 4, 2016 at 20:28 | history | edited | Fernando | CC BY-SA 3.0 |
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| May 4, 2016 at 19:28 | review | First posts | |||
| May 4, 2016 at 20:22 | |||||
| May 4, 2016 at 19:25 | history | asked | Fernando | CC BY-SA 3.0 |