Timeline for Analyticity of the solutions of PDE
Current License: CC BY-SA 3.0
12 events
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| Apr 20, 2013 at 10:18 | history | edited | user9072 |
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| Aug 16, 2012 at 19:18 | comment | added | Otis Chodosh | No harm done! I think that Rafe showed that my comment was partially wrong as it seems as if there is some sort of criteria for linear PDE which have this property. On the other hand, I'd guess that for systems/nonlinear PDE, there is basically no hope of anything more than case by case results, but I could of course be wrong.. | |
| Aug 16, 2012 at 7:03 | vote | accept | Cristi Stoica | ||
| Aug 16, 2012 at 6:57 | comment | added | Cristi Stoica | @Otis: Thanks for the comments. I am interested in the most general settings, systems or singles, linear or nonlinear, elliptic, parabolic, hyperbolic etc. My hope was that if somebody has something to say about the question, in a general or in a particular setting, to be free to say, because I am interested in the general question. So my hope was to receive answers of the form "in general, ...", or, "in the particular case of ... equations, ...". I see that Rafe accuses me of being rude, his interpretation of what I said is wrong, but if I made you feel that I was rude with you I apologize. | |
| Aug 16, 2012 at 0:02 | answer | added | Rafe Mazzeo | timeline score: 5 | |
| Aug 15, 2012 at 23:52 | comment | added | Otis Chodosh | In particular, you don't specify if you're interested in systems, single PDE's, linear PDE's, nonlinear PDE's, etc. It seems that the only place that a nice theory might have a hope of existing is in the setting of linear PDEs for one function. Can you give some more precise details, or motivation for the question? | |
| Aug 15, 2012 at 23:50 | comment | added | Otis Chodosh | You should probably add some motivation as to what sort of PDE's you are interested in, otherwise I doubt there exists a "good" answser. I took a brief glance at the paper you linked to, and if I understand correctly, the paper gives an example of a system of linear PDE's with analytic coefficients and initial data, but non-analytic solutions. This should be contrasted with the Cauchy-Kowalewski theorem, which says that for a linear PDE (for a single function) analytic data and coefficeints imply analyticity of solutions and ( in light of @Bazin's answer) suprisingly existence. | |
| Aug 15, 2012 at 21:17 | answer | added | Deane Yang | timeline score: 1 | |
| Aug 15, 2012 at 12:21 | comment | added | Cristi Stoica | @Otis: I just wanted to know if it is possible to have pde with analytic coefficients and solutions like that in the question. If the answer depends on the particular pde and the initial conditions, I think that the answer should contain the relevant assumptions, not the question. It is more likely that these conditions are known by the person who knows the answer, not by the person who asks the question. | |
| Aug 14, 2012 at 17:28 | comment | added | Otis Chodosh | You certainly need to assume something about the PDE for this question to have a chance of making sense. For example, your PDE could be $\partial_t u =0$, with initial conditions $u(t=0) = f$.... On the other hand, for the heat equation, for example, a solution immediately becomes analytic, even for very rough initial data.. | |
| Aug 14, 2012 at 11:34 | answer | added | Bazin | timeline score: 3 | |
| Aug 14, 2012 at 8:05 | history | asked | Cristi Stoica | CC BY-SA 3.0 |