Timeline for Functional decaying under the heat flow (?)
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2018 at 10:49 | answer | added | Romain Gicquaud | timeline score: 0 | |
| Jun 3, 2018 at 10:19 | comment | added | Romain Gicquaud | Indeed. This seems to be the simplest case to address. | |
| Jun 2, 2018 at 12:06 | comment | added | fedja | For small $p$ you may be fine as far as I know (or it may depend on $a$ then). Let's try to figure out $p=2$. You have posted a separate question about it, right? | |
| Jun 2, 2018 at 6:26 | vote | accept | Romain Gicquaud | ||
| Jun 2, 2018 at 6:23 | comment | added | Romain Gicquaud | Wow, I got your point. Thanks!!!! I will run numerics next week to see it explicitly. This certainly ruins the argument for very large p. But what about small p? | |
| Jun 1, 2018 at 22:30 | comment | added | fedja | OK, I posted the details. Formally that is the end of the story but feel free to modify the question if you can get away with less than you originally requested. | |
| Jun 1, 2018 at 21:18 | answer | added | fedja | timeline score: 2 | |
| Jun 1, 2018 at 18:20 | comment | added | Romain Gicquaud | I do agree with this part of the argument. From the (wrong) argument I gave at the beginning, I was expecting $\lambda = 0$ for the flat torus. | |
| Jun 1, 2018 at 18:15 | comment | added | fedja | Private conversation with Ben in his office 1 year ago. The blackboard has been erased since then, but I can try to recover the details (alas, my memory is not as good as it was when I was 25, so it'll take time). However, do you agree on the main point, which is that if we do not have $\lambda=0$ on the flat torus, we have no $\lambda$ at all. Is that part clear? | |
| Jun 1, 2018 at 18:05 | comment | added | Romain Gicquaud | Can you give me a reference to what you say? | |
| Jun 1, 2018 at 18:03 | comment | added | fedja | Actually, I do know that the answer is bad on a flat torus: you can go up a bit (we figured it out with Ben Jaye once for another problem) but once you can go up, you can accelerate the time derivative as much as you want by considering $v(kx)$ instead of $v$, so no fixed $\lambda$ for you unless you are ready to allow a constant in front of the RHS (i.e., you care about large times only). | |
| Jun 1, 2018 at 17:52 | comment | added | fedja | Would you be interested in the $\mathbb R^n$ case then? (or in the flat torus if you prefer to keep everything compact)? | |
| Jun 1, 2018 at 13:29 | history | edited | Willie Wong |
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| Jun 1, 2018 at 12:40 | history | edited | Romain Gicquaud | CC BY-SA 4.0 |
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| Jun 1, 2018 at 11:57 | history | edited | Romain Gicquaud | CC BY-SA 4.0 |
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| Jun 1, 2018 at 11:56 | comment | added | Romain Gicquaud | @fedja: this is right, the functional is not convex! I did a stupid mistake in my calculation... I removed the remark. | |
| Jun 1, 2018 at 6:05 | comment | added | fedja | I'm not quite sure I follow your argument for $\mathbb R^n$: The mapping $\mathbb R\times\mathbb R^n\ni (u,v)\mapsto [|u|^{a-1}|v|]^p$ is not convex. | |
| May 31, 2018 at 18:26 | history | asked | Romain Gicquaud | CC BY-SA 4.0 |