Timeline for Gradient flows: convex potential vs. contractive flow?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2020 at 18:20 | comment | added | Willie Wong | Ah, I understand. Sorry, cannot help more. I do still think that this is a local and not a global problem, and the key is to understanding the variation of your flow with respect to changes in initial data. | |
| Apr 30, 2020 at 18:16 | comment | added | leo monsaingeon | Thanks a lot anyway, at least now I'm convinced in $\mathbb R^d$, that's a start! ;-) | |
| Apr 30, 2020 at 18:15 | comment | added | leo monsaingeon | yes yes, I get the idea, but the problem is that in abstract metric gradient flows even the $\nabla V(X)$ object does not really exist, only its magnitude somehow (metric slope and upper gradients). So I cannot even try Taylor-expanding anything. Life is hard. | |
| Apr 30, 2020 at 18:15 | comment | added | Willie Wong | ... answer may require quite a bit more work/slightly strengthened assumptions beyond $C^2$. | |
| Apr 30, 2020 at 18:12 | comment | added | Willie Wong | I think the same argument can be made through differences if you are careful with your epsilons and deltas. You have that the difference between solutions $X$ and $Y$ satisfy $$ (X-Y)' = -\nabla V(X) + \nabla V(Y) $$ and if you start with $X(0) - Y(0)$ sufficiently small and only look at sufficiently small times you can approximate using Taylor series and everything goes through. The only part you may have to modify your assumption is when you work in infinite dimensions, and continuity does not guarantee local uniform continuity. This means that the term I denoted by $O(t)$ in the... | |
| Apr 30, 2020 at 18:05 | comment | added | leo monsaingeon | Yes indeed! I should have figured that out sooner. Thank you @Willie Wong. Actually I would be interested in a "non-differential" proof, since I want to go to infinite dimension and metric spaces. | |
| Apr 30, 2020 at 18:03 | vote | accept | leo monsaingeon | ||
| Apr 30, 2020 at 15:33 | history | undeleted | Willie Wong | ||
| Apr 30, 2020 at 15:33 | history | edited | Willie Wong | CC BY-SA 4.0 |
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| Apr 30, 2020 at 15:17 | history | deleted | Willie Wong | via Vote | |
| Apr 30, 2020 at 15:16 | history | answered | Willie Wong | CC BY-SA 4.0 |