Timeline for Lumer-Phillips-type theorem for non-autonomous evolutions
Current License: CC BY-SA 4.0
9 events
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| Jul 20, 2023 at 18:26 | history | edited | András Bátkai | CC BY-SA 4.0 |
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| Jul 13, 2023 at 21:45 | comment | added | Peter Wacken | I ended up using the result by Kato mentioned in the answer that I posted. Nevertheless, thank you for your remarks! | |
| Jul 10, 2023 at 21:23 | comment | added | Jochen Glueck | @Luke: In case that the result is not in the book or in case that you can't read French, you might take Theorem 3.1 in this article as a starting point. | |
| Jul 10, 2023 at 21:16 | comment | added | Jochen Glueck | @Luka: But now that I think about it, in case that you're working in Hilbert spaces and the operators $A(t)$ are associated to (maybe elliptic?) sesquilinears from, I think there is a classical result by Lions about well-posedness. I don't know if there's a standard reference for it - maybe Lions' 1961 book (in French; link to zbMATH)? | |
| Jul 10, 2023 at 20:43 | comment | added | Jochen Glueck | @Luke: I don't have sufficient expertise on the topic to make any definite claims - but it is my understanding that well-posedness is, generally speaking, much more difficult to show in the non-autonomous case, precisely because such nice criteria as Lumer-Phillips break down. But it would probably be better if somebody else with more detailed knowledge of the topic could step in here. | |
| Jul 10, 2023 at 14:56 | comment | added | Peter Wacken | In my situation, I am given operators $A(t)$ and I want to show the well-posedness as well as that the propagators are contractions. I can probably apply the theorem from Pazy's book and then use a renorming rechnique to obtain the contraction property (still have to spell out the details). I was expecting that a general theory exists for the generation of propagators (analogous to the case of semigroups) that would immediately solve my problem, however, it seems that this case is much more involved than the case of semigroups. | |
| Jul 10, 2023 at 14:44 | comment | added | Jochen Glueck | @Luke: Are you interested in results where dissipativity of the operators $A(t)$ facilitates the proof of well-posedness (as the Lumer--Phillips theorem that you cited does in the autonomous case) or are you rather interested in situations where you already know well-posedess and want to chracterize when the propagators are contractive? | |
| Jul 10, 2023 at 13:41 | comment | added | Peter Wacken | Thank you for your answer. If I'm not mistaken, in both books they do not cover propagators of contractions. Do you know of any reference for that case? Or is that usually shown directly in the specific case at hand? | |
| Jul 8, 2023 at 9:24 | history | answered | Jochen Glueck | CC BY-SA 4.0 |