Timeline for Smooth dependence on the initial condition of the integral of an ODE
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 21, 2016 at 9:19 | comment | added | N. Gast | Thanks again for the pointer. It took me some time to take the time to understand your proof of Lemma C.1 of your book but now, I am convinced that it solves my question. | |
| Jul 21, 2016 at 9:17 | vote | accept | N. Gast | ||
| May 27, 2016 at 19:27 | comment | added | Jaap Eldering | Once you know that $\|D\Phi_t(x)\| \le C e^{-\alpha t}$, then $\|D^k\Phi_t(x)\| \le C e^{-\alpha t}$ follows from a Gronwall-like argument. See Lemma C.1 in my book: jaapeldering.nl/includes/get.php?file=NHIM-noncompact-book.pdf | |
| May 27, 2016 at 12:16 | comment | added | N. Gast | Thanks for this answer but I found it hard to generalize it to $C^2$. To me, the main difficulty is to prove that for $x$ close to $x^*$: $\| D^k \Phi_t(x) \| \le C^{-\alpha t}$. Once this is done, you proof seems easy to generalize to $C^k$. But even for the $C^1$ case, this is not obvious for me. Should it be? | |
| May 26, 2016 at 15:08 | history | edited | Jaap Eldering | CC BY-SA 3.0 |
added 836 characters in body
|
| May 26, 2016 at 14:06 | comment | added | Willie Wong | How do you justify $\int_0^\infty D\Phi_t(x) ~\mathrm{d}t$ exists? (Just being nitpicky here, since the proof should depend on the fact that $f$ is $C^k$ at $x^*$, otherwise there are trivial counterexamples.) | |
| May 26, 2016 at 13:58 | history | answered | Jaap Eldering | CC BY-SA 3.0 |