Timeline for Smoothness of solution map for PDE
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16 events
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| Apr 19, 2020 at 5:06 | comment | added | Deane Yang | I just found a nice short proof by Saint Raymond of the Nash-Moser implicit function theorem. pdfs.semanticscholar.org/a21a/… | |
| Apr 19, 2020 at 2:31 | comment | added | Deane Yang | The Nash-Moser theorem does give an inverse between two Banach spaces, just not the two you start with. It does not give you the optimal one. That probably requires arguments specific to a PDE. And, in most versions of Nash-Moser, if the inverse to the linearization loses derivatives, the inverse to the nonlinear PDE loses a lot of derivatives. A near optimal version is in a paper by Hormander. (acadsci.fi/mathematica/Vol10/vol10pp255-259.pdf). | |
| Apr 19, 2020 at 1:53 | comment | added | Quarto Bendir | Also I'm curious why you say that $m$ is usually negative? My impression was that the loss of derivatives is typically somewhat small, at least small enough that you gain some regularity in passing from the RHS to the solution. Is this wrong? | |
| Apr 19, 2020 at 1:50 | comment | added | Quarto Bendir | Like I asked Tobias in my second comment above, isn't there some technical difficulty about which function spaces one identifies for the inverse map to be smooth? In Hamilton's paper, it seems that the only conclusion is that the map $C^\infty\to C^\infty$ is smooth and tame, and it seems like the Hamilton-Nash proof does not (and possibly is incapable of?) finding any kind of optimal $m$ for $C^{k,\alpha}\to C^{k+m,\alpha}$ to be smooth. But maybe I don't understand it properly. | |
| Apr 19, 2020 at 1:32 | comment | added | Deane Yang | So if you can find two Banach spaces, where the linearization of your nonlinear PDE is a smooth map between the two spaces and the inverse of the linearization is bounded, then I'm pretty sure the inverse to the nonlinear map is smooth, too. If the inverse maps back into a weaker space, then you need to use the Nash-Moser theorem. You'll get a smooth inverse, but it will be $C^{k,\alpha} \rightarrow C^{k-m,\alpha$ for some large $m$. The argument here is actually the same as for the Banach implicit function theorem. | |
| Apr 19, 2020 at 1:27 | comment | added | Deane Yang | In the elliptic case, you have a smooth map $F: A \rightarrow B$, where the linearization $F'(a): A \rightarrow B$ is is smooth and has a bounded inverse. In that case, I believe everything follows by the same argument you would use in the finite dimensional inverse function theorem. It basically follows by implicit differentiation. | |
| Apr 18, 2020 at 22:46 | history | edited | Quarto Bendir | CC BY-SA 4.0 |
changed title
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| Apr 18, 2020 at 22:45 | comment | added | Quarto Bendir | @DeaneYang Is it so simple even in the elliptic case? In well-known cases like e.g. the Calabi-Yau complex Monge-Ampère equation, even once one has a solution map $C^{k,\alpha}\to C^{k+2,\alpha}$ and the PDE is known to be uniformly elliptic at the solutions, it doesn't seem totally obvious that the map $C^{k,\alpha}\to C^{k+2,\alpha}$ is smooth. (Is it smooth? It seems like it could follow from estimates for the linearized equation, but it isn't obvious to me how. Is it analytic?) | |
| Apr 18, 2020 at 19:00 | comment | added | Deane Yang | And perhaps you should edit the title, either removing "elliptic" or inserting "degenerate". The situation is much simpler if the PDE is elliptic, because $m = N$. | |
| Apr 18, 2020 at 18:59 | comment | added | Deane Yang | How do you establish the existence of $Q$? Did you use a Nash-Moser argument? Or probably something better, because normally the resulting $m$ would be negative and $k$ would have to be sufficiently. large. | |
| Apr 18, 2020 at 13:26 | comment | added | Quarto Bendir | Aren't there still two differences? 1) Hamilton's conclusions assert tameness, but nothing about the degree of tameness; 2) the conclusion on smoothness is as a map $C^\infty\to C^\infty$, which doesn't (?) automatically extend to smoothness as a map $C^{k,\alpha}\to C^{k+m,\alpha}$ | |
| Apr 18, 2020 at 12:43 | comment | added | Tobias Diez | Yes, he only discusses the linear case but for operators depending on parameters. This allows you then to use the Nash-Moser inverse function theorem to conclude that a non-linear solution operator exists (locally) and is smooth. You might be also interested in my PhD thesis arxiv.org/abs/1909.00744 (non-linear elliptic operators are discussed in section 2.2.4) | |
| Apr 18, 2020 at 12:11 | comment | added | Quarto Bendir | I'm not very familiar with Hamilton's paper, but doesn't that only cover the case of linear $P$? | |
| Apr 18, 2020 at 11:39 | comment | added | Tobias Diez | For elliptic operators, smoothness of the solution operator is established in Section II.3.3 of "Nash-Moser Inverse Fuction Theorem" by R. Hamilton (1982). | |
| Apr 18, 2020 at 11:35 | review | First posts | |||
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| Apr 18, 2020 at 11:30 | history | asked | Quarto Bendir | CC BY-SA 4.0 |