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I don't have much knowledge about Differential Topology neither Differential Geometry, but working on this another question about solutions to differential equations, and someone recommend me to research about the Frobenius theorem (differential topology), so I start to read it and going deeper on the Wikipedia pages, but this abstract framework is hard to understand without previous introduction.

But without diving too deep, I found in Wikipedia that for Differential topology as also for Differential geometry, are both defined over Smooth Manifolds, which are explained as "In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function.".

On the mentioned question I am asking about the properties of the solutions of finite duration to differential equations, like the following example: $$\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1\tag{Eq. 1}\label{Eq. 1}$$ admits the finite duration solution: $$x(t)=\frac{1}{4}(2-t)^2\theta(2-t) \tag{Eq. 2}\label{Eq. 2}$$ as is proved in this answer. Here, is easy to see that \eqref{Eq. 2} is not a smooth function, so following the previous definitions it should't be possible to work with this kind of solutions under the framework of Differential Topology/Geometry, which is indeed what I want to figure out, before lost time going deeper into these theories.

  1. Does Differential Topology/Geometry frameworks only manage to work with smooth solutions to differential equations?
  2. Does Differential Topology/Geometry frameworks manage to work with analytic solutions to differential equations?
  3. It is possible for Differential Topology/Geometry frameworks to work with solutions to differential equations that aren't analytic in their whole domains? (like piecewise defined solutions)
  4. Bonus: It is possible for Differential Topology/Geometry frameworks to work with solutions to differential equations that are of finite duration? Any examples? (see this question for examples of what I mean with solutions of finite duration)

Added later

Recently on another question, a comment leading me for looking for the term finite extinction time, and I found the following paper:

Where is claimed the solutions achieved a finite extinction time, and it is used a language similar to the used on DT/DG.... unfortunately is too advanced to be deciphered by myself.

Could be this an example of what I am asking for?

Being true, I believe it solve all the previous 4 questions. If is true, but somehow it doesn't solve the others previous questions because of any caveats, hope you can mention why (not deep explanations needed since probably I don't gonna understand, better save your time). As example of a possible issue, the paper also claim holding uniqueness of solutions, which should be broken when a solution becomes zero in a non-zero measure domain, but this issue is already asked in this another question.

Also I would like to extend the question if the last paper is indeed a valid example: Are there examples of solutions with finite extinction time being used in Special/General Relativity frameworks?

Last Update:

I found another paper talking about finite extinction times in differential geometry/topology:

But later I found the author is indeed Grigori Perelman (see the last paper $P03b$), the mathematician that refuse the Field Medal and the Millennium Problems' prize for solving the Poincaré conjecture, so maybe the field is still too new for having this finite extinction times applied in solutions to physics problems... but I don't really know (at least I found the previous paper), so I hope it is not the case.

Joako
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    Both Differential Topology and Differential Geometry typically work with "sufficiently" differentiable objects (manifolds, functions, etc). Many authors simply assume infinite differentiability. Others try to find optimal differentiability assumptions. (For Frobenius theorem one assumes $C^1$-smoothness of the distributions.) Some works in DT and DG work in the real-analytic setting. It all depends on what you are reading. As for the bonus question: sure, in DG one sometimes encounters finite-duration solutions. For instance, the issue of "completeness" in DG centers around this problem. – Moishe Kohan May 25 '22 at 11:10
  • @MoisheKohan Thanks you very much for commenting. Do you have any reference to any example of a finite duration solutions to a differential equation under DG analysis??? – Joako May 25 '22 at 11:27
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    Of course: As I said, take the geodesic equation in any incomplete Riemannian (or semi-Riemannian manifold). For instance, take geodesics in a proper open subset $U$ of the plane: They eventually will "escape" $U$, hence, the solution of the geodesic equation will cease to exist in a finite time. More interestingly, the same phenomenon can happen for geodesics in some semi-Riemannian metrics on the 2-dimensional torus, which is a compact space. – Moishe Kohan May 25 '22 at 13:15
  • @MoisheKohan On the other previously mentioned question, other users have explained me that for having a solution of finite duration is mandatory to have a singular point in time where uniqueness is not hold - as example see https://en.wikipedia.org/wiki/Singular_solution#Failure_of_uniqueness . With these, trying to follow your previous comment, I found on the Wiki for Geodesic that the Picard–Lindelöf theorem should be hold, .... – Joako May 25 '22 at 20:30
  • @MoisheKohan with this uniqueness condition as a requirement, finite duration solutions shouldn't be found, so, with the greatest humility possible I ask (I can barely understand the concepts)... Are you sure this framework can work with singular solutions of finite duration in the time variable? (they have differences from being compact-supported solutions in space - better explained on the other question)... from Wiki pages, at least, there're efforts to keep everything smooth, or at least, analytical (maybe they don't touch the general scope) – Joako May 25 '22 at 20:39
  • I think, you might have misunderstood what other users told you or they misunderstood your question. I have hard time understanding what you are asking in the last comment. It's possible that you are asking about a situation when a smooth solution becomes nonsmooth, but still a solution in the sense of distributions (elements of some Sobolev space, do you know what this means?) in a finite amount of time. Then indeed, geodesic equations do not give you such. What is your background? Physics? Chemistry? Engineering? – Moishe Kohan May 25 '22 at 20:47
  • @MoisheKohan I am an electrical engineer, and I am trying to understand solutions to differential equations that by their own dynamics, after a finite ending time $T<\infty$, becomes zero forever after: for every $t>T,,y(t)=0$ with the solution defined on the whole domain (not just in a domain $t\leq T$). As in the example, x(t) is a continuous real-valued solution that exists in the whole $\mathbb{R}$ domain, but it such as $x(t)=0,,t\geq 2$, being a solution to the differential equation in the whole $\mathbb{R}$ domain. – Joako May 25 '22 at 21:05
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    I see. This is a very different question from what was discussed earlier. – Moishe Kohan May 25 '22 at 21:08
  • @MoisheKohan Since this kind of solution is piecewise analytical, but not analytical as a whole (as example $x(t)$ is not smooth at $T=2$), I am trying to figure out if there exists a current framework to study them, from where following comments I reach DT/DG, but from what I have been founding from Wikipedia, it looks it is a more restricted framework (maybe it is not and the webpages, for simplicity, are focused on the general case of usefulness). – Joako May 25 '22 at 21:08
  • Yeh, people in DG/GT prefer not to work with these. However, in GR, this occurs (at least, can occur on the pure math side). Are you aware of the "Cosmic censorship hypothesis"? See here for a nice informal discussion and references to math examples showing that this hypothesis can be violated. I think, these kind of examples is what you are interested in. – Moishe Kohan May 25 '22 at 21:17
  • @MoisheKohan I read it, being interesting, I don't know if its related or not (too advanced for me). Nevertheless, it resembles the problem for the more simple system of the Norton's Dome which could have multiple solutions, and as I asked here its differential equation also supports solutions of finite duration. From your comments, I could read into is that: "in their traditional formulations GT/GM don't work with these issues, but as PDEs, they could be extended to work with them", I am right? – Joako May 25 '22 at 22:13
  • @MoisheKohan I extend the question because I think I find an example of a paper that find solutions which achieve finite extinction times using a framework similar of the DG, but I am not really sure... hope you can see it. Beforhand thanks you very much. – Joako Jun 10 '22 at 23:35
  • @MoisheKohan I have added another related paper I found about having finite extinction times in the freamework of DG/DT, this time by Grigori Perelman. – Joako Jun 16 '22 at 02:50

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