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I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.

When I approached the speaker after the talk whether he could state in a general way what this concept says he was very resilient to state something that is universally true. He rather drifted off into examples in $l^1$ where he talked about non-compact symmetry groups and so on.

Although I appreciated this at the moment, I am a bit unsatisfied now, because I would like to see a very dense and general statement what concentration compactness is about.

To my surprise, also the internet seems to be full of rather vague explanations what this means on a general Banach space. I do not want to give references at this point, because I think many explanations are well written but do not answer my question:

Is there a comprehensive theorem stating the concept of concentration compactness in a most general way?

Put differently: What is the analogue of Banach-Alaoglu for concentration compactness?

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  • $\begingroup$ I think the book Concentration Compactness: Functional-Analytic Grounds and Applications (by Tintarev & Fieseler) was made for you. $\endgroup$
    – Chris Gerig
    Commented Aug 25, 2017 at 10:25
  • $\begingroup$ I am looking for a single theorem and you recommend me an entire book ;) ... $\endgroup$
    – Zinkin
    Commented Aug 25, 2017 at 10:49
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    $\begingroup$ It seems that the answer to your question literally as written ("Is there a comprehensive theorem …") is just "no", but that is not a very useful answer; and yet you seem to be responding to attempts to give a more useful answer by objecting that they aren't comprehensive theorems. (It's not other people's responsibility to do the work of reading books and extracting statements of sufficient generality to satisfy you!) $\endgroup$
    – LSpice
    Commented Aug 25, 2017 at 19:53
  • $\begingroup$ @LSpice I am sorry for making that impression, I just wanted to be sure that it is really a rather implicit concept since this is the main point why I asked here. You see, I suppose if you are not working in the field of nonlinear PDEs you may ask yourself at some point: Is this a concept that I should be interested in (because compactness is of course one of the strongest tools at hand) and apparently the answer to this one is "almost certainly this concept will not be too relevant". $\endgroup$
    – Zinkin
    Commented Aug 25, 2017 at 21:33

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This is really just a longwinded comment.

The earliest instances of concentrated compactness that I know of are for geometric questions such as existence of energy-minimizing harmonic maps (as studied by Sacks and Uhlenbeck), the Yamabe problem (as studied by Trudinger, Aubin, and Schoen), and the existence of self-dual Yang-Mills connections (as studied by Uhlenbeck and Taubes), which all can be cast as nonlinear elliptic PDEs. They can also be formulated as variational problems. The fact that these arise from geometry means that the energy functional has symmetries, including scale invariance, which leads to the observation that the energy functional is critical in the sense that compactness just barely fails when studying a minimizing sequence.

The beautiful and deep insight that I believe was due originally to Sacks and Uhlenbeck is that when compactness just barely fails for a minimizing sequence, one can actually understand quite precisely how compactness fails. In the cases cited above, the local energy (the integrand of the energy functional) can concentrate at a discrete set of points (but not on any larger set) and if one rescales around each point, the limit, often called a "bubble", is a symmetric solution to the original problem. This work led to, for example, new theorems in the differential topology of $4$-manifolds, starting with Donaldson's theorem.

I believe Pierre Louis Lions was the first to formulate all of this into a general principle that he called concentrated compactness and that could be applied to a broad class of nonlinear elliptic PDEs. I'm not familiar with his work, but I think he did state and prove general theorems. I do not know whether his theorems imply the geometric results mentioned above. In general, when studying nonlinear PDEs, it is very hard to formulate a general theorem that can be applied directly to different interesting applications. Usually, the best you can do is to formulate a general principle like concentrated compactness and adapt it to specific situations.

Based on the original examples and formulation, it appeared that concentrated compactness could not be used for non-elliptic PDEs. However, as Tao describes, concentrated compactness, suitably formulated, has indeed proved to be a powerful tool even for nonlinear wave equations. But a precise statement of a more abstract version that encompasses everything is probably even harder to formulate precisely than the original elliptic version.

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    $\begingroup$ The criticality of the energy is actually not an issue. In practice, the lack of compactness is due to the translation invariance of the problem at hand. For instance, P.-L. Lions and I. Catto have proved, using CC, the existence of a ground state for neutral systems of elementary particles, in the Hartree and Thomas-Fermi models. $\endgroup$
    – Denis Serre
    Commented Aug 25, 2017 at 20:43
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    $\begingroup$ Denis, you're right. Concentrated compactness usually appears when the loss of compactness is due to an invariance under a noncompact group action, whether it be rescaling or translations. When it's rescaling, then the loss of compactness causes the energy functional to be critical, and it's studied using blow up arguments. When it's translation invariance, the energy functional does not have to be critical. $\endgroup$
    – Deane Yang
    Commented Aug 25, 2017 at 22:41
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    $\begingroup$ Deane, I am on the same tune as you. $\endgroup$
    – Denis Serre
    Commented Aug 26, 2017 at 8:17
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Well, I think you have to accept that concentration compactness is concept rather than a result. The intro of the mentioned book starts with

The subject of this book, concentration compactness, is a method for establishing convergence, in functional spaces, of sequences that are not a priori located in a compact set.

If you accept that there is no theorem that captures the concept and don't want a whole book, you should read the explanation on concentration compactness here (longer than a theorem, but shorter than a book).

A theorem that may come close to what you want is Theorem 3.1 (page 62) of said book. The basic notion of space is "dislocation space" which is a Hilbert space together with a set of bounded linear operators with certain properties…

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  • $\begingroup$ My question is: To which spaces/sets can this concept be applied so that everything works out? Neither Tao's $l^1$ discussion nor your quote seem to provide an answer to that question. Could you please elaborate on that point? All I want is a discussion of this concept that is not an example. $\endgroup$
    – Zinkin
    Commented Aug 25, 2017 at 12:08
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    $\begingroup$ But such a discussion is exactly what Tao does right before the $l^1$ example starts. If this is not the direction you want, I have no idea what you are looking for. My guess is that the answer to "To which spaces/sets can this concept be applied so that everything works out?" is something like: "You have to check in every case and it may get cumbersome…" $\endgroup$
    – Dirk
    Commented Aug 25, 2017 at 12:38
  • $\begingroup$ thank you, I did not want to bring heat into this debate, I was just curious to understand how far developed this concept is. $\endgroup$
    – Zinkin
    Commented Aug 25, 2017 at 21:27
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    $\begingroup$ I think you may find Gowers' "two cultures" distinction to be useful here: dpmms.cam.ac.uk/~wtg10/2cultures.pdf . Like most other useful things in the analysis of PDE, concentration compactness is definitely a concept coming from the "problem solving" culture of mathematics rather than the "theory building" culture. $\endgroup$
    – Terry Tao
    Commented Aug 26, 2017 at 2:53
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    $\begingroup$ See also Klainerman's essay "PDE as a unified subject", explaining how partial differential equations are best approached not from the perspective of maximum generality, but rather from studying families of PDE clustered around fundamental examples (e.g. heat equation, wave equation, etc..). The same is often true for the tools used in the analysis of PDE. web.math.princeton.edu/~seri/homepage/papers/telaviv.pdf $\endgroup$
    – Terry Tao
    Commented Aug 26, 2017 at 2:59

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