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?



Concentration compactness. Can this concept be stated in a theorem?
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.


