Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Concentration analysis and cocompactness   Loss of compactness that occurs in may significant PDE settings can be
expressed in a well-structured form of profile decomposition for sequences.
Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where
$X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a
set of surjective isometries on both $X$ and $Y$. A profile decomposition is a
representation of a bounded sequence in $X$ as a sum of elementary
concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that
vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any
sequence in $X$ that develops no $D$-concentrations has a subsequence
convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this
property is called $D$-cocompact, a property weaker than, but related to,
compactness. We survey known cocompact imbeddings and their role in profile
decompositions.

29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
18
The categories of this paper are: econ