Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global well-posedness and scattering for the defocusing, energy
  -critical, nonlinear Schr{\"o}dinger equation in the exterior of a convex
  obstacle when $d = 4$   In this paper we prove that the energy - critical nonlinear Schr{\"o}dinger
equation in the domain exterior to a convex obstacle is globally well - posed
and scattering for initial data having finite energy. To prove this we utilize
frequency localized Morawetz estimates adapted to an exterior domain.

22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs