Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global well-posedness and scattering for the higher-dimensional
  energy-critical non-linear Schrodinger equation for radial data   In any dimension $n \geq 3$, we show that spherically symmetric bounded
energy solutions of the defocusing energy-critical non-linear Schr\"odinger
equation $i u_t + \Delta u = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist
globally and scatter to free solutions; this generalizes the three and four
dimensional results of Bourgain and Grillakis. Furthermore we have bounds on
various spacetime norms of the solution which are of exponential type in the
energy, which improves on the tower-type bounds of Bourgain. In higher
dimensions $n \geq 6$ some new technical difficulties arise because of the very
low power of the non-linearity.

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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: eess