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
  $\dot{H}^{\frac{1}{2}}$-critical nonlinear Schr\"odinger equation in
  $\mathbb{R}^2$   In this paper we consider the Cauchy initial value problem for the defocusing
quintic nonlinear Schr\"odinger equation in $\mathbb{R}^2$ with general data in
the critical space $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$. We show that if a
solution remains bounded in $\dot{H}^{\frac{1}{2}} (\mathbb{R}^2)$ in its
maximal interval of existence, then the interval is infinite and the solution
scatters.

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The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin