Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global well-posedness and polynomial bounds for the defocusing
  $L^{2}$-critical nonlinear Schr\"odinger equation in $\R$   We prove global well-posedness for low regularity data for the one
dimensional quintic defocusing nonlinear Schr\"odinger equation. Precisely we
show that a unique and global solution exists for initial data in the Sobolev
space $H^{s}(\mathbb R)$ for any $s>{1/3}$. This improves the result in
\cite{tz}, where global well-posedness was established for any $s>{4/9}$. We
use the $I$-method to take advantage of the conservation laws of the equation.
The new ingredient in our proof is an interaction Morawetz estimate for the
smoothed out solution $Iu$. As a byproduct of our proof we also obtain that the
$H^{s}$ norm of the solution obeys polynomial-in-time bounds.

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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat