Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global existence and uniqueness results for weak solutions of the
  focusing mass-critical non-linear Schr\"odinger equation   We consider the focusing mass-critical NLS $iu_t + \Delta u = - |u|^{4/d} u$
in high dimensions $d \geq 4$, with initial data $u(0) = u_0$ having finite
mass $M(u_0) = \int_{\R^d} |u_0(x)|^2 dx < \infty$. It is well known that this
problem admits unique (but not global) strong solutions in the Strichartz class
$C^0_{t,\loc} L^2_x \cap L^2_{t,\loc} L^{2d/(d-2)}_x$, and also admits global
(but not unique) weak solutions in $L^\infty_t L^2_x$. In this paper we
introduce an intermediate class of solution, which we call a
\emph{semi-Strichartz class solution}, for which one does have global existence
and uniqueness in dimensions $d \geq 4$. In dimensions $d \geq 5$ and assuming
spherical symmetry, we also show the equivalence of the Strichartz class and
the strong solution class (and also of the semi-Strichartz class and the
semi-strong solution class), thus establishing ``unconditional'' uniqueness
results in the strong and semi-strong classes. With these assumptions we also
characterise these solutions in terms of the continuity properties of the mass
function $t \mapsto M(u(t))$.

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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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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-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
0
The categories of this paper are: hep-ph