Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Small data global existence and decay for two dimensional wave maps   We prove the small-data global existence for the wave-map equation on
$\mathbb{R}^{1,2}$ using a variant of the vector field method. The main
innovations lie in the introduction of two new linear estimates. First is the
control of the dispersive decay of the solution $\phi$ itself (as opposed to
its derivatives), via a logarithmic weighted Hardy inequality. This control has
not been previously established using purely physical space methods in two
spatial dimensions. Second is a point-wise decay estimate for a twisted
derivative of $\phi$ associated to the Morawetz $K$ multiplier, that cannot be
reduced to point-wise decay estimates associated to the standard commutator
vector fields. As the linear theory is largely similar between dimensions, and
in view of the novelty of the second innovation even in higher dimensions, we
include a discussion of the method for $\mathbb{R}^{1,d}$ in general. Both
linear estimates are used crucially in our study: the control of the wave-map
equation in the small data regime necessarily requires understanding the
dispersive behavior of the bare solution $\phi$ itself, by virtue of the
equation it satisfies. The point-wise decay for the twisted derivative allows
us to avoid certain top-order logarithmic energy growths; this is indispensable
for extending our argument from the case of compactly supported initial data
(to which our methods are most naturally adapted) to initial data that are
strongly localized but not necessarily of compact support, via an iterative
construction.

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The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin