Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved decay rates with small regularity loss for the wave equation
  about a Schwarzschild black hole   We continue our study of the decoupled wave equation in the exterior of a
spherically symmetric, Schwarzschild, black hole. Because null geodesics on the
photon sphere orbit the black hole, extra effort must be made to show that the
high angular momentum components of a solution decay sufficiently fast,
particularly for low regularity initial data. Previous results are rapid decay
for regular ($H^3$) initial data \cite{BSterbenz} and slower decay for rough
($H^{1+\epsilon}$) initial data \cite{BlueSoffer3}. Here, we combine those
methods to show boundedness of the conformal charge. From this, we conclude
that there are bounds for global in time, space-time norms, in particular
  \int_I |\tilde\phi|^4 d^4vol < C for $H^{1+\epsilon}$ initial data with
additional decay towards infinite and the bifurcation sphere. Here $\tilde\phi$
refers to a solution of the wave equation. $I$ denotes the exterior region of
the Schwarzschild solution, which can be expressed in coordinates as $r>2M$,
$t\in\Reals, \omega\in S^2$, and $d^4\text{vol}$ is the natural 4-dimensional
volume induced by the Schwarzschild pseudo-metric. We also demonstrate that the
photon sphere has the same influence on the wave equation as a closed geodesic
has on the wave equation on a Riemannian manifold. We demonstrate this
similarity by extending our techniques to the wave equation on a class of
Riemannian manifolds. Under further assumptions, the space-time estimates are
sufficient to prove global bounds for small data, nonlinear wave equations on a
class of Riemannian manifolds with closed geodesics. We must use global,
space-time integral estimates since $L^\infty$ estimates cannot hold at this
level of regularity.

20
The categories of this paper are: acc-phys
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: gr-qc
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
3
The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci