788 research outputs found
Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space
We provide a uniform decay estimate of Morawetz type for the local energy of
general solutions to the inhomogeneous wave equation on a Schwarzchild
background. This estimate is both uniform in space and time, so in particular
it implies a uniform bound on the sup norm of solutions which can be given in
terms of certain inverse powers of the radial and advanced/retarded time
coordinate variables. As a model application, we show these estimates give a
very simple proof small amplitude scattering for nonlinear scalar fields with
higher than cubic interactions.Comment: 24 page
Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes
In this paper, we provide an elementary, unified treatment of two distinct
blue-shift instabilities for the scalar wave equation on a fixed Kerr black
hole background: the celebrated blue-shift at the Cauchy horizon (familiar from
the strong cosmic censorship conjecture) and the time-reversed red-shift at the
event horizon (relevant in classical scattering theory).
Our first theorem concerns the latter and constructs solutions to the wave
equation on Kerr spacetimes such that the radiation field along the future
event horizon vanishes and the radiation field along future null infinity
decays at an arbitrarily fast polynomial rate, yet, the local energy of the
solution is infinite near any point on the future event horizon. Our second
theorem constructs solutions to the wave equation on rotating Kerr spacetimes
such that the radiation field along the past event horizon (extended into the
black hole) vanishes and the radiation field along past null infinity decays at
an arbitrarily fast polynomial rate, yet, the local energy of the solution is
infinite near any point on the Cauchy horizon.
The results make essential use of the scattering theory developed in [M.
Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, A scattering theory for the
wave equation on Kerr black hole exteriors, preprint (2014) available at
\url{http://arxiv.org/abs/1412.8379}] and exploit directly the time-translation
invariance of the scattering map and the non-triviality of the transmission
map.Comment: 26 pages, 12 figure
Marginally trapped tubes generated from nonlinear scalar field initial data
We show that the maximal future development of asymptotically flat
spherically symmetric black hole initial data for a self-gravitating nonlinear
scalar field, also called a Higgs field, contains a connected, achronal
marginally trapped tube which is asymptotic to the event horizon of the black
hole, provided the initial data is sufficiently small and decays like
O(r^{-1/2}), and the potential function V is nonnegative with bounded second
derivative. This result can be loosely interpreted as a statement about the
stability of `nice' asymptotic behavior of marginally trapped tubes under
certain small perturbations of Schwarzschild.Comment: 25 pages, 4 figures. Updated to agree with published version; small
but important error in the proof of the main theorem fixed, outline of proof
added in Section 2.5, minor expository change
Sensitivity of wardrop equilibria
We study the sensitivity of equilibria in the well-known game theoretic traffic model due to Wardrop. We mostly consider single-commodity networks. Suppose, given a unit demand flow at Wardrop equilibrium, one increases the demand by Δ or removes an edge carrying only an Δ-fraction of flow. We study how the equilibrium responds to such an Δ-change.
Our first surprising finding is that, even for linear latency functions, for every Δ>â0, there are networks in which an Δ-change causes every agent to change its path in order to recover equilibrium. Nevertheless, we can prove that, for general latency functions, the flow increase or decrease on every edge is at most Δ.
Examining the latency at equilibrium, we concentrate on polynomial latency functions of degree at most p with nonnegative coefficients. We show that, even though the relative increase in the latency of an edge due to an Δ-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1â+âΔ) p . The increase of the price of anarchy is shown to be upper bounded by the same factor. Both bounds are shown to be tight.
Let us remark that all our bounds are tight. For the multi-commodity case, we present examples showing that neither the change in edge flows nor the change in the path latency can be bounded
Redistribution of damping in hyperbolic systems of conservation laws
By a change of variables that redistributes damping among the equations of systems of balance laws in one space dimension, it is demonstrated that dissipation induced by friction, viscosity or other physical mechanism, manifested in the decay of âentropyâ functionals of quadratic order, also controls the total variation of solutions
Inextendibility of expanding cosmological models with symmetry
A new criterion for inextendibility of expanding cosmological models with
symmetry is presented. It is applied to derive a number of new results and to
simplify the proofs of existing ones. In particular it shows that the solutions
of the Einstein-Vlasov system with symmetry, including the vacuum
solutions, are inextendible in the future. The technique introduced adds a
qualitatively new element to the available tool-kit for studying strong cosmic
censorship.Comment: 7 page
Regularity results for the spherically symmetric Einstein-Vlasov system
The spherically symmetric Einstein-Vlasov system is considered in
Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem
is the issue of global existence for initial data without size restrictions.
The main purpose of the present work is to propose a method of approach for
general initial data, which improves the regularity of the terms that need to
be estimated compared to previous methods. We prove that global existence holds
outside the centre in both these coordinate systems. In the Schwarzschild case
we improve the bound on the momentum support obtained in \cite{RRS} for compact
initial data. The improvement implies that we can admit non-compact data with
both ingoing and outgoing matter. This extends one of the results in
\cite{AR1}. In particular our method avoids the difficult task of treating the
pointwise matter terms. Furthermore, we show that singularities never form in
Schwarzschild time for ingoing matter as long as This removes an
additional assumption made in \cite{A1}. Our result in maximal-isotropic
coordinates is analogous to the result in \cite{R1}, but our method is
different and it improves the regularity of the terms that need to be estimated
for proving global existence in general.Comment: 25 pages. To appear in Ann. Henri Poincar\'
Stability and Instability of Extreme Reissner-Nordstr\"om Black Hole Spacetimes for Linear Scalar Perturbations I
We study the problem of stability and instability of extreme
Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we
consider solutions to the linear wave equation on a suitable globally
hyperbolic subset of such a spacetime, arising from regular initial data
prescribed on a Cauchy hypersurface crossing the future event horizon. We
obtain boundedness, decay and non-decay results. Our estimates hold up to and
including the horizon. The fundamental new aspect of this problem is the
degeneracy of the redshift on the event horizon. Several new analytical
features of degenerate horizons are also presented.Comment: 37 pages, 11 figures; published version of results contained in the
first part of arXiv:1006.0283, various new results adde
Stability of Transonic Characteristic Discontinuities in Two-Dimensional Steady Compressible Euler Flows
For a two-dimensional steady supersonic Euler flow past a convex cornered
wall with right angle, a characteristic discontinuity (vortex sheet and/or
entropy wave) is generated, which separates the supersonic flow from the gas at
rest (hence subsonic). We proved that such a transonic characteristic
discontinuity is structurally stable under small perturbations of the upstream
supersonic flow in . The existence of a weak entropy solution and Lipschitz
continuous free boundary (i.e. characteristic discontinuity) is established. To
achieve this, the problem is formulated as a free boundary problem for a
nonstrictly hyperbolic system of conservation laws; and the free boundary
problem is then solved by analyzing nonlinear wave interactions and employing
the front tracking method.Comment: 26 pages, 3 figure
Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes
We initiate the study of the spherically symmetric Einstein-Klein-Gordon
system in the presence of a negative cosmological constant, a model appearing
frequently in the context of high-energy physics. Due to the lack of global
hyperbolicity of the solutions, the natural formulation of dynamics is that of
an initial boundary value problem, with boundary conditions imposed at null
infinity. We prove a local well-posedness statement for this system, with the
time of existence of the solutions depending only on an invariant H^2-type norm
measuring the size of the Klein-Gordon field on the initial data. The proof
requires the introduction of a renormalized system of equations and relies
crucially on r-weighted estimates for the wave equation on asymptotically AdS
spacetimes. The results provide the basis for our companion paper establishing
the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this
system.Comment: 50 pages, v2: minor changes, to appear in Annales Henri Poincar\'
- âŠ