401 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
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 Īµ or removes an edge carrying only an Īµ-fraction of flow. We study how the equilibrium responds to such an Īµ-change.
Our first surprising finding is that, even for linear latency functions, for every Īµ>ā0, there are networks in which an Īµ-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 Īµ.
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 Īµ-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1ā+āĪµ) 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
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
On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics
Using the theory of hyperbolic systems we put in perspective the
mathematical and geometrical structure of the celebrated circularly polarized
waves solutions for isotropic hyperelastic materials determined by Carroll in
Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this
class of solutions yields an infinite family of \emph{linear} solutions for the
equations of isotropic elastodynamics. Moreover, we determine a huge class of
hyperbolic partial differential equations having the same property of the shear
wave system. Restricting the attention to the usual first order asymptotic
approximation of the equations determining transverse waves we provide the
complete integration of this system using generalized symmetries.Comment: 19 page
Weak solutions to problems involving inviscid fluids
We consider an abstract functional-differential equation derived from the
pressure-less Euler system with variable coefficients that includes several
systems of partial differential equations arising in the fluid mechanics. Using
the method of convex integration we show the existence of infinitely many weak
solutions for prescribed initial data and kinetic energy
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
Odd-parity perturbations of self-similar Vaidya spacetime
We carry out an analytic study of odd-parity perturbations of the
self-similar Vaidya space-times that admit a naked singularity. It is found
that an initially finite perturbation remains finite at the Cauchy horizon.
This holds not only for the gauge invariant metric and matter perturbation, but
also for all the gauge invariant perturbed Weyl curvature scalars, including
the gravitational radiation scalars. In each case, `finiteness' refers to
Sobolev norms of scalar quantities on naturally occurring spacelike
hypersurfaces, as well as pointwise values of these quantities.Comment: 28 page
Hyperbolic Balance Laws with a Non Local Source
This paper is devoted to hyperbolic systems of balance laws with non local
source terms. The existence, uniqueness and Lipschitz dependence proved here
comprise previous results in the literature and can be applied to physical
models, such as Euler system for a radiating gas and Rosenau regularization of
the Chapman-Enskog expansion.Comment: 26 page
Time--Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System
Several singular limits are investigated in the context of a
system arising for instance in the modeling of chromatographic processes. In
particular, we focus on the case where the relaxation term and a
projection operator are concentrated on a discrete lattice by means of Dirac
measures. This formulation allows to study more easily some time-splitting
numerical schemes
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