2,475 research outputs found
Energy decay for solutions of the wave equation with general memory boundary conditions
We consider the wave equation in a smooth domain subject to Dirichlet
boundary conditions on one part of the boundary and dissipative boundary
conditions of memory-delay type on the remainder part of the boundary, where a
general borelian measure is involved. Under quite weak assumptions on this
measure, using the multiplier method and a standard integral inequality we show
the exponential stability of the system.
Some examples of measures satisfying our hypotheses are given, recovering and
extending some of the results from the literature.Comment: 14 pages, submitted to Diff. Int. Eq
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress in characteristic
evolution is traced from the early stage of 1D feasibility studies to 2D
axisymmetric codes that accurately simulate the oscillations and gravitational
collapse of relativistic stars and to current 3D codes that provide pieces of a
binary black hole spacetime. Cauchy codes have now been successful at
simulating all aspects of the binary black hole problem inside an artificially
constructed outer boundary. A prime application of characteristic evolution is
to extend such simulations to null infinity where the waveform from the binary
inspiral and merger can be unambiguously computed. This has now been
accomplished by Cauchy-characteristic extraction, where data for the
characteristic evolution is supplied by Cauchy data on an extraction worldtube
inside the artificial outer boundary. The ultimate application of
characteristic evolution is to eliminate the role of this outer boundary by
constructing a global solution via Cauchy-characteristic matching. Progress in
this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note:
updated version of arXiv:gr-qc/050809
Control and stabilization of waves on 1-d networks
We present some recent results on control and stabilization of waves on 1-d networks.The fine time-evolution of solutions of wave equations on networks and, consequently, their control theoretical properties, depend in a subtle manner on the topology of the network under consideration and also on the number theoretical properties of the lengths of the strings entering in it. Therefore, the overall picture is quite complex.In this paper we summarize some of the existing results on the problem of controllability that, by classical duality arguments in control theory, can be reduced to that of observability of the adjoint uncontrolled system. The problem of observability refers to that of recovering the total energy of solutions by means of measurements made on some internal or external nodes of the network. They lead, by duality, to controllability results guaranteeing that L 2-controls located on those nodes may drive sufficiently smooth solutions to equilibrium at a final time. Most of our results in this context, obtained in collaboration with R. DĂĄger, refer to the problem of controlling the network from one single external node. It is, to some extent, the most complex situation since, obviously, increasing the number of controllers enhances the controllability properties of the system. Our methods of proof combine sidewise energy estimates (that in the particular case under consideration can be derived by simply applying the classical d'Alembert's formula), Fourier series representations, non-harmonic Fourier analysis, and number theoretical tools.These control results belong to the class of the so-called open-loop control systems.We then discuss the problem of closed-loop control or stabilization by feedback. We present a recent result, obtained in collaboration with J. Valein, showing that the observability results previously derived, regardless of the method of proof employed, can also be recast a posteriori in the context of stabilization, so to derive explicit decay rates (as) for the energy of smooth solutions. The decay rate depends in a very sensitive manner on the topology of the network and the number theoretical properties of the lengths of the strings entering in it.In the end of the article we also present some challenging open problems
On Stability of Hyperbolic Thermoelastic Reissner-Mindlin-Timoshenko Plates
In the present article, we consider a thermoelastic plate of
Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising
from Cattaneo's law. In the absense of any additional mechanical dissipations,
the system is often not even strongly stable unless restricted to the
rotationally symmetric case, etc. We present a well-posedness result for the
linear problem under general mixed boundary conditions for the elastic and
thermal parts. For the case of a clamped, thermally isolated plate, we show an
exponential energy decay rate under a full damping for all elastic variables.
Restricting the problem to the rotationally symmetric case, we further prove
that a single frictional damping merely for the bending compoment is sufficient
for exponential stability. To this end, we construct a Lyapunov functional
incorporating the Bogovski\u{i} operator for irrotational vector fields which
we discuss in the appendix.Comment: 27 page
Energy decay rate of a transmission system governed by degenerate wave equation with drift and under heat conduction with memory effect
In this paper, we investigate the stabilization of transmission problem of
degenerate wave equation and heat equation under Coleman-Gurtin heat conduction
law or Gurtin-Pipkin law with memory effect. We investigate the polynomial
stability of this system when employing the Coleman-Gurtin heat conduction,
establishing a decay rate of type . Next, we demonstrate exponential
stability in the case when Gurtin-Pipkin heat conduction is applied
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Mathematical Aspects of General Relativity
Mathematical relativity, the subject of this conference, has recently become more and more devoted to the theory of nonlinear evolution equations, with global questions becoming ever more accessible. This is reïŹected by the fact that more than half of the talks given were concerned with the global dynamics of solutions of evolution equations related more or less directly to the Einstein equations of general relativity. Progress was reported in understanding subjects such as black holes, gravitational radiation, cosmology and the relation of general relativity to Newtonian gravitational theory
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Nonlinear Waves and Dispersive Equations
The aim of the workshop was to discuss current developments in nonlinear waves and dispersive equations from a PDE based view. The talks centered around rough initial data, long time and global existence, perturbations of special solutions, and applications
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