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Energy decay for the damped wave equation under a pressure condition, preprint 2009
"... Abstract. We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we s ..."
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Abstract. We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis. hal00415529, version 1 10 Sep 2009 1.
SPECTRAL THEORY OF DAMPED QUANTUM CHAOTIC SYSTEMS
"... Abstract. We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov (very chaotic). The final objective is to obtain conditions (in terms of the geodesic flow ..."
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Abstract. We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov (very chaotic). The final objective is to obtain conditions (in terms of the geodesic flow on X, the structure of the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. The spectrum of the equation amounts to a nonselfadjoint spectral problem. Using semiclassical methods, we derive estimates and upper bounds for the high frequency spectral distribution, in terms of dynamically defined quantities, like the value distribution of the timeaveraged damping. We also consider the toy model of damped quantized chaotic maps, for which we derive similar estimates, as well as a new upper bound for the spectral radius depending on the set of minimally damped trajectories. Contents
EIGENVALUE ASYMPTOTICS, INVERSE PROBLEMS AND A TRACE FORMULA FOR THE LINEAR DAMPED WAVE EQUATION
, 905
"... Abstract. We determine the general form of the asymptotics for Dirichlet eigenvalues of the one–dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a t ..."
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Abstract. We determine the general form of the asymptotics for Dirichlet eigenvalues of the one–dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem. 1.
France.
"... The present study is concerned with the properties of 2D shallow cavities having an irregular boundary. The eigenmodes are calculated numerically on various examples, and it is shown first that, whatever the shape and characteristic sizes of the boundary, irregularity always induces an increase of l ..."
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The present study is concerned with the properties of 2D shallow cavities having an irregular boundary. The eigenmodes are calculated numerically on various examples, and it is shown first that, whatever the shape and characteristic sizes of the boundary, irregularity always induces an increase of localized eigenmodes and a global decrease of the existence surface of the eigenmodes. Besides, irregular cavities are shown to exhibit specific damping properties. As expected, the increased damping, compared to a regular cavity, is related first to the larger perimeter to surface ratio. But more interestingly, there is an enhancement of the dissipation for those modes that are localized near the boundary, modes which are favoured by the geometrical irregularity. PACS numbers: 43.20.Ks, 43.20.Hq, 46.40.Ff, 47.53.+n 2 I.
EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS
"... Abstract. We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of βdamped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperboli ..."
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Abstract. We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of βdamped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of βdamped trajectories of the geodesic flow. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues below the real axis, under a pressure condition on the set of undamped trajectories. hal00673138, version 2 12 Oct 2012 1.