Haut PDF Energy decay for the damped wave equation under a pressure condition

Energy decay for the damped wave equation under a pressure condition

Energy decay for the damped wave equation under a pressure condition

PRESSURE CONDITION EMMANUEL SCHENCK 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 dy- namical 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.
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Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric

Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric

From our point of view one of the most efficient approaches to the question of local en- ergy decay is the theory of resonances. Resonances correspond to the frequencies and rates of dumping of signals emitted by the black hole in the presence of perturbations (see [9, Chapter 4.35]). On the one hand these resonances are today an important hope of effectively detecting the presence of a black hole as we are theoretically able to measure the corresponding gravi- tational waves. On the other hand, the distance of the resonances to the real axis reflects the stability of the system under the perturbation: larger distances correspond to more stability. In particular the knowledge of the localization of resonances gives precise informations about the decay of the local energy and its rate. The aim of the present paper is to show how this method applies to the simplest model of a black hole: the De Sitter–Schwarzschild black hole. In the euclidean space, such results are already known, especially for non trapping geome- tries. The first result is due to Lax and Phillips (see their book [15, Theorem III.5.4]). They have proved that the cut-off propagator associated to the wave equation outside an obstacle in odd dimension ≥ 3 (more precisely the Lax–Phillips semi-group Z(t)) has an expansion in terms of resonances if Z(T ) is compact for a given T . In particular, there is a uniform exponential decay of the local energy. From Melrose–Sj¨ ostrand [18], this assumption is true
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DECAY OF LOCAL ENERGY FOR SOLUTIONS OF THE FREE SCHRÖDINGER EQUATION IN EXTERIOR DOMAINS

DECAY OF LOCAL ENERGY FOR SOLUTIONS OF THE FREE SCHRÖDINGER EQUATION IN EXTERIOR DOMAINS

Since the early works of Lax and Phillips [18, 21] on exterior Dirichlet problem for the wave operator ∂ t 2 − ∆, it is well known that geometry of the boundary plays a crucial role in the time decay properties of the solution whether it is trapping or not. If the boundary is not trapping, it is known [22] that local energy decays at exponential rate. For the same problem, under very general trapping assumptions, N. Burq proved [3] that there exists an exponentially small neighbourhood of the energy real axis, free of resonances, which implies a logarithmic decay rate of local energy.
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Exponential decay for the Schrödinger equation on a dissipative waveguide

Exponential decay for the Schrödinger equation on a dissipative waveguide

We know from [ Har85 , Leb96 ] that a weak assumption on the absorption index a (for instance the dissipation is effective on any open subset of the domain) is enough to ensure that the energy goes to 0 for any initial datum. Uniform exponential decay has been obtained in [ RT74 , BLR92 ] under the now usual geomet- ric control condition. Roughly speaking, the assumption is that any (generalized) bicharacteristic (or classical trajectory, or ray of geometric optics) meets the damping region (in the interior of the domain or at the boundary). For the free wave equation on a subset of R d , the spatial
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CONCENTRATION OF LAPLACE EIGENFUNCTIONS AND STABILIZATION OF WEAKLY DAMPED WAVE EQUATION

CONCENTRATION OF LAPLACE EIGENFUNCTIONS AND STABILIZATION OF WEAKLY DAMPED WAVE EQUATION

It is known that as soon as the damping b ≥ 0 is non trivial, the energy of every solution converge to 0 as t tends to infinity. On the other hand the rate of decay is uniform (and hence exponential) in energy space if and only if the geometric control condition [ 2 , 5 ] is satisfied. Here we want to explore the question when some trajectories are trapped and exhibit decay rates (assuming more regularity on the initial data). This latter question was previously studied in a general setting in [ 19 ] and on tori in [ 11 , 21 , 1 ] (see also [ 12 , 13 ]) and more recently by Leautaud-Lerner [ 18 ]. According to the works by Borichev-Tomilov [ 3 ], stabilization results for the wave equation are equivalent to resolvent estimates. On the other hand, Theorem 1.1 implies easily (see Section 2.2 ) the following resolvent estimate
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Quasimodes and a lower bound for the local energy decay of the Dirac equation in Schwarzschild-Anti-de Sitter spacetime

Quasimodes and a lower bound for the local energy decay of the Dirac equation in Schwarzschild-Anti-de Sitter spacetime

This limit proves that, along radial null geodesic, a particle goes to timelike infinity in finite Boyer-Lindquist time (recall that along these geodesic, t − r ∗ and t + r ∗ are constants). This geometric property will lead to several differences compared to other spacetime. In the massless case, we would have to put boundary condition. In fact, even in the massive case, a confining potential appear near the Anti-de Sitter infinity. Combined with the usual bump created by the photon sphere, we expect that the energy will not decay really fast. In fact, for the wave equation, the local energy is decaying only logarithmically as was proven by G. Holzegel and J. Smulevici [24], [26]. In this work, we obtain a logarithmic lower bound as in [26].
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Complexity and regularity of maximal energy domains for the wave equation with fixed initial data

Complexity and regularity of maximal energy domains for the wave equation with fixed initial data

In this paper we provide a complete mathematical analysis of the optimal design problem (P γ,T ). The article is structured as follows. In Section 2 (see in particular Theorem 2 ) we give a sufficient condition ensuring the existence and uniqueness of an optimal set. More precisely, we prove that, if the initial data under consideration belong to a suitable class of analytic functions, then there always exists a unique optimal domain ω, which has a finite number of connected components for any value of T > 0 but some isolated values. In Section 3 , we investigate the sharpness of the assumptions made in Theorem 2 . More precisely, in Theorem 3 we build initial data (y 0 , y 1 ) of
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Geometric control condition for the wave equation with a time-dependent observation domain

Geometric control condition for the wave equation with a time-dependent observation domain

A glance at inequality ( 1 ), shows that observability is in fact to be understood as occuring in a space-time domain, here (0, T ) × ω. It is then natural to wonder if observability can hold if it is replaced by some other open subset of (0, T ) × Ω. This is the subject of the present article. The motivation for such a study can be seen as fairly theoretical. However, in practical issues, in different industrial contexts, for nondestructive testing, safety applications, as well as tomography techniques used for imaging bodies (human or not), this question becomes quite relevant. In fact, the industrial framework of seismic exploration was the original motivation for this work. In the different fields we mentionned, data are collected to be exploited in an interpretation step which involves the solution of some inverse problem. The point is that the device used to collect data does not fit well with the usual geometric condition which is crucial to obtain an observability result. In some cases it appears of great interest to be able to tackle situations where the observation set is time-dependent. In others, the reduction of data volume may be sought, while preserving the data quality. One may also face a situation in which all sensors cannot be active at the same time. The example of seismic data acquisition can help the reader get a grasp on the industrial need to better design data acquisition procedures. In the case of a towed marine seismic data acquisition campaign, a typical setup consists in six parallel streamers with length 6000 m, separated by a distance of 100 m, floating at a depth of 8 m. The basic receiving elements are pressure sensitive hydrophones composed of piezoelectric ceramic crystal devices that are placed some 20 to 50 m apart along each streamer. A source (a carefully designed air gun array) is shot every 25 m while the boat moves. The seismic data experiment lasts around 8 s. One understands with this description that a huge amount (terabytes) of data is recorded during one such acquisition campaign above 1 In fact, intermediate decay rates have been established in particular geometrical settings, see for instance [ 21 ]
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en fr Decay of semilinear damped wave equations: cases without geometric control condition Décroissance des équations des ondes amorties semilinéaires : des cas sans la condition de contrôle géométrique

tractor in the sense that there is an invariant compact set A ⊂ H 0 1 (Ω) ×L 2 (Ω), which consists of all the bounded trajectories and such that any regular set B bounded in (H 2 (Ω) ∩H 0 1 (Ω)) ×H 0 1 (Ω) is attracted by A in the topology of H 0 1 (Ω) ×L 2 (Ω). Notice that this concept of weak attractor is the one of Babin and Vishik in [3]. At this time, the asymptotic compactness property of the semilinear damped wave equation was not discovered and people thought that a strong attractor (attracting bounded sets of H 0 1 (Ω) × L 2 (Ω)) was impossible due to the lack of regularization property for the damped wave equation. Few years later, Hale [17] and Haraux [19] obtain this asymptotic compactness property and the existence of a strong attractor. Thus this notion of weak attractor has been forgotten. It is noteworthy that it appears again here. Notice that we cannot hope a better attraction property since even in the linear case, {0} is not an attractor in the strong sense.
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EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS

EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS

There is a natural choice of Lagrangian states which is associated to the vertical bundle of the energy layer. These particular states were used by Anantharaman and Nonnenmacher in a selfadjoint setting [1, 4] and also by Schenck in [28] in the context of the damped wave equation. In these references, these Lagrangian states remain under control up to large logarithmic times, due to the global structure of the geodesic flow (it was supposed to be Anosov). Indeed, the Anosov hypothesis implied that the associated Lagrangian submanifolds become uniformly close to the unstable foliation and that they do not develop caustics under the evolution (thanks to the absence of conjugate points) — see [28, §4] for details.
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DECAY FOR THE KELVIN-VOIGT DAMPED WAVE EQUATION: PIECEWISE SMOOTH DAMPING

DECAY FOR THE KELVIN-VOIGT DAMPED WAVE EQUATION: PIECEWISE SMOOTH DAMPING

information of the quasi-modes restricted to the interface by transmission conditions. Then in Section 4, we prove the propagation theorem for the hyperbolic problem in Ω 2 which will lead to a contradiction. We need to analyze two semi-classical scales corresponding to the elliptic and hyperbolic region, connected by the transmission condition on the interface. Finally in Section 5, we construct a sequence of quasi-modes saturating the inequality ( 1.5 ) in a simple geometry. In particular this proves the optimality of the resolvent estimate. We collect various toolboxes in the final section of the appendix.
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Decay rates for the damped wave equation on the torus

Decay rates for the damped wave equation on the torus

We will now study the solutions of the quantization conditions (B.10) and (B.11), taking into account the relations (B.9) between the wavevectors k, k ′ and the energy E. To describe the full spectrum (which we plan to present in a separate publication), we would need to consider several r´egimes, depending on the relative scales of E and h. However, since we are only interested here in proving Proposition B.1, we will focus on the r´egime leading to the smallest possible values of | Im e ζ | = | Re z|. What characterizes the corresponding eigenmodes v(x) ? From (B.3) we see that the mass of v(x) in the damped region, 2 R σ 1/2 |v(x)| 2 dx, should be small compared to its full mass. Intuitively, if such a
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Exponential decay for the damped wave equation in unbounded domains

Exponential decay for the damped wave equation in unbounded domains

The first term of (6.8) is decaying exponentially fast and the integral term is compact since f is compactly supported in x and due to either the compact Sobolev embedding H 1 ,→ L 2p for p < d/(d − 2) or to more technical arguments based on the Strichartz estimates for p ∈ [d/(d − 2), (d + 2)/(d − 2)) (see [9] and see [16]). Thus, following the ideas of [9] and [16], we obtain that any solution is asymptotically compact and converges to a trajectory with constant energy. Now, we would like to show that the energy E associated to (6.5) is a Lyapounov function, that is that it is non-increasing and cannot be constant along a solution u(t), except of course if u(t) is an equilibrium point. If (6.5) admits a Lyapounov function, one says that the corresponding dynamical system is gradient. In particular, it cannot admit periodic orbits, homoclinic orbits. . . The gradient structure of (6.5), together with its asymptotic compactness, will also ensure the existence of a compact global attractor, that is a compact invariant set of X which attracts all the trajectories of (6.5). This set is a central object of the theory of dynamical systems. It contains all the solutions u(t), which exist for all t ∈ R and which are uniformly bounded in H 1 (R d ) × L 2 (R d ) for all t ∈ R (as equilibrium points, heteroclinic orbits etc.). See for example [13] and [31] for a review on the concepts of compact global attractors, of asymptotic compactness or of gradient structure.
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Dispersion for the wave equation outside a ball and counterexamples

Dispersion for the wave equation outside a ball and counterexamples

— I R is the reflected wave: the phase has a singular, Airy type term (easy to deal with). Since difficulties appear near rays issued from Q 0 which hit the boundary without being deviated, only the diffracted wave part (containing I D ) will be dealt with here. Notice that this is the regime which provides counter-examples in higher dimensions. We have I D (τ, Q, Q 0 ) =

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A minimum effort optimal control problem for the wave equation.

A minimum effort optimal control problem for the wave equation.

mizing the tracking error by means of a control which is pointwise as small as possible. The appearance of the L ∞ –control costs leads to nondifferentiability. The analytic and efficient numerical treatment of this nonsmooth problem by a semi-smooth Newton method stands in the focus of this work. We prove superlinear convergence of this iterative method and present numerical exam- ples.

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Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

focusing non-linear wave equation. To be published in Acta Math., 2006. [KV08] Rowan Killip and Monica Vi¸san. The focusing energy-critical nonlinear schr¨ odinger equation in dimen- sions five and higher. Preprint, 2008. [LS95] Hans Lindblad and Christopher D. Sogge. On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal., 130(2):357–426, 1995.

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Wave chaos for the Helmholtz equation

Wave chaos for the Helmholtz equation

TIME-RESPONSE) A. Spectral Rigidity ` a la Berry (diagonal approximation) 1. Form Factor In a complex wave system, either a chaotic cavity or a disordered medium, it is not possible to give a detailed description of the frequency spectrum by providing a determined sequence of numbers. Hence, the frequency spectrum is too complicated to be explained level by level but may nevertheless be studied through a statistical approach, in a way quite analog to the statistical approach of a gas of interacting particles. In the study of spectral properties, a key role is played by the spectral correlations and their description in terms of adequate quantities will be our concern in this section. With these quantities we will be in a position to establish how certain universal features predicted by RMT can be recovered from a global knowledge of chaotic dynamics but also in what respect some non-universal behavior can be related to the shortest POs of the system.
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Time-Domain BEM for the Wave Equation on Distributed-Heterogenous Architectures : a Blocking Approach

Time-Domain BEM for the Wave Equation on Distributed-Heterogenous Architectures : a Blocking Approach

The original formulation relies on the Sparse Matrix-Vector product (SpMV) which has been widely studied on CPU and GPU because this is an essential operation in many scientific ap- plications. Our work is not an optimization or an improvement for the general SpMV on GPU because we use a custom operator that matches our needs and which is at the cross of the SpMV and the general matrix-matrix product. Nevertheless, the optimizations of our implementation on GPU have been inspired by the recent works which include efficient data structures, memory access pattern, global/shared/local memory usage and auto-tunning, see [8] [9] [10] [11]. These studies show that the SpMV on GPU has a very low performance against the hardware capacity and motivate the use of blocking which is crucial in order to improve the performance. The method to compute multiple small matrix/matrix products from [12] has many similarities with our implementation (such as the use of templates for example). In this paper we do not com- pare our CPU and GPU implementations rather we focus on the GPU and propose a system to dynamically balance the work among workers.
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Maximum Decay Rate for the Nonlinear Schrödinger Equation

Maximum Decay Rate for the Nonlinear Schrödinger Equation

As show the results of Section 2 , if we suppose a suitable asymptotic behavior of the initial value (u(0) ∈ X if α 6 N 4 , u(0) ∈ H 1 (R N ) if α > 4 N ), then we have a sharp lower bound. In particular, under the hypotheses of Section 2 , such results do not allow estimates of type ( 2.1 ), for any nontrivial solution of ( 1.1 ), for some r > 2 and δ > 0 (see Remark 2.6 ). In this section, we establish some lower bounds which eventually allow estimates on the above type, only if α is small enough (see Theorem 3.5 below). The loss of sharp estimate is compensated by a weaker assumption on u(0), that is u(0) ∈ L 2 (R N ) if α 6 4
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Solutions of the time-harmonic wave equation in periodic waveguides : asymptotic behaviour and radiation condition

Solutions of the time-harmonic wave equation in periodic waveguides : asymptotic behaviour and radiation condition

ing the whole space : this is the so called Sommerfeld radiation condition that represents the fact that, at infinity, the solution looks like an outgoing spheri- cal wave which decays as |x| − d−1 2 where d is the space dimension. In the case of a homogeneous closed waveguide, that is when the domain of propagation is an infinite cylinder with bounded cross section, the behavior at infinity of the solution is quite different than in the homogeneous whole space : in particular, the solution does not decay any longer at infinity. As a consequence, the radi- ation condition is of quite different nature and relies on the decomposition of the solution as an infinite sum of evanescent modes and a finite sum of (appro- priately chosen) propagative modes : these are the so-called outgoing modes. The situation for a periodic closed waveguide that is considered in this paper is similar in nature to the case of the homogeneous waveguide. However, the notion of outgoing modes is much more delicate (we shall pay a lot of attention to a precise definition and description of such modes) and the analysis relies on quite different mathematical tools (the Floquet-Bloch transform, spectral theory of operators depending analytically on a parameter, complex contour integral techniques etc.).
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