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16
Resonance Expansions Of Scattered Waves
 Comm. Pure Appl. Math
"... The this paper is to describe expansions of solutions to the wave equation on R^n with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in ..."
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Cited by 40 (10 self)
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The this paper is to describe expansions of solutions to the wave equation on R^n with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in time. To avoid the discussion of particular aspects of potential, gravitational or obstacle scattering, the results are stated using the abstract "black box" formalism of Sjöstrand and the second author [19]. When M is a compact Riemannian manifold, and Delta, its Laplacian, then we have a generalized Fourier expansion of the wave group: sin t p p f(x) = X 2 j 2Spec() e i j t w j (x) ; w j = 2 j w j ; (1.1) and the convergence is absolute in the case of smooth data. The simplest case of a noncompact spectral problem is given by taking R^n with the usual Laplacian outside a compact set  we can for instance "glue" any compact Riemannian manifold to R^n or consider the obstacle problem with the Dirichlet Laplacian on R^n\O. Since resonances or scattering poles constitute a natural replacement of discrete spectral data for problems on exterior domains, we expect a similar expansion involving them in place of eigenvalues  this point of view was emphasized early by LaxPhillips [10] (see [18],[32] and [34] for overviews of recent results). The resonances are de ned as poles of the meromorphic continuation of the resolvent or of the scattering matrix but despite the stationary nature of these de nitions, they are fundamentally a dynamical concept: the real part of a resonance describes the rest energy of a state and the imaginary part its rate of decay. Consequently they should be understood in terms of long time behaviour of solutions to evolutio...
Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle., preprint
"... Abstract. In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimens ..."
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Cited by 19 (5 self)
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Abstract. In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. In order to prove the global estimates, we explore weighted Strichartz estimates for solutions of the wave equation when the Cauchy data and forcing term are compactly supported. 1. Introduction. The
Local decay of waves on asymptotically flat stationary spacetimes
"... Abstract. In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establ ..."
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Cited by 16 (1 self)
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Abstract. In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t −3 local uniform decay rate for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds. In the Schwarzschild case, such a decay rate has been heuristically derived by Price [39]. Our results apply to both of these cases. 1.
On the uniform decay of local energy
 Serdica Math. J
, 1999
"... Abstract. It is proved in [1],[2] that in odd dimensional spaces any uniform decay of the local energy implies that it must decay exponentially. We extend this to even dimensional spaces and to more general perturbations (including the transmission problem) showing that any uniform decay of the loc ..."
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Cited by 6 (1 self)
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Abstract. It is proved in [1],[2] that in odd dimensional spaces any uniform decay of the local energy implies that it must decay exponentially. We extend this to even dimensional spaces and to more general perturbations (including the transmission problem) showing that any uniform decay of the local energy implies that it must decay like O(t−2n), t ≫ 1 being the time and n being the space dimension. 1. Introduction and statement of results. It is proved in [1] using the LaxPhillips theory that in the case of obstacle scattering in odd dimensional spaces, if the local energy decays uniformly to zero, it must decay exponentially. This was extended in [2] for more general perturbations but still in odd dimensional space. The purpose of this note is to extend these results
The Exponential Stability Of The Problem Of Transmission Of The Wave Equation
, 1998
"... The problem of exponential stability of the problem of transmission of the wave equation with lowerorder terms is considered. Making use of the classical energy method and multiplier technique, we prove that this problem of transmission is exponentially stable. Key Words: Wave equation, Problem of ..."
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Cited by 5 (0 self)
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The problem of exponential stability of the problem of transmission of the wave equation with lowerorder terms is considered. Making use of the classical energy method and multiplier technique, we prove that this problem of transmission is exponentially stable. Key Words: Wave equation, Problem of transmission, Exponential stability. AMS subject classification: 35B35, 35L05. 1. Introduction Throughout this paper, let# be a bounded domain (open, nonempty, and connected) in lR n ( n # 1) with a boundary # = ## of class C 2 which consists of two parts, S 1 and S 2 (see Figure 1 below). S 1 is assumed to be either empty or to have a nonempty interior and S 2 #= # and relatively open in #. Assume S 1 #S 2 = #. Let S 0 with S 0 #S 1 = S 0 # S 2 = # be a regular hypersurface of class C 2 , which separates# into two domains,# 1 and# 2 , such that S 1 # # 1 = ## 1 and S 2 # # 2 = ## 2 . For T > 0, set Q =# (0, T ), Q 1 =# 1 (0, T ), Q 2 =# 2 (0, ...
Price’s law on nonstationary spacetimes
 Adv. Math
"... Abstract. In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we est ..."
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Cited by 3 (1 self)
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Abstract. In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t−3 local uniform decay rate (Price’s law [54]) for linear waves. As a corollary, we also prove Price’s law for certain small perturbations of the Kerr metric. This result was previously established by the second author in [64] on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates. 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
STRICHARTZ ESTIMATES ON EXTERIOR POLYGONAL DOMAINS
"... Abstract. Using a new local smoothing estimate of the first and third authors, we prove localintime Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and globalintime Strichartz estimates without a loss exterior ..."
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Abstract. Using a new local smoothing estimate of the first and third authors, we prove localintime Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and globalintime Strichartz estimates without a loss exterior to a large class of polygonal obstacles with Dirichlet boundary conditions. In addition, we prove a globalintime local smoothing estimate in exterior wedge domains with Dirichlet boundary conditions and discuss some nonlinear applications. 1.