Decay of solutions to nondissipative hyperbolic systems on compact manifolds

Rauch, Jeffrey; Taylor, Michael E.

doi:10.1002/cpa.3160280405

Cited by 92 publications

(55 citation statements)

References 9 publications

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“…We test these results by performing a Fourier analysis of the transient waves that allows to find a lot of eigenvalues of the Laplace-Beltrami operator ∆ K on K. We compute also the solutions of the damped wave equation ∂ 2 t ψ − ∆ K ψ + a∂ t ψ = 0. When 0 ≤ a ∈ L ∞ (K) and a > 0 on ∂F, the geometric control condition of Rauch and Taylor [11] is satisfied, and our numerical experiments agree with their theoretical results, stating that the energy decays exponentially.…”

Section: Introductionsupporting

confidence: 83%

“…We test our scheme for the damped wave equation (3.1) when the damping function a ≥ 0 is non zero (for deep theoretical results, see, e.g., [7,9,11,12]). We know that the energy of any finite energy solution decays exponentially (uniformly with respect to the initial energy) if the dumping a satisfies the assumption of geometric control introduced by Rauch and Taylor in [11].…”

Section: Damped Wavesmentioning

confidence: 99%

“…We know that the energy of any finite energy solution decays exponentially (uniformly with respect to the initial energy) if the dumping a satisfies the assumption of geometric control introduced by Rauch and Taylor in [11]. This condition means…”

Section: Damped Wavesmentioning

confidence: 99%

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Wave Computation on the Hyperbolic Double Doughnut

Bachelot¹

2010

JCM

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We compute the waves propagating on the compact surface of constant negative curvature and genus 2 that is a toy model in quantum chaos theory and cosmic topology. We adopt a variational approach using finite elements. We have to implement the action of the fuchsian group by suitable boundary conditions of periodic type. Despite the ergodicity of the dynamics that is quantum weak mixing, the computation is very accurate. A spectral analysis of the transient waves allows to compute the spectrum and the eigenfunctions of the Laplace-Beltrami operator. We test the exponential decay due to a localized dumping satisfying the assumption of geometric control.Mathematics subject classification: 65M, 65N, 58J, 37D.

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Section: Introductionsupporting

confidence: 83%

Section: Damped Wavesmentioning

confidence: 99%

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Wave Computation on the Hyperbolic Double Doughnut

Bachelot¹

2010

JCM

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“…The proof is based on the equivalent formulation of the problem in terms of the telegrapher's equation, by following a trick by Kac [8], and on a careful estimate of the time decay of the energy for the non-homogeneous telegrapher's equation, proved by Rauch and Taylor in [13,14] (see also [9]). …”

Section: Introductionmentioning

confidence: 99%

Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model

Bernard

Salvarani

2013

J Stat Phys

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Abstract. In this paper we study the decay to the equilibrium state for the solution of a generalized version of the Goldstein-Taylor system, posed in the one-dimensional torus T = R/Z, by allowing that the nonnegative cross section σ can vanish in a subregion X := {x ∈ T | σ(x) = 0} of the domain with meas (X) ≥ 0 with respect to the Lebesgue measure.We prove that the solution converges in time, with respect to the strong L 2 -topology, to its unique equilibrium with an exponential rate whenever meas (T\ X) ≥ 0 and we give an optimal estimate of the spectral gap.

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“…Then, u satisfies equation (0.1). To get more information on this subject, one can refer to [2,7,10]. With regard to the one-dimensional case, the reader can consult [3,6].…”

Section: Mathematics Subject Classification 35l05 93d15mentioning

confidence: 99%

The geometrical quantity in damped wave equations on a square

Hébrard

Humbert

2006

ESAIM: COCV

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Abstract. The energy in a square membrane Ω subject to constant viscous damping on a subset ω ⊂ Ω decays exponentially in time as soon as ω satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate τ (ω) of this decay satisfies τ (ω) = 2 min(−µ(ω), g(ω)) (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here µ(ω) denotes the spectral abscissa of the damped wave equation operator and g(ω) is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion of a mass-point in Ω subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity g(ω) is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly g(ω) when ω is a finite union of squares. Mathematics Subject Classification. 35L05, 93D15.Received November 14, 2003. Revised July 19, 2004 and June 13, 2005 Let Ω = [0, 1] 2 be the unit square of R 2 and let ω ⊂ Ω be a subdomain of Ω. We are interested here in the problem of uniform stabilization of solutions of the following equationwhere χ ω is the characteristical function of the subset ω. This equation has its origin in a physical problem. Consider a square membrane Ω. We study here the behaviour of a wave in Ω. Let u(x, t) be the vertical position of x ∈ Ω at time t > 0. We assume that we apply on ω a force proportional to the speed of the membrane at x. Then, u satisfies equation (0.1). To get more information on this subject, one can refer to [2,7,10]. With regard to the one-dimensional case, the reader can consult [3,6].We do not deal here with the existence of solutions. We assume that there exists a solution u. Let us define the energy of u at time t by E(t) = Ω |∇u| 2 + u 2 t dx.

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Decay of solutions to nondissipative hyperbolic systems on compact manifolds

Cited by 92 publications

References 9 publications

Wave Computation on the Hyperbolic Double Doughnut

Wave Computation on the Hyperbolic Double Doughnut

Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model

The geometrical quantity in damped wave equations on a square

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