Solutions of the wave equation with localized energy

Ralston, James

doi:10.1002/cpa.3160220605

Cited by 209 publications

(143 citation statements)

References 2 publications

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“…To illustrate this result, we plot in Figure 6 the absolute value of the solutions of the discrete wave equation (1.5) in meshes M h,g for various choices of g corresponding to an initial data given by 16) in the spirit of the Gaussian beams [25]. The oscillatory term localizes the solution in frequency in a neighborhood of ξ 0 /h.…”

Section: A Dynamical System Approachmentioning

confidence: 90%

“…Thus, the projections on the physical space of these rays simply are straight lines traveling at velocity one inside the domain Ω. The construction given in [25] yield solutions of the wave equation which are localized in an arbitrary small neighborhood of these rays. When discretizing the wave equation, the situation is more intricate.…”

Section: A Dynamical System Approachmentioning

confidence: 99%

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Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

2015

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We build non-uniform numerical meshes for the finite difference and finite element approximations of the 1d wave equation, ensuring that all numerical solutions reach the boundary, as continuous solutions do, in the sense that the full discrete energy can be observed by means of boundary measurements, uniformly with respect to the mesh-size. The construction of the nonuniform mesh is achieved by means of a concave diffeomorphic transformation of a uniform grid into a non-uniform one, making the mesh finer and finer when approaching the right boundary. For uniform meshes it is known that high-frequency numerical wave packets propagate very slowly without never getting to the boundary. Our results show that this pathology can be avoided by taking suitable non-uniform meshes. This also allows to build convergent numerical algorithms for the approximation of boundary controls of the wave equation.

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Section: A Dynamical System Approachmentioning

confidence: 90%

Section: A Dynamical System Approachmentioning

confidence: 99%

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

2015

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“…Indeed, if (Φ n ) n is the sequence considered in the proof, let (S(t)Φ n ) n be the sequence of corresponding solutions of (5). By (22) we have that ||R(A : λ n i)Φ n || X 0 0 ≤ ∞ 0 ||S(t)Φ n || X 0 0 dt.…”

Section: Remarkmentioning

confidence: 94%

“…Indeed, the nature of the coupling between the acoustic and elastic components of the system (i.e. the boundary conditions on Γ 0 ) allows to build solutions with arbitrarily slow decay rate and with the energy distributed in all of the domain and not only along some particular ray of geometrical optics as in [22].…”

Section: Introduction and The Mathematical Modelmentioning

confidence: 99%

Stabilization of a hybrid system fo noise reduction

Micu

1998

ESAIM: Proc.

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We consider a simple model arising in the control of noise. We assume that the two-dimensional cavity Ω = (0, 1)×(0, 1) is occupied by an elastic, inviscid, compressible fluid. The potential φ of the velocity field satisfies the linear wave equation. The boundary of Ω is divided in two parts Γ0 and Γ1. The first one, Γ0 is flexible and occupied by a Bernoulli-Euler beam. On Γ0 the continuity of the normal velocities of the fluid and the beam is imposed. The subset Γ1 of the boundary is assumed to be rigid and therefore, the normal velocity of the fluid vanishes. A dissipative term is assumed to act in the one-dimensional beam equation. We prove the existence and the uniqueness of solutions. Each trajectory is proved to converge to an equilibrium as t → ∞. On the other hand we show that the convergence rate of the energy is not exponential. The proof of this r esult uses a perturbation argument allowing to modify the boundary conditions so that separation of variables applies.

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“…It is well known that the decay of energy of a solution to u = 0 at high frequencies is closely tied to the geometry of geodesic rays; in particular, the existence of trapped geodesics is an obstruction to the uniform decay of local energy (see Ralston [18]). Many subsequent results have demonstrated that the decay of high frequency components of the solution persists in a wide variety of geometric settings in which there is no trapping of rays (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Morawetz estimates for the wave equation at low frequency

Vasy

Wunsch

2012

Math. Ann.

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Abstract. We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low frequencies, independent of the geometry of the compact part of the manifold. We further prove a new type of Morawetz estimate in this context, with both hypotheses and conclusion localized inside the forward light cone. This result allows us to gain a 1/2 power of t decay relative to what would be dictated by energy estimates, in a small part of spacetime.

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Solutions of the wave equation with localized energy

Cited by 209 publications

References 2 publications

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

Stabilization of a hybrid system fo noise reduction

Morawetz estimates for the wave equation at low frequency

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