Expansions and eigenfrequencies for damped wave equations

Michael Hitrik

Journées équations aux dérivées partielles (2001)

  • page 1-10
  • ISSN: 0752-0360

Abstract

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We study eigenfrequencies and propagator expansions for damped wave equations on compact manifolds. In the strongly damped case, the propagator is shown to admit an expansion in terms of the finitely many eigenmodes near the real axis, with an error exponentially decaying in time. In the presence of an elliptic closed geodesic not meeting the support of the damping coefficient, we show that there exists a sequence of eigenfrequencies converging rapidly to the real axis. In the case of Zoll manifolds, the set of all eigenfrequencies is shown to exhibit a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. We then show that the propagator can be expanded in terms of the clusters of eigenfrequencies in the entire spectral band.

How to cite

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Hitrik, Michael. "Expansions and eigenfrequencies for damped wave equations." Journées équations aux dérivées partielles (2001): 1-10. <http://eudml.org/doc/93417>.

@article{Hitrik2001,
abstract = {We study eigenfrequencies and propagator expansions for damped wave equations on compact manifolds. In the strongly damped case, the propagator is shown to admit an expansion in terms of the finitely many eigenmodes near the real axis, with an error exponentially decaying in time. In the presence of an elliptic closed geodesic not meeting the support of the damping coefficient, we show that there exists a sequence of eigenfrequencies converging rapidly to the real axis. In the case of Zoll manifolds, the set of all eigenfrequencies is shown to exhibit a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. We then show that the propagator can be expanded in terms of the clusters of eigenfrequencies in the entire spectral band.},
author = {Hitrik, Michael},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-10},
publisher = {Université de Nantes},
title = {Expansions and eigenfrequencies for damped wave equations},
url = {http://eudml.org/doc/93417},
year = {2001},
}

TY - JOUR
AU - Hitrik, Michael
TI - Expansions and eigenfrequencies for damped wave equations
JO - Journées équations aux dérivées partielles
PY - 2001
PB - Université de Nantes
SP - 1
EP - 10
AB - We study eigenfrequencies and propagator expansions for damped wave equations on compact manifolds. In the strongly damped case, the propagator is shown to admit an expansion in terms of the finitely many eigenmodes near the real axis, with an error exponentially decaying in time. In the presence of an elliptic closed geodesic not meeting the support of the damping coefficient, we show that there exists a sequence of eigenfrequencies converging rapidly to the real axis. In the case of Zoll manifolds, the set of all eigenfrequencies is shown to exhibit a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. We then show that the propagator can be expanded in terms of the clusters of eigenfrequencies in the entire spectral band.
LA - eng
UR - http://eudml.org/doc/93417
ER -

References

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