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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  1. [AschLebeau]M. Asch and G. LebeauThe spectrum of the damped wave operator for a bounded domain in , preprint, 2000. MR2016708
  2. [BLR]C. Bardos, G. Lebeau, J. RauchSharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control and Optimization, 30, 1992, 1024-1065. Zbl0786.93009MR1178650
  3. [Burq2]N. BurqMesures semi-classiques et mesures de défaut, Sém. Bourbaki, Asterisque 245, 1997, 167-195. Zbl0954.35102MR1627111
  4. [Burq3]N. BurqSemi-classical estimates for the resolvent in non-trapping geometries, preprint, 2000. MR1876933
  5. [BurqZworski]N. Burq and M. ZworskiResonance expansions in semi-classical propagation, Comm. Math. Phys., to appear. Zbl1042.81582MR1860756
  6. [PopovCardoso]F. Cardoso and G. PopovQuasimodes with exponentially small errors associated with elliptic periodic rays, preprint, 2001. MR1932033
  7. [Hitrik1]M. HitrikEigenfrequencies for damped wave equations on Zoll manifolds, preprint, 2001. MR1937840
  8. [Hitrik2]M. HitrikPropagator expansions for damped wave equations, in preparation. 
  9. [HormIV]L. HörmanderThe analysis of linear partial differential operators IV, Springer Verlag 1985. Zbl0612.35001MR781537
  10. [Lebeau]G. LebeauEquation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli 1993), 73-109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. Zbl0863.58068MR1385677
  11. [Markus] A. S. MarkusIntroduction to the spectral theory of polynomial operator pencils, Stiintsa, Kishinev 1986 (Russian). Engl. transl. in Transl. Math. Monographs 71, Amer. Math. Soc., Providence 1988. Zbl0678.47005MR971506
  12. [Ralston] J. RalstonOn the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys. 51 1976, 219-242. Zbl0333.35066MR426057
  13. [RT1]J. Rauch and M. TaylorExponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J. 24, 1974, 79-86. Zbl0281.35012MR361461
  14. [RT2] J. Rauch and M. TaylorDecay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math. 28, 1975, 501-523. Zbl0295.35048MR397184
  15. [Sjostrand1] J. SjöstrandAsymptotic distribution of eigenfrequencies for damped wave equations, Publ. R.I.M.S., 36 ( 2000), 573-611. Zbl0984.35121MR1798488
  16. [Stefanov]P. StefanovQuasimodes and resonances : sharp lower bounds, Duke Math. J., 99, 1999, 75-92. Zbl0952.47013MR1700740
  17. [StefanovVodev] P. Stefanov and G. VodevNeumann resonances in linear elasticity for an arbitrary body, Comm. Math. Phys., 176 1996, 645-659. Zbl0851.35032MR1376435
  18. [TZ]S. H. Tang and M. ZworskiFrom quasimodes to resonances, Math. Res. Lett., 5, 1998, 261-272. Zbl0913.35101MR1637824
  19. [Weinstein] A. WeinsteinAsymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 1977, 883-892. Zbl0385.58013MR482878
  20. [Zworski]M. ZworskiResonance expansions in wave propagation, Séminaire E.D.P., 1999-2000, École Polytechnique, XXII-1-XXII-9. MR1813184

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