# Expansions and eigenfrequencies for damped wave equations

Journées équations aux dérivées partielles (2001)

- page 1-10
- ISSN: 0752-0360

## Access Full Article

top## Abstract

top## How to cite

topHitrik, 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

top- [AschLebeau]M. Asch and G. LebeauThe spectrum of the damped wave operator for a bounded domain in ${}^{2}$, preprint, 2000. MR2016708
- [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
- [Burq2]N. BurqMesures semi-classiques et mesures de défaut, Sém. Bourbaki, Asterisque 245, 1997, 167-195. Zbl0954.35102MR1627111
- [Burq3]N. BurqSemi-classical estimates for the resolvent in non-trapping geometries, preprint, 2000. MR1876933
- [BurqZworski]N. Burq and M. ZworskiResonance expansions in semi-classical propagation, Comm. Math. Phys., to appear. Zbl1042.81582MR1860756
- [PopovCardoso]F. Cardoso and G. PopovQuasimodes with exponentially small errors associated with elliptic periodic rays, preprint, 2001. MR1932033
- [Hitrik1]M. HitrikEigenfrequencies for damped wave equations on Zoll manifolds, preprint, 2001. MR1937840
- [Hitrik2]M. HitrikPropagator expansions for damped wave equations, in preparation.
- [HormIV]L. HörmanderThe analysis of linear partial differential operators IV, Springer Verlag 1985. Zbl0612.35001MR781537
- [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
- [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
- [Ralston] J. RalstonOn the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys. 51 1976, 219-242. Zbl0333.35066MR426057
- [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
- [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
- [Sjostrand1] J. SjöstrandAsymptotic distribution of eigenfrequencies for damped wave equations, Publ. R.I.M.S., 36 ( 2000), 573-611. Zbl0984.35121MR1798488
- [Stefanov]P. StefanovQuasimodes and resonances : sharp lower bounds, Duke Math. J., 99, 1999, 75-92. Zbl0952.47013MR1700740
- [StefanovVodev] P. Stefanov and G. VodevNeumann resonances in linear elasticity for an arbitrary body, Comm. Math. Phys., 176 1996, 645-659. Zbl0851.35032MR1376435
- [TZ]S. H. Tang and M. ZworskiFrom quasimodes to resonances, Math. Res. Lett., 5, 1998, 261-272. Zbl0913.35101MR1637824
- [Weinstein] A. WeinsteinAsymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 1977, 883-892. Zbl0385.58013MR482878
- [Zworski]M. ZworskiResonance expansions in wave propagation, Séminaire E.D.P., 1999-2000, École Polytechnique, XXII-1-XXII-9. MR1813184