Expansions and eigenfrequencies for damped wave equations
Journées équations aux dérivées partielles (2001), article no. 6, 10 p.

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.

@article{JEDP_2001____A6_0,
     author = {Hitrik, Michael},
     title = {Expansions and eigenfrequencies for damped wave equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {6},
     pages = {1--10},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.590},
     mrnumber = {1843407},
     zbl = {01808682},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.590/}
}
TY  - JOUR
AU  - Hitrik, Michael
TI  - Expansions and eigenfrequencies for damped wave equations
JO  - Journées équations aux dérivées partielles
PY  - 2001
SP  - 1
EP  - 10
PB  - Université de Nantes
UR  - http://www.numdam.org/articles/10.5802/jedp.590/
DO  - 10.5802/jedp.590
LA  - en
ID  - JEDP_2001____A6_0
ER  - 
%0 Journal Article
%A Hitrik, Michael
%T Expansions and eigenfrequencies for damped wave equations
%J Journées équations aux dérivées partielles
%D 2001
%P 1-10
%I Université de Nantes
%U http://www.numdam.org/articles/10.5802/jedp.590/
%R 10.5802/jedp.590
%G en
%F JEDP_2001____A6_0
Hitrik, Michael. Expansions and eigenfrequencies for damped wave equations. Journées équations aux dérivées partielles (2001), article  no. 6, 10 p. doi : 10.5802/jedp.590. http://www.numdam.org/articles/10.5802/jedp.590/

[AschLebeau]M. Asch and G. Lebeau The spectrum of the damped wave operator for a bounded domain in 2 , preprint, 2000. | MR

[BLR]C. Bardos, G. Lebeau, J. Rauch Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control and Optimization, 30, 1992, 1024-1065. | MR | Zbl

[Burq2]N. Burq Mesures semi-classiques et mesures de défaut, Sém. Bourbaki, Asterisque 245, 1997, 167-195. | EuDML | Numdam | MR | Zbl

[Burq3]N. Burq Semi-classical estimates for the resolvent in non-trapping geometries, preprint, 2000. | MR

[BurqZworski]N. Burq and M. Zworski Resonance expansions in semi-classical propagation, Comm. Math. Phys., to appear. | MR | Zbl

[PopovCardoso]F. Cardoso and G. Popov Quasimodes with exponentially small errors associated with elliptic periodic rays, preprint, 2001. | MR

[Hitrik1]M. Hitrik Eigenfrequencies for damped wave equations on Zoll manifolds, preprint, 2001. | MR

[Hitrik2]M. Hitrik Propagator expansions for damped wave equations, in preparation.

[HormIV]L. Hörmander The analysis of linear partial differential operators IV, Springer Verlag 1985. | MR | Zbl

[Lebeau]G. Lebeau Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli 1993), 73-109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. | MR | Zbl

[Markus] A. S. Markus Introduction to the spectral theory of polynomial operator pencils, Stiintsa, Kishinev 1986 (Russian). Engl. transl. in Transl. Math. Monographs 71, Amer. Math. Soc., Providence 1988. | MR | Zbl

[Ralston] J. Ralston On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys. 51 1976, 219-242. | MR | Zbl

[RT1]J. Rauch and M. Taylor Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J. 24, 1974, 79-86. | MR | Zbl

[RT2] J. Rauch and M. Taylor Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math. 28, 1975, 501-523. | MR | Zbl

[Sjostrand1] J. Sjöstrand Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. R.I.M.S., 36 (2000), 573-611. | MR | Zbl

[Stefanov]P. Stefanov Quasimodes and resonances : sharp lower bounds, Duke Math. J., 99, 1999, 75-92. | MR | Zbl

[StefanovVodev] P. Stefanov and G. Vodev Neumann resonances in linear elasticity for an arbitrary body, Comm. Math. Phys., 176 1996, 645-659. | MR | Zbl

[TZ]S. H. Tang and M. Zworski From quasimodes to resonances, Math. Res. Lett., 5, 1998, 261-272. | MR | Zbl

[Weinstein] A. Weinstein Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 1977, 883-892. | MR | Zbl

[Zworski]M. Zworski Resonance expansions in wave propagation, Séminaire E.D.P., 1999-2000, École Polytechnique, XXII-1-XXII-9. | Numdam | MR

Cité par Sources :