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Energy decay for a locally undamped wave equation
Matthieu Léautaud; Nicolas Lerner
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 1, p. 157-205

We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in transversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function. Our method relies on a refined microlocal analysis linked to a second microlocalization procedure to cut the phase space into tiny regions respecting the uncertainty principle but way too small to enter a standard semi-classical analysis localization. Using a multiplier method, we obtain the energy estimates in each region and we then patch the microlocal estimates together.

Nous étudions le taux de décroissance de l’énergie des solutions de l’équation des ondes amorties dans une situation où la Condition de Contrôle Géométrique n’est pas satisfaite. Nous supposons que l’ensemble des trajectoires non amorties forme un sous-tore plat, et que la métrique est localement plate dans un voisinage. Nous supposons aussi que la fonction d’amortissement est localement homogène dans les directions transverses. Nous démontrons la décroissance à un taux polynomial optimal, qui dépend de l’homogénéité de la fonction d’amortissement. Notre méthode repose sur une procédure de deuxième microlocalisation, qui consiste à découper l’espace des phases en toutes petites régions respectant le principe d’incertitude, mais bien trop petites pour entrer dans le cadre de l’analyse microlocale semi-classique standard. Une méthode de multiplicateurs nous permet, dans chaque région, d’obtenir des estimées d’énergie, que nous recollons finalement.

Received : 2015-11-27
Accepted : 2016-03-24
Published online : 2017-02-07
DOI : https://doi.org/10.5802/afst.1528
Keywords: Damped wave equation, polynomial decay, torus, second microlocalization, geometric control condition, non-selfadjoint operators, resolvent estimates
@article{AFST_2017_6_26_1_157_0,
     author = {Matthieu L\'eautaud and Nicolas Lerner},
     title = {Energy decay for a locally undamped wave equation},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {1},
     year = {2017},
     pages = {157-205},
     doi = {10.5802/afst.1528},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_1_157_0}
}
Léautaud, Matthieu; Lerner, Nicolas. Energy decay for a locally undamped wave equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 1, pp. 157-205. doi : 10.5802/afst.1528. afst.centre-mersenne.org/item/AFST_2017_6_26_1_157_0/

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