Sharp polynomial energy decay for locally undamped waves

Matthieu Léautaud[1]; Nicolas Lerner[2]

  • [1] Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris Diderot (Paris VII) Bâtiment Sophie Germain 75205 Paris Cedex 13 France
  • [2] Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Pierre et Marie Curie (Paris VI) 4 Place Jussieu 75252 Paris cedex 05 France

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

  • page 1-13
  • ISSN: 2266-0607

Abstract

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In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. 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 traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.

How to cite

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Léautaud, Matthieu, and Lerner, Nicolas. "Sharp polynomial energy decay for locally undamped waves." Séminaire Laurent Schwartz — EDP et applications (2014-2015): 1-13. <http://eudml.org/doc/275768>.

@article{Léautaud2014-2015,
abstract = {In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. 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 traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.},
affiliation = {Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris Diderot (Paris VII) Bâtiment Sophie Germain 75205 Paris Cedex 13 France; Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Pierre et Marie Curie (Paris VI) 4 Place Jussieu 75252 Paris cedex 05 France},
author = {Léautaud, Matthieu, Lerner, Nicolas},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-13},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Sharp polynomial energy decay for locally undamped waves},
url = {http://eudml.org/doc/275768},
year = {2014-2015},
}

TY - JOUR
AU - Léautaud, Matthieu
AU - Lerner, Nicolas
TI - Sharp polynomial energy decay for locally undamped waves
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2014-2015
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 13
AB - In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. 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 traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.
LA - eng
UR - http://eudml.org/doc/275768
ER -

References

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