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Sharp polynomial decay
rates for the damped wave equation on the torus
Nalini Anantharaman and Matthieu LéautaudAppendix: Stéphane Nonnenmacher |
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Vol. 7 (2014), No. 1, 159–214
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Abstract |
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We address the decay rates of the energy for the damped wave equation when the damping coefficient does not satisfy the geometric control condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove in an abstract setting that the observability of the Schrödinger equation implies that the solutions of the damped wave equation decay at least like (which is a stronger rate than the general logarithmic one predicted by the Lebeau theorem). Second, we focus on the 2-dimensional torus. We prove that the best decay one can expect is , as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients vanishing flatly enough, we show that the semigroup decays at least like , for all . The proof relies on a second microlocalization around trapped directions, and resolvent estimates. In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), Stéphane Nonnenmacher computes in an appendix part of the spectrum of the associated damped wave operator, proving that the semigroup cannot decay faster than . In particular, our study emphasizes that the decay rate highly depends on the way vanishes. |
Keywords
damped wave equation, polynomial decay, observability,
Schrödinger group, torus, two-microlocal semiclassical
measures, spectrum of the damped wave operator.
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Mathematical Subject Classification 2010
Primary: 35A21, 35B35, 35L05, 35P20, 35S05
Secondary: 35B37, 93C20
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Milestones
Received: 11 October 2012
Revised: 21 May 2013
Accepted: 23 July 2013
Published: 7 May 2014
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Authors |
| Nalini Anantharaman | |
| Laboratoire de Mathématiques Université Paris-Sud 11 Bâtiment 425 91405 Orsay Cedex France |
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| Matthieu Léautaud | |
| Département de Mathématiques Université Paris-Sud 11 Bâtiment 425 91405 Orsay Cedex France |
IMJ-PRG, UMR CNRS 7586 Université Paris Diderot–Paris 7 Bâtiment Sophie-Germain 75205 Paris Cedex 13 France |
| Stéphane Nonnenmacher | |
| Institut de Physique Théorique CEA/DSM/IPhT Unité de recherche associée au CNRS,CEA/Saclay 91191 Gif-sur-Yvette France |
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