Some decay properties for the damped wave equation on the torus

Nalini Anantharaman[1]; Matthieu Léautaud[1]

  • [1] Université Paris-Sud 11, Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France

Journées Équations aux dérivées partielles (2012)

  • Volume: 7, Issue: 1, page 1-21
  • ISSN: 0752-0360

Abstract

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This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012).We address the decay rates of the energy of the damped wave equation when the damping coefficient b does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove that the observability of the Schrödinger group implies that the semigroup associated to the damped wave equation decays at rate 1 / t (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 1 / t , as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients b , we show that the semigroup decays at rate 1 / t 1 - ε , for all ε > 0 .In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), we give numerical evidence of decay rates strictly worse than 1 / t . In particular, our study tends to prove that the decay rate highly depends on the way b vanishes.

How to cite

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Anantharaman, Nalini, and Léautaud, Matthieu. "Some decay properties for the damped wave equation on the torus." Journées Équations aux dérivées partielles 7.1 (2012): 1-21. <http://eudml.org/doc/275457>.

@article{Anantharaman2012,
abstract = {This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012).We address the decay rates of the energy of the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove that the observability of the Schrödinger group implies that the semigroup associated to the damped wave equation decays at rate $1/\sqrt\{t\}$ (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 $1/t$, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients $b$, we show that the semigroup decays at rate $1/t^\{1-\varepsilon \}$, for all $\varepsilon &gt;0$.In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), we give numerical evidence of decay rates strictly worse than $1/t$. In particular, our study tends to prove that the decay rate highly depends on the way $b$ vanishes.},
affiliation = {Université Paris-Sud 11, Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France; Université Paris-Sud 11, Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France},
author = {Anantharaman, Nalini, Léautaud, Matthieu},
journal = {Journées Équations aux dérivées partielles},
keywords = {Damped wave equation; polynomial decay; observability; Schrödinger group; torus; two-microlocal semiclassical measures; spectrum of the damped wave operator; geometric control condition; second microlocalization},
language = {eng},
number = {1},
pages = {1-21},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Some decay properties for the damped wave equation on the torus},
url = {http://eudml.org/doc/275457},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Anantharaman, Nalini
AU - Léautaud, Matthieu
TI - Some decay properties for the damped wave equation on the torus
JO - Journées Équations aux dérivées partielles
PY - 2012
PB - Groupement de recherche 2434 du CNRS
VL - 7
IS - 1
SP - 1
EP - 21
AB - This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012).We address the decay rates of the energy of the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove that the observability of the Schrödinger group implies that the semigroup associated to the damped wave equation decays at rate $1/\sqrt{t}$ (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 $1/t$, as soon as the damping region does not satisfy GCC. Conversely, for smooth damping coefficients $b$, we show that the semigroup decays at rate $1/t^{1-\varepsilon }$, for all $\varepsilon &gt;0$.In the case where the damping coefficient is a characteristic function of a strip (hence discontinuous), we give numerical evidence of decay rates strictly worse than $1/t$. In particular, our study tends to prove that the decay rate highly depends on the way $b$ vanishes.
LA - eng
KW - Damped wave equation; polynomial decay; observability; Schrödinger group; torus; two-microlocal semiclassical measures; spectrum of the damped wave operator; geometric control condition; second microlocalization
UR - http://eudml.org/doc/275457
ER -

References

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