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

top
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

top

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

top
  1. N. Anantharaman, M. Léautaud, Decay rates for the damped wave equation on the torus, preprint (2012) Zbl1295.35075
  2. N. Anantharaman, F. Macià, Semiclassical measures for the Schrödinger equation on the torus, preprint (2011) Zbl1298.42028
  3. N. Anantharaman, G. Rivière, Dispersion and controllability for the Schrödinger equation on negatively curved manifolds, to appear in Analysis and PDE (2010) Zbl1267.35176MR2970709
  4. M. Asch, G. Lebeau, The spectrum of the damped wave operator for a bounded domain in R 2 , Experiment. Math. 12 (2003), 227-241 Zbl1061.35064MR2016708
  5. C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), 1024-1065 Zbl0786.93009MR1178650
  6. C. J. K. Batty, T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ. 8 (2008), 765-780 Zbl1185.47043MR2460938
  7. A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010), 455-478 Zbl1185.47044MR2606945
  8. N. Burq, Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math. 180 (1998), 1-29 Zbl0918.35081MR1618254
  9. N. Burq, M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Res. Lett. 14 (2007), 35-47 Zbl1122.35015MR2289618
  10. N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17 (2004), 443-471 Zbl1050.35058MR2051618
  11. N. Burq, M. Zworski, Control for Schrödinger operators on tori, preprint. (2011) Zbl1281.35011MR2955763
  12. H. Christianson, Corrigendum to “Semiclassical non-concentration near hyperbolic orbits” [J. Funct. Anal. 246 (2) (2007) 145–195], J. Funct. Anal. 258 (2010), 1060-1065 Zbl1181.58019MR2321040
  13. C. Fermanian-Kammerer, Mesures semi-classiques 2-microlocales, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 515-518 Zbl0964.35009MR1794090
  14. C. Fermanian-Kammerer, Analyse à deux échelles d’une suite bornée de L 2 sur une sous-variété du cotangent, C. R. Math. Acad. Sci. Paris 340 (2005), 269-274 Zbl1067.35083MR2121889
  15. P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), 1761-1794 Zbl0770.35001MR1135919
  16. P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), 559-607 Zbl0788.35103MR1233448
  17. A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures Appl. (9) 68 (1989), 457-465 (1990) Zbl0685.93039MR1046761
  18. S. Jaffard, Contrôle interne exact des vibrations d’une plaque rectangulaire, Portugal. Math. 47 (1990), 423-429 Zbl0718.49026MR1090480
  19. V. Komornik, On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl. (9) 71 (1992), 331-342 Zbl0889.34055MR1176015
  20. G. Lebeau, Contrôle de l’équation de Schrödinger, J. Math. Pures Appl. (9) 71 (1992), 267-291 Zbl0838.35013
  21. G. Lebeau, Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993) 19 (1996), 73-109, Kluwer Acad. Publ., Dordrecht Zbl0863.58068MR1385677
  22. G. Lebeau, L. Robbiano, Stabilisation de l’équation des ondes par le bord, Duke Math. J. 86 (1997), 465-491 Zbl0884.58093MR1432305
  23. Z. Liu, B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys. 56 (2005), 630-644 Zbl1100.47036MR2185299
  24. F. Macià, High-frequency propagation for the Schrödinger equation on the torus, J. Funct. Anal. 258 (2010), 933-955 Zbl1180.35438MR2558183
  25. L. Miller, Short waves through thin interfaces and 2-microlocal measures, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1997) (1997), École Polytech., Palaiseau Zbl1005.35077MR1482278
  26. L. Miller, Controllability cost of conservative systems: resolvent condition and transmutation, J. Funct. Anal. 218 (2005), 425-444 Zbl1122.93011MR2108119
  27. K.-D. Phung, Polynomial decay rate for the dissipative wave equation, J. Differential Equations 240 (2007), 92-124 Zbl1130.35017MR2349166
  28. K. Ramdani, T. Takahashi, G. Tenenbaum, M. Tucsnak, A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator, J. Funct. Anal. 226 (2005), 193-229 Zbl1140.93395MR2158180
  29. J. Rauch, M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J. 24 (1974), 79-86 Zbl0281.35012MR361461
  30. E. Schenck, Exponential stabilization without geometric control, Math. Res. Lett. 18 (2011), 379-388 Zbl1244.93144MR2784679

NotesEmbed ?

top

You must be logged in to post comments.