Analysis & PDE

• Anal. PDE
• Volume 7, Number 1 (2014), 159-214.

Sharp polynomial decay rates for the damped wave equation on the torus

Abstract

We address the decay rates of the energy for 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 in an abstract setting that the observability of the Schrödinger equation implies that the solutions of the damped wave equation decay at least like $1∕t$ (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 $b$ vanishing flatly enough, we show that the semigroup decays at least like $1∕t1−ε$, for all $ε>0$. 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 $1∕t2∕3$. In particular, our study emphasizes that the decay rate highly depends on the way $b$ vanishes.

Article information

Source
Anal. PDE, Volume 7, Number 1 (2014), 159-214.

Dates
Revised: 21 May 2013
Accepted: 23 July 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731469

Digital Object Identifier
doi:10.2140/apde.2014.7.159

Mathematical Reviews number (MathSciNet)
MR3219503

Zentralblatt MATH identifier
1295.35075

Citation

Anantharaman, Nalini; Léautaud, Matthieu. Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE 7 (2014), no. 1, 159--214. doi:10.2140/apde.2014.7.159. https://projecteuclid.org/euclid.apde/1513731469

References

• N. Anantharaman and M. Léautaud, “Some decay properties for the damped wave equation on the torus”, pp. [exposé] no. 6 in Journées “Équations aux Dérivées Partielles” (Biarritz, 2012), Cedram, 2012.
• N. Anantharaman and F. Macià, “Semiclassical measures for the Schrödinger equation on the torus”, 2010. to appear in J. Europ. Math. Soc.
• N. Anantharaman and G. Rivière, “Dispersion and controllability for the Schrödinger equation on negatively curved manifolds”, Anal. PDE 5:2 (2012), 313–338.
• M. Asch and G. Lebeau, “The spectrum of the damped wave operator for a bounded domain in ${\bf R}\sp 2$”, Experiment. Math. 12:2 (2003), 227–241.
• C. Bardos, G. Lebeau, and J. Rauch, “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30:5 (1992), 1024–1065.
• C. J. K. Batty and T. Duyckaerts, “Non-uniform stability for bounded semi-groups on Banach spaces”, J. Evol. Equ. 8:4 (2008), 765–780.
• A. Borichev and Y. Tomilov, “Optimal polynomial decay of functions and operator semigroups”, Math. Ann. 347:2 (2010), 455–478.
• A. Boulkhemair, “$L\sp 2$ estimates for Weyl quantization”, J. Funct. Anal. 165:1 (1999), 173–204. http://msp.org/idx/mr/2001f:35454MR 2001f:35454
• 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. 180:1 (1998), 1–29.
• N. Burq and P. Gérard, “Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes”, C. R. Acad. Sci. Paris Sér. I Math. 325:7 (1997), 749–752.
• N. Burq and M. Hitrik, “Energy decay for damped wave equations on partially rectangular domains”, Math. Res. Lett. 14:1 (2007), 35–47.
• N. Burq and M. Zworski, “Geometric control in the presence of a black box”, J. Amer. Math. Soc. 17:2 (2004), 443–471.
• N. Burq and M. Zworski, “Control for Schrödinger operators on tori”, Math. Res. Lett. 19:2 (2012), 309–324.
• H. Christianson, “Semiclassical non-concentration near hyperbolic orbits”, J. Funct. Anal. 246:2 (2007), 145–195.
• M. Dimassi and J. Sj östrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999.
• C. Fermanian-Kammerer, “Mesures semi-classiques 2-microlocales”, C. R. Acad. Sci. Paris Sér. I Math. 331:7 (2000), 515–518.
• C. Fermanian Kammerer, “Propagation and absorption of concentration effects near shock hypersurfaces for the heat equation”, Asymptot. Anal. 24:2 (2000), 107–141.
• C. Fermanian Kammerer, “Analyse à deux échelles d'une suite bornée de $L\sp 2$ sur une sous-variété du cotangent”, C. R. Math. Acad. Sci. Paris 340:4 (2005), 269–274.
• C. Fermanian-Kammerer and P. Gérard, “Mesures semi-classiques et croisement de modes”, Bull. Soc. Math. France 130:1 (2002), 123–168.
• P. Gérard, “Microlocal defect measures”, Comm. Partial Differential Equations 16:11 (1991), 1761–1794.
• P. Gérard and É. Leichtnam, “Ergodic properties of eigenfunctions for the Dirichlet problem”, Duke Math. J. 71:2 (1993), 559–607.
• A. Haraux, “Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire”, J. Math. Pures Appl. $(9)$ 68:4 (1989), 457–465.
• A. Haraux, “Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps”, Portugal. Math. 46:3 (1989), 245–258.
• F. L. Huang, “Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces”, Ann. Differential Equations 1:1 (1985), 43–56.
• S. Jaffard, “Contrôle interne exact des vibrations d'une plaque rectangulaire”, Portugal. Math. 47:4 (1990), 423–429.
• V. Komornik, “On the exact internal controllability of a Petrowsky system”, J. Math. Pures Appl. $(9)$ 71:4 (1992), 331–342.
• \hskip-0.5pt G. Lebeau, \hskip-0.5pt“Contrôle de l'équation de Schrödinger”\hskip-0.5pt, J. \hskip-0.5pt Math. Pures Appl. $(9)$ 71:3 (1992), 267–291.\hskip-0.5pt
• G. Lebeau, “Équation des ondes amorties”, pp. 73–109 in Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), edited by A. B. de Monvel and V. Marchenko, Math. Phys. Stud. 19, Kluwer Acad. Publ., Dordrecht, 1996.
• G. Lebeau and L. Robbiano, “Stabilisation de l'équation des ondes par le bord”, Duke Math. J. 86:3 (1997), 465–491.
• N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications 3, Birkhäuser, Basel, 2010.
• Z. Liu and B. Rao, “Characterization of polynomial decay rate for the solution of linear evolution equation”, Z. Angew. Math. Phys. 56:4 (2005), 630–644.
• F. Macià, “High-frequency propagation for the Schrödinger equation on the torus”, J. Funct. Anal. 258:3 (2010), 933–955.
• L. Miller, Propagation d'ondes semiclassiques à travers une interface et mesures 2-microlocales, Ph.D. thesis, École Polytechnique, Palaiseau, 1996.
• L. Miller, “Short waves through thin interfaces and 2-microlocal measures”, pp. exposé no. 12 in Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1997), École Polytech., Palaiseau, 1997.
• L. Miller, “Controllability cost of conservative systems: resolvent condition and transmutation”, J. Funct. Anal. 218:2 (2005), 425–444.
• F. Nier, “A semi-classical picture of quantum scattering”, Ann. Sci. École Norm. Sup. $(4)$ 29:2 (1996), 149–183.
• H. Nishiyama, “Polynomial decay for damped wave equations on partially rectangular domains”, Math. Res. Lett. 16:5 (2009), 881–894.
• A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences 44, Springer, New York, 1983.
• K. D. Phung, “Polynomial decay rate for the dissipative wave equation”, J. Differential Equations 240:1 (2007), 92–124.
• K. Ramdani, T. Takahashi, G. Tenenbaum, and M. Tucsnak, “A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator”, J. Funct. Anal. 226:1 (2005), 193–229.
• J. Rauch and M. Taylor, “Exponential decay of solutions to hyperbolic equations in bounded domains”, Indiana Univ. Math. J. 24 (1974), 79–86.
• M. Reed and B. Simon, Methods of modern mathematical physics, I: Functional analysis, 2nd ed., Academic Press, New York, 1980.
• E. Schenck, “Exponential stabilization without geometric control”, Math. Res. Lett. 18:2 (2011), 379–388.
• J. Sj östrand, “Wiener type algebras of pseudodifferential operators”, pp. [exposé] no. 4 in Séminaire “Équations aux Dérivées Partielles”, 1994–1995, École Polytech., Palaiseau, 1995.
• M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Birkhäuser, Basel, 2009.