Non-uniform stability for bounded semi-groups on Banach spaces

Batty, C. J. K.; Duyckaerts, Thomas

doi:10.1007/s00028-008-0424-1

Cited by 259 publications

(324 citation statements)

References 30 publications

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“…See also [BD08] for general decay rates in Banach spaces. Note in particular that the proof of a decay rate is reduced to the proof of a resolvent estimate on the imaginary axes.…”

Section: The Damped Wave Equation In An Abstract Settingmentioning

confidence: 99%

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

2014

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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 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. 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/t 2/3 . In particular, our study shows that the decay rate highly depends on the way b vanishes. KeywordsDamped wave equation, polynomial decay, observability, Schrödinger group, torus, two-microlocal semiclassical measures, spectrum of the damped wave operator.

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“…See also [BD08] for general decay rates in Banach spaces. Note in particular that the proof of a decay rate is reduced to the proof of a resolvent estimate on the imaginary axes.…”

Section: The Damped Wave Equation In An Abstract Settingmentioning

confidence: 99%

“…that does not intersect supp(b)). More precisely, in such geometry, Theorem 2.5 combined with Lemma 4.6 and [BD08, Proposition 1.3] shows that m 1 (t) ≥ C 1+t , for some C > 0 (with the notation of [BD08] where m 1 (t) denotes the best decay rate).…”

Section: Decay Rates For the Damped Wave Equation On The Torusmentioning

confidence: 99%

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

2014

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“…According to [1,5,8,29], proving a decay rate for solutions of (1.1) reduces to proving a high-energy estimate for the operators…”

Section: E(u(t))mentioning

confidence: 99%

Energy decay for a locally undamped wave equation

Léautaud¹,

Lerner²

2017

Ann. Fac. Sci. Toulouse Math

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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. Our method relies on a refined microlocal analysis linked to a second microlocalization procedure to cut the phase space into tiny regions respecting the uncertainty principle but way too small to enter a standard semiclassical analysis localization. Using a multiplier method, we obtain the energy estimates in each region and we then patch the microlocal estimates together.

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“…See also [6] for general decay rates in Banach spaces. Note in particular that the proof of a decay rate is reduced to the proof of a resolvent estimate on the imaginary axes.…”

Section: Theorem 2 Suppose That There Existmentioning

confidence: 99%

Some decay properties for the damped wave equation on the torus

Anantharaman¹,

Léautaud²

2013

Journées Équations Aux Dérivées Partielles

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Quelques propriétés de décroissance pour l'équation des ondes amorties sur le tore RésuméCet article est la version courte d'un travail en cours [1], et a fait l'objet d'un exposé du second auteur au cours des Journées "Équations aux Dérivées Partielles" (Biarritz, 2012).On s'intéresse aux taux de décroissance de l'énergie pour l'équation des ondes amorties dans des situations où le coefficient d'amortissement b ne satisfait pas la condition de contrôle géométrique. On donne tout d'abord un lien avec la contrôlabilité de l'équation de Schrödinger associée. On montre que l'observabilité du groupe de Schrödinger implique la décroissance à taux 1/ √ t du semigroupe des ondes amorties (taux meilleur que le taux logarithmique a priori fourni par le théorème de Lebeau).Dans un second temps, on se focalise sur le tore 2-D. Toujours en supposant que le contrôle géométrique n'est pas réalisé, on montre que le semigroupe décroît au mieux à taux 1/t. Réciproquement, pour des coefficients d'amortissements b réguliers, on prouve la décroissance à taux 1/t 1−ε , pour tout ε > 0.Dans le cas où le le coefficient d'amortissement est la fonction caractéristique d'une bande (donc discontinu), on effectue des simulations numériques qui semblent exhiber un taux de décroissance strictement pire que 1/t.En particulier, notre étude tend à montrer que le taux de décroissance dépend fortement du taux d'annulation de b. VI-1 AbstractThis 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.

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Non-uniform stability for bounded semi-groups on Banach spaces

Cited by 259 publications

References 30 publications

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

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

Energy decay for a locally undamped wave equation

Some decay properties for the damped wave equation on the torus

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