Decay rates for the damped wave equation on the torus
With an appendix by Stéphane Nonnenmacher^{1}^{1}1snonnenmacher@cea.fr
Abstract
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 group implies that the semigroup associated to the damped wave equation decays at rate (which is a stronger rate than the general logarithmic one predicted by the Lebeau Theorem).
Second, we focus on the 2dimensional 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 , we show that the semigroup decays at rate , 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 shows that the decay rate highly depends on the way vanishes.
Keywords
Damped wave equation, polynomial decay, observability, Schrödinger group, torus, twomicrolocal semiclassical measures, spectrum of the damped wave operator.
Contents
 I The damped wave equation
 II Resolvent estimates and stabilization in the abstract setting

III Proof of Theorem 2.6: smooth damping coefficients on the torus
 5 Semiclassical measures
 6 Zeroth and first order informations on
 7 Geometry on the torus and decomposition of invariant measures
 8 Change of quasimode and construction of an invariant cutoff function
 9 Second microlocalization on a resonant affine subspace
 10 Propagation laws for the twomicrolocal measures and
 11 The measures and vanish identically. End of the proof of Theorem 2.6
 12 Proof of Proposition 8.2
 13 Proof of Proposition 8.3: existence of the cutoff function
 IV An a priori lower bound for decay rates on the torus: proof of Theorem 2.5
Part I The damped wave equation
1 Decay of energy: a survey of existing results
Let be a smooth compact connected Riemannian dimensional manifold with or without boundary . We denote by the (nonpositive) LaplaceBeltrami operator on for the metric . Given a bounded nonnegative function, , on , we want to understand the asymptotic behaviour as of the solution of the problem
(1.1) 
The energy of a solution is defined by
(1.2) 
Multiplying (1.1) by and integrating on yields the following dissipation identity
which, as is nonnegative, implies a decay of the energy. As soon as on a nonempty open subset of , the decay is strict and as . The question is then to know at which rate the energy goes to zero.
The first interesting issue concerns uniform stabilization: under which condition does there exist a function , , such that
(1.3) 
The answer was given by Rauch and Taylor [RT74] in the case and by Bardos, Lebeau and Rauch [BLR92] in the general case (see also [BG97] for the necessity of this condition): assuming that , uniform stabilisation occurs if and only if the set satisfies the Geometric Control Condition (GCC). Recall that a set is said to satisfy GCC if there exists such that every geodesic (resp. generalised geodesic in the case ) of with length larger than satisfies . Under this condition, one can take (for some constants ) in (1.3), and the energy decays exponentially. Finally, Lebeau gives in [Leb96] the explicit (and optimal) value of the best decay rate in terms of the spectral abscissa of the generator of the semigroup and the mean value of the function along the rays of geometrical optics.
In the case where does not satisfy GCC, i.e. in the presence of “trapped rays” that do not meet , what can be said about the decay rate of the energy? As soon as on a nonempty open subset of , Lebeau shows in [Leb96] that the energy (of smoother initial data) goes at least logarithmically to zero (see also [Bur98]):
(1.4) 
with (where and have to be replaced by and respectively if ). Note that here, characterizes the decay of the energy, whereas is that of the associated semigroup. Moreover, the author constructed a series of explicit examples of geometries for which this rate is optimal, including for instance the case where is the twodimensional sphere and , where is a neighbourhood of an equator of . This result is generalised in [LR97] for a wave equation damped on a (small) part of the boundary. In this paper, the authors also make the following comment about the result they obtain:
“Notons toutefois qu’une étude plus approfondie de la localisation spectrale et des taux de décroissance de l’énergie pour des données régulières doit faire intervenir la dynamique globale du flot géodésique généralisé sur . Les théorèmes [LR97, Théorème 1] et [LR97, Théorème 2] ne fournissent donc que les bornes a priori qu’on peut obtenir sans aucune hypothèse sur la dynamique, en n’utilisant que les inégalités de Carleman qui traduisent “l’effet tunnel”.”
In all examples where the optimal decay rate is logarithmic, the trapped ray is a stable trajectory from the point of view of the dynamics of the geodesic flow. This means basically that an important amount of the energy can stay concentrated, for a long time, in a neighbourhood of the trapped ray, i.e. away from the damping region.
If the trapped trajectories are less stable, or unstable, one can expect to obtain an intermediate decay rate, between exponential and logarithmic. We shall say that the energy decays at rate if (1.4) is satisfied (more generally, see Definition 2.2 below in the abstract setting). This problem has already been adressed and, in some particular geometries, several different behaviours have been exhibited. Two main directions have been investigated.
On the one hand, Liu and Rao considered in [LR05] the case where is a square and the set contains a vertical strip. In this situation, the trapped trajectories consist in a family of parallel vertical geodesics; these are unstable, in the sense that nearby geodesics diverge at a linear rate. They proved that the energy decays at rate (i.e., that (1.4) is satisfied with ). This was extended by Burq and Hitrik [BH07] (see also [Nis09]) to the case of partially rectangular twodimensional domains, if the set contains a neighbourhood of the nonrectangular part. In [Phu07], Phung proved a decay at rate for some (unprecised) in a threedimensional domain having two parallel faces. In all these situations, the only obstruction to GCC is due to a “cylinder of periodic orbits”. The geometry is flat and the unstabilities of the geodesic flow around the trapped rays are relatively weak (geodesics diverge at a linear rate).
In [BH07], the authors argue that the optimal decay in their geometry should be of the form , for all . They provide conditions on the damping coefficient under which one can obtain such decay rates, and wonder whether this is true in general. Our main theorem (see Theorem 2.6 below) extends these results to more general damping functions on the twodimensional torus.
On the other hand, Christianson [Chr10] proved that the energy decays at rate for some , in the case where the trapped set is a hyperbolic closed geodesic. Schenck [Sch11] proved an energy decay at rate on manifolds with negative sectional curvature, if the trapped set is “small enough” in terms of topological pressure (for instance, a small neighbourhood of a closed geodesic), and if the damping is “large enough” (that is, starting from a damping function , will work for any sufficiently large). In these two papers, the geodesic flow near the trapped set enjoys strong instability properties: the flow on the trapped set is uniformly hyperbolic, in particular all trajectories are exponentially unstable.
These cases confirm the idea that the decay rates of the energy strongly depends on the stability of trapped trajectories.
One may now want to compare these geometric situations to situations where the Schrödinger group is observable (or, equivalently, controllable), i.e. for which there exist and such that, for all , we have
(1.5) 
The conditions under which this property holds are also known to be related to stability of the geodesic flow. In particular, the works [BLR92], [LR05], [BH07, Nis09] and [Chr10, Sch11] can be seen as counterparts for damped wave equations of the articles [Leb92], [Har89a, Jaf90], [BZ04] and [AR10], respectively, in the context of observation of the Schrödinger group.
Our main results are twofold. First, we clarify (in an abstract setting) the link between the observability (or the controllability) of the Schrödinger equation and polynomial decay for the damped wave equation. This follows the spirit of [Har89b], [Mil05], exploring the links between the different equations and their control properties (e.g. observability, controllability, stabilization…). More precisely, we prove that the controllability of the Schrödinger equation implies a polynomial decay at rate for the damped wave equation (Theorem 2.3).
Second, we study precisely the damped wave equation on the flat torus in case GCC fails. We give the following a priori lower bound on the decay rate, revisiting the argument of [BH07]: (1.1) is not stable at a better rate than , provided that GCC is not satisfied. In this situation, the Schrödinger group is known to be controllable (see [Jaf90], [Kom92] and the more recent works [AM11] and [BZ11]). Thus, one cannot hope to have a decay better than polynomial in our previous result, i.e. under the mere assumption that the Schrödinger flow is observable.
The remainder of the paper is devoted to studying the gap between the a priori lower and upper bounds given respectively by and on flat tori. For smooth nonvanishing damping coefficient , we prove that the energy decays at rate for all . This result holds without making any dynamical assumption on the damping coefficient, but only on the order of vanishing of . It generalises a result of [BH07], which holds in the case where is invariant in one direction. Our analysis is, again, inspired by the recent microlocal approach proposed in [AM11] and [BZ11] for the observability of the Schrödinger group. More precisely, we follow here several ideas and tools introduced in [Mac10] and [AM11].
In the situation where is a characteristic function of a vertical strip of the torus (hence discontinuous), Stéphane Nonnenmacher proves in Appendix B that the decay rate cannot be faster than . This is done by explicitly computing the high frequency eigenvalues of the damped wave operator which are closest to the imaginary axis (see for instance the figures in [AL03, AL12]). The fact that the decay rate is not achieved in this situation was observed in the numerical computations presented in [AL12].
In contrast to the control problem for the Schödinger equation, this result shows that the stabilization of the wave equation is not only sensitive to the global properties of the geodesic flow, but also to the rate at which the damping function vanishes.
2 Main results of the paper
Our first result can be stated in a general abstract setting that we now introduce. We come back to the case of the torus afterwards.
2.1 The damped wave equation in an abstract setting
Let and be two Hilbert spaces (resp. the state space and the observation/control space) with norms and , and associated inner products and .
We denote by a nonnegative selfadjoint operator with compact resolvent, and a control operator. We recall that is defined by for all and .
Definition 2.1.
We say that the system
(2.1) 
is observable in time if there exists a constant such that, for all solution of (2.1), we have
We recall that the observability of (2.1) in time is equivalent to the exact controllability in time of the adjoint problem
(2.2) 
(see for instance [Leb92] or [RTTT05]). More precisely, given , the exact controllability in time is the ability of finding for any a control function so that the solution of (2.2) satisfies .
We equip with the graph norm
and define the seminorm
Of course, if is coercive on , is a norm on equivalent to .
We also introduce in this abstract setting the damped wave equation on the space ,
(2.3) 
which can be recast on as a first order system
(2.4) 
The compact injections imply that compactly, and that the operator has a compact resolvent.
We define the energy of solutions of (2.3) by
Definition 2.2.
Theorem 2.3.
Note that the gain of the with respect to [LR05, BH07] is not essential in our work. It is due to the optimal characterization of polynomially decaying semigroups obtained by Borichev and Tomilov [BT10].
This Theorem may be compared with the works (both presented in a similar abstract setting) [Har89b] by Haraux, proving that the controllability of wavetype equations in some time is equivalent to uniform stabilization of (2.3), and [Mil05] by Miller, showing that the controllability of wavetype equations in some time implies the controllability of Schrödingertype equations in any time.
Note that the link between this abstract setting and that of Problem (1.1) is , with if and otherwise, is the multiplication in by the bounded function .
As a first application of Theorem 2.3 we obtain a different proof of the polynomial decay results for wave equations of [LR05] and [BH07] as consequences of the associated control results for the Schrödinger equation of [Har89a] and [BZ04] respectively.
Moreover, Theorem 2.3 provides also several new stability results for System (1.1) in particular geometric situations; namely, in all following situations, the Schrödinger group is proved to be observable, and Theorem 2.3 gives the polynomial stability at rate for (1.1):

If is the Bunimovich stadium and on the neighbourhood of one half disc and on one point of the opposite side, as a consequence of [BZ04];
Moreover, Lebeau gives in [Leb96, Théorème 1 (ii)] several dimensional examples for which the decay rate is optimal. For all these geometrical situations, Theorem 2.3 implies that the Schrödinger group is not observable.
The proof of Theorem 2.3 relies on the following characterization of polynomial decay for System (2.3). For , we define on the operator , with domain . We prove in Lemma 4.2 below that is invertible for all , .
Proposition 2.4.
Suppose that
(2.5) 
Then, for all , the five following assertions are equivalent:
(2.6) 
(2.7) 
(2.8) 
(2.9) 
(2.10) 
This proposition is proved as a consequence of the characterization of polynomial decay for general semigroups in terms of resolvent estimates given in [BT10], providing the equivalence between (2.6) and (2.7). 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. By the way, this estimate implies the existence of a “spectral gap” between the spectrum of and the imaginary axis, given by (2.8).
2.2 Decay rates for the damped wave equation on the torus
The main results of this article deal with the decay rate for Problem (1.1) on the torus . In this setting, as well as in the abstract setting, we shall write .
First, we give an a priori lower bound for the decay rate of the damped wave equation, on , when GCC is “strongly violated”, i.e. assuming that does not satisfy GCC (instead of ). This theorem is proved by constructing explicit quasimodes for the operator .
Theorem 2.5.
Suppose that there exists , , such that
Then there exist two constants and such that for all ,
(2.11) 
As a consequence of Proposition 2.4, polynomial stabilization at rate for is not possible if there is a strongly trapped ray (i.e. that does not intersect ). More precisely, in such geometry, Theorem 2.5 combined with Lemma 4.6 and [BD08, Proposition 1.3] shows that , for some (with the notation of [BD08] where denotes the best decay rate).
Then, the main goal of this paper is to explore the gap between the a priori upper bound for the decay rate, given by Theorem 2.3, and the a priori lower bound of Theorem 2.5. Our results are twofold (somehow in two opposite directions) and concern either the case of smooth damping functions , or the case , with .
2.2.1 The case of smooth damping coefficients
Our main result deals with the case of smooth damping coefficients. Without any geometric assumption, but with an additional hypothesis on the order of vanishing of the damping function , we prove a weak converse of Theorem 2.5.
Theorem 2.6.
Let with the standard flat metric. There exists satisfying the following property. Suppose that is a nonnegative nonvanishing function on satisfying and that there exist and such that
(2.12) 
Then, there exist and such that for all , ,
(2.13) 
As a consequence of Proposition 2.4, in this situation, the damped wave equation (1.1) is stable at rate .
Following carefully the steps of the proof, one sees that works, but the proof is not optimized with respect to this parameter, and it is likely that it could be much improved.
One of the main difficulties in understanding the decay rates is that there exists no general monotonicity property of the type “ for all the decay rate associated to the damping is larger (or smaller) than the decay rate associated to the damping ”. This makes a significant difference with observability or controllability problems of the type (1.5).
Assumption (2.12) is only a local assumption in a neighbourhood of (even if it is stated here globally on ). Far from this set, i.e. on each compact set for , the constant can be choosen uniformly, depending only on , and not on . Hence, somehow quantifies the vanishing rate of the damping function .
An interesting situation is when the smooth function vanishes like in smooth local coordinates, for some . In this case, Assumption (2.12) is satisfied for any , and the associated damped wave equation (1.1) is stable at rate for any . This shows that the lower bound given by Theorem 2.5, as well as the decay rate , are sharp in general. This phenomenon had already been remarked by Burq and Hitrik in [BH07] in the case where is invariant in one direction.
Typical smooth functions not satisfying Assumption (2.12) are for instance functions vanishing like . We do not have any idea of the decay rate achieved in this case (except for the a priori bounds and ).
Theorem 2.6 generalises the result of [BH07], which only holds if is assumed to be invariant in one direction. Our proof is based on ideas and tools developped in [Mac10, AM11] and especially on twomicrolocal semiclassical measures. One of the key technical points appears in Section 13: we have to construct, for each trapped direction, a cutoff function invariant in that direction and adapted to the damping coefficient . We do not know how to adapt this technical construction to tori of higher dimension, ; hence we do not know whether Theorem 2.6 holds in higher dimension (although we have no reason to suspect it should not hold). Only in the particular case where is invariant in directions can our methods (or those of [BH07]) be applied to prove the analogue of Theorem 2.6.
Note that if GCC is satisfied, one has (on a general compact manifold ) for some and all the estimate
(2.14) 
instead of (2.13). Estimate (2.14) is in turn equivalent to uniform stabilization (see [Hua85] together with Lemma 4.6 below).
Remark 2.7.
As a consequence of Theorem 2.6 on the torus, we can deduce that the decay rate also holds for Equation (1.1) if is the square, one takes with Dirichlet or Neumann boundary conditions, and the damping function is smooth, vanishes near and satisfies Assumption (2.12). First, we extend the function as an even (with respect to both variables) smooth function on the larger square , and using the injection , as a smooth function on , still satisfying (2.12). Moreover, (resp. ) on can be identified as the closed subspace of odd (resp. even) functions of (resp. ) on . Using again the injection , it can also be identified with a closed subspace of . The estimate
is thus also true on the square for Dirichlet or Neumann boundary conditions. In particular, this strongly improves the results of [LR05].
The lower bound of Theorem 2.5 can be similarly extended to the case of a square with Dirichlet or Neumann boundary conditions, implying that the rate is optimal if GCC is strongly violated.
2.2.2 The case of discontinuous damping functions
Appendix B (by Stéphane Nonnenmacher) deals with the case where is the characteristic function of a vertical strip, i.e. , for some and . Due to the invariance of in one direction, the spectrum of the damped wave operator splits into countably many “branches” of eigenvalues. This structure of the spectrum is illustrated in the numerics of [AL03, AL12].
The branch closest to the imaginary axis is explicitly computed, it contains a sequence of eigenvalues such that and . This result is in agreement with the numerical tests given in [AL12].
As a consequence, for any and , the strip contains infinitely many poles of the resolvent , so item (2.8) in Proposition 2.4 implies the following obstruction to the stability of this damped system :
Corollary 2.8.
For any , the damped wave equation (1.1) on with the damping function (B.1) cannot be stable at the rate .
The same result holds on the square with Dirichlet or Neumann boundary conditions.
More precisely, in this situation, Lemma 4.6 and [BD08, Proposition 1.3] yield that , for some (with the notation of [BD08] where denotes the best decay rate).
This corollary shows in particular that the regularity conditions in Theorem 2.6 cannot be completely disposed of if one wants a stability at the rate for small .
2.3 Some related open questions
The various results obtained in this article lead to several open questions.

In the case where is the characteristic function of a vertical strip, our analysis shows that the best decay rate lies somewhere between and , but the “true” decay rate is not yet clear.

It would also be interesting to investigate the spectrum and the decay rates for damping functions invariant in one direction, but having a less singular behaviour than a characteristic function. In particular, is it possible to give a precise link between the vanishing rate of and the decay rate?

In the general setting of Section 2.1 (as well as in the case of the damped wave equation on the torus), is the a priori upper bound for the decay rate optimal?

For smooth damping functions vanishing like , Theorem 2.6 yields stability at rate for all . Is the decay rate reached in this situation? Can one find a damping function such that the decay rate is exactly ?
Part II Resolvent estimates and stabilization in the abstract setting
3 Proof of Theorem 2.3 assuming Proposition 2.4
To prove Theorem 2.3, we express the observability condition as a resolvent estimate (also known as the Hautus test), as introduced by Burq and Zworski [BZ04], and further developed by Miller [Mil05] and Ramdani, Takahashi, Tenenbaum and Tucsnak [RTTT05]. For a survey of this notion, we refer to the book [TW09, Section 6.6].
4 Proof of Proposition 2.4
Our proof strongly relies on the characterization of polynomially stable semigroups, given in [BT10, Theorem 2.4], which can be reformulated as follows.
Theorem 4.1 ([Bt10], Theorem 2.4).
Let be a bounded semigroup on a Hilbert space , generated by . Suppose that . Then, the following conditions are equivalent:
(4.1) 
(4.2) 
Let us first describe some spectral properties of the operator defined in (2.4).
Lemma 4.2.
The spectrum of contains only isolated eigenvalues and we have
with .
Moreover, the operator is an isomorphism from onto if and only if . If this is satisfied, we have
(4.3) 
The localization properties for the spectrum of , stated in the first part of this lemma are illustrated for instance in [AL03] or [AL12].
This Lemma leads us to introduce the spectral projector of on , given by
where denotes a positively oriented circle centered on with a radius so small that is the single eigenvalue of in the interior of . We set and equip this space with the norm
and associated inner product. This is indeed a norm on since is equivalent to .
Besides, we set with domain . A first remark is that , so that .
The remainder of the proof consists in applying Theorem 4.1 to the operator in . We first check the assumptions of Theorem 4.1 and describe the solutions of the evolution problem (2.4) (or equivalently (2.3)).
Lemma 4.3.
As a consequence, we can apply Theorem 4.1 to the semigroup generated by . The proof of Proposition 2.4 will be achieved when the following lemmata are proved.
Lemma 4.6.
Proof of Lemma 4.2.
As has compact resolvent, its spectrum contains only isolated eigenvalues. Suppose that , then we have, for some ,
and in particular
(4.7) 
with .
Suppose that , then, this yields . Following [Leb96], taking the inner product of this equation with yields . Hence, either , or . In the first case, , i.e. , and . This yields (and the other inclusion is clear). In the second case, is an eigenvector of associated to the eigenvalue and satisfies , which is absurd, according to Assumption (2.5). Thus, .
Now, for a general eigenvalue , taking the inner product of (4.7) with yields
(4.8) 
If , then, the second equation of (4.8) together with gives
If , then, the first equation of (4.8) together with gives , which yields
Following [Leb96], we now give the link between and for . Taking , and , we have
(4.9) 
As a consequence, we obtain that is invertible if and only if is invertible, i.e. if and only if . Moreover, for such values of , System (4.9) is equivalent to
which can be rewritten as (4.3). This concludes the proof of Lemma 4.2. ∎
Proof of Lemma 4.3.
Let us check that is a maximal dissipative operator on [Paz83]. First, it is dissipative since, for ,
Next, the fact that is onto is a consequence of Lemma 4.2. Hence, for all , there exists such that . Applying to this identity yields , so that is onto. According to the LumerPhillips Theorem (see for instance [Paz83, Chapter 1, Theorem 4.3]) generates a contraction semigroup on . Then, Formula (4.4) directly comes from the linearity of Equation (2.4) (or equivalently (2.3)) together with the decomposition of the initial condition . ∎