Haut PDF Decay rates for the damped wave equation on the torus

Decay rates for the damped wave equation on the torus

Decay rates for the damped wave equation on the torus

two-dimensional sphere and {b > 0} ∩ N ε = ∅, where N ε is a neighbourhood of an equator of S 2 . 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 ´etude plus approfondie de la localisation spectrale et des taux de d´ecroissance de l’´energie pour des donn´ees r´eguli`eres doit faire intervenir la dynamique globale du flot g´eod´esique g´en´eralis´e sur M . Les th´eor`emes [LR97, Th´eor`eme 1] et [LR97, Th´eor`eme 2] ne four- nissent donc que les bornes a priori qu’on peut obtenir sans aucune hypoth`ese sur la dynamique, en n’utilisant que les in´egalit´es de Carleman qui traduisent “l’effet tunnel”.”
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Exponential decay for the damped wave equation in unbounded domains

Exponential decay for the damped wave equation in unbounded domains

If Ω is a bounded manifold, the uniform positivity of hγi T (x, ξ) in Σ for some T > 0 implies that the exponential decay (1.5) holds, as shown in the celebrated articles [30], [3] and [4] of Bardos, Lebeau, Rauch and Taylor. The assumption that there exists T > 0 such that hγi T (x, ξ) > 0 in Σ is called the geometric control condition. The article [20] underlines in addition the importance of the value of min (x,ξ)∈Σ hγi T (x, ξ) in order to control the rate of decay of the high frequencies. In the case of an unbounded manifold, two situations have been investigated. First, some authors have considered the free wave equation (1.1) in an exterior domain (with γ ≡ 0 or γ > 0 only on a compact subset the exterior domain). They have shown that the local energy decays to zero in the sense that, under suitable assumptions, the energy of any solution escapes away from any compact set, see [18], [26] and [2] and the references therein. Secondly, several works have studied the damped wave equation in an unbounded manifold and with a non-linearity, but assuming that the damping satisfies γ(x) ≥ α > 0 outside a compact set, see [33], [10], [9] and [16].
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Energy decay for the damped wave equation under a pressure condition

Energy decay for the damped wave equation under a pressure condition

PRESSURE CONDITION EMMANUEL SCHENCK Abstract. We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dy- namical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.
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DECAY FOR THE KELVIN-VOIGT DAMPED WAVE EQUATION: PIECEWISE SMOOTH DAMPING

DECAY FOR THE KELVIN-VOIGT DAMPED WAVE EQUATION: PIECEWISE SMOOTH DAMPING

implies the exponential stabilization, which holds for the wave equation with localized viscous damping a(x)∂ t u (see [ LR05 , Zh18 ] for results using the multiplier methods) 1.2. The main result. To state our main result, we first make some geometric assumptions. Let Ω ⊂ R d with d ≥ 2. We consider the piecewise smooth damping a ∈ C ∞ (Ω 1 ), a| Ω\Ω 1 = 0, such that there exists α 0 > 0,

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One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates

then it is extended to the full X 2 by a density argument, cf. [1]. As regards UGAS, it is proved that under a linear cone condition in addition to the damping one (i.e., there exist positive a, b such that as 2 ≤ sσ(s) ≤ bs 2 for every s ∈ R), and exponential stability is achieved, i.e., one can choose β(s, t) = Cse −µt for some positive constants C, µ, cf. [27] for instance. If the linear cone condition only holds in a neighborhood of zero then exponential stability cannot hold in general, as shown in [27] where σ is chosen as a saturation function, i.e., such as σ(s) = arctan(s). Besides the linear cone condition, several results establishing UGAS have been obtained, cf. [18, 20, 21, 28] where σ verifies a linear cone condition for large s and is either of polynomial type or weaker than any polynomial in a neighborhood of the origin, see also [27] for a extensive list of references. It has to be noticed that many of these studies deal with wave equations in dimension not necessarily equal to one and, for all of them, the estimates are obtained by refined arguments based on the multiplier method or highly nontrivial Lyapunov functionals.
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QUANTITATIVE RATES OF CONVERGENCE TO EQUILIBRIUM FOR THE DEGENERATE LINEAR BOLTZMANN EQUATION ON THE TORUS

QUANTITATIVE RATES OF CONVERGENCE TO EQUILIBRIUM FOR THE DEGENERATE LINEAR BOLTZMANN EQUATION ON THE TORUS

An equation related to ( 1 ), the 1d Goldstein-Taylor type model, has been studied in [ 7 ] where the authors do get explicit rates via comparing this equation to a damped wave equation for which explicit rates were obtained by Lebeau in [ 26 ]. The case where V is unbounded is treated in [ 22 ] by Han-Kwan and L´eautaud, where the authors study linear Boltzmann type equations for a general class of collision operators and external confining potential terms on a closed, smooth, connected and compact Riemannian manifold M (and in particular the torus). In this context, the authors indentify geometric control conditions in the natural phase space T ∗ M (similar to Definition 2 in the case M = T d ) allowing to completely
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Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation

Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation

where Ω is a bounded domain in R n , n ∈ N ∗ , with smooth boundary ∂Ω := Γ. While there is a great number of papers regarding the Kirchhoff equation subject to a frictional damping, in contrast, there is just a few number of papers concerned with the Kirchhoff equation subject to a dissipation given by a memory term. We are aware solely the paper [ 22 ], where stronger conditions were considered on the kernel of the memory term. The assumption given in ( 1.7 ), firstly introduced in [ 20 ], is much more general and allows us to consider a wide class of kernels, and consequently, get new and optimal decay rate estimates then those ones considered previously in the literature for the linear viscoelastic wave equation. In the present paper, we combine techniques given in [ 20 ] with new ingredients inherent to the nonlinear character of the Kirchhoff equation ( 1.2 ).
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Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation

Well-posedness and optimal decay rates for the viscoelastic Kirchhoff equation

where Ω is a bounded domain in R n , n ∈ N ∗ , with smooth boundary ∂Ω := Γ. While there is a great number of papers regarding the Kirchhoff equation subject to a frictional damping, in contrast, there is just a few number of papers concerned with the Kirchhoff equation subject to a dissipation given by a memory term. We are aware solely the paper [ 22 ], where stronger conditions were considered on the kernel of the memory term. The assumption given in ( 1.7 ), firstly introduced in [ 20 ], is much more general and allows us to consider a wide class of kernels, and consequently, get new and optimal decay rate estimates then those ones considered previously in the literature for the linear viscoelastic wave equation. In the present paper, we combine techniques given in [ 20 ] with new ingredients inherent to the nonlinear character of the Kirchhoff equation ( 1.2 ).
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Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric

Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric

From our point of view one of the most efficient approaches to the question of local en- ergy decay is the theory of resonances. Resonances correspond to the frequencies and rates of dumping of signals emitted by the black hole in the presence of perturbations (see [9, Chapter 4.35]). On the one hand these resonances are today an important hope of effectively detecting the presence of a black hole as we are theoretically able to measure the corresponding gravi- tational waves. On the other hand, the distance of the resonances to the real axis reflects the stability of the system under the perturbation: larger distances correspond to more stability. In particular the knowledge of the localization of resonances gives precise informations about the decay of the local energy and its rate. The aim of the present paper is to show how this method applies to the simplest model of a black hole: the De Sitter–Schwarzschild black hole. In the euclidean space, such results are already known, especially for non trapping geome- tries. The first result is due to Lax and Phillips (see their book [15, Theorem III.5.4]). They have proved that the cut-off propagator associated to the wave equation outside an obstacle in odd dimension ≥ 3 (more precisely the Lax–Phillips semi-group Z(t)) has an expansion in terms of resonances if Z(T ) is compact for a given T . In particular, there is a uniform exponential decay of the local energy. From Melrose–Sj¨ ostrand [18], this assumption is true
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Quantitative rates of convergence to equilibrium for the degenreate linear Boltzman equation on the Torus

Quantitative rates of convergence to equilibrium for the degenreate linear Boltzman equation on the Torus

An equation related to ( 1 ), the 1d Goldstein-Taylor type model, has been studied in [7] where the authors do get explicit rates via comparing this equation to a damped wave equation for which explicit rates were obtained by Lebeau in [26]. The case where V is unbounded is treated in [22] by Han-Kwan and L´eautaud, where the authors study linear Boltzmann type equations for a general class of collision operators and external confining potential terms on a closed, smooth, connected and compact Riemannian manifold M (and in particular the torus). In this context, the authors indentify geometric control conditions in the natural phase space T ∗ M (similar to Definition 2 in the case M = T d ) allowing to completely
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Stabilisation of wave equations on the torus with rough dampings

Stabilisation of wave equations on the torus with rough dampings

The second micro-localization procedure has a well established history starting with the works by Laurent [ 30 , 31 ], Kashiwara-Kawai [ 28 ], Sjöstrand [ 47 ], Lebeau [ 33 ] in the analytic context, (see also Bony-Lerner [ 7 ] in the C ∞ framework and Sjöstrand-Zworski [ 49 ] in the semi-classical setting) and in the framework of defect measures by Fermanian–Kammerer [ 20 ], Miller [ 37 , 38 , 39 ], Nier [ 41 ], Fermanian–Kammerer-Gérard [ 22 , 23 , 24 ]. Notice that most of these previous works in the framework of measures dealt with lagrangian or involutive sub-manifolds, and it is worth comparing our contribution with these previous works, in par- ticular [ 41 , 2 ]. Here we are interested in the wave equation while the authors in [ 41 , 2 ] were interested in the Schrödinger equation, and (compared to [ 2 ]) we are dealing with worse quasi- modes (o(h) instead of o(h 2 )). Another difference is that we perform a second microlocalization along a symplectic submanifold (namely {(x = 0, y, ξ = 0, η) ∈ T ∗ T 2 }), while they consider an isotropic submanifold {x = 0} in [ 41 ] or {(x 0 , x 00 , ξ 0 = 0, ξ 00 ) ∈ T ∗ T d } in [ 2 ]. An exception is the note by Fermanian–Kammerer [ 21 ], to which our construction is very close. On the other hand, a feature shared by the present work and [ 41 , 2 ] is that in all cases the analy- sis requires to work at the edges of uncertainty principle and use refinements of some exotic Weyl-Hörmander classes (S 1,1 in [ 41 ], S 0,0 in [ 2 ] and S 1/2,1/2 in the present work), see [ 27 ] and Léautaud-Lerner [ 32 ] for related work. Another worthwhile comparison is with the series of works by Burq-Hitrik [ 12 ] and Anantharaman-Leautaud [ 1 ] on the damped wave equation on the torus when the control domain is arbitrary (in this case (WGCC) is in general not satisfied). However, though both works use some kind of second microlocalisation and deal with the wave equation, in [ 12 , 1 ] the approaches use Schrödinger equations methods (strong quasi-modes) transposed to get wave equations result and consequently leads to much weaker results (polynomial decay v.s. exponential decay) under much weaker assumptions (arbitrary open sets).
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Exponential decay for the Schrödinger equation on a dissipative waveguide

Exponential decay for the Schrödinger equation on a dissipative waveguide

For the dissipative equations, this non-trapping assumption can be replaced by the same damping condition (geometric control condition) on bounded trajectories as in the compact case (where all the trajectories were bounded). This means that the energy of the wave (or at least the contribution of high frequencies) escapes at infinity or is dissipated by the medium. This has been used in [ AK07 ] for a dissipation in the interior of the domain and in [ Alo02 , AK10 ] for dissipation at the boundary, for the free equations on an exterior domain in both cases. See also [ Roy10b ] for the corresponding resolvent estimates, [ BR14 ] for the damped wave equation with a Laplace-Beltrami operator corresponding to a metric which is a long-range perturbation of the flat one, and [ AKV13 ] where the damping condition is not satisfied but the dissipation is stronger. In this paper we consider a domain which is neither bounded nor the complement of a bounded obstacle. In particular, compared to the situations mentioned above, the boundary of the wave- guide is not compact.
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CONCENTRATION OF LAPLACE EIGENFUNCTIONS AND STABILIZATION OF WEAKLY DAMPED WAVE EQUATION

CONCENTRATION OF LAPLACE EIGENFUNCTIONS AND STABILIZATION OF WEAKLY DAMPED WAVE EQUATION

It is known that as soon as the damping b ≥ 0 is non trivial, the energy of every solution converge to 0 as t tends to infinity. On the other hand the rate of decay is uniform (and hence exponential) in energy space if and only if the geometric control condition [ 2 , 5 ] is satisfied. Here we want to explore the question when some trajectories are trapped and exhibit decay rates (assuming more regularity on the initial data). This latter question was previously studied in a general setting in [ 19 ] and on tori in [ 11 , 21 , 1 ] (see also [ 12 , 13 ]) and more recently by Leautaud-Lerner [ 18 ]. According to the works by Borichev-Tomilov [ 3 ], stabilization results for the wave equation are equivalent to resolvent estimates. On the other hand, Theorem 1.1 implies easily (see Section 2.2 ) the following resolvent estimate
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NULL CONTROLLABILITY OF THE STRUCTURALLY DAMPED WAVE EQUATION ON THE TWO-DIMENSIONAL TORUS

NULL CONTROLLABILITY OF THE STRUCTURALLY DAMPED WAVE EQUATION ON THE TWO-DIMENSIONAL TORUS

for some constant C obs > 0 and all v T , w T ∈ L 2 (T 2 ). In order to establish the observability inequality ( 1.38 ), following [ 1 , 4 ], we derive Carleman estimates for the backward heat equation ( 1.34 ) and for the transport equation ( 1.35 ) with the same weights functions. To get “almost sharp” results for the geometry of the control region, the construction of the weight functions in the Carleman estimates turns out to be much more delicate than in [ 4 ]. To derive Theorem 1.2 , we need to prove the existence of a function ψ 0 ∈ C ∞ (T 2 ) such that
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Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

tially understood in [4, 8]. In particular, the existence of a spectral gap was established for all α 6= α ∗ = (2 − d)/2, but its value was not stated for all values of α. J. Denzler and R.J. McCann in [17, 18] formally linearized the fast diffusion flow (considered as a gradient flow of the en- tropy with respect to the Wasserstein distance) in the frame- work of mass transportation, in order to guess the asymptotic behaviour of the solutions of [ 1 ]. This leads to a different functional setting, with a different linearized operator, H α,d .
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Global infinite energy solutions for the cubic wave equation

Global infinite energy solutions for the cubic wave equation

Let us now mention two possible extensions of our result. In the case d = 4 one may expect to get uniqueness by combining the analysis of [8] with the critical H 1 theory for (1.1). One may also expect to include the case s = 0 by elaborating on the arguments developed in [8] to treat this case. It is not clear to us what happens for s < 0 (and in [8] as well). In particular we do not know whether s = 0 is the optimal regularity one may achieve by our approach. Invariant Gibbs measures for dispersive equations were extensively studied (see e.g. [20, 3, 2, 19, 18, 16, 17, 6] ). In these papers the Gibbs measure is combined with a suitable local in time result (which can sometimes be quite involved) to get global existence and uniqueness on the support of the measure. By an extension of the method (using in particular Skorohod and Prokhorov theorems) we use in this paper one may construct a dynamics (without any uniqueness) on the support of a Gibbs measure and prove its invariance. We plan to give several relevant examples of this observation in [7]. We however do not see how to make work such an approach in the context of (1.1). Indeed, the present methods of renormalization of Gibbs measures are restricted to dimensions ≤ 2 (see [3]). Let us also recall that as mentioned above a global existence based on Gibbs measures only works for a very specific choice of the initial distribution. On the other hand, it has of course the advantage to give a quite remarkable dynamical property of the flow.
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1-D non-periodic homogenization for the seismic wave equation

1-D non-periodic homogenization for the seismic wave equation

In the static case, this kind of problem has been studied for a long time and a large number of results have been obtained us- ing the so-called homogenization theory applied to media showing rapid and periodical variations of their physical properties. Since the pioneering work of Auriault & Sanchez-Palencia (1977), nu- merous studies have been devoted either to the mathematical foun- dations of the homogenization theory in the static context (e.g. Bensoussan et al. 1978; Murat & Tartar 1985; Allaire 1992), to applications to the effective static behaviour of composite materials (e.g. Dumontet 1986; Francfort & Murat 1986; Abdelmoula & Marigo 2000; Haboussi et al. 2001a,b), to the application to the heat diffusion (e.g. Marchenko & Khruslov 2005), to porous me- dia (e.g. Hornung 1996), etc. In contrast, fewer studies have been devoted to the theory and its applications in the general dynamical context or to the non-periodic cases. However, one can for example refer to Sanchez-Palencia (1980), Willis (1981), Auriault & Bonnet (1985), Moskow & Vogelius (1997), Allaire & Conca (1998), Fish et al. (2002), Fish & Chen (2004), Parnell & Abrahams (2006), Milton & Willis (2007), Lurie (2009) or Allaire et al. (2009) for the dynamical context, to Briane (1994), Nguetseng (2003) or to Marchenko & Khruslov (2005) for the non-periodic case. Moczo et al. (2002) have also used a kind of local homogenization to take into account interfaces with the finite differences method. The specific case of a long wave propagating in finely layered media has been studied by Backus (1962) and the same results can be extracted from the 0th-order term of the asymptotic expansion im- plied in homogenization theory. Higher order homogenization in the non-periodic case has been studied by Capdeville & Marigo (2007) and Capdeville & Marigo (2008) for wave propagation in strati- fied media, but the extension to media characterized by 3-D rapid variation is not obvious from these works. Indeed, the non-periodic homogenization strategy suggested in Capdeville & Marigo (2007)
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Decay rates for radiative transitions in the Pr IV spectrum

Decay rates for radiative transitions in the Pr IV spectrum

2. Level structure of Pr IV The ground state of Pr IV is [Xe]4f 2 3 H 4 . All of the term energies experimentally deduced so far are listed in the NIST compilation (Martin et al 1978 ) which contains 88 levels identified as belonging to the 4f 2 , 4f5f, 4f6p, 5d 2 even configurations and to the 4f5d, 4f6d, 4f6s, 4f7s, 5d6p odd configurations. This critical compilation was essentially based on the works of Sugar ( 1965 , 1971a ) and Crosswhite et al ( 1965 ) who observed the Pr IV spectrum emitted by sliding-spark discharges and reported line lists covering the wavelength region from 69.1 to 302.1 nm. The level compositions in the 4f 2 configuration were taken from the theoretical analysis of Goldschmidt ( 1968 ) later refined to include additional magnetic interactions by Goldschmidt et al ( 1968 ) and Pasternak ( 1970 ). Theoretical percentages for levels of the 4f6s, 4f5d, 4f6d, 4f6p and 4f5f configurations were taken from Sugar ( 1965 , 1971a ) supplemented by some of his unpublished results (Sugar 1971b ).
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Alfvén Wave Propagation in the Io Plasma Torus

Alfvén Wave Propagation in the Io Plasma Torus

The plasma in the Io torus is close to rotating with Jupiter's 10 ‐hr spin period. This means that the sulfur and oxygen ions experience a considerable centrifugal force that tends to con fine them to the farthest point from the planet's spin axis —the centrifugal equator. For a tilted dipole magnetic field, the centrifugal equator is located approximately 2/3 of the ~10° tilt of the magnetic equator from the jovigraphic equator. The lighter electrons experience a much weaker centrifugal force but are attracted to the positive ions. Thus, under steady state conditions, there is an equilibrium between the ambipolar electric force, the centrifugal force, and the pressure gradient force among all the species in the plasma. The equations of this diffusive equili- brium are straightforward to solve, particularly if we assume the gravity of Jupiter can be neglected and that the ions are isotropic (e.g., see the appendix of Dougherty et al., 2017). For a single ion species, the equations reduce to a simple Gaussian distribution about the centrifugal equator with a scale height that at Jupiter is H (R J ) = 0.64 [T i /A i ] 1/2 , where R J is the radius of Jupiter (71,492 km), T i is the ion temperature (in eV), and A i is the ion atomic mass (in units of the proton mass).
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Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

is the trivial one u ≡ 0. Such a property is very important in Control Theory, as it is equivalent to the approximate controllability for linear PDE, and it is involved in the classical unique- ness/compactness approach in the proof of the stability for a PDE with a localized damping. The UCP is usually proved with the aid of some Carleman estimate (see e.g. [ 46 ]). The UCP for KdV was established in [ 48 ] by the inverse scattering approach, in [ 12 , 40 , 46 ] by means of Carleman estimates, and in [ 5 ] by a perturbative approach and Fourier analysis. For BBM, the study of the UCP is only at its early age. The main reason is that both x = const and t = const are characteristic lines for (1.1). Thus, the Cauchy problem in the UCP (assuming e.g. that u = 0 for x ≤ 0, and solving BBM for x ≥ 0) is characteristic, which prevents from applying Holmgren’s theorem, even for the linearized equation. The Carleman approach for the UCP of BBM was developed in [ 9 ] and in [ 47 ]. Unfortunately, Theorems 3.1-3.4 in [ 9 ] are not correct without further assumptions, as noticed in [ 49 ]. On the other hand, the UCP in [ 47 ] for the BBM-like equation
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