We show that we can reconstruct two coefficients of a wave equation by a single boundary measurement of the solution. The identification and reconstruction are based on Krein’s inverse spectral theory for the first coefficient and on the Gelfand−Levitan theory for the second. To do so we use spectral estimation to extract the first spectrum and then interpolation to map the second one. The control of the solution is also studied.

In this paper we consider the problem of estimating the singular support of the Green’s function of the wave equation by using ambient noise signals recorded by passive sensors. We assume that noise sources emit stationary random signals into the medium which are recorded by sensors. We explain how the cross correlation of the signals recorded by two sensors is related to the Green’s function between the sensors. By looking at the singular support of the cross correlation we can obtain an estimate...

We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not...

We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not...

On étudie un opérateur de la forme $-\Delta +V$ sur ${ℝ}^d$, où $V$ est un potentiel admettant plusieurs pôles en $a/r^2$. Plus précisément, on démontre l’estimation de résolvante tronquée à hautes fréquences, classique dans les cas non-captifs, et qui implique l’effet régularisant standard pour l’équation de Schrödinger correspondante. La preuve est basée sur l’introduction d’une mesure de défaut micro-locale semi-classique. On démontre également, dans le même contexte, des inégalités de Strichartz pour l’équation de Schrödinger....

Ingham [6] ha migliorato un risultato precedente di Wiener [23] sulle serie di Fourier non armoniche. Modificando la sua funzione di peso noi otteniamo risultati ottimali, migliorando precedenti teoremi di Kahane [9], Castro e Zuazua [3], Jaffard, Tucsnak e Zuazua [7] e di Ullrich [21]. Applichiamo poi questi risultati a problemi di osservabilità simultanea.

We mainly study initial boundary value problems for the Degasperis-Procesi equation on the half line and on a compact interval. By the symmetry of the equation, we can convert these boundary value problems into Cauchy problems on the line and on the circle, respectively. Applying thus known results for the equation on the line and on the circle, we first obtain the local well-posedness of the initial boundary value problems. Then we present some blow-up and global existence results for strong solutions....

On démontre dans cet article des versions probabilistes des injections de Sobolev sur une variété riemannienne compacte, $(M,g)$. Plus précisément on démontre que pour des mesures de probabilité naturelles sur l’espace $L^2\left(M\right)$, presque toute fonction appartient à tous les espaces $L^p\left(M\right)$, $p\<+\infty$. On donne ensuite des applications à l’étude des harmoniques sphériques sur la sphère ${𝕊}^d$ : on démontre (encore pour des mesures de probabilité naturelles) que presque toute base hilbertienne de $L^2\left({𝕊}^d\right)$ formée d’harmoniques sphériques...

We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension 3 with linear multiplicative spatially homogeneous gaussian noise that is white in time. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison,...

Inertial manifold with delay (IMD) for dissipative systems of second order in time is constructed. This result is applied to the study of different asymptotic properties of solutions. Using IMD, we construct approximate inertial manifolds containing all the stationary solutions and give a new characterization of the K-invariant manifold.

These past few years, new types of computational architectures based on graphics processors have emerged. These technologies provide important computational resources at low cost and low energy consumption. Lots of developments have been done around GPU and many tools and libraries are now available to implement efficiently softwares on those architectures.This article contains the two contributions of the mini-symposium about GPU organized by Loïc Gouarin (Laboratoire de Mathématiques d’Orsay),...

We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations.

We establish an upper bound on the Kolmogorov’s entropy of the locally compact attractor for strongly damped wave equation posed in locally uniform spaces in subcritical case using the method of trajectories.

We consider a potential type perturbation of the three dimensional wave equation and we establish a dispersive estimate for the associated propagator. The main estimate is proved under the assumption that the potential $V\ge 0$ satisfies $$\left|V\left(x\right)\right|\le \frac{C}{{\left(1+\left|x\right|\right)}^{2+{ϵ}_0}},$$ where ${ϵ}_0>0$.