Research Interests:
Research Interests:
Research Interests:
... Lions a remarqué qu'une hypothèse d'unicité entraîne la densité de l'espace F des états contrôlables dans l'espace d'énergie . ... d'Holmgren entraînant l'unicité, l'idée étant d'utiliser... more
... Lions a remarqué qu'une hypothèse d'unicité entraîne la densité de l'espace F des états contrôlables dans l'espace d'énergie . ... d'Holmgren entraînant l'unicité, l'idée étant d'utiliser conjointement le théorème de Cauchy-Kowalewski précisé par Leray, l'argument de déformation ...
Research Interests:
Research Interests:
Research Interests:
Research Interests:
... for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Croisille, Jean-Pierre: Diffraction by an immersed elastic wedge / Jean-Pierre Croisille; Gilles Lebeau. ... Duplication of this publication or parts thereof is permitted only under... more
... for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Croisille, Jean-Pierre: Diffraction by an immersed elastic wedge / Jean-Pierre Croisille; Gilles Lebeau. ... Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of ...
Research Interests:
We consider the two and three-dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as t->∞ . First... more
We consider the two and three-dimensional system of linear thermoelasticity in a bounded smooth domain with Dirichlet boundary conditions. We analyze whether the energy of solutions decays exponentially uniformly to zero as t->∞ . First of all, by a decoupling method, we reduce the problem to an observability inequality for the Lamé system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component. We show that when the domain is convex, the decay rate is never uniform. In fact, the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary. We also show that, in three space dimensions, the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement. In two space dimensions we prove a sufficient (and almost necessary) condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays, under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents. We also give an example, due to D. Hulin, in which these geometric properties hold. In three space dimensions we indicate (without proof) how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold. In two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite-dimensional subspace of solutions except when the domain is a ball or an annulus. Finally we discuss some closely related controllability and spectral issues.
Research Interests:
Research Interests:
... Code Matière AMS. 35 L 05, 35 S 15. 1. Introduction On s' intéresse ici à un problème modèle de stabilisation pour l'équation des ondes. Soit (M, g) une variété C riemannienne compacte, connexe, à bord CdM, A= Ag le... more
... Code Matière AMS. 35 L 05, 35 S 15. 1. Introduction On s' intéresse ici à un problème modèle de stabilisation pour l'équation des ondes. Soit (M, g) une variété C riemannienne compacte, connexe, à bord CdM, A= Ag le laplacien sur M pour la métrique g, et a (x) G C (M, R+). ...
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
For the observation or control of solutions of second-order hyperbolic equation in $\ mathbb {R} _t\ times\ Omega $, Ralston's construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. 807823] showed that... more
For the observation or control of solutions of second-order hyperbolic equation in $\ mathbb {R} _t\ times\ Omega $, Ralston's construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. 807823] showed that it is necessary that the region of control meet every ...
Research Interests:
SUPG-stabilized finite element formulations of compressible Euler equations based on the conservation and entropy variables are investigated and compared. The formulation based on the conservation variables consists of the formulation... more
SUPG-stabilized finite element formulations of compressible Euler equations based on the conservation and entropy variables are investigated and compared. The formulation based on the conservation variables consists of the formulation introduced by Tezduyar and Hughes plus a shock ...
Research Interests:
Research Interests:
Research Interests:
In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2(M)$, almost every function belong to all spaces $L^p(M)$,... more
In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold $(M,g)$. More precisely, we prove that for natural probability measures on $L^2(M)$, almost every function belong to all spaces $L^p(M)$, $p<+\infty$. We then give applications to the study of the growth of the $L^p$ norms of spherical harmonics on spheres $\mathbb{S}^d$: we prove (again for natural probability measures) that almost every Hilbert base of $L^2(\mathbb{S}^d)$ made of spherical harmonics has all its elements uniformly bounded in all $L^p(\mathbb{S}^d), p<+\infty$ spaces. We also prove similar results on tori $\mathbb{T}^d$. We give then an application to the study of the decay rate of damped wave equations in a frame-work where the geometric control property on Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated for a measure 0 set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the $H^1$-supercritical wave equation, for which we prove that for almost all initial data, the weak solutions are strong and unique, locally in time.
We prove a sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold $(M,g)$. The proof includes a detailed study of the spectral theory of the associated operator.
Research Interests:
Research Interests:
We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $H^1_0(\Omega) \times L^2(\Omega)$ for any smooth (compact) domain $\Omega \subset \mathbb{R}^3$. The main ingredient in the... more
We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $H^1_0(\Omega) \times L^2(\Omega)$ for any smooth (compact) domain $\Omega \subset \mathbb{R}^3$. The main ingredient in the proof is an $L^5$ spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.