Temps de vie des solutions d'un problème de Cauchy non linéaire

We solve the global Cauchy problem, with small initial data, in the space of the holomorphic functions with respect to t and Gevrey class with respect to x. We establish the existence and the stability of the solution to Cauchy problem with nul initial data without hyperbolicity hypothesis. In the stationary case, we give estimates of life span of the solutions with respect to size of the initial data.     

Analyse spectrale et singularités d'un problème de transmission non coercif

This Note is devoted to the spectral analysis of an unbounded operator associated with a non-coercive transmission problem. Using an integral equation method, we show that, if the interface is regular, this operator is selfadjoint and has compact resolvent. If the interface has a corner, the study of the singularities using Mellin transform allows us to derive a necessary and sufficient condition on the contrast between the two media for selfadjointness. If the operator is not selfadjoint, a characterization of its selfadjoint extensions is given.     

Existence de solutions d'un problème de couplage fluide-structure bidimensionnel instationnaire

We study the well-posedness of a steady stale problem: we consider a two-dimensional viscous incompressible flow, which is modeled bv the Navier-Stokes equations. The structure is a rigid moving disc. The fluid domain depends on time and is defined by the position of the structure, itself resulting from a stress distribution coming from the fluid. The problem is then nonlinear and the equations we deal with are coupled. We prove its local solvability in time though two fixed point procedures.     

Contrôlabilité exacte d'un problème de transmission

By using HUM, we prove the exact controllability for the transmission wave equation in the case a₁<a₂.     

Stabilisation d'un système hybride par un feedback non linéaire,non monotone

For a hybrid system composed of a cable with masses at both ends, we prove strong stability for a nonlinear and nonmonotone feedback law applied at one end.     

Éléments finis non conformes pour un problème de contact unilatéral avec frottement

In a previous study, we considered it nonconforming finite element method for a frictionless contact problem. The aim of this note is to establish the convergence of the method for a frictional contact problem between two deformable bodies.     

Homogénéisation d'un problème de capillarité

We study the behavior of the height of a liquid contained in a vertical cylindrical tube where are immerged a periodic distribution (period ɛ) of cylindrical fibers with radius r < ɛ and whose material-liquid coefficient is λ. It is showed that the homogenized problem depends on the limit of λr/ɛ² as (ɛ, λ, r) → (0,0, 0). The second problem is when the border is made of two alternate periodic bands, the first of length r being incrusted in the second of length ɛ − r. Here, the homogenized problem depends on the limit of r/ɛ as (, r) → (0, 0). For this, we use epi-convergence techniques.     

Sur un type de problème singulier au sein d'un groupement

Résumé Origine épistémologique de cette étude: on envisage, dans un groupement, des problèmes infiniment voisins d'un problème impossible. A ce dernier titre, est retenue la recherche de surfaces vérifiant · dM=0 pour V non nul. Intérêt du guide offert en ce cas par deux règles heuristigues, ayant la constructivité, la stabilité, pour object respectifs. Usage, plus efficace, de divers recours à des groupements de problèmes. Je veux, avant toute chose, rendre hommage au professeur Enrico Bompiani pour son oeuvre mathématique ample et belle. Je m'excuse de m'en écarter beaucoup ici, en reprenant une étude qui me tenait en échec depuis 1937. Je commence par un cas un peu simplifié, livrant un relais vers le cas général. Je terminerai par des remarques relatives aux règles heuristiques, grace auxquelles j'ai finalement dominé le cas général. à M. Enrico Bompiani pour son Jubilé scientifique.     

Modélisation d'un problème de diffusion avec pyrolyse et ablation

We discuss existence and uniqueness for the solutions of an unusual parabolic quasilinear problem, coming from a heat conduction model problem (atmosphere re-entry of a body) with pyrolysis and weak ablation.     

Sur un problème de Steiner

Sans résumé     

Caractère bien posé du problème de Cauchy pour le système de Bérenger

The Bérenger perfectly matched layer is used in computational electromagnetism as an absorbing layer in scattering problems. It raises delicate mathematical issues. In this Note we show, for regular data, the existence and uniqueness of strong solutions to the Cauchy problem derived from the PML method. The result is presented in the 2-D case. The key to the proof is an appropriate control of a mixed H¹- L² norm of the solution by the same norm of the initial data. Beside a paper is in preparation about extensions of this results (L² estimates, 3-D case) (see also [5]).     

Stabilisation frontière de problèmes de Ventcel

We study Ventcel's problems for the wave equation and the isotropic linear elastodynamic system. The boundary observability and the exact controlabillity were established in [3]. We prove the energy decay for the elastodynamic system with stationary Ventcel's conditions. We also give a boundary feedback leading to arbitrarily large energy decay rates for the elastodynamic system with evolutive Ventcel's conditions. A spectral study proves, finally, that the “natural” feedback is not sufficient for the exponential decay in the case of the wave equation with Ventcel's conditions.     

Existence et unicité de solutions d'un problème de couplage fluide-structure bidimensionnel stationnaire

We study the well-posedness of a steady-state problem: we consider a two-dimensional viscous incompressible flow, which is modeled by the Stokes equations. The structure is monodimensional and the equations describing the behaviour of elastic medium are beam equations. The fluid domain is defined by the shape of the structure function, itself resulting from a stress distribution coming from the fluid. The problem is thus nonlinear and the equations we deal with are coupled. We prove its solvability through Schauder's fixed point theorem.     

Analyse d'un élément mixte pour le problème de Stokes

Résumé Nous considérons quelques extensions des résultats classiques d'estimation d'erreur pour les problèmes mixtes. Nous étudions en particulier l'effet de l'emploi d'une approximation non conforme et celui de la décomposition de la formeb(·,·) enb ₁(·,·) etb ₂(·,·). Nous employons cette domposition pour affiner certaines estimations dans le cas où la formea(·,·) n'est pas coercive dans la bonne norme. Nous considérons aussi des résultats de dualité dans le cas non conforme.

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Analysis of a mixed finite element for the Stokes problemI. General results

Summary We consider some extensions of the classical error estimates for mixed problems. We take into account, in particular, nonconforming approximations and the splitting of the bilinear formb(·,·), intob ₁(·,·) andb ₂(·,·). We use this last fact to sharpen some estimates in a case where the forma(·,·) does not have coercivity in the right norm. We also consider duality results in the nonconforming case.

Ce travail a été réalisé grâce à des subventions du fonds FCAR et du CRSNG.     

Sur un problème de Lord Rayleigh

Sans résumé     

Caractère multiplicatif d'un module de Fredholm

L'objectif essential de cet article est de définir un accouplement

Sur un problème de Rényi

Let (n) be the number of all prime divisors ofn and (n) the number of distinct prime divisors ofn. We definev _q (x)=|{nx(n)–(n)=q}|. In this paper, we give an asymptotic development ofv _q (x); this improves on previous results.