Einstein equations under polarized U(1)\mathbb U(1) symmetry in an elliptic gauge

We prove local existence of solutions to the Einstein--null dust system under polarized $\mathbb U(1)$ symmetry in an elliptic gauge. Using in particular the previous work of the first author on the constraint equations, we show that one can identify freely prescribable data, solve the constraints equations, and construct a unique local in time solution in an elliptic gauge. Our main motivation for this work, in addition to merely constructing solutions in an elliptic gauge, is to provide a setup for our companion paper in which we study high frequency backreaction for the Einstein equations. In that work, the elliptic gauge we consider here plays a crucial role to handle high frequency terms in the equations. The main technical difficulty in the present paper, in view of the application in our companion paper, is that we need to build a framework consistent with the solution being high frequency, and therefore having large higher order norms. This difficulty is handled by exploiting a reductive structure in the system of equations

High-frequency backreaction for the Einstein equations under polarized U(1)\mathbb U(1) symmetry

Known examples in plane symmetry or Gowdy symmetry show that given a $1$-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a non-trivial stress-energy-momentum tensor. We consider this phenomenon under polarized $\mathbb U(1)$ symmetry - a much weaker symmetry than most of the known examples - such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized-$\mathbb U(1)$-symmetric solution to the Einstein-multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large, and appears to be the first construction of such examples with more than two families

Limit equation for vacuum Einstein constraints with a translational Killing vector field in the compact hyperbolic case

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a \bS^1-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from CMC results.Comment: 14 page

The global stability of the Kaluza-Klein spacetime

In this paper we show the classical global stability of the flat Kaluza-Klein spacetime, which corresponds to Minkowski spacetime in \m R^{1+4} with one direction compactified on a circle. We consider small perturbations which are allowed to vary in all directions including the compact direction. These perturbations lead to the creation of massless modes and Klein-Gordon modes. On the analytic side, this leads to a PDE system coupling wave equations to an infinite sequence of Klein-Gordon equations with different masses. The techniques we use are based purely in physical space using the vectorfield method.Comment: 80 page

In-situ SEM study of fatigue crack growth mechanism in carbon black-filled natural rubber

International audienceA micro-tensile testing machine placed in the chamber of a scanning electron microscope is used to perform in-situ fatigue tests on a 43 phr carbon black-filled cis-1,4-polyisoprene rubber; the crack tip is observed in realtime during crack propagation. These observations lead to a detailed description of the crack tip morphology, to the understanding of the fatigue crack growth mechanism and to the microstructural explanation of crack branching phenomenon. Finally these results are related to the great fatigue properties of natural rubber and to strain-induced crystallization

Autour des équations d’Einstein dans le vide avec un champ de Killing spatial de translation.

This thesis aim sat studying vacuum Einstein equations with a space-like Killing vector field. With this symmetry, 3+1 vacuum Einstein equations reduce, in the polarized case, to Einstein equations coupled to a scalar field in 2+ 1 dimensions. In the first part of this thesis, we study the constraint equations in the asymptotically flat case. The constraint equations correspond to computability conditions that the initial data must satisfy. We show the existence of solutions for small data, and we introduce an asymptotic expansion involving quantities which are the 2 dimensional equivalents for the global charges. In the second part, we show the stability of Minkowski space-time with a translation space-like Killing vector field in exponential time with respect to the smallness of initial data. We introduce a family of Ricci flat metrics, and we impose the asymptotic behaviour of our solutions in the exterior of the light cone by picking the right element in the family. This choice allows for the convergence to Minkowski solution in the interior of the light cone. In the last part of this thesis, we study the constraint equations in the compact hyperbolic case. We show the existence of a limit equation associated to the constraint equations.Dans cette thèse, nous étudions les équations d’Einstein dans le vide avec un champ de Killing de translation. En présence de cette symétrie, les équations d’Einstein dans le vide en dimension 3+1 peuvent s’écrire, dans le cas polarisé, comme un système d’équations d’Einstein couplées à un champ scalaire en dimension 2+1. Dans la première partie de cette thèse, nous étudions les équations de contraintes dans le cas asymptotiquement plat. Les équations de contraintes sont des équations de compatibilité qui doivent être satisfaites par les données initiales. Nous montrons l’existence de solutions pour des données assez petites, et introduisons un développement asymptotique faisant intervenir des quantités correspondant aux charges globales. Dans une deuxième partie nous montrons la stabilité de l’espace-temps de Minkowski avec un champ de Killing de translation, en temps exponentiellement grand par rapport à la petitesse de la donnée initiale. Nous travaillons dans les coordonnées d’onde généralisées. Nous introduisons une famille de métriques Ricci plates, et imposons le comportement asymptotique de nos solutions à l’extérieur du cône de lumière en choisissant un élément de cette famille de manière adéquate. Ce choix permet la convergence de nos solutions à l’intérieur du cône de lumière vers la solution de Minkowski. Dans la dernière partie de cette thèse nous étudions les équations de contraintes dans le cas compact hyperbolique. Nous montrons l’existence d’une équation limite associée aux équations de contraintes

Estimation des couples durant un mouvement de squat

International audienceAbstract : A repetitive movement over a long period of time, habituation to poor posture, handling heavy load can lead to musculoskeletal disorders. The most important thing is to prevent MSDs by automating repetitive tasks. But it is also necessary to help a patient with rehabilitation when an MDS has appeared. Regarding the human musculoskeletal system of movement analyses, joint torques developed by muscles are useful to prevent or treat MSDs. The knowledge of these torques is important to evaluate the possibilities of assisting a joint with an orthosis for instance. In this study, we analyze a specific half squat motion considered as a planar movement. Joint torques are estimated with two models-a 3D Opensim and a sagittal Matlab model. Torque variations from the two models are consistent even if the amplitudes differ. This result is attributable to the different anthropometric tables which the two models are based on. Another strategy developed here consists in estimating joint torques without the measured ground reaction forces. In that case, global vertical reaction force is well estimated. The use of the sagittal Matlab model is a simple and efficient way to preliminarily analyze squat trajectories.Un mouvement répété sur une longue période, l'accoutumance à une mauvaise posture, la manipulation de charges lourdes peuvent entraîner des troubles musculosquelettiques (TMS). Il est important de pré-venir les TMS en automatisant les tâches répétitives. Mais il est également nécessaire d'aider un patient en phase de rééducation lorsqu'une TMS apparait. En ce qui concerne le système musculosquelettique humain, les couples articulaires développés par les muscles sont utiles pour prévenir ou traiter les TMS. La connaissance de ces couples est importante pour évaluer les possibilités d'assister une articulation avec une orthèse par exemple. Dans cette étude, nous analysons un mouvement de demi-squat spécifique considéré comme un mouvement planaire. Les couples articulaires sont estimés à l'aide de deux mo-dèles : un modèle 3D sur Opensim et un modèle sagittal sur Matlab. Les variations de couple issues des deux modèles sont cohérentes même si les amplitudes peuvent différer. Ce résultat s'explique par le fait que les modèles se basent sur des tables anthropométriques différentes. Une autre stratégie développée ici consiste à estimer les couples articulaires sans utiliser la mesure des forces de réaction du sol sur les pieds. Dans ce cas, la force de réaction verticale totale est bien estimée. L'utilisation du modèle sagittal Matlab est un moyen simple et efficace d'analyser préalablement les trajectoires de squat