Weakly regular Einstein-Euler spacetimes with Gowdy symmetry. The global areal foliation

We consider weakly regular Gowdy-symmetric spacetimes on T3 satisfying the Einstein-Euler equations of general relativity, and we solve the initial value problem when the initial data set has bounded variation, only, so that the corresponding spacetime may contain impulsive gravitational waves as well as shock waves. By analyzing, both, future expanding and future contracting spacetimes, we establish the existence of a global foliation by spacelike hypersurfaces so that the time function coincides with the area of the surfaces of symmetry and asymptotically approaches infinity in the expanding case and zero in the contracting case. More precisely, the latter property in the contracting case holds provided the mass density does not exceed a certain threshold, which is a natural assumption since certain exceptional data with sufficiently large mass density are known to give rise to a Cauchy horizon, on which the area function attains a positive value. An earlier result by LeFloch and Rendall assumed a different class of weak regularity and did not determine the range of the area function in the contracting case. Our method of proof is based on a version of the random choice scheme adapted to the Einstein equations for the symmetry and regularity class under consideration. We also analyze the Einstein constraint equations under weak regularity.Comment: 36 page

On the existence of stationary splash singularities for the Euler equations

In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.Comment: 19 page

Semi-Riemannian manifolds with distributional curvature

Die vorliegende Magisterarbeit befasst sich mit den Grundlagen der Semi-Riemannschen Geometrie im Fall niedriger Regularität. Genauer gesagt behandeln wir Zusammenhänge bzw. Semi-Riemannsche Metriken die in einem geeigneten lokalen Sobolev Raum liegen. Das Interesse an solchen Geometrien niedrieger Regularität wird durch Anwedungen in der Allgemeinen Relativität motiviert, wo sie zur Beschreibung von Raumzeiten, deren Energie-Materie Inhalt auf einem niedrieger-dimensionalen Bereich konzentriert ist, eingesetzt werden. Solche Raumzeiten werden ihrerseits zur Modellierung dünner Schalen von Materie oder Strahlung, kosmischer Strings und impulsiver Gravitationswellen verwendet. Nachdem wir die nötigen Grundkenntnise (i.e. Distributionen und Sobolev Räume auf Mannigfaltigkeiten) wiederholt haben, diskutieren wir die grundlegenden Begriffe der Semi-Riemannschen Geometrie im Rahmen der distributionellen Geometrie. Wir befassen uns insbesondere mit der Frage nach der minimalen Regularität distributioneller Metriken, welche es erlaubt, den Levi-Civita Zusammenhang bzw. die Krümmung zu definieren. Zum Abschluss, betrachten wir den Spezialfall von Metriken die außerhalb einer Hyperfläche ’glatt’ sind, aber ’Sprünge’ längs dieser Hyperfläche aufweisen. In diesem Kontext prsentieren wir einige ’Sprungformeln’ für die entsprechenden Krümmungsgrößen.This thesis is concerned with a low-regularity formulation of Semi-Riemannian geometry. More precisely, we deal with connections and Semi-Riemannian metrics which belong to some appropriate local Sobolev space. The interest in geometries of low-regularity is motivated by applications in general relativity, where they are used to describe space-times with energy-matter concentration on some lower dimensional region. Such space-times are frequently used to model thin shells of matter and radiation, cosmic strings and impulsive gravitational waves. After collecting all the necessary prerequisites (distributions and Sobolev spaces on manifolds), we discuss the basic notions of Semi-Riemannian geometry within the framework of distributional geometry. In particular, we study what regularity assumptions have to be imposed on a distributional metric, for its Levi-Civita connection resp. curvature to be defined. Finally, we specialize to the case of connections resp. metrics which are ’smooth’ everywhere except on some hypersurface, across which, they suffer a ’jump discontinuity’. In this context we discuss several ’jump formulas’ for the respective curvature quantities

Finite-time singularity formation for angled-crested water waves

We show that the gravity water waves system is locally wellposed in weighted Sobolev spaces which allow for interfaces with corners. No symmetry assumptions are required. These singular points are not rigid: if the initial interface exhibits a corner, it remains a corner but generically its angle changes. Using a characterization of the asymptotic behavior of the fluid near a corner that follows from our a priori energy estimates, we show the existence of initial data in these spaces for which the fluid becomes singular in finite time

Local wellposedness for the free boundary incompressible Euler equations with interfaces that exhibit cusps and corners of nonconstant angle

We prove that free boundary incompressible Euler equations are locally well posed in a class of solutions in which the interfaces can exhibit corners and cusps. Contrary to what happens in all the previously known non-$C^1$ water waves, the angle of these crests can change in time.Comment: 80 page

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).Comment: 43 page

Modèles de fluides et de corps élastiques sur des espaces courbes

In this thesis, we consider some problems related to Einstein-Euler equations of general relativity and to models of nonlinear elasticity in a curved space. In the first part we study the evolution of a perfect fluid in a Gowdy symmetric spacetime satisfying Einstein-Euler equations. We solve the corresponding initial value problem for a given initial data set of bounded variation. Analyzing both future expanding and future contracting spacetimes, we establish the existence of a global foliation by spacelike hypersurfaces when the time coordinate is chosen to coincide with the area of symmetry orbits. The proof relies on a version of the Glimm scheme adapted to the symmetry and the regularity class under consideration. In the future contracting case, we give geometric conditions on the initial data that ensure that the area function asymptotically approaches zero. The second part is dedicated to the study of equations of nonlinear elasticity within the framework of Riemannian manifolds. They generalize, in a natural way, the equations of classical elasticity set in a three-dimensional Euclidean space. Our approach is based on the principle of least action, stating that the deformation of the elastic body arising in response to given external forces minimizes the total energy of the elastic body. From there, we first derive the principle of virtual work and then the corresponding boundary value problem. Finally, we show that the latter admits a solution provided the external forces are sufficiently small.Dans cette thèse, nous abordons quelques problèmes liés aux équations d'Einstein-Euler de la relativité générale et aux modèles d'élasticité non linéaire sur des espaces courbes. Dans la première partie, nous étudions l'évolution d'un fluide parfait dans un espace-temps courbe à symétrie de Gowdy, satisfaisant aux équations d'Einstein-Euler. Nous cherchons à résoudre le problème de Cauchy correspondant pour une donnée initiale à variation bornée. Analysant à la fois les espace-temps en expansion et les espace-temps en contraction, nous démontrons l'existence d'un feuilletage global lorsque la coordonnée de temps coïncide avec l'aire des orbites de symétrie. La démonstration repose sur le schéma de Glimm adapté à la symétrie et la classe de régularité considérées. Dans le cas des espace-temps en contraction, nous donnons les conditions géométriques sur la donnée initiale assurant que la fonction d'aire tend asymptotiquement vers zéro. La deuxième partie est consacrée à l'étude des équations de l'élasticité non-linéaire dans le cadre des variétés riemanniennes. Ces dernières généralisent de manière naturelle les équations de l'élasticité classique posées dans l'espace euclidien de dimension 3. Notre approche est basée sur le principe de moindre action qui affirme que la déformation du corps élastique sous l'action des forces extérieures minimise l'énergie totale du corps élastique. À partir de là, nous déduisons le principe des travaux virtuels, puis le problème aux limites correspondant. Enfin, nous démontrons que ce dernier admet une solution pourvu que les forces soient suffisamment petitesPARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

Self-intersecting interfaces for stationary solutions of the two-fluid Euler equations

We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, possibly with a small gravity constant, that feature a splash singularity. More precisely, in the solutions we construct the interface is a C2,¿ smooth curve that intersects itself at one point, and the vorticity density on the interface is of class C¿. The proof consists in perturbing Crapper¿s family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid. These estimates will also be applied to interface evolution problems in a forthcoming paper.A.E. and N.G. are supported by the ERC Starting Grant 633152. This work is supported in part by the Spanish Ministry of Economy under the ICMAT–Severo Ochoa grant SEV- 2015-0554 and the MTM2017-89976-P. 788250. D.C. was partially supported by the ERC Advanced Grant 788250

The equations of elastostatics in a Riemannian manifold

Abstract To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify). Résumé Dans un premier temps, nous identifions leséquations de l'élastostatique dans une variété riemannienne, qui généralisent celles de la théorie classique de l'élasticité dans l'espace euclidien tridimensionnel. Notre approche repose sur le principe de moindre action, qui affirme que la déformation du corpsélastique sous l'action des forces externes minimise sur l'ensemble des déformations admissibles l'énergie totale du corpsélastique, définie comme la différence entre l'énergie de déformation et le potentiel des forces externes. Sous l'hypothèse que l'énérgie de déformation est une fonction du champ de tenseurs métriques induit par la déformation, on déduit dans un premier temps le principe des travaux virtuels et le problème aux limites associéà partir de l'expression de l'énergie totale du corpsélastique. On démontre ensuite que ce problème aux limites admet une solution si les forces externes sont suffisamment petites (en un sens que nous précisons)