Mathematical general relativity, compressible fluids, and more

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Seminar on

Mathematical General Relativity

Organizers: Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Univ. Pierre et Marie Curie) Gabriele Veneziano (Collège de France)

With the financial support of  the Fondation des Sciences Mathématiques de Paris

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Monday March 28, 2011

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie

Lecture rooms: 15/25 101  (morning) and 15-/25 102 (afternoon)

11h  Rafe MAZZEO (Stanford) The constraint equation and cylindrical ends

Abstract. This talk will give a closer look at some of the results discussed in last month’s seminar by Piotr Chrusciel concerning our joint work on the existence and classification of solutions of the constraint equations on manifolds with cylindrical ends. I will also describe some new work, with Akutagawa and Carron, concerning the Yamabe equation on stratified spaces, and some regularity theorems for all of these problems.

14h  Jared SPECK (Princeton and MIT) The global stability of the Minkowski spacetime solutions to the Einstein-nonlinear electromagnetic system in wave coordinates

Abstract. The Einstein-nonlinear electromagnetic system is a coupling of the Einstein field equations of general relativity to nonlinear electromagnetic field equations. In this talk, I will discuss the family of covariant electromagnetic models that satisfy the following criteria: i) they are derivable from a sufficiently regular Lagrangian, ii) they reduce to the familiar Maxwell model in the weak-field limit, and iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. I will mention several specific electromagnetic models that are of interest to researchers working in the foundations of physics and in string theory. I will then discuss my main result, which is a proof of the global nonlinear stability of the 1 + 3 dimensional Minkowski spacetime solution to the coupled system. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in a wave coordinate gauge. The analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows one to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. The analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor that I develop. Throughout the analysis, I work directly with the electromagnetic fields, thus avoiding the introduction of electromagnetic potentials.

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Seminar on

Mathematical General Relativity

Organizers: Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Univ. Pierre et Marie Curie) Gabriele Veneziano (Collège de France)

With the financial support of  the Fondation des Sciences Mathématiques de Paris

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Monday March 21, 2011

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie

Lecture rooms: 15/25  1-03 (morning) and 1-02 (afternoon)

11h   Ghani Zeghib (Ecole Normale Supérieure, Lyon) Sur les métriques riemanniennes dégénérées

Abstract. Une métrique riemannienne dégénérée sur une variété est un tenseur qui est une forme quadratique positive (non nécessairement définie) sur l’espace tangent de tout point de la variété. Le noyau de cette forme est un champ de plans(discontinue, e.g. de dimension variable). Le cas riemannien correspond au cas où ce champ est trivial. Le cas le plus proche est celui où ce noya est partout de dimension 1. On appellera une telle structure métrique de lumière. En fait, une métrique sous- riemannienne (de type contact) sur est essentiellement une métrique de lumière sur le fibré cotangent. On se pose ici des questions autour de la rigidité locale d’une telle structure, e.g. du fait qu’elle soit une G-structure de type fini au sens de Cartan.

14h   Willie Wong (Cambridge University, UK) A local characterization of Kerr-Newman spacetimes and some applications

Abstract. In 2000, M. Mars gave a local tensorial characterisation of Kerr space-times based on earlier work of W. Simon. This characterisation was recently used by A. Ionescu and S. Klainerman in their program to study black hole uniqueness (in the vacuum case) without the assumption that the space-time is real analytic. In this talk I will describe a generalisation of the characterisation that applies to electro-vacuum space-times, and survey some results making use of it. Parts of this work is based on joint work with Pin Yu.

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Seminar on

Mathematical General Relativity

Organizers: Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Univ. Pierre et Marie Curie) Gabriele Veneziano (Collège de France)

With the financial support of  the Fondation des Sciences Mathématiques de Paris

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Monday March 7, 2011

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie

Lecture rooms: 15/25  1-03 (morning) and 1-02 (afternoon)

11h   Jacques Smulevici (AEI, Potsdam)  Weakly regular T2 symmetric spacetimes

Abstract. Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2–symmetry, establish the global well-posedness of the initial value problem for Einstein’s field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity assumptions are the minimal ones allowing to give a meaning to the Einstein equations under the assumed symmetry and to solve the initial value problem. First of all, we introduce a frame adapted to the symmetry in which each Christoffel symbol can be checked to belong to Lp for some p. We identify certain cancellation properties taking place in the expression of the Riemann and Ricci curvatures, and this leads us to a reformulation of the initial value problem for the Einstein field equations when the initial data set has weak regularity. Second, we investigate the future development of a weakly regular initial data set. We check that the area R of the orbits of symmetry must grow to infinity in the future timelike directions, and we establish the existence of a global foliation by the level sets of the function R. Our weak regularity assumptions only require that R is Lipschitz continuous while the metric coefficients describing the initial geometry of the orbits of symmetry are in the Sobolev space H1 and the remaining coefficients have even weaker regularity. We develop here the compactness arguments required to cover the natural level of regularity associated with the energy of the system of partial differential equations determined from Einstein’s field equations. This is a joint work in collaboration with P.G. LeFloch (Paris).

14h   Erwann Delay (Univ. Avignon) Recollement des TT tenseurs

Abstract. La méthode de Corvino-Schoen permet de recoller localement et de façon lisse deux solutions des équations de contraintes. Nous verrons que l’on peut d’une façon générale recoller deux éléments du noyau de certains opérateurs à symbole surjectif et non injectif. Par exemple on peut recoller deux champs de vecteurs à divergence nulle ou deux TT-tenseurs. On en déduit par exemple que sur toute boule riemannienne ouverte, l’ensemble des TT-tenseurs lisses à support compact est de dimension infinie. On simplifie ainsi la construction de données initiales CMC.