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Seminar on
Mathematical General Relativity
Organizers:
Philippe G. LeFloch (Univ. Pierre et Marie Curie)
Ghani Zeghib (Ecole Normale Supérieure, Lyon)
With the financial support of the ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Thursday November 3, 2011
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Lecture room 15-25 326 (third floor)
14h Charles Boubel (Strasbourg) Germs of Lorentzian metrics and holonomy
Abstract. The holonomy group of a pseudo-Riemannian metric g -so e.g. a Riemannian or a Lorentzian metric- is a subgroup of O(g) which indicates, in a certain sense, how much its Levi-Civita connection fails to be flat. A central task related to those groupes is to determine the list of the subgroups of O(g) arising as holonomy, and for each item, to parametrize the set of corresponding metrics and build global examples (i.e. complete or compact). In the Riemannian case, this work is now done. We will see that, regarding holonomy matters, Lorentzian metrics behave totally differenly from Riemannian ones. I will review works of L. Bérard-Bergery, A. Ikemakhen, T. Leistner, A. Galaev, and myself, that together deal with the local aspect of the question.
15h30 David Parlongue (Nice) Breakdown criteria and extendibility in general relativity
Abstract. We will begin this talk by reviewing a geometric breakdown criterion for Einstein’s vacuum equations introduced by S. Klainerman and I. Rodnianski and various improvements (non-vacuum case, integral conditions, various gauge choices). We will then examine a spacetime localization of these criteria. We will focus on consequences in terms of formation of singularities, extendibility of spacetimes, and local regularity of foliations.
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Seminar on
Mathematical General Relativity
Organizers:
Philippe G. LeFloch (Univ. Pierre et Marie Curie)
Ghani Zeghib (Ecole Normale Supérieure, Lyon)
With the financial support of the ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Thursday September 29, 2011
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Lecture room 15-25 1-01 (first floor)
14h Robert Beig (Vienna) Elastic perturbations of static perfect fluid bodies: Newtonian theory
Abstract. We describe a model for static self-gravitating bodies with both fluid and elastic properties. This should allow for a rigorous existence theory of solutions near a perfect fluid configuration, e.g. ‘putting mountains on a neutron star’.
15h30 Jacques Smulevici (Orsay) Wave equations on asymptotically AdS black hole
Abstract. This is joint work with Gustav Holzelgel. We prove a logarithmic decay estimate for solutions of the linear wave equation on a slowly rotating Kerr-Anti-de-Sitter spacetime. This estimate is expected to be sharp in view of heuristics and numerics from the physics litterature. The underlying reason for the slow decay rate established here can be traced back to a stable trapping phenomenon for asymptotically anti de Sitter black holes near infinity.
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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
Monday June 13, 2011
Institut Henri Poincaré, Paris, Lecture room 314
11h Lars Andersson (Potsdam) Linear fields on Kerr spacetime
Abstract. The problem of nonlinear stability for the Kerr model of a rotating black hole is one of the central problems in general relativity. The analysis of linear fields of spin 0, 1, 2, on the Kerr spacetime is an important model problem for full nonlinear stability. I will report on recent progress on this problem.
14h Yvonne Choquet-Bruhat (IHES, Bures-sur-Yvette) Positive gravitational energy in arbitrary dimensions
Abstract. The most elegant and convincing proof of the positive energy theorem is based on spinors, as did Witten in dimension 3 +1, inspired by heuristic work by Deser and Grisaru originating from supergravity. Our aim here is to present a streamlined and complete proof, that is valid in arbitrary space dimension and uses only spinors on an oriented Riemannian space without referring to spacetime spinors.
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Seminar on Compressible Fluids
Wednesday May 18, 2011
Laboratoire Jacques-Louis Lions
Université Pierre et Marie Curie
4 Place Jussieu, 75258 PARIS
Jussieu campus. Building 15/16. Lecture room 309.
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- 11h – Siddhartha MISHRA (ETH, Zurich) Entropy stable high-order schemes for systems of conservation laws
Abstract. We design arbitrarily high-order schemes for systems of conservation laws that satisfy a discrete version of the entropy inequality. Consequently, these schemes are stable in L2. The proposed schemes are based on a combination of arbitrarily high-order entropy conservative schemes together with numerical diffusion operators. The numerical diffusion operators require an ENO reconstruction of the entropy variables. The resulting schemes are shown to be entropy stable for conservation laws in several space dimensions. Recent work extending these schemes to the fully discrete case and to unstructured meshes based on a shock-capturing space time Discontinuous Galerkin (DG) method will be mentioned. Numerical experiments illustrating the robust performance of the proposed schemes are presented. The talk is based on joint work with U. S. Fjordholm, A. Hiltebrand (ETH, Zurich) and E. Tadmor (University of Maryland, U.S.A).
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Organizers. Frédéric Coquel, Edwige Godlewski, Philippe LeFloch
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Seminar on Compressible Fluids
Wednesday March 30, 2011
Laboratoire Jacques-Louis Lions
Université Pierre et Marie Curie
4 Place Jussieu, 75258 PARIS
Jussieu campus. Building 15/16. Lecture room 309.
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- 10h00 : François Gay-Balmaz (ENS, Paris) Intégrateurs variationels pour les fluides géophysiques
Abstract. Récemment, des nouvelles méthodes numériques ont été formulées pour résoudre les équations d’Euler incompressible. Il s’agit d’intégrateurs variationels basés sur la discrétisation du groupe des difféomorphismes qui préservent le volume. Ces nouvelles méthodes ont des propriétés attrayantes : – elles conservent l’énergie et les théorèmes de circulation de Kelvin, – elles ne sont pas plus coûteuses que des méthodes de différences finies ordinaires, – elles respectent la structure géométrique et Hamiltonienne des équations, – elles sont applicables sur des grilles 2D ou 3D non structurées. Dans ce séminaire nous présentons ces nouvelles méthodes et montrons comment les généraliser aux modèles de fluides géophysiques (Boussinesq, équations primitives) tout en respectant les bonnes propriétés énoncées ci-dessus.
Abstract. We consider a nonlinear diffusion equation with a cubic-like diffusion function arising in the context of phase transitions (in the spirit of the Cahn–Hilliard equation). Because of the non-monotonicity of the diffusion function, the Cauchy problem is ill-posed. To restore the well-posedness, it is possible to take into consideration a generalized formulation determined by considering the forward-backward equation as the singular limit of a corresponding higher order equation, given by the addition of a third order term (two space and one time derivative) of Sobolev type, different with respect to the fourth-space derivative term considered in the case of Cahn–Hilliard equations. Because of some analogies with the case of hyperbolic conservation laws, such kind of solutions has been called entropy solutions for the forward-backward diffusion equation. The aim of the talk is to present and discuss the entropy framework for this equation, with particular attention given to solutions taking values in the zones where the diffusion function is monotone increasing. Joint works with A.Terracina, A. Tesei and P.Lafitte.
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Organizers: Frédéric Coquel, Edwige Godlewski, et Philippe LeFloch
