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Seminar at the

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie, Paris

Organizers

 Philippe G. LeFloch (Paris)

 Jacques Smulevici (Orsay)

Jérémie Szeftel (Paris)

 This Fall: October 10, November 21, and December 12, 2016

 

Monday October 10, 2016

room 15/25-104

 

 14h Peter Hintz (Berkeley)

Nonlinear stability of Kerr-de Sitter black holes

Abstract. In joint work with András Vasy, we recently established the stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will explain the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges; I will also address the issue of finding the mass and the angular momentum of the final black hole.

 15h30 Stefan Czimek (Paris)

An extension procedure for the constraint equations

Abstract. In this talk we present a new extension procedure for the maximal constraint equations of general relativity, motivated by applications to the Cauchy problem. Given a small solution on the unit ball, we can extend it to an asymptotically flat global solution. The main features are that our extension procedure does not need a gluing region, preserves regularity and works in weak regularity. For the proof, we use new methods to solve the prescribed divergence equation for the second fundamental form and the prescribed scalar curvature equation for the metric. We use the under-determinedness of the constraint equations to conserve regularity.

 

 

Monday November 21, 2016

room 15/16-413

 

14h The-Cang Nguyen (Paris)

Progress and recent results for the conformal equations

Abstract. The presentation will be divided into two parts. First, I will introduce the conformal equations and present recent results for these equations as well as questions arising naturally. In a second part, I will talk about the “half-continuity method” and explain how to use this method for giving answers to the questions posed in the first part.

15h30 Volker Schlue (Paris)

On the nonlinear stability of expanding black hole cosmologies

 

 

Monday December 12, 2016

room 15/25-102

 

14h Michał Wrochna (Grenoble)

The quantum stress-energy tensor and its intricate relationship with spacetime geometry

Abstract. It is widely believed that at low energies, quantum gravity should yield an effective theory described by Einstein equations with a stress-energy tensor made of averaged fluctuations of quantum fields. The construction of that stress-energy tensor is however very problematic and its intricate dependence on spacetime geometry results in highly non-linear equations that possess no qualitative theory to date. In this talk I will review this problem as a motivation for improving the construction of linear Klein-Gordon quantum fields, and discuss recent progress that allows for a better control of the dependence on the spacetime metric (partly based on joint work with Christian Gérard).

15h30 Guillaume Idelon-Riton (Regensburg)

Some results about the scattering theory for the massive Dirac fields in the Schwarzschild-Anti-de Sitter space-time

Abstract.  I will first give a brief presentation of the Schwarzschild-Anti-de Sitter spacetime and of some of its geometrical properties that will concern us. Then I will present the massive Dirac equation in this space-time and first study the Cauchy problem which is not completely obvious since our spacetime is not globally hyperbolic. I will then give a result concerning the asymptotic completeness for these fields. By means of a Mourre estimate, it is possible to obtain that the minimal velocity for these fields is 1. I will then show that our dynamics behaves in asymptotic regions like a transport at unit speed in the direction of the black hole. In a third part, I will study the local energy decay for these fields. First, using the existence of exponentially accurate quasi-modes, I will show a logarithmic lower bound on the local energy decay which is in accordance with the results of G. Holzegel and J. Smulevici in the Kerr-Anti-de Sitter spacetime for the Klein-Gordon fields. In order to obtain an upper bound, I will prove the existence of resonances and give some tools in order to localize them.