Tag Archive

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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)

Dates of the Seminar:

January 30, February 27, March 20, April 10, May 22, June 6, June 19, July 4

Monday January 30, 2017

room 15/25-104

 14h Georgios Moschidis (Princeton, USA)

The scalar wave equation on general asymptotically flat spacetimes. Stability and instability results

Abstract. We will examine how certain geometric conditions on general asymptotically flat spacetimes (M,g) are related to stability or instability properties of solutions to the scalar wave equation on M. First, in the case when (M,g) possesses an event horizon with positive surface gravity and an ergo-region which is sufficiently small in terms of the near-horizon geometry, we will prove a logarithmic decay result for solutions to  the wave equation, provided a uniform energy boundedness estimate holds. This result, applicable also in the absence of a horizon and an ergo-region, generalizes a result of Burq for the wave equation on the complement of an arbitrary compact obstacle in flat space. We will then apply the methods developed for the proof of this result in obtaining a rigorous proof of Friedman’s ergosphere instability for scalar waves in the case when (M,g) possesses an ergo-region and no event horizon.

 15h30 Xavier Lachaume (Tours)

The constraint equations of scalar tensor and Lovelock theories

Abstract. The ADM decomposition is the projection of the Einstein field equations on a spacelike foliation of the spacetime. It gives the constraint equations that must necessarily be satisfied by a riemannian metric and a 2-form to be the initial data of an Einstein spacetime. In this talk, we shall introduce some modified gravity theories: the scalar-tensor and Lovelock theories, and see how they behave under the ADM decomposition. We shall examine their constraint equations, and solve them in particular cases. This involves the study of whether a certain function of the elementary symmetric polynomials is concave or not.

Monday February 27, 2017

room 15/16-309

14h Mokdad Mokdad (Brest)

Conformal scattering for Maxwell fields on Reissner-Nordstrøm-de Sitter spacetimes

Abstract. The Reissner-Nordstrøm-de Sitter spacetime models a spherically symmetric charged and non-rotating black hole in the presence of a positive cosmological constant. Depending on the parameters of the metric, this spacetime can have up tothree distinct event horizons. In the case of three horizons, we develop a scattering theory for Maxwell fields using the conformal geometric approach initiated by Penrose and Friedlander and referred to as conformal scattering. The idea is that a complete scattering theory is equivalent to the well-posedness of the Goursat problem (characteristic Cauchy problem) at the null boundary of the conformal manifold. Decay estimates obtained by geometric energy inequalities are essential tools for closing the estimates that allow the construction of the scattering operator : their role is to prove that energy cannot accumulate at timelike infinity, which can be understood as a weak form of Huygens’ principle.

15h30 Annalaura Stingo (Paris 13)

Global existence and asymptotic behavior for small solutions to 1D quasi-linear cubic Klein-Gordon equations

Abstract. Let u be a solution to a quasi-linear cubic Klein-Gordon equation, with smooth, small Cauchy data. It is known that, under a suitable condition on the nonlinearity, the solution is global-in-time for compactly supported Cauchy data. We prove that the result holds even when data are not compactly supported but only decay like 1/r at infinity, combining the method of Klainerman vector fields with a semiclassical normal forms method introduced by Delort. Moreover, we get a one-term asymptotic expansion for the solutions and establish a modified scattering property.

Monday March 20, 2017

room 15/16-309

14h Dominic Dold (Cambridge, UK)

Exponentially growing mode solutions to the Klein-Gordon equation in Kerr-AdS spacetimes

Abstract. We consider solutions to the Klein-Gordon equation in the black hole exterior of Kerr-AdS spacetimes. It is known that, if the spacetime parameters satisfy the Hawking-Reall bound, solutions (with Dirichlet boundary conditions at infinity) decay logarithmically. We shall present our recent result of the existence of exponentially growing mode solutions in the parameter range where the Hawking-Reall bound is violated. We will discuss various boundary conditions at infinity.

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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.