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Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday April 15, 2015

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25–326

 

14h Shiwu Yang (Cambridge) Decay properties of solutions of Maxwell Klein-Gordon equations

Abstract.  I will present some recent progress on the asymptotic behavior of global solutions to Maxwell-Klein-Gordon equations. I will show that the integrated local energy and the energy flux through the outgoing null hypersurfaces decays polynomially in the retarded time in Minkowski space with data merely bounded in some gauge invariant weighted Sobolev space. This in particular includes the case with large charge. One novelty of this work is that these decay estimates precisely capture the asymptotic properties for the non-linear fields with arbitrarily large data. If in addition that the initial data for the scalar field is sufficiently small, then we show the pointwise decay of the solutions. This result improves the previous result of Lindblad and Sterbenz in which smallness is required for both the scalar field and the Maxwell field.

15h30 Gustav Holzegel (London) Local and global dynamics in asymptotically anti de Sitter spacetimes

Abstract.  Asymptotically anti de Sitter (aAdS) spacetimes play a prominent role in theoretical physics and mathematics.  Due to the presence of a timelike hypersurface at infinity these spacetimes are not globally hyperbolic, a fact that leads to intricate initial boundary value problems when studying global dynamics of hyperbolic equations on these backgrounds. In this talk, I will present several local and global results for the massive wave equation on aAdS spacetimes (including black hole spacetimes) with emphasis on how different boundary conditions (Dirichlet, Neumann or dissipative) influence the global dynamics. In particular, I will outline a recent proof (obtained in collaboration with J. Luk, J. Smulevici and C. Warnick) of linear stability and decay for gravitational perturbations on anti de Sitter space under dissipative boundary conditions. The proof unravels an interesting trapping phenomenon near the conformal boundary which necessarily leads to a degeneration in the decay estimates. Time permitting some future applications will also be discussed.

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Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday March 4, 2015

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25–326

 

14h François Fillastre (Cergy-Pontoise) Minkowski problem in Minkowski space

Abstract. T. Barbot, F. Beguin and A. Zeghib solved a smooth Lorentzian version of the Minkowski problem in dimension (2+1). More precisely they proved that if M is a flat 3-dimensional maximal globally hyperbolic spatially compact spacetime, then there exists a unique strictly convex smooth space-like surface in M with a prescribed smooth positive Gauss curvature. We will look at this problem for any dimensions. The existence part is solved in a generalized way (a measure is prescribed rather than a function). Concerning the regularity of the solution, the 2+1 case is specific. The arguments are based on tools from the geometry of convex sets. Joint work with Francesco Bonsante (Pisa).

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Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday Feb. 11, 2015

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25–326

 


14h Bruno Premoselli (Cergy-Pontoise) Robustness of the conformal constraints in a scalar-field setting

Abstract. The constraint equations arise in the initial-value formulation of the Einstein equations. The conformal method allows one to rewrite the constraint equations into a determined system of nonlinear, supercritical, elliptic PDE’s. In this talk, we will investigate some stability properties for this elliptic system. The notion of stability under consideration, defined as the continuous dependence of the set of solutions of the conformal constraint system with respect to its coefficients, is reformulated for the conformal method. The analysis of these stability properties involves blow-up techniques concerning defects of compactness for supercritical nonlinear elliptic equations. This is a joint work with Olivier Druet.

15h30 Christophe Bavard (Bordeaux) Points conjugués des tores lorentziens

Abstract. Les points conjugués jouent un rôle important dans l’étude des variétés riemanniennes et lorentziennes, en particulier pour l’étude du rayon d’injectivité. Dans le cadre riemannien, l’absence de points conjugués impose des contraintes assez fortes sur la topologie de la variété, et parfois même sur sa géométrie. Ainsi, un résultat de Hopf (1948), généralisé par Burago et Ivanov (1994), affirme qu’un tore riemannien sans points conjugués est nécessairement plat. Dans cet exposé, je montrerai l’existence de métriques sans points conjugués dans toute composante connexe de l’espace des métriques lorentziennes sur le tore de dimension 2 ; cela prouve en particulier l’existence de tores lorentziens sans points conjugués et non plats. Il s’agit d’un travail conjoint avec Pierre Mounoud.

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Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday Nov. 26, 2014

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25–104

 


14h Qian Wang (Oxford)  A geometric approach to the sharp local well-posedness theory for quasilinear wave equations

Abstract. The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces Hs with s>2+(2-\sqrt 3)/2 for general quasilinear wave equation in (1+3) spacetime by him and Rodnianski. Via this approach they obtained the local well-posedness with s>2 for (1+3) vacuum Einstein equations. A proof of the sharp H2+ local well-posedness result for general quasilinear wave equation was provided by Smith and Tataru by constructing a parametrix using wave packets. The difficulty of the problem is that one has to face the major hurdle caused by the Ricci tensor of the metric for the quasilinear wave equations. I will present my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in (1+3) spacetime by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.

15h30 Jonathan Luk (Cambridge, UK) Stability of the Kerr Cauchy horizon

Abstract. The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

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Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris) 

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday Sept. 17, 2014

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25-321

 


14h Arick Shao (Imperial College) Unique continuation from infinity for linear waves

Abstract. We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. These results rely on a new family of geometrically robust Carleman estimates near null cones and on an adaptation of the standard conformal inversion of Minkowski spacetime. Also, the results are nearly optimal in many respects. This is joint work with Spyros Alexakis and Volker Schlue.

15h30 Claude Warnick (Warwick)  Dynamics in anti-de Sitter spacetimes

Abstract. When solving Einstein’s equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike hypersurface. Questions of local well-posedness and global stability are sensitive to the choices of boundary conditions. I will present recent work exploring the effects of non-trivial boundary data for the asymptotically AdS initial-boundary value problem, including a recent result in collaboration with Holzegel. I will also outline some interesting open problems in the area.

Philippe LeFloch, DIRECTOR OF RESEARCH AT CNRS Email address: pglefloch [at] gmail.com

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