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#### Seminar at the

#### Laboratoire Jacques-Louis Lions

#### Sorbonne Université, Paris

*Organizers*

#### Philippe G. LeFloch (Sorbonne, Paris)

#### Jacques Smulevici (Sorbonne, Paris)

#### Jérémie Szeftel (Sorbonne, Paris)

**Seminar organized during the Winter-Spring 2019
**

#### Tuesday February 19, 2019

*lecture room 15/16-309*

#### 14h João Costa (Lisbon)

#### Strong cosmic censorship, linear waves, and quasi-normal modes

Abstract. I will present some recent results concerning the Strong Cosmic Censorship Conjecture (SCCC) in the presence of a positive cosmological constant. I will start by reviewing some of the progress made in the context of the Einstein-Maxwell-scalar field system in spherical symmetry and the linear wave equation in the black hole interior of Reissner-Nordström de Sitter. These results show that the validity of the SCCC hinges on the precise decay rates of perturbations along the event horizon, which are known to be determined by the black hole’s quasi-normal spectrum. I will also discuss recent numerical computations of quasi-normal modes that suggest the failure of the SCCC in a near extremal regime of charged de Sitter black holes.

#### 15h30 Shijie Dong (Paris)

#### Evolution of the U(1) Higgs Boson: global nonlinear stability with energy bounds

Abstract. Relying on the hyperboloidal foliation method, we establish the nonlinear stability of the ground state for the so-called U(1) standard model of electroweak interactions. This amounts to establishing a global-in-time theory for the initial value problem for a nonlinear wave-Klein-Gordon system coupling massive (Dirac, scalar, gauge) equations together. In particular, we need to investigate here for the Dirac operator and the properties of energy functionals associated with the hyperboloidal foliation of Minkowski spacetime. We also provide a unified decay result for the Dirac equation when the mass coefficient can be arbitrarily small. Our energy bounds are uniform with respect to our (hyperboloidal) time variable, except for a mild log-growth. This is a joint work with P.G. LeFloch (Paris) and Z. Wyatt (Edinburgh).

#### Monday March 18, 2019

*lecture room 15/16-101*

#### 14h Maria-Caterina Valcu (Lyon)

#### Des équations de contrainte en relativité générale

Abstract. On s’intéresse à la caractérisation des données initiales en relativité générale, c’est à dire aux solutions des équations de contrainte. On s’appuie sur une version modifiée de la méthode conforme, introduite cette fois par David Maxwell, qui semble mieux adaptée à l’étude du système dans le cas où la courbure moyenne n’est pas constante. Par contre, le système devient bien plus compliqué du point de vue analytique. On travaille sous des conditions de petitesse sur nos paramètres, en petite dimension (n=3,4,5) et en présence d’un champ scalaire avec potentiel positif, ce qui mène à un terme non-linéaire dominant focalisant. L’analyse est assez fine et implique une série d’outils différents, dont des résultats de compacité et un théorème du point fixe.

#### 15h30 Léo Bigorgne (Orsay)

#### Sharp asymptotic behavior of solutions of the 3d Vlasov-Maxwell system with small data

Abstract. The Vlasov-Maxwell system is a classical model in plasma physics. Glassey and Strauss proved global existence for the small data solutions of this system under a compact support assumption on the initial data. They used in particular that under these hypotheses, the velocity support of the particle density remains compact. This allows a clean separation between the characteristics of the wave equations (which are null geodesics) and those of the transport equation (which are time-like). We will see how vector field methods can be applied to revisit this problem. In particular, it allows to remove all compact support assumptions on the initial data and obtain sharp asymptotics on the solutions and their derivatives. We will also study the null structure of the system, which constitutes a crucial element of the proof and allows us to deal with high velocities.

#### Monday May 6, 2019

*lecture room 15/16-309*

#### 14h Erwann Delay (Avignon)

#### Le théorème d’énergie positive hyperbolique

Abstract. Le théorème d’énergie positive hyperbolique affirme que toute variété rieman-nienne complète, asymptotique à l’espace hyperbolique réel, et dont la courbure scalaire est minorée par celle du modèle, possède un vecteur énergie-impulsion de genre temps dirigé vers le futur, ce vecteur étant nul seulement pour le modèle. Nous verrons une preuve de ce résultat en toutes dimensions et sans condition spin. Il s’agit d’un travail en collaboration avec Piotr Chrusciel.

#### 15h30 Olivier Graf (Sorbonne)

#### The spacelike-characteristic Cauchy problem with L2 bounded curvature

Abstract. The bounded L2 curvature theorem by Klainerman, Rodnianski, and Szeftel states that the time of existence of a solution to Einstein’s vacuum equations is controled by the L2 norm of its curvature on spacelike Cauchy hypersurfaces. I will present a version of this result where the curvature is bounded in L2 on null hypersurfaces. This provides a first breakdown criterion on characteristic hypersurfaces at this level of regularity. The proof relies on an extension procedure, as well as on the existence and control at low regularity of a new parabolic foliation of null hypersurfaces. This is a joint work with Stefan Czimek (Toronto).

#### Monday June 24, 2018

*lecture **room 15/16-309*

#### 14h Oscar J. C. Campos-Dias (Southampton)

#### Strong cosmic censorship (in de Sitter backgrounds)

Abstract. Generically, strong cosmic censorship (SCC) is the statement that physics within general relativity should be predicted from initial data prescribed on a Cauchy hypersurface. In this talk I will review how fine-tuned versions of SCC have been formulated and evolved along the last decades up to the point where we believe that Christodoulou’s version is true in asymptotically flat spacetimes. However, I will also describe that in the last 2 years it was found that this is no longer necessarily true for some other backgrounds, namely in de Sitter (with a positive cosmological) spacetimes.** **

#### 15h30 Shiyuan Ma (Sorbonne)

#### Linear stability for the Kerr spacetime

Abstract. The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. In this talk, based on recent works on basic energy and Morawetz estimates for solutions of the Teukolsky equation, I shall show how to derive improved decay estimates for the Teukolsky equation and explain how such results can be used to prove linear stability for the Kerr spacetime. The proof relies on using a radiation gauge. This is joint work with Lars Andersson, Thomas Bäckdahl, and Pieter Blue.