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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.

###### _____________________________**_______________****_________________________________________****_______________**_________

**_________________________________________**

#### Seminar at the

#### Laboratoire Jacques-Louis Lions

#### Sorbonne Université, Paris

*Organizers*

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

#### Jacques Smulevici (Sorbonne, Paris)

#### Jérémie Szeftel (Sorbonne, Paris)

**Dates of the Seminar this Fall 2018**

**September 24, October 29, November 19**

#### Monday September 24, 2018

*room 15/16-309*

#### 14h Dietrich Häfner (Grenoble)

#### Scattering for Dirac and Klein-Gordon fields on the (De Sitter) Kerr metric and the Hawking effect

Abstract. We consider Dirac and Klein-Gordon fields on the (De Sitter) Kerr metric which describes rotating black holes. Whereas there exists a conserved L2 norm for the Dirac field, no positive conserved quantity exists for the Klein-Gordon field, which makes the analysis more difficult for the latter. We obtain asymptotic completeness results for the Dirac field on the Kerr and for the Klein-Gordon field on the De Sitter Kerr metric. We then present a rigorous result about the Hawking effect for fermions in the setting of a collapse of a rotating charged star. This effect predicts the creation of particles by black holes.

#### 15h30 Jean-Philippe Nicolas (Brest)

#### Peeling for scalar fields on the Kerr metric

Abstract. The peeling is an asymptotic behavior of massless fields along outgoing null geodesics in asymptotically flat spacetimes, initially observed by Sachs at the beginning of the 1960’s, then reformulated in very simple terms by Penrose in 1965 using conformal geometry. The question of its genericity, especially when talking about the peeling of the Weyl tensor of an Einstein spacetime, was controversial for several decades after Penrose’s paper. For Einstein’s equations, the question is now essentially settled, but given an Einstein spacetime, it is not clear whether there is a large class of Cauchy data giving rise to solutions with a good peeling. Lionel Mason and the speaker answered the question for fields of spin 0, 1/2 and 1 on Schwarzschild’s spacetime in 2009 and 2012. We extended recently the results to linear and non linear scalar fields on the Kerr geometry in a joint work with Pham Truong Xuan. We shall recall the history of the subject, describe the principles of the approach developed with Lionel Mason and talk about the specific features of our work for Kerr metrics.

#### Monday October 29, 2018

*room 15/16-101*

#### 14h Joe Keir (Cambridge)

#### The weak null condition and the p-weighted energy method

Abstract. The Einstein equations in wave coordinates are an example of a system which does not obey Klainerman’s “null condition”. Their failure to satisfy this condition leads to many difficulties, both in Lindblad-Rodnianski’s proof of global existence and in any attempt to apply other techniques to these equations. One such technique is the “p-weighted energy method” of Dafermos- Rodnianski, which is a very powerful and robust method that can easily be adapted to understand the behavior of waves in many interesting situations, including black holes. In this talk I will explain how to modify this method to systems which only obey the “weak null condition”, including the Einstein equations. This involves adapting the p-weighted energy method, and combining it with the many of the geometric methods used by Christodoulou and Klainerman. Among other things, this allows us to enlarge the class of wave equations which are known to admit small-data global solutions, and it also yields a detailed description of null infinity. In particular, in some situations we can understand the geometric origin of the slow decay towards null infinity exhibited by these systems: it is due to the formation of “shocks at infinity”.

#### 15h30 Volker Schlue (Paris)

#### Scattering from infinity for semi-linear wave equations

Abstract. I will discuss the construction of global solutions from scattering data (at null infinity) for various semi-linear wave equations on Minkowski space satisfying the (weak) null condition. I will elaborate on the proof which relies, i) on a fractional Morawetz estimate, and (ii) on the construction of suitable approximate solutions from the scattering data. Finally I will outline the application of these results to Einstein’s equations in harmonic coordinates. This is joint work with Hans Lindblad.

#### Monday November 19, 2018

*room 15/16-309*

#### 14h Adam Layne (Stockholm)

#### Stability within T2-symmetric expanding spacetimes

Abstract. We present a recently completed, non-polarized analogue of the asymptotic characterization of T2-symmetric Einstein flow solutions by P. LeFloch and J. Smulevici. We impose a far weaker condition, but obtain similar rates of decay for the normalized energy and associated quantities. Critical to this work have been novel numerical simulations which indicate that there is locally attractive behavior for those T2-symmetric solutions not subject to this weakened condition. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarized asymptotics are on one hand stable within a larger class than merely polarized solutions, but unstable within all T2-symmetric solutions.

#### 15h30 Grigorios Fournodavlos (Sorbonne)

#### Dynamics of the Einstein vacuum equations about the Schwarzschild black hole interior

Abstract. We will talk about the dynamical behavior of the Schwarzschild black hole singularity, in the context of the Einstein equations in vacuum, from the point of view of the Cauchy problem in general relativity. As it is well known, the Schwarzschild singularity is highly unstable under arbitrarily small perturbations, which makes the study of its dynamics in full generality a difficult problem. We will begin by giving an overview of the current status of the near-Schwarzschild-black hole interior problem and we will compare it to the dynamics observed near other singularity models, in black hole interiors or Big Bangs. Then we will discuss linear and non-linear partial results in the near-Schwarzschild case, both backwards and forwards in time, with and without symmetries.