Seminar at the
Laboratoire Jacques-Louis Lions
Université Pierre et Marie Curie, Paris
Philippe G. LeFloch (Paris)
Jacques Smulevici (Orsay)
Jérémie Szeftel (Paris)
Dates of the Seminar: January 30, February 27, March 20, April 10, May 22
Monday January 30, 2017
14h Georgios Moschidis (Cambridge, UK)
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, generalises 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
Monday February 27, 2017
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 just decaying as 1/<x> 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 u, showing that there is modified scattering.
Monday March 20, 2017
14h Dominic Dold (Cambridge)
Monday April 10, 2017
Monday May 22, 2017
14h Jan Sbierski (Cambridge)
15h30 Grigorios Fournodavlos (Cambridge)
Dynamics of the Einstein equations near a Schwarzschild singularity
Abstract. We will discuss dynamical properties of the Schwarzschild interior, backwards and forwards (in time) with respect to the initial value problem for the Einstein vacuum equations.