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.