12 décembre 2019

Olivier Graf (UPMC)
[Reporté] Low regularity characteristic Cauchy problem in general relativity

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Lieu : Salle 3L8 - IMO

Résumé : Einstein equations of general relativity describe the coupling between the gravitational field represented by a Lorentzian metric g and matter. They are covariant by a change of frame, i.e. a change of coordinates, which can therefore be freely prescribed. In the so-called wave coordinates, Einstein equations reduce to a system of quasilinear wave equations for g for which the d’Alembertian is the wave operator associated to the metric g. In particular, the characteristic hypersurfaces for this system of equations are the null hypersurfaces for the Lorentzian metric g (i.e. the hypersurfaces on which the induced metric is degenerate). These hypersurfaces are independent of the coordinates and choosing these to be adapted to them (e.g. by prescribing generalised u, v coordinates to be constant on the null hypersurfaces) is a powerful tool to capture the propagation features of Einstein equations. In particular, it is natural to consider the Cauchy problem for data given on initial null hypersurfaces, i.e. on surfaces u = cst or v = cst, rather than on an initial spacelike hypersurface t = cst. A local existence result for such a Cauchy problem for weakly regular initial data turned out to be crucial in the proof of the weak cosmic censorship conjecture in spherical symmetry by Christodoulou. In a joint work with Stefan Czimek (Toronto), we recently obtained a local existence result for Einstein equations in vacuum, without symmetries and for weakly regular initial data posed on null hypersurfaces. The weak regularity is measured by an L2 control of the curvature (roughly speaking an H2-control on the metric g). The proof relies on the bounded L2 theorem obtained by Klainerman-Rodnianski-Szeftel which is the corresponding result for data posed on spacelike hypersurfaces.
In this talk, I will introduce Einstein equations, its geometric features (such as the general covariance) from a PDE perspective and discuss the associated (geometric) Cauchy problem. Then, I will review on the Cauchy problem for low regular initial data posed on a spacelike hypersurface and present the recent result that we obtained for initial data posed on null hypersurfaces.

Notes de dernières minutes : Reporté

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