We have completed a large part of our objective concerning understanding small data solutions for Kinetic/wave systems and the non-linear interactions occurring in several systems of mixed kinetic and wave equations. In particular, we have achieved a proof of the stability of the Minkowski space for the Einstein-Vlasov system of General Relativity and a detailed description of the solutions of the Vlasov-Maxwell system (i.e. the equations describing plasmas) with small data in several different situations. For this, we needed in particular to develop several mathematical techniques to control the contributions of the kinetic terms. Some of these techniques have found applications also in classical systems (Vlasov-Poisson, Vlasov-Yukawa). With the progress done so far, we are starting to look at solutions to wave/kinetic systems in more complicated situations. We are also working on improving our results concerning the Einstein-Vlasov system for more general initial data. We developed further the analysis of the Einstein-Vlasov system, with the introduction of novel methods for the construction of stationary states, including an important of one of the team members (Jabiri), concerning the construction of stationary states outside from axisymmetric black holes.
Our study of the Einstein equations with boundaries has led to the publication of 3 articles on the subject, including the first result for the Einstein equations in the maximal gauge with boundary and the first results for which geometric uniqueness holds. We made considerable progress concerning the stability of AdS with dissipative boundary conditions, in particular, introducing a new gauge for the study of this problem. On the way, we also team members of the project also made significant progress on several related topics, such as the study of singularities for the Einstein equations (by Fournodavlos) and the problem of unique continuation from the boundary for asymptotically AdS spacetimes by Chatzikaleas.
Team member Chatzikaleas and the PI have completed several papers concerning the construction of periodic non-linear waves on AdS, thus reaching one of the main goal of the project.
Those results have been published in high quality peer reviewed journals and the PI and the team members have given many talks in international conferences, in particular in places such as the Newton Institute in Cambridge, the Institut Henri Poincaré in Paris, the Fields Institute in Toronto, Princeton University, Oxford University, Oberwolfach mathematical research center. We also animated a monthly seminar on topics related to mathematical relativity, and have participated to the organisation of one conference on mathematical GR in May 2018, held at the Institut Henri Poincaré, Paris.
