Periodic Reporting for period 4 - GEOWAKI (The analysis of geometric non-linear wave and kinetic equations)
Período documentado: 2021-09-01 hasta 2022-07-31
In a gravitational system and in plasma models, particles generate waves and in return the waves dictate how the particles should move. Thus non-linear interactions occur and our aim is to have a better theoretical control of those non-linear interactions.
Another aspect of our project concerned the study of gravitational systems in the presence of boundaries or in the compact case, where the spatial volume is finite. In this setting, gravitational waves are not allowed to escape to infinity and again, due to the nature of the equations, complex non-linear interactions can occur which we want to better understand.
In this context, our overall objectives are : 1) Understand small data solutions for Kinetic/wave systems and the non-linear interactions occurring in several systems mixing kinetic and wave equations 2) understand General Relativity in the presence of either boundaries, in particular the so-called AdS case, or in the cosmological case (finite volume). On top of the study of the local existence problem, we want to understand several important global in time problems, such as the existence of nonlinear periodic waves on AdS 3) In the presence of boundaries, we also want to study the case where the waves are dissipated at the boundary.
Conclusions of the action: We proved several small data results for systems of Kinetic/wave equations and have greatly improved both the understanding of those non-linear interactions and their mathematical analysis, those completing objective 1) above. 2) We proved several local existence results for the Einstein equations with boundaries. This provides some answers in the direction of objective 2) but the general case is still open, some there is still room for improvement of our results. We also made the first contributions to the construction of periodic non-linear waves on AdS, completing one of our main goals 3) We made considerable progress concerning objective 3) and we are still working on a full completion of the objective.
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.
Concerning the study of time-periodic solutions in AdS, these solutions had been constructed numerically before, but there was so far no rigorous proof of their existence. We have now completed several papers which rigorously establish their existence, using a variety of methods inspired by KAM theory.
Concerning the study of the stability of the AdS solution with dissipative boundary conditions, we have found a better set-up to go from the linear analysis that we already carried before to the non-linear analysis. This set-up will be essential for completing a full proof of stability.
Motivated by the analysis of the Einstein equations with boundary, we have established existence of solutions to these equations in the presence of a boundary in the so-called maximal gauge. This is a set-up which has found many applications before for the long time analysis of solutions, so this analysis should fine many applications.
We have also obtained the first set-up for the Einstein equations with boundary such that geometric uniqueness holds. Our result is still considerably restricted, since we can only address the case of totally geodesic or umbilic boundary, but nonetheless there was no result of this kind prior to our work.