The analysis of geometric non-linear wave and kinetic equations

Project Information

GEOWAKI

Grant agreement ID: 714408

DOI

10.3030/714408

Project closed

EC signature date 8 November 2016

Start date 1 February 2017

End date 31 July 2022

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 071 008,00

EU contribution

€ 1 071 008,00

Coordinated by

SORBONNE UNIVERSITE
France

Periodic Reporting for period 4 - GEOWAKI (The analysis of geometric non-linear wave and kinetic equations)

Reporting period: 2021-09-01 to 2022-07-31

Summary of the context and overall objectives of the project

The project is addressing several mathematical problems concerned with the asymptotic behaviour of solutions to geometric wave and kinetic equations. These mathematical models arise naturally in a variety of physical contexts, such as in General Relativity or in the Plasma theory.

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Concerning the study of Kinetic/Wave systems, prior to our work, most results were restricted to special situations, such as spherically symmetric systems, or special types of initial data. We have recently completed several works which go far beyond the previous state of the art. Namely, our proof of the stability of the Minkowski space for the Einstein-Vlasov system does not require any symmetry and the initial data for the kinetic fields are quite general (in particular no compactness assumption). 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. Prior to her work, there was no result of this kind in axisymmetry, so this work goes well beyond the state of the art.



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.

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