A proof of the instability of AdS for the Einstein-massless Vlasov system

In recent years, the conjecture on the instability of Anti-de Sitter spacetime, put forward by Dafermos-Holzegel (Dynamic instability of solitons in 4 + 1 dimesnional gravity with negative cosmological constant, 2006. https://www.dpmms.cam.ac.uk/similar to md384/ADSinstability.pdf) and Dafermos (The Black Hole Stability problem, Talk at the Newton Institute, Cambridge, 2006. http://www-old.newton.ac.uk/webseminars/pg+ws/2006/gmx/1010/dafermos/) in 2006, has attracted a substantial amount of numerical and heuristic studies. Following the pioneering work (Phys Rev Lett 107(3):031102, 2011) of Bizon-Rostworowski, research efforts have been mainly focused on the study of the spherically symmetric Einstein-scalar field system. The first rigorous proof of the instability of AdS in the simplest spherically symmetric setting, namely for the Einstein-null dust system, was obtained in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). In order to circumvent problems associated with the trivial break down of the Einstein-null dust system occuring at the center r=0, Moschidis (2017) studied the evolution of the system in the exterior of an inner mirror placed at r=r0, r0>0. However, in view of additional considerations on the nature of the instability, it was necessary for Moschidis (2017) to allow the mirror radius r0 to shrink to 0 with the size of the initial perturbation; well-posedness in the resulting complicated setup (involving low-regularity estimates of uniform modulus with respect to r0) was obtained in Moschidis (The Einstein-null dust system in spherical symmetry with an inner mirror: structure of the maximal development and Cauchy stability, 2017. arXiv:1704.08685). In this paper, we establish the instability of AdS for the Einstein-massless Vlasov system in spherical symmetry; this will be the first proof of the AdS instability conjecture for an Einstein-matter system which is well-posed for regular initial data in the standard sense, without the addition of an inner mirror. The necessary well-posedness results for this system are obtained in our companion paper (Moschidis in The characteristic initial-boundary value problem for the Einstein-massless Vlasov system in spherical symmetry, 2018. arXiv:1812.04274). Our proof utilises an instability mechanism based on beam interactions which is superficially similar to the one appearing in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). However, new difficulties associated with the Einstein-massless Vlasov system (such as the need for control on the paths of non-radial geodesics in a large curvature regime) will force us to develop a different strategy of proof involving a novel configuration of beam interactions. One of the main novelties of our construction is the introduction of a multi-scale hierarchy of domains in phase space, on which the ini