Semi-Global Existence and Trapped Surface Formation for the Einstein-Vlasov System

Abstract

In this article, we study the Einstein-Vlasov system for massless particles. Our main contribution is the provision of a novel method for controlling the Vlasov matter in a double null gauge, relying purely on vector field commutation and bypassing the need for introducing Jacobi fields, which has so far been the only existing technique outside of symmetry in this gauge. To obtain it, we have to overcome stringent regularity issues that exist along this path. Because it relies solely on vector field commutation, our technique is fundamentally simple and flexible enough to be applicable in any data regime. We thus anticipate this to be a helpful tool for many subsequent problems regarding the Einstein-Vlasov system, including, for instance, simplified approaches to the proof of nonlinear stability of Minkowski spacetime with massless Vlasov matter, scattering problems for data close to black holes among others. Within the present double-null commutator framework, we show that two derivatives of curvature suffice to close the coupled Einstein--Vlasov hierarchy, without derivative loss, given smooth initial data. Using this method, we obtain a large data semi-global existence theorem and a dynamical trapped surface formation statement for our system.