The characteristic initial-boundary value problem for the Einstein--massless Vlasov system in spherical symmetry

Abstract

In this paper, we initiate the study of the asymptotically AdS initial-boundary value problem for the Einstein-massless Vlasov system with $\Lambda<0$ in spherical symmetry. We will establish the existence and uniqueness of a maximal future development for the characteristic initial-boundary value problem in the case when smooth initial data are prescribed on a future light cone $\mathcal{C}^{+}$ emanating from a point at $\{r=0\}$ and a reflecting boundary condition is imposed on conformal infinity $\mathcal{I}$. We will then prove a number of continuation criteria for smooth solutions of the spherically symmetric Einstein-massless Vlasov system, under the condition that the ratio $2m/r$ remains small in a neighborhood of $\{r=0\}$. Finally, we will establish a Cauchy stability statement for Anti-de Sitter spacetime as a solution of the spherically symmetric Einstein-massless Vlasov system under initial perturbations which are small only with respect to a low regularity, scale invariant norm $||\cdot||$. This result will imply, in particular, a long time of existence statement for $||\cdot||$-small initial data. This paper provides the necessary tools for addressing the AdS instability conjecture in the setting of the spherically symmetric Einstein--massless Vlasov system, a task which is carried out in our companion paper. However, the results of this paper are also of independent interest.