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

TL;DR

This paper develops the mathematical framework for analyzing the Einstein-massless Vlasov system in spherical symmetry with negative cosmological constant, establishing existence, uniqueness, and stability results for solutions with specific boundary conditions.

Contribution

It introduces the initial-boundary value problem for the Einstein-massless Vlasov system in AdS spacetime, proving key existence, continuation, and stability results under spherical symmetry.

Findings

Existence and uniqueness of maximal future development.

Continuation criteria based on the ratio 2m/r.

Cauchy stability of Anti-de Sitter spacetime.

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.