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In this paper, we study the evolution of asymptotically AdS initial data forthe spherically symmetric Einstein--massless Vlasov system for $\Lambda<0$,with reflecting boundary conditions imposed on timelike infinity $\mathcal{I}$,in the case when the Vlasov field is supported only on radial geodesics. Thissystem is equivalent to the spherically symmetric Einstein--null dust system,allowing for both ingoing and outgoing dust. In general, solutions to thissystem break down in finite time (independent of the size of the initial data);we highlight this fact by showing that, at the first point where the ingoingdust reaches the axis of symmetry, solutions become $C^{0}$ inextendible,although the spacetime metric remains regular up to that point.  One way to overcome this trivial obstacle to well-posedness is to place aninner mirror on a timelike hypersurface of the form $\{r=r_{0}\}$, $r_{0}>0$,and study the evolution on the exterior domain $\{r\ge r_{0}\}$. In thissetting, we prove the existence and uniqueness of maximal developments forgeneral smooth and asymptotically AdS initial data sets, and study the basicgeometric properties of these developments. Furthermore, we establish thewell-posedness and Cauchy stabilty of solutions with respect to a rough initialdata norm, measuring the concentration of energy at scales proportional to themirror radius $r_{0}$.  The above well-posedness and Cauchy stability estimates are used in ourcompanion paper for the proof of the AdS instability conjecture for theEinstein--null dust system. However, the results of the present paper mightalso be of independent interest.

The Einstein--null dust system in spherical symmetry with an inner mirror: structure of the maximal development and Cauchy stability