The phenomenon of weak turbulence is ubiquitous in the world of nonlinear dispersive equations. On backgrounds with confining geometry, the nonlinear interaction of small amplitude waves often leads to a non-trivial gradual energy flux to modes of high frequency, a process that is otherwise suppressed in the presence of strong dispersion at the linear level. In the case of the vacuum Einstein equations, an example of weakly turbulent dynamics is given by the scenario postulated by the AdS instability conjecture (put forward by Dafermos and Holzegel in 2006); according to the conjecture, generic small perturbations of the AdS initial data lead to the formation of trapped surfaces when reflecting (hence confining) boundary conditions are imposed at conformal infinity. However, whether a similar scenario also holds in the exterior region of an asymptotically AdS black hole spacetime, where the presence of the event horizon acts as a dispersive mechanism, has been the subject of debate.

In this talk, we will show that weakly turbulent dynamics do indeed emerge in the evolution of a quasilinear model for the vacuum Einstein equations on the Schwarzschild-AdS exterior spacetimes for an open and dense set of black hole mass parameters. In particular, solutions arising from data which are small in Sobolev norms of arbitrarily high regularity exhibit norm inflation over sufficiently long time intervals. This is joint work with Christoph Kehle.