Quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes

Understand

We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes in general relativity.

• There are two distinguishing aspects of our approach.
• The first aspect is its dyadically localised nature: The nontrivial part of the analysis is reduced entirely to time-translation invariant $r^p$-weighted estimates, in the spirit of [DR09], to be applied on dyadic time-slabs which for large $r$ are outgoing.
• Global existence and decay then both immediately follow by elementary iteration on consecutive such time-slabs, without further global bootstrap.