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We prove global existence and decay for small-data solutions to a class ofquasilinear wave equations on a wide variety of asymptotically flat spacetimebackgrounds, allowing in particular for the presence of horizons, ergoregionsand trapped null geodesics, and including as a special case the Schwarzschildand very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes ingeneral relativity. There are two distinguishing aspects of our approach. Thefirst aspect is its dyadically localised nature: The nontrivial part of theanalysis is reduced entirely to time-translation invariant $r^p$-weightedestimates, in the spirit of [DR09], to be applied on dyadic time-slabs whichfor large $r$ are outgoing. Global existence and decay then both immediatelyfollow by elementary iteration on consecutive such time-slabs, without furtherglobal bootstrap. The second, and more fundamental, aspect is our direct use ofa "blackbox" linear inhomogeneous energy estimate on exactly stationarymetrics, together with a novel but elementary physical space top order identitythat need not capture the structure of trapping and is robust to perturbation.In the specific example of Kerr black holes, the required linear inhomogeneousestimate can then be quoted directly from the literature [DRSR16], while theadditional top order physical space identity can be shown easily in many cases(we include in the Appendix a proof for the Kerr case $\vert a \vert \ll M$,which can in fact be understood in this context simply as a perturbation ofSchwarzschild). In particular, the approach circumvents the need either forproducing a purely physical space identity capturing trapping or for a carefulanalysis of the commutation properties of frequency projections with the waveoperator of time-dependent metrics.

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