Quasilinear wave equations on Kerr black holes in the full subextremal range $|a|<M$

TL;DR

This paper proves global existence, boundedness, and decay for small data solutions to quasilinear wave equations on Kerr black holes for all subextremal spins, extending previous results to the full range of rotation.

Contribution

It extends prior work by establishing results for the entire subextremal Kerr range using refined estimates and frequency decompositions tailored to Kerr geometry.

Findings

Proved global existence and decay for small data solutions on Kerr backgrounds.

Developed a novel frequency decomposition based on azimuthal and stationary frequencies.

Refined linear estimates to handle the full subextremal Kerr range.

Abstract

We prove global existence, boundedness and decay for small data solutions $\psi$ to a general class of quasilinear wave equations on Kerr black hole backgrounds in the full sub-extremal range $|a|<M$. The method extends our previous [DHRT22], which considered such equations on a wide class of background spacetimes, including Kerr, but restricted in that case to the very slowly rotating regime $|a|\ll M$ (which may be treated simply as a perturbation of Schwarzschild $a=0$). To handle the general $|a|<M$ case, our present proof is based on two ingredients: (i) the linear inhomogeneous estimates on Kerr backgrounds proven in [DRSR16], further refined however in order to gain a derivative in elliptic frequency regimes, and (ii) the existence of appropriate physical space currents satisfying degenerate coercivity properties, but which now must be tailored to a finite number of wave packets defined by suitable frequency projection. The above ingredients can be thought of as adaptations of the two basic ingredients of [DHRT22], exploiting however the peculiarities of the Kerr geometry. The novel frequency decomposition in (ii), inspired by the boundedness arguments of [DR11, DRSR16], is defined using only azimuthal and stationary frequencies, and serves both to separate the superradiant and non-superradiant parts of the solution and to localise trapping to small regions of spacetime. The strengthened (i), on the other hand, allows us to relax the required coercivity properties of our wave-packet dependent currents, so as in particular to accept top order errors provided that they are localised to the elliptic frequency regime. These error terms are analysed with the help of the full Carter separation.